diff options
Diffstat (limited to 'sourcecodes/bnt-master/docs')
155 files changed, 40442 insertions, 0 deletions
diff --git a/sourcecodes/bnt-master/docs/ChangeLog.Sourceforge.txt b/sourcecodes/bnt-master/docs/ChangeLog.Sourceforge.txt new file mode 100644 index 00000000..17ec3df1 --- /dev/null +++ b/sourcecodes/bnt-master/docs/ChangeLog.Sourceforge.txt @@ -0,0 +1,4436 @@ + + +2007-02-11 17:12 nsaunier + + * BNT/learning/learn_struct_pdag_pc.m: Bug submitted by Imme Ebert-Uphoff (ebert@tree.com) (see Thu Feb 8, 2007 email on the BNT mailing list). + +2005-11-26 12:12 yozhik + + * BNT/inference/dynamic/@hmm_inf_engine/fwdback_twoslice.m: merged + fwdback_twoslice.m to release branch + +2005-11-25 17:24 nsaunier + + * BNT/inference/dynamic/@hmm_inf_engine/fwdback_twoslice.m: adding + old missing fwdback_twoslice.m + +2005-11-25 17:24 yozhik + + * BNT/inference/dynamic/@hmm_inf_engine/fwdback_twoslice.m: file + fwdback_twoslice.m was added on branch release-1_0 on 2005-11-26 + 20:12:05 +0000 + +2005-09-25 15:54 yozhik + + * BNT/add_BNT_to_path.m: fix paths + +2005-09-25 15:30 yozhik + + * BNT/add_BNT_to_path.m: Restored directories to path. + +2005-09-25 15:29 yozhik + + * HMM/fwdback_twoslice.m: added missing fwdback_twoslice + +2005-09-17 11:14 yozhik + + * ChangeLog, BNT/add_BNT_to_path.m, BNT/test_BNT.m, + BNT/examples/static/cmp_inference_static.m, + BNT/inference/dynamic/@hmm_inf_engine/enter_evidence.m: Merged + bug fixes from HEAD. + +2005-09-17 11:11 yozhik + + * ChangeLog: added change log + +2005-09-17 11:11 yozhik + + * ChangeLog: file ChangeLog was added on branch release-1_0 on + 2005-09-17 18:14:47 +0000 + +2005-09-17 10:00 yozhik + + * BNT/inference/dynamic/@hmm_inf_engine/enter_evidence.m: Temporary + rollback to fix error, per Kevin. + +2005-09-17 09:59 yozhik + + * BNT/examples/static/cmp_inference_static.m: Commented out + erroneous line, per Kevin. + +2005-09-17 09:58 yozhik + + * BNT/add_BNT_to_path.m: Changed to require BNT_HOME to be + predefined. + +2005-09-17 09:56 yozhik + + * BNT/test_BNT.m: Commented out problematic tests. + +2005-09-17 09:38 yozhik + + * BNT/test_BNT.m: renable tests + +2005-09-12 22:18 yozhik + + * KPMtools/pca_kpm.m: Initial import of code base from Kevin + Murphy. + +2005-09-12 22:18 yozhik + + * KPMtools/pca_kpm.m: Initial revision + +2005-08-29 10:44 yozhik + + * graph/: README.txt, Old/best_first_elim_order.m, + Old/dag_to_jtree.m, Old/dfs.m, Old/dsep_test.m, + Old/mk_2D_lattice_slow.m, acyclic.m, assignEdgeNums.m, + best_first_elim_order.m, check_jtree_property.m, + check_triangulated.m, children.m, cliques_to_jtree.m, + cliques_to_strong_jtree.m, connected_graph.m, + dag_to_essential_graph.m, dfs.m, dfs_test.m, dijkstra.m, + family.m, graph_separated.m, graph_to_jtree.m, + min_subtree_con_nodes.m, minimum_spanning_tree.m, minspan.m, + mk_2D_lattice.m, mk_2D_lattice_slow.m, mk_adj_mat.m, + mk_adjmat_chain.m, mk_all_dags.m, mk_nbrs_of_dag.m, + mk_nbrs_of_digraph.m, mk_nbrs_of_digraph_broken.m, + mk_nbrs_of_digraph_not_vectorized.m, mk_rnd_dag.m, + mk_rnd_dag_given_edge_prob.m, mk_rooted_tree.m, mk_undirected.m, + moralize.m, neighbors.m, parents.m, pred2path.m, + reachability_graph.m, scc.m, strong_elim_order.m, test.m, + test_strong_root.m, topological_sort.m, trees.txt, triangulate.c, + triangulate.m, triangulate_2Dlattice_demo.m, triangulate_test.m: + Initial import of code base from Kevin Murphy. + +2005-08-29 10:44 yozhik + + * graph/: README.txt, Old/best_first_elim_order.m, + Old/dag_to_jtree.m, Old/dfs.m, Old/dsep_test.m, + Old/mk_2D_lattice_slow.m, acyclic.m, assignEdgeNums.m, + best_first_elim_order.m, check_jtree_property.m, + check_triangulated.m, children.m, cliques_to_jtree.m, + cliques_to_strong_jtree.m, connected_graph.m, + dag_to_essential_graph.m, dfs.m, dfs_test.m, dijkstra.m, + family.m, graph_separated.m, graph_to_jtree.m, + min_subtree_con_nodes.m, minimum_spanning_tree.m, minspan.m, + mk_2D_lattice.m, mk_2D_lattice_slow.m, mk_adj_mat.m, + mk_adjmat_chain.m, mk_all_dags.m, mk_nbrs_of_dag.m, + mk_nbrs_of_digraph.m, mk_nbrs_of_digraph_broken.m, + mk_nbrs_of_digraph_not_vectorized.m, mk_rnd_dag.m, + mk_rnd_dag_given_edge_prob.m, mk_rooted_tree.m, mk_undirected.m, + moralize.m, neighbors.m, parents.m, pred2path.m, + reachability_graph.m, scc.m, strong_elim_order.m, test.m, + test_strong_root.m, topological_sort.m, trees.txt, triangulate.c, + triangulate.m, triangulate_2Dlattice_demo.m, triangulate_test.m: + Initial revision + +2005-08-26 18:08 yozhik + + * KPMtools/fullfileKPM.m: Initial import of code base from Kevin + Murphy. + +2005-08-26 18:08 yozhik + + * KPMtools/fullfileKPM.m: Initial revision + +2005-08-21 13:00 yozhik + + * + BNT/inference/static/@pearl_inf_engine/private/parallel_protocol.m: + Initial import of code base from Kevin Murphy. + +2005-08-21 13:00 yozhik + + * + BNT/inference/static/@pearl_inf_engine/private/parallel_protocol.m: + Initial revision + +2005-07-11 12:07 yozhik + + * KPMtools/plotcov2New.m: Initial import of code base from Kevin + Murphy. + +2005-07-11 12:07 yozhik + + * KPMtools/plotcov2New.m: Initial revision + +2005-07-06 12:32 yozhik + + * KPMtools/montageKPM2.m: Initial import of code base from Kevin + Murphy. + +2005-07-06 12:32 yozhik + + * KPMtools/montageKPM2.m: Initial revision + +2005-06-27 18:35 yozhik + + * KPMtools/montageKPM3.m: Initial import of code base from Kevin + Murphy. + +2005-06-27 18:35 yozhik + + * KPMtools/montageKPM3.m: Initial revision + +2005-06-27 18:30 yozhik + + * KPMtools/cell2matPad.m: Initial import of code base from Kevin + Murphy. + +2005-06-27 18:30 yozhik + + * KPMtools/cell2matPad.m: Initial revision + +2005-06-15 14:13 yozhik + + * BNT/CPDs/@gaussian_CPD/gaussian_CPD.m: Initial import of code + base from Kevin Murphy. + +2005-06-15 14:13 yozhik + + * BNT/CPDs/@gaussian_CPD/gaussian_CPD.m: Initial revision + +2005-06-08 18:56 yozhik + + * Kalman/testKalman.m: Initial import of code base from Kevin + Murphy. + +2005-06-08 18:56 yozhik + + * Kalman/testKalman.m: Initial revision + +2005-06-08 18:25 yozhik + + * HMM/: testHMM.m, fixed_lag_smoother_demo.m: Initial import of + code base from Kevin Murphy. + +2005-06-08 18:25 yozhik + + * HMM/: testHMM.m, fixed_lag_smoother_demo.m: Initial revision + +2005-06-08 18:22 yozhik + + * HMM/: README.txt, dhmm_em.m: Initial import of code base from + Kevin Murphy. + +2005-06-08 18:22 yozhik + + * HMM/: README.txt, dhmm_em.m: Initial revision + +2005-06-08 18:17 yozhik + + * HMM/fwdback.m: Initial import of code base from Kevin Murphy. + +2005-06-08 18:17 yozhik + + * HMM/fwdback.m: Initial revision + +2005-06-05 11:46 yozhik + + * KPMtools/: rectintLoopC.c, rectintLoopC.dll: Initial import of + code base from Kevin Murphy. + +2005-06-05 11:46 yozhik + + * KPMtools/: rectintLoopC.c, rectintLoopC.dll: Initial revision + +2005-06-01 12:39 yozhik + + * KPMtools/montageKPM.m: Initial import of code base from Kevin + Murphy. + +2005-06-01 12:39 yozhik + + * KPMtools/montageKPM.m: Initial revision + +2005-05-31 21:49 yozhik + + * KPMtools/initFigures.m: Initial import of code base from Kevin + Murphy. + +2005-05-31 21:49 yozhik + + * KPMtools/initFigures.m: Initial revision + +2005-05-31 11:19 yozhik + + * KPMstats/unidrndKPM.m: Initial import of code base from Kevin + Murphy. + +2005-05-31 11:19 yozhik + + * KPMstats/unidrndKPM.m: Initial revision + +2005-05-30 15:08 yozhik + + * KPMtools/filepartsLast.m: Initial import of code base from Kevin + Murphy. + +2005-05-30 15:08 yozhik + + * KPMtools/filepartsLast.m: Initial revision + +2005-05-29 23:01 yozhik + + * KPMtools/plotBox.m: Initial import of code base from Kevin + Murphy. + +2005-05-29 23:01 yozhik + + * KPMtools/plotBox.m: Initial revision + +2005-05-25 18:31 yozhik + + * KPMtools/plotColors.m: Initial import of code base from Kevin + Murphy. + +2005-05-25 18:31 yozhik + + * KPMtools/plotColors.m: Initial revision + +2005-05-25 12:11 yozhik + + * KPMtools/genpathKPM.m: Initial import of code base from Kevin + Murphy. + +2005-05-25 12:11 yozhik + + * KPMtools/genpathKPM.m: Initial revision + +2005-05-23 17:03 yozhik + + * netlab3.3/demhmc1.m: Initial import of code base from Kevin + Murphy. + +2005-05-23 17:03 yozhik + + * netlab3.3/demhmc1.m: Initial revision + +2005-05-23 16:44 yozhik + + * netlab3.3/gmminit.m: Initial import of code base from Kevin + Murphy. + +2005-05-23 16:44 yozhik + + * netlab3.3/gmminit.m: Initial revision + +2005-05-23 16:07 yozhik + + * netlab3.3/metrop.m: Initial import of code base from Kevin + Murphy. + +2005-05-23 16:07 yozhik + + * netlab3.3/metrop.m: Initial revision + +2005-05-22 23:23 yozhik + + * netlab3.3/demmet1.m: Initial import of code base from Kevin + Murphy. + +2005-05-22 23:23 yozhik + + * netlab3.3/demmet1.m: Initial revision + +2005-05-22 16:32 yozhik + + * KPMstats/: dirichletrnd.m, dirichletpdf.m, test_dir.m, + multirnd.m, multipdf.m: Initial import of code base from Kevin + Murphy. + +2005-05-22 16:32 yozhik + + * KPMstats/: dirichletrnd.m, dirichletpdf.m, test_dir.m, + multirnd.m, multipdf.m: Initial revision + +2005-05-13 13:52 yozhik + + * KPMtools/: asort.m, dirKPM.m: Initial import of code base from + Kevin Murphy. + +2005-05-13 13:52 yozhik + + * KPMtools/: asort.m, dirKPM.m: Initial revision + +2005-05-09 18:32 yozhik + + * netlab3.3/dem2ddat.m: Initial import of code base from Kevin + Murphy. + +2005-05-09 18:32 yozhik + + * netlab3.3/dem2ddat.m: Initial revision + +2005-05-09 15:20 yozhik + + * KPMtools/: mkdirKPM.m, optimalMatching.m, optimalMatchingTest.m, + subsets1.m: Initial import of code base from Kevin Murphy. + +2005-05-09 15:20 yozhik + + * KPMtools/: mkdirKPM.m, optimalMatching.m, optimalMatchingTest.m, + subsets1.m: Initial revision + +2005-05-09 09:47 yozhik + + * KPMtools/bipartiteMatchingDemo.m: Initial import of code base + from Kevin Murphy. + +2005-05-09 09:47 yozhik + + * KPMtools/bipartiteMatchingDemo.m: Initial revision + +2005-05-08 22:25 yozhik + + * KPMtools/bipartiteMatchingIntProg.m: Initial import of code base + from Kevin Murphy. + +2005-05-08 22:25 yozhik + + * KPMtools/bipartiteMatchingIntProg.m: Initial revision + +2005-05-08 21:45 yozhik + + * KPMtools/bipartiteMatchingDemoPlot.m: Initial import of code base + from Kevin Murphy. + +2005-05-08 21:45 yozhik + + * KPMtools/bipartiteMatchingDemoPlot.m: Initial revision + +2005-05-08 19:55 yozhik + + * KPMtools/subsetsFixedSize.m: Initial import of code base from + Kevin Murphy. + +2005-05-08 19:55 yozhik + + * KPMtools/subsetsFixedSize.m: Initial revision + +2005-05-08 15:48 yozhik + + * KPMtools/centeringMatrix.m: Initial import of code base from + Kevin Murphy. + +2005-05-08 15:48 yozhik + + * KPMtools/centeringMatrix.m: Initial revision + +2005-05-08 10:51 yozhik + + * netlab3.3/demgmm1.m: Initial import of code base from Kevin + Murphy. + +2005-05-08 10:51 yozhik + + * netlab3.3/demgmm1.m: Initial revision + +2005-05-06 18:09 yozhik + + * BNT/add_BNT_to_path.m: Initial import of code base from Kevin + Murphy. + +2005-05-06 18:09 yozhik + + * BNT/add_BNT_to_path.m: Initial revision + +2005-05-03 21:35 yozhik + + * KPMstats/standardize.m: Initial import of code base from Kevin + Murphy. + +2005-05-03 21:35 yozhik + + * KPMstats/standardize.m: Initial revision + +2005-05-03 13:18 yozhik + + * KPMstats/histCmpChi2.m: Initial import of code base from Kevin + Murphy. + +2005-05-03 13:18 yozhik + + * KPMstats/histCmpChi2.m: Initial revision + +2005-05-03 12:01 yozhik + + * KPMtools/strsplit.m: Initial import of code base from Kevin + Murphy. + +2005-05-03 12:01 yozhik + + * KPMtools/strsplit.m: Initial revision + +2005-05-02 13:19 yozhik + + * KPMtools/hsvKPM.m: Initial import of code base from Kevin Murphy. + +2005-05-02 13:19 yozhik + + * KPMtools/hsvKPM.m: Initial revision + +2005-04-27 11:34 yozhik + + * BNT/potentials/@dpot/: subsasgn.m, subsref.m: Initial import of + code base from Kevin Murphy. + +2005-04-27 11:34 yozhik + + * BNT/potentials/@dpot/: subsasgn.m, subsref.m: Initial revision + +2005-04-27 10:58 yozhik + + * KPMtools/mahal2conf.m, nethelp3.3/conffig.htm, + nethelp3.3/confmat.htm, nethelp3.3/conjgrad.htm, + nethelp3.3/consist.htm, nethelp3.3/convertoldnet.htm, + nethelp3.3/datread.htm, nethelp3.3/datwrite.htm, + nethelp3.3/dem2ddat.htm, nethelp3.3/demard.htm, + nethelp3.3/demev1.htm, nethelp3.3/demev2.htm, + nethelp3.3/demev3.htm, nethelp3.3/demgauss.htm, + nethelp3.3/demglm1.htm, nethelp3.3/demglm2.htm, + nethelp3.3/demgmm1.htm, nethelp3.3/demgmm2.htm, + nethelp3.3/demgmm3.htm, nethelp3.3/demgmm4.htm, + nethelp3.3/demgmm5.htm, nethelp3.3/demgp.htm, + nethelp3.3/demgpard.htm, nethelp3.3/demgpot.htm, + nethelp3.3/demgtm1.htm, nethelp3.3/demgtm2.htm, + nethelp3.3/demhint.htm, nethelp3.3/demhmc1.htm, + nethelp3.3/demhmc2.htm, nethelp3.3/demhmc3.htm, + nethelp3.3/demkmn1.htm, nethelp3.3/demknn1.htm, + nethelp3.3/demmdn1.htm, nethelp3.3/demmet1.htm, + nethelp3.3/demmlp1.htm, nethelp3.3/demmlp2.htm, + nethelp3.3/demnlab.htm, nethelp3.3/demns1.htm, + nethelp3.3/demolgd1.htm, nethelp3.3/demopt1.htm, + nethelp3.3/dempot.htm, nethelp3.3/demprgp.htm, + nethelp3.3/demprior.htm, nethelp3.3/demrbf1.htm, + nethelp3.3/demsom1.htm, nethelp3.3/demtrain.htm, + nethelp3.3/dist2.htm, nethelp3.3/eigdec.htm, + nethelp3.3/errbayes.htm, nethelp3.3/evidence.htm, + nethelp3.3/fevbayes.htm, nethelp3.3/gauss.htm, + nethelp3.3/gbayes.htm, nethelp3.3/glm.htm, + nethelp3.3/glmderiv.htm, nethelp3.3/glmerr.htm, + nethelp3.3/glmevfwd.htm, nethelp3.3/glmfwd.htm, + nethelp3.3/glmgrad.htm, nethelp3.3/glmhess.htm, + nethelp3.3/glminit.htm, nethelp3.3/glmpak.htm, + nethelp3.3/glmtrain.htm, nethelp3.3/glmunpak.htm, + nethelp3.3/gmm.htm, nethelp3.3/gmmactiv.htm, + nethelp3.3/gmmem.htm, nethelp3.3/gmminit.htm, + nethelp3.3/gmmpak.htm, nethelp3.3/gmmpost.htm, + nethelp3.3/gmmprob.htm, nethelp3.3/gmmsamp.htm, + nethelp3.3/gmmunpak.htm, nethelp3.3/gp.htm, + nethelp3.3/gpcovar.htm, nethelp3.3/gpcovarf.htm, + nethelp3.3/gpcovarp.htm, nethelp3.3/gperr.htm, + nethelp3.3/gpfwd.htm, nethelp3.3/gpgrad.htm, + nethelp3.3/gpinit.htm, nethelp3.3/gppak.htm, + nethelp3.3/gpunpak.htm, nethelp3.3/gradchek.htm, + nethelp3.3/graddesc.htm, nethelp3.3/gsamp.htm, + nethelp3.3/gtm.htm, nethelp3.3/gtmem.htm, nethelp3.3/gtmfwd.htm, + nethelp3.3/gtminit.htm, nethelp3.3/gtmlmean.htm, + nethelp3.3/gtmlmode.htm, nethelp3.3/gtmmag.htm, + nethelp3.3/gtmpost.htm, nethelp3.3/gtmprob.htm, + nethelp3.3/hbayes.htm, nethelp3.3/hesschek.htm, + nethelp3.3/hintmat.htm, nethelp3.3/hinton.htm, + nethelp3.3/histp.htm, nethelp3.3/hmc.htm, nethelp3.3/index.htm, + nethelp3.3/kmeans.htm, nethelp3.3/knn.htm, nethelp3.3/knnfwd.htm, + nethelp3.3/linef.htm, nethelp3.3/linemin.htm, + nethelp3.3/maxitmess.htm, nethelp3.3/mdn.htm, + nethelp3.3/mdn2gmm.htm, nethelp3.3/mdndist2.htm, + nethelp3.3/mdnerr.htm, nethelp3.3/mdnfwd.htm, + nethelp3.3/mdngrad.htm, nethelp3.3/mdninit.htm, + nethelp3.3/mdnpak.htm, nethelp3.3/mdnpost.htm, + nethelp3.3/mdnprob.htm, nethelp3.3/mdnunpak.htm, + nethelp3.3/metrop.htm, nethelp3.3/minbrack.htm, + nethelp3.3/mlp.htm, nethelp3.3/mlpbkp.htm, + nethelp3.3/mlpderiv.htm, nethelp3.3/mlperr.htm, + nethelp3.3/mlpevfwd.htm, nethelp3.3/mlpfwd.htm, + nethelp3.3/mlpgrad.htm, nethelp3.3/mlphdotv.htm, + nethelp3.3/mlphess.htm, nethelp3.3/mlphint.htm, + nethelp3.3/mlpinit.htm, nethelp3.3/mlppak.htm, + nethelp3.3/mlpprior.htm, nethelp3.3/mlptrain.htm, + nethelp3.3/mlpunpak.htm, nethelp3.3/netderiv.htm, + nethelp3.3/neterr.htm, nethelp3.3/netevfwd.htm, + nethelp3.3/netgrad.htm, nethelp3.3/nethelp3.3.zip, + nethelp3.3/nethess.htm, nethelp3.3/netinit.htm, + nethelp3.3/netopt.htm, nethelp3.3/netpak.htm, + nethelp3.3/netunpak.htm, nethelp3.3/olgd.htm, nethelp3.3/pca.htm, + nethelp3.3/plotmat.htm, nethelp3.3/ppca.htm, + nethelp3.3/quasinew.htm, nethelp3.3/rbf.htm, + nethelp3.3/rbfbkp.htm, nethelp3.3/rbfderiv.htm, + nethelp3.3/rbferr.htm, nethelp3.3/rbfevfwd.htm, + nethelp3.3/rbffwd.htm, nethelp3.3/rbfgrad.htm, + nethelp3.3/rbfhess.htm, nethelp3.3/rbfjacob.htm, + nethelp3.3/rbfpak.htm, nethelp3.3/rbfprior.htm, + nethelp3.3/rbfsetbf.htm, nethelp3.3/rbfsetfw.htm, + nethelp3.3/rbftrain.htm, nethelp3.3/rbfunpak.htm, + nethelp3.3/rosegrad.htm, nethelp3.3/rosen.htm, + nethelp3.3/scg.htm, nethelp3.3/som.htm, nethelp3.3/somfwd.htm, + nethelp3.3/sompak.htm, nethelp3.3/somtrain.htm, + nethelp3.3/somunpak.htm, netlab3.3/Contents.m, netlab3.3/LICENSE, + netlab3.3/conffig.m, netlab3.3/confmat.m, netlab3.3/conjgrad.m, + netlab3.3/consist.m, netlab3.3/convertoldnet.m, + netlab3.3/datread.m, netlab3.3/datwrite.m, netlab3.3/demard.m, + netlab3.3/demev1.m, netlab3.3/demev2.m, netlab3.3/demev3.m, + netlab3.3/demgauss.m, netlab3.3/demglm1.m, netlab3.3/demglm2.m, + netlab3.3/demgmm2.m, netlab3.3/demgmm3.m, netlab3.3/demgmm4.m, + netlab3.3/demgmm5.m, netlab3.3/demgp.m, netlab3.3/demgpard.m, + netlab3.3/demgpot.m, netlab3.3/demgtm1.m, netlab3.3/demgtm2.m, + netlab3.3/demhint.m, netlab3.3/demhmc2.m, netlab3.3/demhmc3.m, + netlab3.3/demkmn1.m, netlab3.3/demknn1.m, netlab3.3/demmdn1.m, + netlab3.3/demmlp1.m, netlab3.3/demmlp2.m, netlab3.3/demnlab.m, + netlab3.3/demns1.m, netlab3.3/demolgd1.m, netlab3.3/demopt1.m, + netlab3.3/dempot.m, netlab3.3/demprgp.m, netlab3.3/demprior.m, + netlab3.3/demrbf1.m, netlab3.3/demsom1.m, netlab3.3/demtrain.m, + netlab3.3/dist2.m, netlab3.3/eigdec.m, netlab3.3/errbayes.m, + netlab3.3/evidence.m, netlab3.3/fevbayes.m, netlab3.3/gauss.m, + netlab3.3/gbayes.m, netlab3.3/glm.m, netlab3.3/glmderiv.m, + netlab3.3/glmerr.m, netlab3.3/glmevfwd.m, netlab3.3/glmfwd.m, + netlab3.3/glmgrad.m, netlab3.3/glmhess.m, netlab3.3/glminit.m, + netlab3.3/glmpak.m, netlab3.3/glmtrain.m, netlab3.3/glmunpak.m, + netlab3.3/gmm.m, netlab3.3/gmmactiv.m, netlab3.3/gmmem.m, + netlab3.3/gmmpak.m, netlab3.3/gmmpost.m, netlab3.3/gmmprob.m, + netlab3.3/gmmsamp.m, netlab3.3/gmmunpak.m, netlab3.3/gp.m, + netlab3.3/gpcovar.m, netlab3.3/gpcovarf.m, netlab3.3/gpcovarp.m, + netlab3.3/gperr.m, netlab3.3/gpfwd.m, netlab3.3/gpgrad.m, + netlab3.3/gpinit.m, netlab3.3/gppak.m, netlab3.3/gpunpak.m, + netlab3.3/gradchek.m, netlab3.3/graddesc.m, netlab3.3/gsamp.m, + netlab3.3/gtm.m, netlab3.3/gtmem.m, netlab3.3/gtmfwd.m, + netlab3.3/gtminit.m, netlab3.3/gtmlmean.m, netlab3.3/gtmlmode.m, + netlab3.3/gtmmag.m, netlab3.3/gtmpost.m, netlab3.3/gtmprob.m, + netlab3.3/hbayes.m, netlab3.3/hesschek.m, netlab3.3/hintmat.m, + netlab3.3/hinton.m, netlab3.3/histp.m, netlab3.3/hmc.m, + netlab3.3/kmeansNetlab.m, netlab3.3/knn.m, netlab3.3/knnfwd.m, + netlab3.3/linef.m, netlab3.3/linemin.m, netlab3.3/maxitmess.m, + netlab3.3/mdn.m, netlab3.3/mdn2gmm.m, netlab3.3/mdndist2.m, + netlab3.3/mdnerr.m, netlab3.3/mdnfwd.m, netlab3.3/mdngrad.m, + netlab3.3/mdninit.m, netlab3.3/mdnnet.mat, netlab3.3/mdnpak.m, + netlab3.3/mdnpost.m, netlab3.3/mdnprob.m, netlab3.3/mdnunpak.m, + netlab3.3/minbrack.m, netlab3.3/mlp.m, netlab3.3/mlpbkp.m, + netlab3.3/mlpderiv.m, netlab3.3/mlperr.m, netlab3.3/mlpevfwd.m, + netlab3.3/mlpfwd.m, netlab3.3/mlpgrad.m, netlab3.3/mlphdotv.m, + netlab3.3/mlphess.m, netlab3.3/mlphint.m, netlab3.3/mlpinit.m, + netlab3.3/mlppak.m, netlab3.3/mlpprior.m, netlab3.3/mlptrain.m, + netlab3.3/mlpunpak.m, netlab3.3/netderiv.m, netlab3.3/neterr.m, + netlab3.3/netevfwd.m, netlab3.3/netgrad.m, netlab3.3/nethess.m, + netlab3.3/netinit.m, netlab3.3/netlab3.3.zip, + netlab3.3/netlogo.mat, netlab3.3/netopt.m, netlab3.3/netpak.m, + netlab3.3/netunpak.m, netlab3.3/oilTrn.dat, netlab3.3/oilTst.dat, + netlab3.3/olgd.m, netlab3.3/pca.m, netlab3.3/plotmat.m, + netlab3.3/ppca.m, netlab3.3/quasinew.m, netlab3.3/rbf.m, + netlab3.3/rbfbkp.m, netlab3.3/rbfderiv.m, netlab3.3/rbferr.m, + netlab3.3/rbfevfwd.m, netlab3.3/rbffwd.m, netlab3.3/rbfgrad.m, + netlab3.3/rbfhess.m, netlab3.3/rbfjacob.m, netlab3.3/rbfpak.m, + netlab3.3/rbfprior.m, netlab3.3/rbfsetbf.m, netlab3.3/rbfsetfw.m, + netlab3.3/rbftrain.m, netlab3.3/rbfunpak.m, netlab3.3/rosegrad.m, + netlab3.3/rosen.m, netlab3.3/scg.m, netlab3.3/som.m, + netlab3.3/somfwd.m, netlab3.3/sompak.m, netlab3.3/somtrain.m, + netlab3.3/somunpak.m, netlab3.3/xor.dat, netlabKPM/README.txt, + netlabKPM/demgmm1_movie.m, netlabKPM/evidence_weighted.m, + netlabKPM/glmerr_weighted.m, netlabKPM/glmgrad_weighted.m, + netlabKPM/glmhess_weighted.m, netlabKPM/glmtrain_weighted.m, + netlabKPM/gmm1.avi, netlabKPM/gmmem2.m, + netlabKPM/gmmem_multi_restart.m, netlabKPM/kmeans_demo.m, + netlabKPM/mlperr_weighted.m, netlabKPM/mlpgrad_weighted.m, + netlabKPM/mlphdotv_weighted.m, netlabKPM/mlphess_weighted.m, + netlabKPM/neterr_weighted.m, netlabKPM/netgrad_weighted.m, + netlabKPM/nethess_weighted.m, netlabKPM/netopt_weighted.m, + netlabKPM/process_options.m: Initial import of code base from + Kevin Murphy. + +2005-04-27 10:58 yozhik + + * KPMtools/mahal2conf.m, nethelp3.3/conffig.htm, + nethelp3.3/confmat.htm, nethelp3.3/conjgrad.htm, + nethelp3.3/consist.htm, nethelp3.3/convertoldnet.htm, + nethelp3.3/datread.htm, nethelp3.3/datwrite.htm, + nethelp3.3/dem2ddat.htm, nethelp3.3/demard.htm, + nethelp3.3/demev1.htm, nethelp3.3/demev2.htm, + nethelp3.3/demev3.htm, nethelp3.3/demgauss.htm, + nethelp3.3/demglm1.htm, nethelp3.3/demglm2.htm, + nethelp3.3/demgmm1.htm, nethelp3.3/demgmm2.htm, + nethelp3.3/demgmm3.htm, nethelp3.3/demgmm4.htm, + nethelp3.3/demgmm5.htm, nethelp3.3/demgp.htm, + nethelp3.3/demgpard.htm, nethelp3.3/demgpot.htm, + nethelp3.3/demgtm1.htm, nethelp3.3/demgtm2.htm, + nethelp3.3/demhint.htm, nethelp3.3/demhmc1.htm, + nethelp3.3/demhmc2.htm, nethelp3.3/demhmc3.htm, + nethelp3.3/demkmn1.htm, nethelp3.3/demknn1.htm, + nethelp3.3/demmdn1.htm, nethelp3.3/demmet1.htm, + nethelp3.3/demmlp1.htm, nethelp3.3/demmlp2.htm, + nethelp3.3/demnlab.htm, nethelp3.3/demns1.htm, + nethelp3.3/demolgd1.htm, nethelp3.3/demopt1.htm, + nethelp3.3/dempot.htm, nethelp3.3/demprgp.htm, + nethelp3.3/demprior.htm, nethelp3.3/demrbf1.htm, + nethelp3.3/demsom1.htm, nethelp3.3/demtrain.htm, + nethelp3.3/dist2.htm, nethelp3.3/eigdec.htm, + nethelp3.3/errbayes.htm, nethelp3.3/evidence.htm, + nethelp3.3/fevbayes.htm, nethelp3.3/gauss.htm, + nethelp3.3/gbayes.htm, nethelp3.3/glm.htm, + nethelp3.3/glmderiv.htm, nethelp3.3/glmerr.htm, + nethelp3.3/glmevfwd.htm, nethelp3.3/glmfwd.htm, + nethelp3.3/glmgrad.htm, nethelp3.3/glmhess.htm, + nethelp3.3/glminit.htm, nethelp3.3/glmpak.htm, + nethelp3.3/glmtrain.htm, nethelp3.3/glmunpak.htm, + nethelp3.3/gmm.htm, nethelp3.3/gmmactiv.htm, + nethelp3.3/gmmem.htm, nethelp3.3/gmminit.htm, + nethelp3.3/gmmpak.htm, nethelp3.3/gmmpost.htm, + nethelp3.3/gmmprob.htm, nethelp3.3/gmmsamp.htm, + nethelp3.3/gmmunpak.htm, nethelp3.3/gp.htm, + nethelp3.3/gpcovar.htm, nethelp3.3/gpcovarf.htm, + nethelp3.3/gpcovarp.htm, nethelp3.3/gperr.htm, + nethelp3.3/gpfwd.htm, nethelp3.3/gpgrad.htm, + nethelp3.3/gpinit.htm, nethelp3.3/gppak.htm, + nethelp3.3/gpunpak.htm, nethelp3.3/gradchek.htm, + nethelp3.3/graddesc.htm, nethelp3.3/gsamp.htm, + nethelp3.3/gtm.htm, nethelp3.3/gtmem.htm, nethelp3.3/gtmfwd.htm, + nethelp3.3/gtminit.htm, nethelp3.3/gtmlmean.htm, + nethelp3.3/gtmlmode.htm, nethelp3.3/gtmmag.htm, + nethelp3.3/gtmpost.htm, nethelp3.3/gtmprob.htm, + nethelp3.3/hbayes.htm, nethelp3.3/hesschek.htm, + nethelp3.3/hintmat.htm, nethelp3.3/hinton.htm, + nethelp3.3/histp.htm, nethelp3.3/hmc.htm, nethelp3.3/index.htm, + nethelp3.3/kmeans.htm, nethelp3.3/knn.htm, nethelp3.3/knnfwd.htm, + nethelp3.3/linef.htm, nethelp3.3/linemin.htm, + nethelp3.3/maxitmess.htm, nethelp3.3/mdn.htm, + nethelp3.3/mdn2gmm.htm, nethelp3.3/mdndist2.htm, + nethelp3.3/mdnerr.htm, nethelp3.3/mdnfwd.htm, + nethelp3.3/mdngrad.htm, nethelp3.3/mdninit.htm, + nethelp3.3/mdnpak.htm, nethelp3.3/mdnpost.htm, + nethelp3.3/mdnprob.htm, nethelp3.3/mdnunpak.htm, + nethelp3.3/metrop.htm, nethelp3.3/minbrack.htm, + nethelp3.3/mlp.htm, nethelp3.3/mlpbkp.htm, + nethelp3.3/mlpderiv.htm, nethelp3.3/mlperr.htm, + nethelp3.3/mlpevfwd.htm, nethelp3.3/mlpfwd.htm, + nethelp3.3/mlpgrad.htm, nethelp3.3/mlphdotv.htm, + nethelp3.3/mlphess.htm, nethelp3.3/mlphint.htm, + nethelp3.3/mlpinit.htm, nethelp3.3/mlppak.htm, + nethelp3.3/mlpprior.htm, nethelp3.3/mlptrain.htm, + nethelp3.3/mlpunpak.htm, nethelp3.3/netderiv.htm, + nethelp3.3/neterr.htm, nethelp3.3/netevfwd.htm, + nethelp3.3/netgrad.htm, nethelp3.3/nethelp3.3.zip, + nethelp3.3/nethess.htm, nethelp3.3/netinit.htm, + nethelp3.3/netopt.htm, nethelp3.3/netpak.htm, + nethelp3.3/netunpak.htm, nethelp3.3/olgd.htm, nethelp3.3/pca.htm, + nethelp3.3/plotmat.htm, nethelp3.3/ppca.htm, + nethelp3.3/quasinew.htm, nethelp3.3/rbf.htm, + nethelp3.3/rbfbkp.htm, nethelp3.3/rbfderiv.htm, + nethelp3.3/rbferr.htm, nethelp3.3/rbfevfwd.htm, + nethelp3.3/rbffwd.htm, nethelp3.3/rbfgrad.htm, + nethelp3.3/rbfhess.htm, nethelp3.3/rbfjacob.htm, + nethelp3.3/rbfpak.htm, nethelp3.3/rbfprior.htm, + nethelp3.3/rbfsetbf.htm, nethelp3.3/rbfsetfw.htm, + nethelp3.3/rbftrain.htm, nethelp3.3/rbfunpak.htm, + nethelp3.3/rosegrad.htm, nethelp3.3/rosen.htm, + nethelp3.3/scg.htm, nethelp3.3/som.htm, nethelp3.3/somfwd.htm, + nethelp3.3/sompak.htm, nethelp3.3/somtrain.htm, + nethelp3.3/somunpak.htm, netlab3.3/Contents.m, netlab3.3/LICENSE, + netlab3.3/conffig.m, netlab3.3/confmat.m, netlab3.3/conjgrad.m, + netlab3.3/consist.m, netlab3.3/convertoldnet.m, + netlab3.3/datread.m, netlab3.3/datwrite.m, netlab3.3/demard.m, + netlab3.3/demev1.m, netlab3.3/demev2.m, netlab3.3/demev3.m, + netlab3.3/demgauss.m, netlab3.3/demglm1.m, netlab3.3/demglm2.m, + netlab3.3/demgmm2.m, netlab3.3/demgmm3.m, netlab3.3/demgmm4.m, + netlab3.3/demgmm5.m, netlab3.3/demgp.m, netlab3.3/demgpard.m, + netlab3.3/demgpot.m, netlab3.3/demgtm1.m, netlab3.3/demgtm2.m, + netlab3.3/demhint.m, netlab3.3/demhmc2.m, netlab3.3/demhmc3.m, + netlab3.3/demkmn1.m, netlab3.3/demknn1.m, netlab3.3/demmdn1.m, + netlab3.3/demmlp1.m, netlab3.3/demmlp2.m, netlab3.3/demnlab.m, + netlab3.3/demns1.m, netlab3.3/demolgd1.m, netlab3.3/demopt1.m, + netlab3.3/dempot.m, netlab3.3/demprgp.m, netlab3.3/demprior.m, + netlab3.3/demrbf1.m, netlab3.3/demsom1.m, netlab3.3/demtrain.m, + netlab3.3/dist2.m, netlab3.3/eigdec.m, netlab3.3/errbayes.m, + netlab3.3/evidence.m, netlab3.3/fevbayes.m, netlab3.3/gauss.m, + netlab3.3/gbayes.m, netlab3.3/glm.m, netlab3.3/glmderiv.m, + netlab3.3/glmerr.m, netlab3.3/glmevfwd.m, netlab3.3/glmfwd.m, + netlab3.3/glmgrad.m, netlab3.3/glmhess.m, netlab3.3/glminit.m, + netlab3.3/glmpak.m, netlab3.3/glmtrain.m, netlab3.3/glmunpak.m, + netlab3.3/gmm.m, netlab3.3/gmmactiv.m, netlab3.3/gmmem.m, + netlab3.3/gmmpak.m, netlab3.3/gmmpost.m, netlab3.3/gmmprob.m, + netlab3.3/gmmsamp.m, netlab3.3/gmmunpak.m, netlab3.3/gp.m, + netlab3.3/gpcovar.m, netlab3.3/gpcovarf.m, netlab3.3/gpcovarp.m, + netlab3.3/gperr.m, netlab3.3/gpfwd.m, netlab3.3/gpgrad.m, + netlab3.3/gpinit.m, netlab3.3/gppak.m, netlab3.3/gpunpak.m, + netlab3.3/gradchek.m, netlab3.3/graddesc.m, netlab3.3/gsamp.m, + netlab3.3/gtm.m, netlab3.3/gtmem.m, netlab3.3/gtmfwd.m, + netlab3.3/gtminit.m, netlab3.3/gtmlmean.m, netlab3.3/gtmlmode.m, + netlab3.3/gtmmag.m, netlab3.3/gtmpost.m, netlab3.3/gtmprob.m, + netlab3.3/hbayes.m, netlab3.3/hesschek.m, netlab3.3/hintmat.m, + netlab3.3/hinton.m, netlab3.3/histp.m, netlab3.3/hmc.m, + netlab3.3/kmeansNetlab.m, netlab3.3/knn.m, netlab3.3/knnfwd.m, + netlab3.3/linef.m, netlab3.3/linemin.m, netlab3.3/maxitmess.m, + netlab3.3/mdn.m, netlab3.3/mdn2gmm.m, netlab3.3/mdndist2.m, + netlab3.3/mdnerr.m, netlab3.3/mdnfwd.m, netlab3.3/mdngrad.m, + netlab3.3/mdninit.m, netlab3.3/mdnnet.mat, netlab3.3/mdnpak.m, + netlab3.3/mdnpost.m, netlab3.3/mdnprob.m, netlab3.3/mdnunpak.m, + netlab3.3/minbrack.m, netlab3.3/mlp.m, netlab3.3/mlpbkp.m, + netlab3.3/mlpderiv.m, netlab3.3/mlperr.m, netlab3.3/mlpevfwd.m, + netlab3.3/mlpfwd.m, netlab3.3/mlpgrad.m, netlab3.3/mlphdotv.m, + netlab3.3/mlphess.m, netlab3.3/mlphint.m, netlab3.3/mlpinit.m, + netlab3.3/mlppak.m, netlab3.3/mlpprior.m, netlab3.3/mlptrain.m, + netlab3.3/mlpunpak.m, netlab3.3/netderiv.m, netlab3.3/neterr.m, + netlab3.3/netevfwd.m, netlab3.3/netgrad.m, netlab3.3/nethess.m, + netlab3.3/netinit.m, netlab3.3/netlab3.3.zip, + netlab3.3/netlogo.mat, netlab3.3/netopt.m, netlab3.3/netpak.m, + netlab3.3/netunpak.m, netlab3.3/oilTrn.dat, netlab3.3/oilTst.dat, + netlab3.3/olgd.m, netlab3.3/pca.m, netlab3.3/plotmat.m, + netlab3.3/ppca.m, netlab3.3/quasinew.m, netlab3.3/rbf.m, + netlab3.3/rbfbkp.m, netlab3.3/rbfderiv.m, netlab3.3/rbferr.m, + netlab3.3/rbfevfwd.m, netlab3.3/rbffwd.m, netlab3.3/rbfgrad.m, + netlab3.3/rbfhess.m, netlab3.3/rbfjacob.m, netlab3.3/rbfpak.m, + netlab3.3/rbfprior.m, netlab3.3/rbfsetbf.m, netlab3.3/rbfsetfw.m, + netlab3.3/rbftrain.m, netlab3.3/rbfunpak.m, netlab3.3/rosegrad.m, + netlab3.3/rosen.m, netlab3.3/scg.m, netlab3.3/som.m, + netlab3.3/somfwd.m, netlab3.3/sompak.m, netlab3.3/somtrain.m, + netlab3.3/somunpak.m, netlab3.3/xor.dat, netlabKPM/README.txt, + netlabKPM/demgmm1_movie.m, netlabKPM/evidence_weighted.m, + netlabKPM/glmerr_weighted.m, netlabKPM/glmgrad_weighted.m, + netlabKPM/glmhess_weighted.m, netlabKPM/glmtrain_weighted.m, + netlabKPM/gmm1.avi, netlabKPM/gmmem2.m, + netlabKPM/gmmem_multi_restart.m, netlabKPM/kmeans_demo.m, + netlabKPM/mlperr_weighted.m, netlabKPM/mlpgrad_weighted.m, + netlabKPM/mlphdotv_weighted.m, netlabKPM/mlphess_weighted.m, + netlabKPM/neterr_weighted.m, netlabKPM/netgrad_weighted.m, + netlabKPM/nethess_weighted.m, netlabKPM/netopt_weighted.m, + netlabKPM/process_options.m: Initial revision + +2005-04-25 19:29 yozhik + + * KPMstats/KLgauss.m, KPMstats/README.txt, KPMstats/beta_sample.m, + KPMstats/chisquared_histo.m, KPMstats/chisquared_prob.m, + KPMstats/chisquared_readme.txt, KPMstats/chisquared_table.m, + KPMstats/clg_Mstep.m, KPMstats/clg_Mstep_simple.m, + KPMstats/clg_prob.m, KPMstats/condGaussToJoint.m, + KPMstats/cond_indep_fisher_z.m, + KPMstats/condgaussTrainObserved.m, KPMstats/condgauss_sample.m, + KPMstats/convertBinaryLabels.m, KPMstats/cwr_demo.m, + KPMstats/cwr_em.m, KPMstats/cwr_predict.m, KPMstats/cwr_prob.m, + KPMstats/cwr_readme.txt, KPMstats/cwr_test.m, + KPMstats/dirichlet_sample.m, KPMstats/distchck.m, + KPMstats/eigdec.m, KPMstats/est_transmat.m, + KPMstats/fit_paritioned_model_testfn.m, + KPMstats/fit_partitioned_model.m, KPMstats/gamma_sample.m, + KPMstats/gaussian_prob.m, KPMstats/gaussian_sample.m, + KPMstats/linear_regression.m, KPMstats/logist2.m, + KPMstats/logist2Apply.m, KPMstats/logist2ApplyRegularized.m, + KPMstats/logist2Fit.m, KPMstats/logist2FitRegularized.m, + KPMstats/logistK.m, KPMstats/logistK_eval.m, + KPMstats/marginalize_gaussian.m, KPMstats/matrix_T_pdf.m, + KPMstats/matrix_normal_pdf.m, KPMstats/mc_stat_distrib.m, + KPMstats/mixgauss_Mstep.m, KPMstats/mixgauss_classifier_apply.m, + KPMstats/mixgauss_classifier_train.m, KPMstats/mixgauss_em.m, + KPMstats/mixgauss_init.m, KPMstats/mixgauss_prob.m, + KPMstats/mixgauss_prob_test.m, KPMstats/mixgauss_sample.m, + KPMstats/mkPolyFvec.m, KPMstats/mk_unit_norm.m, + KPMstats/multinomial_prob.m, KPMstats/multinomial_sample.m, + KPMstats/normal_coef.m, KPMstats/partial_corr_coef.m, + KPMstats/parzen.m, KPMstats/parzenC.c, KPMstats/parzenC.dll, + KPMstats/parzenC.mexglx, KPMstats/parzenC_test.m, + KPMstats/parzen_fit_select_unif.m, KPMstats/pca.m, + KPMstats/rndcheck.m, KPMstats/sample.m, + KPMstats/sample_discrete.m, KPMstats/sample_gaussian.m, + KPMstats/student_t_logprob.m, KPMstats/student_t_prob.m, + KPMstats/unif_discrete_sample.m, KPMstats/weightedRegression.m, + KPMtools/README.txt, KPMtools/approx_unique.m, + KPMtools/approxeq.m, KPMtools/argmax.m, KPMtools/argmin.m, + KPMtools/assert.m, KPMtools/assignEdgeNums.m, + KPMtools/assign_cols.m, KPMtools/axis_pct.m, KPMtools/block.m, + KPMtools/cell2num.m, KPMtools/chi2inv.m, KPMtools/choose.m, + KPMtools/collapse_mog.m, KPMtools/colmult.c, + KPMtools/colmult.mexglx, KPMtools/computeROC.m, + KPMtools/compute_counts.m, KPMtools/conf2mahal.m, + KPMtools/cross_entropy.m, KPMtools/div.m, KPMtools/draw_circle.m, + KPMtools/draw_ellipse.m, KPMtools/draw_ellipse_axes.m, + KPMtools/em_converged.m, KPMtools/entropy.m, + KPMtools/exportfig.m, KPMtools/extend_domain_table.m, + KPMtools/factorial.m, KPMtools/find_equiv_posns.m, + KPMtools/hash_add.m, KPMtools/hash_del.m, KPMtools/hash_lookup.m, + KPMtools/hungarian.m, KPMtools/image_rgb.m, + KPMtools/imresizeAspect.m, KPMtools/ind2subv.c, + KPMtools/ind2subv.m, KPMtools/installC_KPMtools.m, + KPMtools/is_psd.m, KPMtools/is_stochastic.m, + KPMtools/isemptycell.m, KPMtools/isposdef.m, KPMtools/isscalar.m, + KPMtools/isvector.m, KPMtools/junk.c, KPMtools/loadcell.m, + KPMtools/logb.m, KPMtools/logdet.m, KPMtools/logsum.m, + KPMtools/logsum_simple.m, KPMtools/logsum_test.m, + KPMtools/logsumexp.m, KPMtools/logsumexpv.m, + KPMtools/marg_table.m, KPMtools/marginalize_table.m, + KPMtools/matprint.m, KPMtools/max_mult.c, KPMtools/max_mult.m, + KPMtools/mexutil.c, KPMtools/mexutil.h, + KPMtools/mk_multi_index.m, KPMtools/mk_stochastic.m, + KPMtools/mult_by_table.m, KPMtools/myintersect.m, + KPMtools/myismember.m, KPMtools/myones.m, KPMtools/myplot.m, + KPMtools/myrand.m, KPMtools/myrepmat.m, KPMtools/myreshape.m, + KPMtools/mysetdiff.m, KPMtools/mysize.m, KPMtools/mysubset.m, + KPMtools/mysymsetdiff.m, KPMtools/bipartiteMatchingHungarian.m, + KPMtools/myunion.m, KPMtools/nchoose2.m, KPMtools/ncols.m, + KPMtools/nonmaxsup.m, KPMtools/normalise.m, + KPMtools/normaliseC.c, KPMtools/normaliseC.dll, + KPMtools/normalize.m, KPMtools/nrows.m, KPMtools/num2strcell.m, + KPMtools/partitionData.m, KPMtools/partition_matrix_vec.m, + KPMtools/pca_netlab.m, KPMtools/pick.m, KPMtools/plotROC.m, + KPMtools/plotROCkpm.m, KPMtools/plot_axis_thru_origin.m, + KPMtools/plot_ellipse.m, KPMtools/plot_matrix.m, + KPMtools/plot_polygon.m, KPMtools/plotcov2.m, + KPMtools/plotcov3.m, KPMtools/plotgauss1d.m, + KPMtools/plotgauss2d.m, KPMtools/plotgauss2d_old.m, + KPMtools/polygon_area.m, KPMtools/polygon_centroid.m, + KPMtools/polygon_intersect.m, KPMtools/previewfig.m, + KPMtools/process_options.m, KPMtools/rand_psd.m, + KPMtools/rectintC.m, KPMtools/rectintLoopC.mexglx, + KPMtools/rectintSparse.m, KPMtools/rectintSparseC.m, + KPMtools/rectintSparseLoopC.c, KPMtools/rectintSparseLoopC.dll, + KPMtools/repmatC.c, KPMtools/repmatC.dll, + KPMtools/repmatC.mexglx, KPMtools/rgb2grayKPM.m, + KPMtools/rnd_partition.m, KPMtools/rotate_xlabel.m, + KPMtools/safeStr.m, KPMtools/sampleUniformInts.m, + KPMtools/sample_discrete.m, KPMtools/set_xtick_label.m, + KPMtools/set_xtick_label_demo.m, KPMtools/setdiag.m, + KPMtools/softeye.m, KPMtools/sort_evec.m, + KPMtools/splitLongSeqIntoManyShort.m, KPMtools/sprintf_intvec.m, + KPMtools/sqdist.m, KPMtools/strmatch_multi.m, + KPMtools/strmatch_substr.m, KPMtools/subplot2.m, + KPMtools/subplot3.m, KPMtools/subsets.m, KPMtools/subv2ind.c, + KPMtools/subv2ind.m, KPMtools/sumv.m, KPMtools/suptitle.m, + KPMtools/unaryEncoding.m, KPMtools/wrap.m, + KPMtools/xticklabel_rotate90.m, KPMtools/zipload.m, + KPMtools/zipsave.m: Initial import of code base from Kevin + Murphy. + +2005-04-25 19:29 yozhik + + * KPMstats/KLgauss.m, KPMstats/README.txt, KPMstats/beta_sample.m, + KPMstats/chisquared_histo.m, KPMstats/chisquared_prob.m, + KPMstats/chisquared_readme.txt, KPMstats/chisquared_table.m, + KPMstats/clg_Mstep.m, KPMstats/clg_Mstep_simple.m, + KPMstats/clg_prob.m, KPMstats/condGaussToJoint.m, + KPMstats/cond_indep_fisher_z.m, + KPMstats/condgaussTrainObserved.m, KPMstats/condgauss_sample.m, + KPMstats/convertBinaryLabels.m, KPMstats/cwr_demo.m, + KPMstats/cwr_em.m, KPMstats/cwr_predict.m, KPMstats/cwr_prob.m, + KPMstats/cwr_readme.txt, KPMstats/cwr_test.m, + KPMstats/dirichlet_sample.m, KPMstats/distchck.m, + KPMstats/eigdec.m, KPMstats/est_transmat.m, + KPMstats/fit_paritioned_model_testfn.m, + KPMstats/fit_partitioned_model.m, KPMstats/gamma_sample.m, + KPMstats/gaussian_prob.m, KPMstats/gaussian_sample.m, + KPMstats/linear_regression.m, KPMstats/logist2.m, + KPMstats/logist2Apply.m, KPMstats/logist2ApplyRegularized.m, + KPMstats/logist2Fit.m, KPMstats/logist2FitRegularized.m, + KPMstats/logistK.m, KPMstats/logistK_eval.m, + KPMstats/marginalize_gaussian.m, KPMstats/matrix_T_pdf.m, + KPMstats/matrix_normal_pdf.m, KPMstats/mc_stat_distrib.m, + KPMstats/mixgauss_Mstep.m, KPMstats/mixgauss_classifier_apply.m, + KPMstats/mixgauss_classifier_train.m, KPMstats/mixgauss_em.m, + KPMstats/mixgauss_init.m, KPMstats/mixgauss_prob.m, + KPMstats/mixgauss_prob_test.m, KPMstats/mixgauss_sample.m, + KPMstats/mkPolyFvec.m, KPMstats/mk_unit_norm.m, + KPMstats/multinomial_prob.m, KPMstats/multinomial_sample.m, + KPMstats/normal_coef.m, KPMstats/partial_corr_coef.m, + KPMstats/parzen.m, KPMstats/parzenC.c, KPMstats/parzenC.dll, + KPMstats/parzenC.mexglx, KPMstats/parzenC_test.m, + KPMstats/parzen_fit_select_unif.m, KPMstats/pca.m, + KPMstats/rndcheck.m, KPMstats/sample.m, + KPMstats/sample_discrete.m, KPMstats/sample_gaussian.m, + KPMstats/student_t_logprob.m, KPMstats/student_t_prob.m, + KPMstats/unif_discrete_sample.m, KPMstats/weightedRegression.m, + KPMtools/README.txt, KPMtools/approx_unique.m, + KPMtools/approxeq.m, KPMtools/argmax.m, KPMtools/argmin.m, + KPMtools/assert.m, KPMtools/assignEdgeNums.m, + KPMtools/assign_cols.m, KPMtools/axis_pct.m, KPMtools/block.m, + KPMtools/cell2num.m, KPMtools/chi2inv.m, KPMtools/choose.m, + KPMtools/collapse_mog.m, KPMtools/colmult.c, + KPMtools/colmult.mexglx, KPMtools/computeROC.m, + KPMtools/compute_counts.m, KPMtools/conf2mahal.m, + KPMtools/cross_entropy.m, KPMtools/div.m, KPMtools/draw_circle.m, + KPMtools/draw_ellipse.m, KPMtools/draw_ellipse_axes.m, + KPMtools/em_converged.m, KPMtools/entropy.m, + KPMtools/exportfig.m, KPMtools/extend_domain_table.m, + KPMtools/factorial.m, KPMtools/find_equiv_posns.m, + KPMtools/hash_add.m, KPMtools/hash_del.m, KPMtools/hash_lookup.m, + KPMtools/hungarian.m, KPMtools/image_rgb.m, + KPMtools/imresizeAspect.m, KPMtools/ind2subv.c, + KPMtools/ind2subv.m, KPMtools/installC_KPMtools.m, + KPMtools/is_psd.m, KPMtools/is_stochastic.m, + KPMtools/isemptycell.m, KPMtools/isposdef.m, KPMtools/isscalar.m, + KPMtools/isvector.m, KPMtools/junk.c, KPMtools/loadcell.m, + KPMtools/logb.m, KPMtools/logdet.m, KPMtools/logsum.m, + KPMtools/logsum_simple.m, KPMtools/logsum_test.m, + KPMtools/logsumexp.m, KPMtools/logsumexpv.m, + KPMtools/marg_table.m, KPMtools/marginalize_table.m, + KPMtools/matprint.m, KPMtools/max_mult.c, KPMtools/max_mult.m, + KPMtools/mexutil.c, KPMtools/mexutil.h, + KPMtools/mk_multi_index.m, KPMtools/mk_stochastic.m, + KPMtools/mult_by_table.m, KPMtools/myintersect.m, + KPMtools/myismember.m, KPMtools/myones.m, KPMtools/myplot.m, + KPMtools/myrand.m, KPMtools/myrepmat.m, KPMtools/myreshape.m, + KPMtools/mysetdiff.m, KPMtools/mysize.m, KPMtools/mysubset.m, + KPMtools/mysymsetdiff.m, KPMtools/bipartiteMatchingHungarian.m, + KPMtools/myunion.m, KPMtools/nchoose2.m, KPMtools/ncols.m, + KPMtools/nonmaxsup.m, KPMtools/normalise.m, + KPMtools/normaliseC.c, KPMtools/normaliseC.dll, + KPMtools/normalize.m, KPMtools/nrows.m, KPMtools/num2strcell.m, + KPMtools/partitionData.m, KPMtools/partition_matrix_vec.m, + KPMtools/pca_netlab.m, KPMtools/pick.m, KPMtools/plotROC.m, + KPMtools/plotROCkpm.m, KPMtools/plot_axis_thru_origin.m, + KPMtools/plot_ellipse.m, KPMtools/plot_matrix.m, + KPMtools/plot_polygon.m, KPMtools/plotcov2.m, + KPMtools/plotcov3.m, KPMtools/plotgauss1d.m, + KPMtools/plotgauss2d.m, KPMtools/plotgauss2d_old.m, + KPMtools/polygon_area.m, KPMtools/polygon_centroid.m, + KPMtools/polygon_intersect.m, KPMtools/previewfig.m, + KPMtools/process_options.m, KPMtools/rand_psd.m, + KPMtools/rectintC.m, KPMtools/rectintLoopC.mexglx, + KPMtools/rectintSparse.m, KPMtools/rectintSparseC.m, + KPMtools/rectintSparseLoopC.c, KPMtools/rectintSparseLoopC.dll, + KPMtools/repmatC.c, KPMtools/repmatC.dll, + KPMtools/repmatC.mexglx, KPMtools/rgb2grayKPM.m, + KPMtools/rnd_partition.m, KPMtools/rotate_xlabel.m, + KPMtools/safeStr.m, KPMtools/sampleUniformInts.m, + KPMtools/sample_discrete.m, KPMtools/set_xtick_label.m, + KPMtools/set_xtick_label_demo.m, KPMtools/setdiag.m, + KPMtools/softeye.m, KPMtools/sort_evec.m, + KPMtools/splitLongSeqIntoManyShort.m, KPMtools/sprintf_intvec.m, + KPMtools/sqdist.m, KPMtools/strmatch_multi.m, + KPMtools/strmatch_substr.m, KPMtools/subplot2.m, + KPMtools/subplot3.m, KPMtools/subsets.m, KPMtools/subv2ind.c, + KPMtools/subv2ind.m, KPMtools/sumv.m, KPMtools/suptitle.m, + KPMtools/unaryEncoding.m, KPMtools/wrap.m, + KPMtools/xticklabel_rotate90.m, KPMtools/zipload.m, + KPMtools/zipsave.m: Initial revision + +2005-04-03 18:39 yozhik + + * BNT/learning/score_dags.m: Initial import of code base from Kevin + Murphy. + +2005-04-03 18:39 yozhik + + * BNT/learning/score_dags.m: Initial revision + +2005-03-31 11:20 yozhik + + * BNT/installC_BNT.m: Initial import of code base from Kevin + Murphy. + +2005-03-31 11:20 yozhik + + * BNT/installC_BNT.m: Initial revision + +2005-03-30 11:59 yozhik + + * KPMtools/asdemo.html: Initial import of code base from Kevin + Murphy. + +2005-03-30 11:59 yozhik + + * KPMtools/asdemo.html: Initial revision + +2005-03-26 18:51 yozhik + + * KPMtools/asdemo.m: Initial import of code base from Kevin Murphy. + +2005-03-26 18:51 yozhik + + * KPMtools/asdemo.m: Initial revision + +2005-01-15 18:27 yozhik + + * BNT/CPDs/@tabular_CPD/: get_field.m, set_fields.m, tabular_CPD.m: + Initial import of code base from Kevin Murphy. + +2005-01-15 18:27 yozhik + + * BNT/CPDs/@tabular_CPD/: get_field.m, set_fields.m, tabular_CPD.m: + Initial revision + +2004-11-24 12:12 yozhik + + * BNT/inference/dynamic/@cbk_inf_engine/junk: Initial import of + code base from Kevin Murphy. + +2004-11-24 12:12 yozhik + + * BNT/inference/dynamic/@cbk_inf_engine/junk: Initial revision + +2004-11-22 14:41 yozhik + + * BNT/examples/dynamic/orig_water1.m: Initial import of code base + from Kevin Murphy. + +2004-11-22 14:41 yozhik + + * BNT/examples/dynamic/orig_water1.m: Initial revision + +2004-11-22 14:15 yozhik + + * BNT/inference/dynamic/@cbk_inf_engine/cbk_inf_engine.m: Initial + import of code base from Kevin Murphy. + +2004-11-22 14:15 yozhik + + * BNT/inference/dynamic/@cbk_inf_engine/cbk_inf_engine.m: Initial + revision + +2004-11-06 13:52 yozhik + + * BNT/examples/static/StructLearn/model_select2.m: Initial import + of code base from Kevin Murphy. + +2004-11-06 13:52 yozhik + + * BNT/examples/static/StructLearn/model_select2.m: Initial revision + +2004-11-06 12:55 yozhik + + * BNT/examples/static/StructLearn/model_select1.m: Initial import + of code base from Kevin Murphy. + +2004-11-06 12:55 yozhik + + * BNT/examples/static/StructLearn/model_select1.m: Initial revision + +2004-10-22 18:18 yozhik + + * HMM/viterbi_path.m: Initial import of code base from Kevin + Murphy. + +2004-10-22 18:18 yozhik + + * HMM/viterbi_path.m: Initial revision + +2004-09-29 20:09 yozhik + + * BNT/inference/static/@var_elim_inf_engine/marginal_nodes.m: + Initial import of code base from Kevin Murphy. + +2004-09-29 20:09 yozhik + + * BNT/inference/static/@var_elim_inf_engine/marginal_nodes.m: + Initial revision + +2004-09-12 20:21 yozhik + + * BNT/examples/limids/amnio.m: Initial import of code base from + Kevin Murphy. + +2004-09-12 20:21 yozhik + + * BNT/examples/limids/amnio.m: Initial revision + +2004-09-12 19:27 yozhik + + * BNT/examples/limids/oil1.m: Initial import of code base from + Kevin Murphy. + +2004-09-12 19:27 yozhik + + * BNT/examples/limids/oil1.m: Initial revision + +2004-09-12 14:01 yozhik + + * BNT/examples/static/sprinkler1.m: Initial import of code base + from Kevin Murphy. + +2004-09-12 14:01 yozhik + + * BNT/examples/static/sprinkler1.m: Initial revision + +2004-08-29 05:41 yozhik + + * HMM/transmat_train_observed.m: Initial import of code base from + Kevin Murphy. + +2004-08-29 05:41 yozhik + + * HMM/transmat_train_observed.m: Initial revision + +2004-08-05 08:25 yozhik + + * BNT/potentials/: @dpot/divide_by_pot.m, Tables/divide_by_table.m: + Initial import of code base from Kevin Murphy. + +2004-08-05 08:25 yozhik + + * BNT/potentials/: @dpot/divide_by_pot.m, Tables/divide_by_table.m: + Initial revision + +2004-08-04 12:59 yozhik + + * BNT/potentials/@dpot/: marginalize_pot.m, multiply_by_pot.m: + Initial import of code base from Kevin Murphy. + +2004-08-04 12:59 yozhik + + * BNT/potentials/@dpot/: marginalize_pot.m, multiply_by_pot.m: + Initial revision + +2004-08-04 12:36 yozhik + + * BNT/@assocarray/subsref.m: Initial import of code base from Kevin + Murphy. + +2004-08-04 12:36 yozhik + + * BNT/@assocarray/subsref.m: Initial revision + +2004-08-04 08:54 yozhik + + * BNT/potentials/@dpot/normalize_pot.m: Initial import of code base + from Kevin Murphy. + +2004-08-04 08:54 yozhik + + * BNT/potentials/@dpot/normalize_pot.m: Initial revision + +2004-08-04 08:51 yozhik + + * BNT/potentials/Tables/: marg_table.m, mult_by_table.m, + extend_domain_table.m: Initial import of code base from Kevin + Murphy. + +2004-08-04 08:51 yozhik + + * BNT/potentials/Tables/: marg_table.m, mult_by_table.m, + extend_domain_table.m: Initial revision + +2004-08-02 15:23 yozhik + + * BNT/CPDs/@noisyor_CPD/CPD_to_CPT.m: Initial import of code base + from Kevin Murphy. + +2004-08-02 15:23 yozhik + + * BNT/CPDs/@noisyor_CPD/CPD_to_CPT.m: Initial revision + +2004-08-02 15:05 yozhik + + * BNT/general/noisyORtoTable.m: Initial import of code base from + Kevin Murphy. + +2004-08-02 15:05 yozhik + + * BNT/general/noisyORtoTable.m: Initial revision + +2004-06-29 10:46 yozhik + + * BNT/learning/learn_struct_pdag_pc.m: Initial import of code base + from Kevin Murphy. + +2004-06-29 10:46 yozhik + + * BNT/learning/learn_struct_pdag_pc.m: Initial revision + +2004-06-15 10:50 yozhik + + * GraphViz/graph_to_dot.m: Initial import of code base from Kevin + Murphy. + +2004-06-15 10:50 yozhik + + * GraphViz/graph_to_dot.m: Initial revision + +2004-06-11 14:16 yozhik + + * BNT/CPDs/@tabular_CPD/log_marg_prob_node.m: Initial import of + code base from Kevin Murphy. + +2004-06-11 14:16 yozhik + + * BNT/CPDs/@tabular_CPD/log_marg_prob_node.m: Initial revision + +2004-06-09 18:56 yozhik + + * BNT/README.txt: Initial import of code base from Kevin Murphy. + +2004-06-09 18:56 yozhik + + * BNT/README.txt: Initial revision + +2004-06-09 18:53 yozhik + + * BNT/CPDs/@generic_CPD/learn_params.m: Initial import of code base + from Kevin Murphy. + +2004-06-09 18:53 yozhik + + * BNT/CPDs/@generic_CPD/learn_params.m: Initial revision + +2004-06-09 18:42 yozhik + + * BNT/examples/static/nodeorderExample.m: Initial import of code + base from Kevin Murphy. + +2004-06-09 18:42 yozhik + + * BNT/examples/static/nodeorderExample.m: Initial revision + +2004-06-09 18:33 yozhik + + * BNT/: learning/score_family.m, test_BNT.m: Initial import of code + base from Kevin Murphy. + +2004-06-09 18:33 yozhik + + * BNT/: learning/score_family.m, test_BNT.m: Initial revision + +2004-06-09 18:28 yozhik + + * BNT/: learning/learn_params.m, CPDs/@gaussian_CPD/learn_params.m, + examples/static/gaussian2.m: Initial import of code base from + Kevin Murphy. + +2004-06-09 18:28 yozhik + + * BNT/: learning/learn_params.m, CPDs/@gaussian_CPD/learn_params.m, + examples/static/gaussian2.m: Initial revision + +2004-06-09 18:25 yozhik + + * BNT/CPDs/@tabular_CPD/learn_params.m: Initial import of code base + from Kevin Murphy. + +2004-06-09 18:25 yozhik + + * BNT/CPDs/@tabular_CPD/learn_params.m: Initial revision + +2004-06-09 18:17 yozhik + + * BNT/general/sample_bnet.m: Initial import of code base from Kevin + Murphy. + +2004-06-09 18:17 yozhik + + * BNT/general/sample_bnet.m: Initial revision + +2004-06-07 12:45 yozhik + + * BNT/examples/static/discrete1.m: Initial import of code base from + Kevin Murphy. + +2004-06-07 12:45 yozhik + + * BNT/examples/static/discrete1.m: Initial revision + +2004-06-07 12:04 yozhik + + * BNT/: inference/static/@global_joint_inf_engine/marginal_nodes.m, + inference/static/@global_joint_inf_engine/enter_evidence.m, + examples/dynamic/mk_bat_dbn.m: Initial import of code base from + Kevin Murphy. + +2004-06-07 12:04 yozhik + + * BNT/: inference/static/@global_joint_inf_engine/marginal_nodes.m, + inference/static/@global_joint_inf_engine/enter_evidence.m, + examples/dynamic/mk_bat_dbn.m: Initial revision + +2004-06-07 08:53 yozhik + + * BNT/examples/limids/asia_dt1.m: Initial import of code base from + Kevin Murphy. + +2004-06-07 08:53 yozhik + + * BNT/examples/limids/asia_dt1.m: Initial revision + +2004-06-07 08:48 yozhik + + * BNT/general/: solve_limid.m, compute_joint_pot.m: Initial import + of code base from Kevin Murphy. + +2004-06-07 08:48 yozhik + + * BNT/general/: solve_limid.m, compute_joint_pot.m: Initial + revision + +2004-06-07 07:39 yozhik + + * Kalman/README.txt: Initial import of code base from Kevin Murphy. + +2004-06-07 07:39 yozhik + + * Kalman/README.txt: Initial revision + +2004-06-07 07:33 yozhik + + * GraphViz/README.txt: Initial import of code base from Kevin + Murphy. + +2004-06-07 07:33 yozhik + + * GraphViz/README.txt: Initial revision + +2004-05-31 15:19 yozhik + + * HMM/dhmm_sample.m: Initial import of code base from Kevin Murphy. + +2004-05-31 15:19 yozhik + + * HMM/dhmm_sample.m: Initial revision + +2004-05-25 17:32 yozhik + + * HMM/mhmm_sample.m: Initial import of code base from Kevin Murphy. + +2004-05-25 17:32 yozhik + + * HMM/mhmm_sample.m: Initial revision + +2004-05-24 15:26 yozhik + + * HMM/mc_sample.m: Initial import of code base from Kevin Murphy. + +2004-05-24 15:26 yozhik + + * HMM/mc_sample.m: Initial revision + +2004-05-18 07:50 yozhik + + * BNT/installC_graph.m: Initial import of code base from Kevin + Murphy. + +2004-05-18 07:50 yozhik + + * BNT/installC_graph.m: Initial revision + +2004-05-13 18:13 yozhik + + * BNT/inference/static/@gaussian_inf_engine/gaussian_inf_engine.m: + Initial import of code base from Kevin Murphy. + +2004-05-13 18:13 yozhik + + * BNT/inference/static/@gaussian_inf_engine/gaussian_inf_engine.m: + Initial revision + +2004-05-11 12:23 yozhik + + * BNT/examples/dynamic/mk_chmm.m: Initial import of code base from + Kevin Murphy. + +2004-05-11 12:23 yozhik + + * BNT/examples/dynamic/mk_chmm.m: Initial revision + +2004-05-11 11:45 yozhik + + * BNT/examples/dynamic/mk_water_dbn.m: Initial import of code base + from Kevin Murphy. + +2004-05-11 11:45 yozhik + + * BNT/examples/dynamic/mk_water_dbn.m: Initial revision + +2004-05-05 06:32 yozhik + + * GraphViz/draw_dot.m: Initial import of code base from Kevin + Murphy. + +2004-05-05 06:32 yozhik + + * GraphViz/draw_dot.m: Initial revision + +2004-03-30 09:18 yozhik + + * BNT/: general/mk_named_CPT.m, + CPDs/@softmax_CPD/convert_to_table.m: Initial import of code base + from Kevin Murphy. + +2004-03-30 09:18 yozhik + + * BNT/: general/mk_named_CPT.m, + CPDs/@softmax_CPD/convert_to_table.m: Initial revision + +2004-03-22 14:32 yozhik + + * GraphViz/draw_graph.m: Initial import of code base from Kevin + Murphy. + +2004-03-22 14:32 yozhik + + * GraphViz/draw_graph.m: Initial revision + +2004-03-12 15:21 yozhik + + * GraphViz/dot_to_graph.m: Initial import of code base from Kevin + Murphy. + +2004-03-12 15:21 yozhik + + * GraphViz/dot_to_graph.m: Initial revision + +2004-03-04 14:34 yozhik + + * BNT/examples/static/burglary.m: Initial import of code base from + Kevin Murphy. + +2004-03-04 14:34 yozhik + + * BNT/examples/static/burglary.m: Initial revision + +2004-03-04 14:27 yozhik + + * BNT/examples/static/burglar-alarm-net.lisp.txt: Initial import of + code base from Kevin Murphy. + +2004-03-04 14:27 yozhik + + * BNT/examples/static/burglar-alarm-net.lisp.txt: Initial revision + +2004-02-28 09:25 yozhik + + * BNT/examples/static/learn1.m: Initial import of code base from + Kevin Murphy. + +2004-02-28 09:25 yozhik + + * BNT/examples/static/learn1.m: Initial revision + +2004-02-22 11:43 yozhik + + * BNT/examples/static/brainy.m: Initial import of code base from + Kevin Murphy. + +2004-02-22 11:43 yozhik + + * BNT/examples/static/brainy.m: Initial revision + +2004-02-20 14:00 yozhik + + * BNT/CPDs/@discrete_CPD/convert_to_pot.m: Initial import of code + base from Kevin Murphy. + +2004-02-20 14:00 yozhik + + * BNT/CPDs/@discrete_CPD/convert_to_pot.m: Initial revision + +2004-02-18 17:12 yozhik + + * + BNT/inference/dynamic/@jtree_dbn_inf_engine/enter_soft_evidence.m: + Initial import of code base from Kevin Murphy. + +2004-02-18 17:12 yozhik + + * + BNT/inference/dynamic/@jtree_dbn_inf_engine/enter_soft_evidence.m: + Initial revision + +2004-02-13 18:06 yozhik + + * HMM/mhmmParzen_train_observed.m: Initial import of code base from + Kevin Murphy. + +2004-02-13 18:06 yozhik + + * HMM/mhmmParzen_train_observed.m: Initial revision + +2004-02-12 15:08 yozhik + + * HMM/gausshmm_train_observed.m: Initial import of code base from + Kevin Murphy. + +2004-02-12 15:08 yozhik + + * HMM/gausshmm_train_observed.m: Initial revision + +2004-02-12 04:57 yozhik + + * BNT/examples/static/HME/hmemenu.m: Initial import of code base + from Kevin Murphy. + +2004-02-12 04:57 yozhik + + * BNT/examples/static/HME/hmemenu.m: Initial revision + +2004-02-07 20:52 yozhik + + * HMM/mhmm_em.m: Initial import of code base from Kevin Murphy. + +2004-02-07 20:52 yozhik + + * HMM/mhmm_em.m: Initial revision + +2004-02-04 15:53 yozhik + + * BNT/examples/dynamic/mk_orig_bat_dbn.m: Initial import of code + base from Kevin Murphy. + +2004-02-04 15:53 yozhik + + * BNT/examples/dynamic/mk_orig_bat_dbn.m: Initial revision + +2004-02-03 23:42 yozhik + + * BNT/inference/dynamic/@cbk_inf_engine/enter_soft_evidence.m: + Initial import of code base from Kevin Murphy. + +2004-02-03 23:42 yozhik + + * BNT/inference/dynamic/@cbk_inf_engine/enter_soft_evidence.m: + Initial revision + +2004-02-03 09:15 yozhik + + * GraphViz/Old/graphToDot.m: Initial import of code base from Kevin + Murphy. + +2004-02-03 09:15 yozhik + + * GraphViz/Old/graphToDot.m: Initial revision + +2004-01-30 18:57 yozhik + + * BNT/examples/dynamic/mk_orig_water_dbn.m: Initial import of code + base from Kevin Murphy. + +2004-01-30 18:57 yozhik + + * BNT/examples/dynamic/mk_orig_water_dbn.m: Initial revision + +2004-01-27 13:08 yozhik + + * GraphViz/: my_call.m, editGraphGUI.m: Initial import of code base + from Kevin Murphy. + +2004-01-27 13:08 yozhik + + * GraphViz/: my_call.m, editGraphGUI.m: Initial revision + +2004-01-27 13:01 yozhik + + * GraphViz/Old/: dot_to_graph.m, draw_graph.m: Initial import of + code base from Kevin Murphy. + +2004-01-27 13:01 yozhik + + * GraphViz/Old/: dot_to_graph.m, draw_graph.m: Initial revision + +2004-01-27 12:47 yozhik + + * GraphViz/Old/pre_pesha_graph_to_dot.m: Initial import of code + base from Kevin Murphy. + +2004-01-27 12:47 yozhik + + * GraphViz/Old/pre_pesha_graph_to_dot.m: Initial revision + +2004-01-27 12:42 yozhik + + * GraphViz/Old/draw_dot.m: Initial import of code base from Kevin + Murphy. + +2004-01-27 12:42 yozhik + + * GraphViz/Old/draw_dot.m: Initial revision + +2004-01-14 17:06 yozhik + + * BNT/examples/static/Models/mk_hmm_bnet.m: Initial import of code + base from Kevin Murphy. + +2004-01-14 17:06 yozhik + + * BNT/examples/static/Models/mk_hmm_bnet.m: Initial revision + +2004-01-12 12:53 yozhik + + * BNT/inference/dynamic/@cbk_inf_engine/enter_evidence.m: Initial + import of code base from Kevin Murphy. + +2004-01-12 12:53 yozhik + + * BNT/inference/dynamic/@cbk_inf_engine/enter_evidence.m: Initial + revision + +2004-01-04 17:23 yozhik + + * BNT/inference/static/@belprop_mrf2_inf_engine/bp_mrf2.m: Initial + import of code base from Kevin Murphy. + +2004-01-04 17:23 yozhik + + * BNT/inference/static/@belprop_mrf2_inf_engine/bp_mrf2.m: Initial + revision + +2003-12-15 22:17 yozhik + + * BNT/inference/dynamic/@cbk_inf_engine/marginal_nodes.m: Initial + import of code base from Kevin Murphy. + +2003-12-15 22:17 yozhik + + * BNT/inference/dynamic/@cbk_inf_engine/marginal_nodes.m: Initial + revision + +2003-10-31 14:37 yozhik + + * BNT/inference/static/@jtree_inf_engine/jtree_inf_engine.m: + Initial import of code base from Kevin Murphy. + +2003-10-31 14:37 yozhik + + * BNT/inference/static/@jtree_inf_engine/jtree_inf_engine.m: + Initial revision + +2003-09-05 07:06 yozhik + + * BNT/learning/learn_struct_mcmc.m: Initial import of code base + from Kevin Murphy. + +2003-09-05 07:06 yozhik + + * BNT/learning/learn_struct_mcmc.m: Initial revision + +2003-08-18 14:50 yozhik + + * BNT/learning/learn_params_dbn_em.m: Initial import of code base + from Kevin Murphy. + +2003-08-18 14:50 yozhik + + * BNT/learning/learn_params_dbn_em.m: Initial revision + +2003-07-30 06:37 yozhik + + * BNT/potentials/: @mpot/set_domain_pot.m, + @cgpot/Old/set_domain_pot.m, @cgpot/set_domain_pot.m: Initial + import of code base from Kevin Murphy. + +2003-07-30 06:37 yozhik + + * BNT/potentials/: @mpot/set_domain_pot.m, + @cgpot/Old/set_domain_pot.m, @cgpot/set_domain_pot.m: Initial + revision + +2003-07-28 19:44 yozhik + + * BNT/inference/dynamic/@cbk_inf_engine/: dbn_init_bel.m, + dbn_marginal_from_bel.m, dbn_update_bel.m, dbn_update_bel1.m, + marginal_family.m, update_engine.m: Initial import of code base + from Kevin Murphy. + +2003-07-28 19:44 yozhik + + * BNT/inference/dynamic/@cbk_inf_engine/: dbn_init_bel.m, + dbn_marginal_from_bel.m, dbn_update_bel.m, dbn_update_bel1.m, + marginal_family.m, update_engine.m: Initial revision + +2003-07-28 15:44 yozhik + + * GraphViz/: approxeq.m, process_options.m: Initial import of code + base from Kevin Murphy. + +2003-07-28 15:44 yozhik + + * GraphViz/: approxeq.m, process_options.m: Initial revision + +2003-07-24 06:41 yozhik + + * BNT/CPDs/@hhmmQ_CPD/update_ess.m: Initial import of code base + from Kevin Murphy. + +2003-07-24 06:41 yozhik + + * BNT/CPDs/@hhmmQ_CPD/update_ess.m: Initial revision + +2003-07-22 15:55 yozhik + + * BNT/CPDs/@gaussian_CPD/update_ess.m: Initial import of code base + from Kevin Murphy. + +2003-07-22 15:55 yozhik + + * BNT/CPDs/@gaussian_CPD/update_ess.m: Initial revision + +2003-07-06 13:57 yozhik + + * BNT/inference/static/@pearl_inf_engine/bethe_free_energy.m: + Initial import of code base from Kevin Murphy. + +2003-07-06 13:57 yozhik + + * BNT/inference/static/@pearl_inf_engine/bethe_free_energy.m: + Initial revision + +2003-05-21 06:49 yozhik + + * BNT/potentials/@scgpot/: complement_pot.m, normalize_pot.m, + recursive_combine_pots.m: Initial import of code base from Kevin + Murphy. + +2003-05-21 06:49 yozhik + + * BNT/potentials/@scgpot/: complement_pot.m, normalize_pot.m, + recursive_combine_pots.m: Initial revision + +2003-05-20 07:10 yozhik + + * BNT/CPDs/@gaussian_CPD/maximize_params.m: Initial import of code + base from Kevin Murphy. + +2003-05-20 07:10 yozhik + + * BNT/CPDs/@gaussian_CPD/maximize_params.m: Initial revision + +2003-05-13 09:11 yozhik + + * HMM/mhmm_em_demo.m: Initial import of code base from Kevin + Murphy. + +2003-05-13 09:11 yozhik + + * HMM/mhmm_em_demo.m: Initial revision + +2003-05-13 07:35 yozhik + + * BNT/examples/dynamic/viterbi1.m: Initial import of code base from + Kevin Murphy. + +2003-05-13 07:35 yozhik + + * BNT/examples/dynamic/viterbi1.m: Initial revision + +2003-05-11 16:31 yozhik + + * BNT/CPDs/@gaussian_CPD/convert_to_table.m: Initial import of code + base from Kevin Murphy. + +2003-05-11 16:31 yozhik + + * BNT/CPDs/@gaussian_CPD/convert_to_table.m: Initial revision + +2003-05-11 16:13 yozhik + + * BNT/CPDs/@gaussian_CPD/gaussian_CPD_params_given_dps.m: Initial + import of code base from Kevin Murphy. + +2003-05-11 16:13 yozhik + + * BNT/CPDs/@gaussian_CPD/gaussian_CPD_params_given_dps.m: Initial + revision + +2003-05-11 08:39 yozhik + + * BNT/inference/static/@stab_cond_gauss_inf_engine/README: Initial + import of code base from Kevin Murphy. + +2003-05-11 08:39 yozhik + + * BNT/inference/static/@stab_cond_gauss_inf_engine/README: Initial + revision + +2003-05-04 15:31 yozhik + + * BNT/uninstallC_BNT.m: Initial import of code base from Kevin + Murphy. + +2003-05-04 15:31 yozhik + + * BNT/uninstallC_BNT.m: Initial revision + +2003-05-04 15:23 yozhik + + * BNT/examples/dynamic/: dhmm1.m, ghmm1.m, mhmm1.m: Initial import + of code base from Kevin Murphy. + +2003-05-04 15:23 yozhik + + * BNT/examples/dynamic/: dhmm1.m, ghmm1.m, mhmm1.m: Initial + revision + +2003-05-04 15:11 yozhik + + * HMM/mhmm_logprob.m: Initial import of code base from Kevin + Murphy. + +2003-05-04 15:11 yozhik + + * HMM/mhmm_logprob.m: Initial revision + +2003-05-04 15:01 yozhik + + * HMM/: dhmm_logprob.m, dhmm_em_online.m, dhmm_em_online_demo.m: + Initial import of code base from Kevin Murphy. + +2003-05-04 15:01 yozhik + + * HMM/: dhmm_logprob.m, dhmm_em_online.m, dhmm_em_online_demo.m: + Initial revision + +2003-05-04 14:58 yozhik + + * HMM/: pomdp_sample.m, dhmm_sample_endstate.m, dhmm_em_demo.m: + Initial import of code base from Kevin Murphy. + +2003-05-04 14:58 yozhik + + * HMM/: pomdp_sample.m, dhmm_sample_endstate.m, dhmm_em_demo.m: + Initial revision + +2003-05-04 14:47 yozhik + + * + BNT/inference/online/@hmm_2TBN_inf_engine/private/mk_hmm_obs_lik_vec.m: + Initial import of code base from Kevin Murphy. + +2003-05-04 14:47 yozhik + + * + BNT/inference/online/@hmm_2TBN_inf_engine/private/mk_hmm_obs_lik_vec.m: + Initial revision + +2003-05-04 14:42 yozhik + + * + BNT/inference/dynamic/@hmm_inf_engine/private/mk_hmm_obs_lik_matrix.m: + Initial import of code base from Kevin Murphy. + +2003-05-04 14:42 yozhik + + * + BNT/inference/dynamic/@hmm_inf_engine/private/mk_hmm_obs_lik_matrix.m: + Initial revision + +2003-04-22 14:00 yozhik + + * BNT/CPDs/@tabular_CPD/display.m: Initial import of code base from + Kevin Murphy. + +2003-04-22 14:00 yozhik + + * BNT/CPDs/@tabular_CPD/display.m: Initial revision + +2003-03-28 09:22 yozhik + + * BNT/examples/dynamic/ho1.m: Initial import of code base from + Kevin Murphy. + +2003-03-28 09:22 yozhik + + * BNT/examples/dynamic/ho1.m: Initial revision + +2003-03-28 09:12 yozhik + + * + BNT/inference/static/@stab_cond_gauss_inf_engine/stab_cond_gauss_inf_engine.m: + Initial import of code base from Kevin Murphy. + +2003-03-28 09:12 yozhik + + * + BNT/inference/static/@stab_cond_gauss_inf_engine/stab_cond_gauss_inf_engine.m: + Initial revision + +2003-03-28 08:35 yozhik + + * GraphViz/arrow.m: Initial import of code base from Kevin Murphy. + +2003-03-28 08:35 yozhik + + * GraphViz/arrow.m: Initial revision + +2003-03-25 16:06 yozhik + + * BNT/examples/static/Models/mk_asia_bnet.m: Initial import of code + base from Kevin Murphy. + +2003-03-25 16:06 yozhik + + * BNT/examples/static/Models/mk_asia_bnet.m: Initial revision + +2003-03-20 07:07 yozhik + + * BNT/potentials/@scgpot/README: Initial import of code base from + Kevin Murphy. + +2003-03-20 07:07 yozhik + + * BNT/potentials/@scgpot/README: Initial revision + +2003-03-14 01:45 yozhik + + * + BNT/inference/dynamic/@stable_ho_inf_engine/stable_ho_inf_engine.m: + Initial import of code base from Kevin Murphy. + +2003-03-14 01:45 yozhik + + * + BNT/inference/dynamic/@stable_ho_inf_engine/stable_ho_inf_engine.m: + Initial revision + +2003-03-12 02:38 yozhik + + * + BNT/inference/static/@stab_cond_gauss_inf_engine/enter_evidence.m: + Initial import of code base from Kevin Murphy. + +2003-03-12 02:38 yozhik + + * + BNT/inference/static/@stab_cond_gauss_inf_engine/enter_evidence.m: + Initial revision + +2003-03-11 10:07 yozhik + + * BNT/potentials/@scgpot/reduce_pot.m: Initial import of code base + from Kevin Murphy. + +2003-03-11 10:07 yozhik + + * BNT/potentials/@scgpot/reduce_pot.m: Initial revision + +2003-03-11 09:49 yozhik + + * BNT/potentials/@scgpot/combine_pots.m: Initial import of code + base from Kevin Murphy. + +2003-03-11 09:49 yozhik + + * BNT/potentials/@scgpot/combine_pots.m: Initial revision + +2003-03-11 09:37 yozhik + + * BNT/potentials/@scgcpot/reduce_pot.m: Initial import of code base + from Kevin Murphy. + +2003-03-11 09:37 yozhik + + * BNT/potentials/@scgcpot/reduce_pot.m: Initial revision + +2003-03-11 09:06 yozhik + + * BNT/potentials/@scgpot/marginalize_pot.m: Initial import of code + base from Kevin Murphy. + +2003-03-11 09:06 yozhik + + * BNT/potentials/@scgpot/marginalize_pot.m: Initial revision + +2003-03-11 06:04 yozhik + + * BNT/potentials/@scgpot/scgpot.m: Initial import of code base from + Kevin Murphy. + +2003-03-11 06:04 yozhik + + * BNT/potentials/@scgpot/scgpot.m: Initial revision + +2003-03-09 15:03 yozhik + + * BNT/CPDs/@gaussian_CPD/convert_to_pot.m: Initial import of code + base from Kevin Murphy. + +2003-03-09 15:03 yozhik + + * BNT/CPDs/@gaussian_CPD/convert_to_pot.m: Initial revision + +2003-03-09 14:44 yozhik + + * BNT/CPDs/@tabular_CPD/maximize_params.m: Initial import of code + base from Kevin Murphy. + +2003-03-09 14:44 yozhik + + * BNT/CPDs/@tabular_CPD/maximize_params.m: Initial revision + +2003-02-21 03:20 yozhik + + * + BNT/inference/static/@stab_cond_gauss_inf_engine/marginal_difclq_nodes.m: + Initial import of code base from Kevin Murphy. + +2003-02-21 03:20 yozhik + + * + BNT/inference/static/@stab_cond_gauss_inf_engine/marginal_difclq_nodes.m: + Initial revision + +2003-02-21 03:13 yozhik + + * + BNT/inference/static/@stab_cond_gauss_inf_engine/marginal_nodes.m: + Initial import of code base from Kevin Murphy. + +2003-02-21 03:13 yozhik + + * + BNT/inference/static/@stab_cond_gauss_inf_engine/marginal_nodes.m: + Initial revision + +2003-02-19 01:52 yozhik + + * BNT/inference/dynamic/@stable_ho_inf_engine/: enter_evidence.m, + marginal_family.m, marginal_nodes.m, test_ho_inf_enginge.m, + update_engine.m: Initial import of code base from Kevin Murphy. + +2003-02-19 01:52 yozhik + + * BNT/inference/dynamic/@stable_ho_inf_engine/: enter_evidence.m, + marginal_family.m, marginal_nodes.m, test_ho_inf_enginge.m, + update_engine.m: Initial revision + +2003-02-10 07:38 yozhik + + * BNT/inference/static/@stab_cond_gauss_inf_engine/push.m: Initial + import of code base from Kevin Murphy. + +2003-02-10 07:38 yozhik + + * BNT/inference/static/@stab_cond_gauss_inf_engine/push.m: Initial + revision + +2003-02-06 18:25 yozhik + + * KPMtools/checkpsd.m: Initial import of code base from Kevin + Murphy. + +2003-02-06 18:25 yozhik + + * KPMtools/checkpsd.m: Initial revision + +2003-02-05 19:16 yozhik + + * GraphViz/draw_hmm.m: Initial import of code base from Kevin + Murphy. + +2003-02-05 19:16 yozhik + + * GraphViz/draw_hmm.m: Initial revision + +2003-02-01 16:23 yozhik + + * BNT/: general/dbn_to_hmm.m, learning/learn_params_dbn.m: Initial + import of code base from Kevin Murphy. + +2003-02-01 16:23 yozhik + + * BNT/: general/dbn_to_hmm.m, learning/learn_params_dbn.m: Initial + revision + +2003-02-01 11:42 yozhik + + * BNT/general/mk_dbn.m: Initial import of code base from Kevin + Murphy. + +2003-02-01 11:42 yozhik + + * BNT/general/mk_dbn.m: Initial revision + +2003-01-30 16:13 yozhik + + * BNT/CPDs/@gaussian_CPD/maximize_params_debug.m: Initial import of + code base from Kevin Murphy. + +2003-01-30 16:13 yozhik + + * BNT/CPDs/@gaussian_CPD/maximize_params_debug.m: Initial revision + +2003-01-30 14:38 yozhik + + * BNT/CPDs/@gaussian_CPD/Old/maximize_params.m: Initial import of + code base from Kevin Murphy. + +2003-01-30 14:38 yozhik + + * BNT/CPDs/@gaussian_CPD/Old/maximize_params.m: Initial revision + +2003-01-29 03:23 yozhik + + * + BNT/inference/static/@stab_cond_gauss_inf_engine/marginal_singleclq_nodes.m: + Initial import of code base from Kevin Murphy. + +2003-01-29 03:23 yozhik + + * + BNT/inference/static/@stab_cond_gauss_inf_engine/marginal_singleclq_nodes.m: + Initial revision + +2003-01-24 11:36 yozhik + + * Kalman/sample_lds.m: Initial import of code base from Kevin + Murphy. + +2003-01-24 11:36 yozhik + + * Kalman/sample_lds.m: Initial revision + +2003-01-24 04:52 yozhik + + * BNT/potentials/@scgpot/extension_pot.m: Initial import of code + base from Kevin Murphy. + +2003-01-24 04:52 yozhik + + * BNT/potentials/@scgpot/extension_pot.m: Initial revision + +2003-01-23 10:49 yozhik + + * BNT/: general/convert_dbn_CPDs_to_tables1.m, + inference/dynamic/@hmm_inf_engine/private/mk_hmm_obs_lik_vec.m: + Initial import of code base from Kevin Murphy. + +2003-01-23 10:49 yozhik + + * BNT/: general/convert_dbn_CPDs_to_tables1.m, + inference/dynamic/@hmm_inf_engine/private/mk_hmm_obs_lik_vec.m: + Initial revision + +2003-01-23 10:44 yozhik + + * BNT/general/convert_dbn_CPDs_to_tables.m: Initial import of code + base from Kevin Murphy. + +2003-01-23 10:44 yozhik + + * BNT/general/convert_dbn_CPDs_to_tables.m: Initial revision + +2003-01-22 13:38 yozhik + + * BNT/inference/dynamic/@hmm_inf_engine/enter_evidence.m: Initial + import of code base from Kevin Murphy. + +2003-01-22 13:38 yozhik + + * BNT/inference/dynamic/@hmm_inf_engine/enter_evidence.m: Initial + revision + +2003-01-22 12:32 yozhik + + * HMM/mc_sample_endstate.m: Initial import of code base from Kevin + Murphy. + +2003-01-22 12:32 yozhik + + * HMM/mc_sample_endstate.m: Initial revision + +2003-01-22 09:56 yozhik + + * HMM/fixed_lag_smoother.m: Initial import of code base from Kevin + Murphy. + +2003-01-22 09:56 yozhik + + * HMM/fixed_lag_smoother.m: Initial revision + +2003-01-20 08:56 yozhik + + * GraphViz/draw_graph_test.m: Initial import of code base from + Kevin Murphy. + +2003-01-20 08:56 yozhik + + * GraphViz/draw_graph_test.m: Initial revision + +2003-01-18 15:10 yozhik + + * BNT/general/dsep_test.m: Initial import of code base from Kevin + Murphy. + +2003-01-18 15:10 yozhik + + * BNT/general/dsep_test.m: Initial revision + +2003-01-18 15:00 yozhik + + * BNT/copyright.txt: Initial import of code base from Kevin Murphy. + +2003-01-18 15:00 yozhik + + * BNT/copyright.txt: Initial revision + +2003-01-18 14:49 yozhik + + * Kalman/tracking_demo.m: Initial import of code base from Kevin + Murphy. + +2003-01-18 14:49 yozhik + + * Kalman/tracking_demo.m: Initial revision + +2003-01-18 14:22 yozhik + + * BNT/: examples/dummy, inference/dummy, inference/dynamic/dummy, + inference/online/dummy, inference/static/dummy: Initial import of + code base from Kevin Murphy. + +2003-01-18 14:22 yozhik + + * BNT/: examples/dummy, inference/dummy, inference/dynamic/dummy, + inference/online/dummy, inference/static/dummy: Initial revision + +2003-01-18 14:16 yozhik + + * BNT/examples/dynamic/: ehmm1.m, jtree_clq_test.m: Initial import + of code base from Kevin Murphy. + +2003-01-18 14:16 yozhik + + * BNT/examples/dynamic/: ehmm1.m, jtree_clq_test.m: Initial + revision + +2003-01-18 14:11 yozhik + + * BNT/inference/static/: + @jtree_sparse_inf_engine/jtree_sparse_inf_engine.m, + @jtree_mnet_inf_engine/jtree_mnet_inf_engine.m: Initial import of + code base from Kevin Murphy. + +2003-01-18 14:11 yozhik + + * BNT/inference/static/: + @jtree_sparse_inf_engine/jtree_sparse_inf_engine.m, + @jtree_mnet_inf_engine/jtree_mnet_inf_engine.m: Initial revision + +2003-01-18 13:17 yozhik + + * GraphViz/draw_dbn_test.m: Initial import of code base from Kevin + Murphy. + +2003-01-18 13:17 yozhik + + * GraphViz/draw_dbn_test.m: Initial revision + +2003-01-11 10:53 yozhik + + * BNT/inference/static/@pearl_inf_engine/pearl_inf_engine.m: + Initial import of code base from Kevin Murphy. + +2003-01-11 10:53 yozhik + + * BNT/inference/static/@pearl_inf_engine/pearl_inf_engine.m: + Initial revision + +2003-01-11 10:48 yozhik + + * BNT/examples/dynamic/HHMM/Map/learn_map.m: Initial import of code + base from Kevin Murphy. + +2003-01-11 10:48 yozhik + + * BNT/examples/dynamic/HHMM/Map/learn_map.m: Initial revision + +2003-01-11 10:41 yozhik + + * BNT/inference/dynamic/@jtree_dbn_inf_engine/enter_evidence.m: + Initial import of code base from Kevin Murphy. + +2003-01-11 10:41 yozhik + + * BNT/inference/dynamic/@jtree_dbn_inf_engine/enter_evidence.m: + Initial revision + +2003-01-11 10:13 yozhik + + * BNT/inference/dynamic/@bk_inf_engine/enter_soft_evidence.m: + Initial import of code base from Kevin Murphy. + +2003-01-11 10:13 yozhik + + * BNT/inference/dynamic/@bk_inf_engine/enter_soft_evidence.m: + Initial revision + +2003-01-07 08:25 yozhik + + * BNT/CPDs/@softmax_CPD/softmax_CPD.m: Initial import of code base + from Kevin Murphy. + +2003-01-07 08:25 yozhik + + * BNT/CPDs/@softmax_CPD/softmax_CPD.m: Initial revision + +2003-01-03 14:01 yozhik + + * + BNT/inference/static/@belprop_mrf2_inf_engine/belprop_mrf2_inf_engine.m: + Initial import of code base from Kevin Murphy. + +2003-01-03 14:01 yozhik + + * + BNT/inference/static/@belprop_mrf2_inf_engine/belprop_mrf2_inf_engine.m: + Initial revision + +2003-01-02 09:49 yozhik + + * BNT/inference/static/@belprop_mrf2_inf_engine/find_mpe.m: Initial + import of code base from Kevin Murphy. + +2003-01-02 09:49 yozhik + + * BNT/inference/static/@belprop_mrf2_inf_engine/find_mpe.m: Initial + revision + +2003-01-02 09:28 yozhik + + * BNT/inference/static/@belprop_mrf2_inf_engine/: set_params.m, + enter_soft_evidence.m: Initial import of code base from Kevin + Murphy. + +2003-01-02 09:28 yozhik + + * BNT/inference/static/@belprop_mrf2_inf_engine/: set_params.m, + enter_soft_evidence.m: Initial revision + +2002-12-31 14:06 yozhik + + * BNT/general/mk_mrf2.m: Initial import of code base from Kevin + Murphy. + +2002-12-31 14:06 yozhik + + * BNT/general/mk_mrf2.m: Initial revision + +2002-12-31 13:24 yozhik + + * BNT/inference/static/@belprop_mrf2_inf_engine/marginal_nodes.m: + Initial import of code base from Kevin Murphy. + +2002-12-31 13:24 yozhik + + * BNT/inference/static/@belprop_mrf2_inf_engine/marginal_nodes.m: + Initial revision + +2002-12-31 11:00 yozhik + + * BNT/inference/static/@belprop_inf_engine/belprop_inf_engine.m: + Initial import of code base from Kevin Murphy. + +2002-12-31 11:00 yozhik + + * BNT/inference/static/@belprop_inf_engine/belprop_inf_engine.m: + Initial revision + +2002-12-16 11:16 yozhik + + * BNT/examples/dynamic/HHMM/remove_hhmm_end_state.m: Initial import + of code base from Kevin Murphy. + +2002-12-16 11:16 yozhik + + * BNT/examples/dynamic/HHMM/remove_hhmm_end_state.m: Initial + revision + +2002-12-16 09:57 yozhik + + * BNT/general/unroll_set.m: Initial import of code base from Kevin + Murphy. + +2002-12-16 09:57 yozhik + + * BNT/general/unroll_set.m: Initial revision + +2002-11-26 14:14 yozhik + + * BNT/examples/dynamic/HHMM/Mgram/mgram3.m: Initial import of code + base from Kevin Murphy. + +2002-11-26 14:14 yozhik + + * BNT/examples/dynamic/HHMM/Mgram/mgram3.m: Initial revision + +2002-11-26 14:04 yozhik + + * BNT/examples/dynamic/HHMM/Mgram/mgram2.m: Initial import of code + base from Kevin Murphy. + +2002-11-26 14:04 yozhik + + * BNT/examples/dynamic/HHMM/Mgram/mgram2.m: Initial revision + +2002-11-22 16:44 yozhik + + * BNT/examples/dynamic/HHMM/Mgram/Old/mgram2.m: Initial import of + code base from Kevin Murphy. + +2002-11-22 16:44 yozhik + + * BNT/examples/dynamic/HHMM/Mgram/Old/mgram2.m: Initial revision + +2002-11-22 15:59 yozhik + + * BNT/examples/dynamic/HHMM/Mgram/mgram1.m: Initial import of code + base from Kevin Murphy. + +2002-11-22 15:59 yozhik + + * BNT/examples/dynamic/HHMM/Mgram/mgram1.m: Initial revision + +2002-11-22 15:51 yozhik + + * BNT/inference/dynamic/@jtree_dbn_inf_engine/marginal_nodes.m: + Initial import of code base from Kevin Murphy. + +2002-11-22 15:51 yozhik + + * BNT/inference/dynamic/@jtree_dbn_inf_engine/marginal_nodes.m: + Initial revision + +2002-11-22 15:07 yozhik + + * BNT/examples/dynamic/HHMM/Mgram/: num2letter.m, letter2num.m: + Initial import of code base from Kevin Murphy. + +2002-11-22 15:07 yozhik + + * BNT/examples/dynamic/HHMM/Mgram/: num2letter.m, letter2num.m: + Initial revision + +2002-11-22 14:35 yozhik + + * BNT/general/convert_dbn_CPDs_to_pots.m: Initial import of code + base from Kevin Murphy. + +2002-11-22 14:35 yozhik + + * BNT/general/convert_dbn_CPDs_to_pots.m: Initial revision + +2002-11-22 13:45 yozhik + + * HMM/mk_rightleft_transmat.m: Initial import of code base from + Kevin Murphy. + +2002-11-22 13:45 yozhik + + * HMM/mk_rightleft_transmat.m: Initial revision + +2002-11-14 12:33 yozhik + + * BNT/examples/dynamic/water2.m: Initial import of code base from + Kevin Murphy. + +2002-11-14 12:33 yozhik + + * BNT/examples/dynamic/water2.m: Initial revision + +2002-11-14 12:07 yozhik + + * BNT/examples/dynamic/water1.m: Initial import of code base from + Kevin Murphy. + +2002-11-14 12:07 yozhik + + * BNT/examples/dynamic/water1.m: Initial revision + +2002-11-14 12:02 yozhik + + * BNT/inference/: online/@hmm_2TBN_inf_engine/marginal_nodes.m, + dynamic/@hmm_inf_engine/marginal_nodes.m, + online/@hmm_2TBN_inf_engine/hmm_2TBN_inf_engine.m, + dynamic/@hmm_inf_engine/hmm_inf_engine.m, + dynamic/@hmm_inf_engine/marginal_family.m, + online/@hmm_2TBN_inf_engine/marginal_family.m: Initial import of + code base from Kevin Murphy. + +2002-11-14 12:02 yozhik + + * BNT/inference/: online/@hmm_2TBN_inf_engine/marginal_nodes.m, + dynamic/@hmm_inf_engine/marginal_nodes.m, + online/@hmm_2TBN_inf_engine/hmm_2TBN_inf_engine.m, + dynamic/@hmm_inf_engine/hmm_inf_engine.m, + dynamic/@hmm_inf_engine/marginal_family.m, + online/@hmm_2TBN_inf_engine/marginal_family.m: Initial revision + +2002-11-14 08:31 yozhik + + * BNT/inference/: + online/@jtree_2TBN_inf_engine/jtree_2TBN_inf_engine.m, + dynamic/@jtree_dbn_inf_engine/jtree_dbn_inf_engine.m: Initial + import of code base from Kevin Murphy. + +2002-11-14 08:31 yozhik + + * BNT/inference/: + online/@jtree_2TBN_inf_engine/jtree_2TBN_inf_engine.m, + dynamic/@jtree_dbn_inf_engine/jtree_dbn_inf_engine.m: Initial + revision + +2002-11-13 17:01 yozhik + + * BNT/examples/: static/qmr2.m, dynamic/arhmm1.m: Initial import of + code base from Kevin Murphy. + +2002-11-13 17:01 yozhik + + * BNT/examples/: static/qmr2.m, dynamic/arhmm1.m: Initial revision + +2002-11-03 08:44 yozhik + + * BNT/examples/static/Models/mk_alarm_bnet.m: Initial import of + code base from Kevin Murphy. + +2002-11-03 08:44 yozhik + + * BNT/examples/static/Models/mk_alarm_bnet.m: Initial revision + +2002-11-01 16:32 yozhik + + * Kalman/kalman_forward_backward.m: Initial import of code base + from Kevin Murphy. + +2002-11-01 16:32 yozhik + + * Kalman/kalman_forward_backward.m: Initial revision + +2002-10-23 08:17 yozhik + + * Kalman/learning_demo.m: Initial import of code base from Kevin + Murphy. + +2002-10-23 08:17 yozhik + + * Kalman/learning_demo.m: Initial revision + +2002-10-18 13:05 yozhik + + * BNT/inference/static/@pearl_inf_engine/marginal_family.m: Initial + import of code base from Kevin Murphy. + +2002-10-18 13:05 yozhik + + * BNT/inference/static/@pearl_inf_engine/marginal_family.m: Initial + revision + +2002-10-10 16:45 yozhik + + * BNT/examples/dynamic/jtree_clq_test2.m: Initial import of code + base from Kevin Murphy. + +2002-10-10 16:45 yozhik + + * BNT/examples/dynamic/jtree_clq_test2.m: Initial revision + +2002-10-10 16:14 yozhik + + * BNT/examples/dynamic/: mk_mildew_dbn.m, mk_uffe_dbn.m: Initial + import of code base from Kevin Murphy. + +2002-10-10 16:14 yozhik + + * BNT/examples/dynamic/: mk_mildew_dbn.m, mk_uffe_dbn.m: Initial + revision + +2002-10-09 13:36 yozhik + + * BNT/examples/dynamic/mk_ps_from_clqs.m: Initial import of code + base from Kevin Murphy. + +2002-10-09 13:36 yozhik + + * BNT/examples/dynamic/mk_ps_from_clqs.m: Initial revision + +2002-10-07 06:26 yozhik + + * BNT/CPDs/@deterministic_CPD/deterministic_CPD.m: Initial import + of code base from Kevin Murphy. + +2002-10-07 06:26 yozhik + + * BNT/CPDs/@deterministic_CPD/deterministic_CPD.m: Initial revision + +2002-10-02 08:39 yozhik + + * BNT/potentials/Tables/marg_tableC.c: Initial import of code base + from Kevin Murphy. + +2002-10-02 08:39 yozhik + + * BNT/potentials/Tables/marg_tableC.c: Initial revision + +2002-10-02 08:28 yozhik + + * BNT/potentials/Tables/: mult_by_tableM.m, mult_by_table2.m: + Initial import of code base from Kevin Murphy. + +2002-10-02 08:28 yozhik + + * BNT/potentials/Tables/: mult_by_tableM.m, mult_by_table2.m: + Initial revision + +2002-10-01 14:33 yozhik + + * BNT/potentials/Tables/mult_by_tableC.c: Initial import of code + base from Kevin Murphy. + +2002-10-01 14:33 yozhik + + * BNT/potentials/Tables/mult_by_tableC.c: Initial revision + +2002-10-01 14:23 yozhik + + * BNT/potentials/Tables/mult_by_table.c: Initial import of code + base from Kevin Murphy. + +2002-10-01 14:23 yozhik + + * BNT/potentials/Tables/mult_by_table.c: Initial revision + +2002-10-01 14:20 yozhik + + * BNT/potentials/Tables/repmat_and_mult.c: Initial import of code + base from Kevin Murphy. + +2002-10-01 14:20 yozhik + + * BNT/potentials/Tables/repmat_and_mult.c: Initial revision + +2002-10-01 12:04 yozhik + + * BNT/potentials/@dpot/dpot.m: Initial import of code base from + Kevin Murphy. + +2002-10-01 12:04 yozhik + + * BNT/potentials/@dpot/dpot.m: Initial revision + +2002-10-01 11:21 yozhik + + * BNT/examples/static/Belprop/belprop_polytree_discrete.m: Initial + import of code base from Kevin Murphy. + +2002-10-01 11:21 yozhik + + * BNT/examples/static/Belprop/belprop_polytree_discrete.m: Initial + revision + +2002-10-01 11:16 yozhik + + * BNT/examples/static/cmp_inference_static.m: Initial import of + code base from Kevin Murphy. + +2002-10-01 11:16 yozhik + + * BNT/examples/static/cmp_inference_static.m: Initial revision + +2002-10-01 10:39 yozhik + + * BNT/potentials/Tables/marg_tableM.m: Initial import of code base + from Kevin Murphy. + +2002-10-01 10:39 yozhik + + * BNT/potentials/Tables/marg_tableM.m: Initial revision + +2002-09-29 03:21 yozhik + + * BNT/potentials/Tables/mult_by_table_global.m: Initial import of + code base from Kevin Murphy. + +2002-09-29 03:21 yozhik + + * BNT/potentials/Tables/mult_by_table_global.m: Initial revision + +2002-09-26 01:39 yozhik + + * BNT/learning/learn_struct_K2.m: Initial import of code base from + Kevin Murphy. + +2002-09-26 01:39 yozhik + + * BNT/learning/learn_struct_K2.m: Initial revision + +2002-09-24 15:43 yozhik + + * BNT/: CPDs/@hhmm2Q_CPD/update_ess.m, + CPDs/@hhmm2Q_CPD/maximize_params.m, + examples/dynamic/HHMM/Map/disp_map_hhmm.m: Initial import of code + base from Kevin Murphy. + +2002-09-24 15:43 yozhik + + * BNT/: CPDs/@hhmm2Q_CPD/update_ess.m, + CPDs/@hhmm2Q_CPD/maximize_params.m, + examples/dynamic/HHMM/Map/disp_map_hhmm.m: Initial revision + +2002-09-24 15:34 yozhik + + * BNT/CPDs/@hhmm2Q_CPD/: hhmm2Q_CPD.m, reset_ess.m: Initial import + of code base from Kevin Murphy. + +2002-09-24 15:34 yozhik + + * BNT/CPDs/@hhmm2Q_CPD/: hhmm2Q_CPD.m, reset_ess.m: Initial + revision + +2002-09-24 15:13 yozhik + + * BNT/examples/dynamic/HHMM/Map/mk_rnd_map_hhmm.m: Initial import + of code base from Kevin Murphy. + +2002-09-24 15:13 yozhik + + * BNT/examples/dynamic/HHMM/Map/mk_rnd_map_hhmm.m: Initial revision + +2002-09-24 06:10 yozhik + + * BNT/CPDs/@hhmmQ_CPD/maximize_params.m: Initial import of code + base from Kevin Murphy. + +2002-09-24 06:10 yozhik + + * BNT/CPDs/@hhmmQ_CPD/maximize_params.m: Initial revision + +2002-09-24 06:02 yozhik + + * BNT/examples/dynamic/HHMM/Map/sample_from_map.m: Initial import + of code base from Kevin Murphy. + +2002-09-24 06:02 yozhik + + * BNT/examples/dynamic/HHMM/Map/sample_from_map.m: Initial revision + +2002-09-24 05:46 yozhik + + * BNT/CPDs/@hhmm2Q_CPD/CPD_to_CPT.m: Initial import of code base + from Kevin Murphy. + +2002-09-24 05:46 yozhik + + * BNT/CPDs/@hhmm2Q_CPD/CPD_to_CPT.m: Initial revision + +2002-09-24 03:49 yozhik + + * BNT/examples/dynamic/HHMM/Map/mk_map_hhmm.m: Initial import of + code base from Kevin Murphy. + +2002-09-24 03:49 yozhik + + * BNT/examples/dynamic/HHMM/Map/mk_map_hhmm.m: Initial revision + +2002-09-24 00:02 yozhik + + * BNT/examples/dynamic/HHMM/Map/Old/mk_map_hhmm.m: Initial import + of code base from Kevin Murphy. + +2002-09-24 00:02 yozhik + + * BNT/examples/dynamic/HHMM/Map/Old/mk_map_hhmm.m: Initial revision + +2002-09-23 21:19 yozhik + + * BNT/CPDs/@hhmmQ_CPD/hhmmQ_CPD.m: Initial import of code base from + Kevin Murphy. + +2002-09-23 21:19 yozhik + + * BNT/CPDs/@hhmmQ_CPD/hhmmQ_CPD.m: Initial revision + +2002-09-23 19:58 yozhik + + * BNT/CPDs/@hhmmQ_CPD/update_CPT.m: Initial import of code base + from Kevin Murphy. + +2002-09-23 19:58 yozhik + + * BNT/CPDs/@hhmmQ_CPD/update_CPT.m: Initial revision + +2002-09-23 19:30 yozhik + + * BNT/CPDs/@hhmmQ_CPD/Old/update_CPT.m: Initial import of code base + from Kevin Murphy. + +2002-09-23 19:30 yozhik + + * BNT/CPDs/@hhmmQ_CPD/Old/update_CPT.m: Initial revision + +2002-09-21 14:37 yozhik + + * BNT/examples/dynamic/HHMM/abcd_hhmm.m: Initial import of code + base from Kevin Murphy. + +2002-09-21 14:37 yozhik + + * BNT/examples/dynamic/HHMM/abcd_hhmm.m: Initial revision + +2002-09-21 13:58 yozhik + + * BNT/examples/dynamic/HHMM/mk_hhmm.m: Initial import of code base + from Kevin Murphy. + +2002-09-21 13:58 yozhik + + * BNT/examples/dynamic/HHMM/mk_hhmm.m: Initial revision + +2002-09-10 10:44 yozhik + + * BNT/CPDs/@gaussian_CPD/log_prob_node.m: Initial import of code + base from Kevin Murphy. + +2002-09-10 10:44 yozhik + + * BNT/CPDs/@gaussian_CPD/log_prob_node.m: Initial revision + +2002-07-28 16:09 yozhik + + * BNT/learning/: learn_struct_pdag_pc_constrain.m, CovMat.m: + Initial import of code base from Kevin Murphy. + +2002-07-28 16:09 yozhik + + * BNT/learning/: learn_struct_pdag_pc_constrain.m, CovMat.m: + Initial revision + +2002-07-24 07:48 yozhik + + * BNT/general/hodbn_to_bnet.m: Initial import of code base from + Kevin Murphy. + +2002-07-24 07:48 yozhik + + * BNT/general/hodbn_to_bnet.m: Initial revision + +2002-07-23 06:17 yozhik + + * BNT/general/mk_higher_order_dbn.m: Initial import of code base + from Kevin Murphy. + +2002-07-23 06:17 yozhik + + * BNT/general/mk_higher_order_dbn.m: Initial revision + +2002-07-20 18:25 yozhik + + * BNT/inference/online/@jtree_2TBN_inf_engine/set_fields.m: Initial + import of code base from Kevin Murphy. + +2002-07-20 18:25 yozhik + + * BNT/inference/online/@jtree_2TBN_inf_engine/set_fields.m: Initial + revision + +2002-07-20 17:32 yozhik + + * BNT/inference/online/@jtree_2TBN_inf_engine/back_mpe.m: Initial + import of code base from Kevin Murphy. + +2002-07-20 17:32 yozhik + + * BNT/inference/online/@jtree_2TBN_inf_engine/back_mpe.m: Initial + revision + +2002-07-02 15:56 yozhik + + * BNT/examples/dynamic/HHMM/Motif/learn_motif_hhmm.m: Initial + import of code base from Kevin Murphy. + +2002-07-02 15:56 yozhik + + * BNT/examples/dynamic/HHMM/Motif/learn_motif_hhmm.m: Initial + revision + +2002-06-27 13:34 yozhik + + * BNT/general/add_ev_to_dmarginal.m: Initial import of code base + from Kevin Murphy. + +2002-06-27 13:34 yozhik + + * BNT/general/add_ev_to_dmarginal.m: Initial revision + +2002-06-24 16:54 yozhik + + * BNT/CPDs/@hhmmF_CPD/update_ess.m: Initial import of code base + from Kevin Murphy. + +2002-06-24 16:54 yozhik + + * BNT/CPDs/@hhmmF_CPD/update_ess.m: Initial revision + +2002-06-24 16:38 yozhik + + * BNT/CPDs/@hhmmF_CPD/hhmmF_CPD.m: Initial import of code base from + Kevin Murphy. + +2002-06-24 16:38 yozhik + + * BNT/CPDs/@hhmmF_CPD/hhmmF_CPD.m: Initial revision + +2002-06-24 15:45 yozhik + + * BNT/CPDs/@hhmmF_CPD/update_CPT.m: Initial import of code base + from Kevin Murphy. + +2002-06-24 15:45 yozhik + + * BNT/CPDs/@hhmmF_CPD/update_CPT.m: Initial revision + +2002-06-24 15:35 yozhik + + * BNT/CPDs/@hhmmF_CPD/Old/: hhmmF_CPD.m, log_prior.m, + maximize_params.m, reset_ess.m, update_CPT.m, update_ess.m: + Initial import of code base from Kevin Murphy. + +2002-06-24 15:35 yozhik + + * BNT/CPDs/@hhmmF_CPD/Old/: hhmmF_CPD.m, log_prior.m, + maximize_params.m, reset_ess.m, update_CPT.m, update_ess.m: + Initial revision + +2002-06-24 15:23 yozhik + + * BNT/CPDs/@hhmmQ_CPD/Old/update_ess4.m: Initial import of code + base from Kevin Murphy. + +2002-06-24 15:23 yozhik + + * BNT/CPDs/@hhmmQ_CPD/Old/update_ess4.m: Initial revision + +2002-06-24 15:08 yozhik + + * BNT/CPDs/@hhmmQ_CPD/Old/update_ess3.m: Initial import of code + base from Kevin Murphy. + +2002-06-24 15:08 yozhik + + * BNT/CPDs/@hhmmQ_CPD/Old/update_ess3.m: Initial revision + +2002-06-24 14:20 yozhik + + * BNT/CPDs/@hhmmQ_CPD/Old/update_ess2.m: Initial import of code + base from Kevin Murphy. + +2002-06-24 14:20 yozhik + + * BNT/CPDs/@hhmmQ_CPD/Old/update_ess2.m: Initial revision + +2002-06-24 11:56 yozhik + + * BNT/: general/mk_fgraph_given_ev.m, + CPDs/mk_isolated_tabular_CPD.m: Initial import of code base from + Kevin Murphy. + +2002-06-24 11:56 yozhik + + * BNT/: general/mk_fgraph_given_ev.m, + CPDs/mk_isolated_tabular_CPD.m: Initial revision + +2002-06-24 11:19 yozhik + + * BNT/CPDs/@hhmmQ_CPD/Old/: hhmmQ_CPD.m, log_prior.m, + maximize_params.m, reset_ess.m, update_ess.m: Initial import of + code base from Kevin Murphy. + +2002-06-24 11:19 yozhik + + * BNT/CPDs/@hhmmQ_CPD/Old/: hhmmQ_CPD.m, log_prior.m, + maximize_params.m, reset_ess.m, update_ess.m: Initial revision + +2002-06-20 13:30 yozhik + + * BNT/examples/dynamic/mildew1.m: Initial import of code base from + Kevin Murphy. + +2002-06-20 13:30 yozhik + + * BNT/examples/dynamic/mildew1.m: Initial revision + +2002-06-19 17:18 yozhik + + * BNT/: inference/dynamic/@hmm_inf_engine/find_mpe.m, + examples/dynamic/HHMM/Square/learn_square_hhmm_cts.m: Initial + import of code base from Kevin Murphy. + +2002-06-19 17:18 yozhik + + * BNT/: inference/dynamic/@hmm_inf_engine/find_mpe.m, + examples/dynamic/HHMM/Square/learn_square_hhmm_cts.m: Initial + revision + +2002-06-19 17:03 yozhik + + * BNT/examples/static/fgraph/fg1.m: Initial import of code base + from Kevin Murphy. + +2002-06-19 17:03 yozhik + + * BNT/examples/static/fgraph/fg1.m: Initial revision + +2002-06-19 16:59 yozhik + + * BNT/: examples/static/softev1.m, + inference/static/@belprop_fg_inf_engine/find_mpe.m: Initial + import of code base from Kevin Murphy. + +2002-06-19 16:59 yozhik + + * BNT/: examples/static/softev1.m, + inference/static/@belprop_fg_inf_engine/find_mpe.m: Initial + revision + +2002-06-19 15:11 yozhik + + * BNT/inference/static/@var_elim_inf_engine/find_mpe.m: Initial + import of code base from Kevin Murphy. + +2002-06-19 15:11 yozhik + + * BNT/inference/static/@var_elim_inf_engine/find_mpe.m: Initial + revision + +2002-06-19 15:08 yozhik + + * BNT/: inference/static/@belprop_inf_engine/find_mpe.m, + examples/static/mpe1.m, examples/static/mpe2.m: Initial import of + code base from Kevin Murphy. + +2002-06-19 15:08 yozhik + + * BNT/: inference/static/@belprop_inf_engine/find_mpe.m, + examples/static/mpe1.m, examples/static/mpe2.m: Initial revision + +2002-06-19 15:04 yozhik + + * BNT/inference/static/@var_elim_inf_engine/: + var_elim_inf_engine.m, enter_evidence.m: Initial import of code + base from Kevin Murphy. + +2002-06-19 15:04 yozhik + + * BNT/inference/static/@var_elim_inf_engine/: + var_elim_inf_engine.m, enter_evidence.m: Initial revision + +2002-06-19 14:56 yozhik + + * BNT/inference/static/@global_joint_inf_engine/find_mpe.m: Initial + import of code base from Kevin Murphy. + +2002-06-19 14:56 yozhik + + * BNT/inference/static/@global_joint_inf_engine/find_mpe.m: Initial + revision + +2002-06-17 16:49 yozhik + + * BNT/inference/online/: @jtree_2TBN_inf_engine/back1_mpe.m, + @smoother_engine/find_mpe.m: Initial import of code base from + Kevin Murphy. + +2002-06-17 16:49 yozhik + + * BNT/inference/online/: @jtree_2TBN_inf_engine/back1_mpe.m, + @smoother_engine/find_mpe.m: Initial revision + +2002-06-17 16:46 yozhik + + * BNT/inference/online/: @jtree_2TBN_inf_engine/fwd.m, + @jtree_2TBN_inf_engine/fwd1.m, @smoother_engine/enter_evidence.m: + Initial import of code base from Kevin Murphy. + +2002-06-17 16:46 yozhik + + * BNT/inference/online/: @jtree_2TBN_inf_engine/fwd.m, + @jtree_2TBN_inf_engine/fwd1.m, @smoother_engine/enter_evidence.m: + Initial revision + +2002-06-17 16:38 yozhik + + * BNT/inference/online/@jtree_2TBN_inf_engine/backT_mpe.m: Initial + import of code base from Kevin Murphy. + +2002-06-17 16:38 yozhik + + * BNT/inference/online/@jtree_2TBN_inf_engine/backT_mpe.m: Initial + revision + +2002-06-17 16:34 yozhik + + * BNT/inference/online/@jtree_2TBN_inf_engine/: back.m, backT.m, + back1.m: Initial import of code base from Kevin Murphy. + +2002-06-17 16:34 yozhik + + * BNT/inference/online/@jtree_2TBN_inf_engine/: back.m, backT.m, + back1.m: Initial revision + +2002-06-17 16:14 yozhik + + * BNT/inference/static/@jtree_inf_engine/: find_mpe.m, + find_max_config.m: Initial import of code base from Kevin Murphy. + +2002-06-17 16:14 yozhik + + * BNT/inference/static/@jtree_inf_engine/: find_mpe.m, + find_max_config.m: Initial revision + +2002-06-17 14:58 yozhik + + * BNT/general/Old/calc_mpe.m: Initial import of code base from + Kevin Murphy. + +2002-06-17 14:58 yozhik + + * BNT/general/Old/calc_mpe.m: Initial revision + +2002-06-17 13:59 yozhik + + * BNT/inference/static/@jtree_inf_engine/: enter_evidence.m, + distribute_evidence.m: Initial import of code base from Kevin + Murphy. + +2002-06-17 13:59 yozhik + + * BNT/inference/static/@jtree_inf_engine/: enter_evidence.m, + distribute_evidence.m: Initial revision + +2002-06-17 13:29 yozhik + + * BNT/inference/static/@jtree_mnet_inf_engine/: find_mpe.m, + enter_evidence.m: Initial import of code base from Kevin Murphy. + +2002-06-17 13:29 yozhik + + * BNT/inference/static/@jtree_mnet_inf_engine/: find_mpe.m, + enter_evidence.m: Initial revision + +2002-06-16 13:01 yozhik + + * BNT/general/is_mnet.m: Initial import of code base from Kevin + Murphy. + +2002-06-16 13:01 yozhik + + * BNT/general/is_mnet.m: Initial revision + +2002-06-16 12:52 yozhik + + * BNT/general/mk_mnet.m: Initial import of code base from Kevin + Murphy. + +2002-06-16 12:52 yozhik + + * BNT/general/mk_mnet.m: Initial revision + +2002-06-16 12:34 yozhik + + * BNT/inference/static/@jtree_inf_engine/init_pot.m: Initial import + of code base from Kevin Murphy. + +2002-06-16 12:34 yozhik + + * BNT/inference/static/@jtree_inf_engine/init_pot.m: Initial + revision + +2002-06-16 12:06 yozhik + + * BNT/potentials/@dpot/find_most_prob_entry.m: Initial import of + code base from Kevin Murphy. + +2002-06-16 12:06 yozhik + + * BNT/potentials/@dpot/find_most_prob_entry.m: Initial revision + +2002-05-31 03:25 yozhik + + * BNT/general/unroll_higher_order_topology.m: Initial import of + code base from Kevin Murphy. + +2002-05-31 03:25 yozhik + + * BNT/general/unroll_higher_order_topology.m: Initial revision + +2002-05-29 08:59 yozhik + + * BNT/@assocarray/assocarray.m, + BNT/CPDs/@boolean_CPD/boolean_CPD.m, + BNT/CPDs/@discrete_CPD/CPD_to_lambda_msg.m, + BNT/CPDs/@discrete_CPD/CPD_to_pi.m, + BNT/CPDs/@discrete_CPD/CPD_to_scgpot.m, + BNT/CPDs/@discrete_CPD/README, + BNT/CPDs/@discrete_CPD/convert_CPD_to_table_hidden_ps.m, + BNT/CPDs/@discrete_CPD/convert_obs_CPD_to_table.m, + BNT/CPDs/@discrete_CPD/convert_to_sparse_table.c, + BNT/CPDs/@discrete_CPD/convert_to_table.m, + BNT/CPDs/@discrete_CPD/discrete_CPD.m, + BNT/CPDs/@discrete_CPD/dom_sizes.m, + BNT/CPDs/@discrete_CPD/log_prob_node.m, + BNT/CPDs/@discrete_CPD/prob_node.m, + BNT/CPDs/@discrete_CPD/sample_node.m, + BNT/CPDs/@discrete_CPD/Old/convert_to_pot.m, + BNT/CPDs/@discrete_CPD/Old/convert_to_table.m, + BNT/CPDs/@discrete_CPD/Old/prob_CPD.m, + BNT/CPDs/@discrete_CPD/Old/prob_node.m, + BNT/CPDs/@discrete_CPD/private/prod_CPT_and_pi_msgs.m, + BNT/CPDs/@gaussian_CPD/CPD_to_lambda_msg.m, + BNT/CPDs/@gaussian_CPD/CPD_to_pi.m, + BNT/CPDs/@gaussian_CPD/CPD_to_scgpot.m, + BNT/CPDs/@gaussian_CPD/adjustable_CPD.m, + BNT/CPDs/@gaussian_CPD/convert_CPD_to_table_hidden_ps.m, + BNT/CPDs/@gaussian_CPD/display.m, + BNT/CPDs/@gaussian_CPD/get_field.m, + BNT/CPDs/@gaussian_CPD/reset_ess.m, + BNT/CPDs/@gaussian_CPD/sample_node.m, + BNT/CPDs/@gaussian_CPD/set_fields.m, + BNT/CPDs/@gaussian_CPD/Old/CPD_to_lambda_msg.m, + BNT/CPDs/@gaussian_CPD/Old/gaussian_CPD.m, + BNT/CPDs/@gaussian_CPD/Old/log_prob_node.m, + BNT/CPDs/@gaussian_CPD/Old/update_ess.m, + BNT/CPDs/@gaussian_CPD/Old/update_tied_ess.m, + BNT/CPDs/@gaussian_CPD/private/CPD_to_linear_gaussian.m, + BNT/CPDs/@generic_CPD/README, + BNT/CPDs/@generic_CPD/adjustable_CPD.m, + BNT/CPDs/@generic_CPD/display.m, + BNT/CPDs/@generic_CPD/generic_CPD.m, + BNT/CPDs/@generic_CPD/log_prior.m, + BNT/CPDs/@generic_CPD/set_clamped.m, + BNT/CPDs/@generic_CPD/Old/BIC_score_CPD.m, + BNT/CPDs/@generic_CPD/Old/CPD_to_dpots.m, + BNT/CPDs/@gmux_CPD/CPD_to_lambda_msg.m, + BNT/CPDs/@gmux_CPD/convert_to_pot.m, + BNT/CPDs/@gmux_CPD/CPD_to_pi.m, BNT/CPDs/@gmux_CPD/display.m, + BNT/CPDs/@gmux_CPD/gmux_CPD.m, BNT/CPDs/@gmux_CPD/sample_node.m, + BNT/CPDs/@gmux_CPD/Old/gmux_CPD.m, + BNT/CPDs/@hhmmF_CPD/log_prior.m, + BNT/CPDs/@hhmmF_CPD/maximize_params.m, + BNT/CPDs/@hhmmF_CPD/reset_ess.m, BNT/CPDs/@hhmmQ_CPD/log_prior.m, + BNT/CPDs/@hhmmQ_CPD/reset_ess.m, + BNT/CPDs/@mlp_CPD/convert_to_table.m, + BNT/CPDs/@mlp_CPD/maximize_params.m, BNT/CPDs/@mlp_CPD/mlp_CPD.m, + BNT/CPDs/@mlp_CPD/reset_ess.m, BNT/CPDs/@mlp_CPD/update_ess.m, + BNT/CPDs/@noisyor_CPD/CPD_to_lambda_msg.m, + BNT/CPDs/@noisyor_CPD/CPD_to_pi.m, + BNT/CPDs/@noisyor_CPD/noisyor_CPD.m, + BNT/CPDs/@noisyor_CPD/private/sum_prod_CPD_and_pi_msgs.m, + BNT/CPDs/@root_CPD/CPD_to_pi.m, + BNT/CPDs/@root_CPD/convert_to_pot.m, + BNT/CPDs/@root_CPD/log_marg_prob_node.m, + BNT/CPDs/@root_CPD/log_prob_node.m, + BNT/CPDs/@root_CPD/root_CPD.m, BNT/CPDs/@root_CPD/sample_node.m, + BNT/CPDs/@root_CPD/Old/CPD_to_CPT.m, + BNT/CPDs/@softmax_CPD/convert_to_pot.m, + BNT/CPDs/@softmax_CPD/display.m, + BNT/CPDs/@softmax_CPD/get_field.m, + BNT/CPDs/@softmax_CPD/maximize_params.m, + BNT/CPDs/@softmax_CPD/reset_ess.m, + BNT/CPDs/@softmax_CPD/sample_node.m, + BNT/CPDs/@softmax_CPD/set_fields.m, + BNT/CPDs/@softmax_CPD/update_ess.m, + BNT/CPDs/@softmax_CPD/private/extract_params.m, + BNT/CPDs/@tabular_CPD/CPD_to_CPT.m, + BNT/CPDs/@tabular_CPD/bayes_update_params.m, + BNT/CPDs/@tabular_CPD/log_nextcase_prob_node.m, + BNT/CPDs/@tabular_CPD/log_prior.m, + BNT/CPDs/@tabular_CPD/reset_ess.m, + BNT/CPDs/@tabular_CPD/update_ess.m, + BNT/CPDs/@tabular_CPD/update_ess_simple.m, + BNT/CPDs/@tabular_CPD/Old/BIC_score_CPD.m, + BNT/CPDs/@tabular_CPD/Old/bayesian_score_CPD.m, + BNT/CPDs/@tabular_CPD/Old/log_marg_prob_node_case.m, + BNT/CPDs/@tabular_CPD/Old/mult_CPD_and_pi_msgs.m, + BNT/CPDs/@tabular_CPD/Old/prob_CPT.m, + BNT/CPDs/@tabular_CPD/Old/prob_node.m, + BNT/CPDs/@tabular_CPD/Old/sample_node.m, + BNT/CPDs/@tabular_CPD/Old/sample_node_single_case.m, + BNT/CPDs/@tabular_CPD/Old/tabular_CPD.m, + BNT/CPDs/@tabular_CPD/Old/update_params.m, + BNT/CPDs/@tabular_decision_node/CPD_to_CPT.m, + BNT/CPDs/@tabular_decision_node/display.m, + BNT/CPDs/@tabular_decision_node/get_field.m, + BNT/CPDs/@tabular_decision_node/set_fields.m, + BNT/CPDs/@tabular_decision_node/tabular_decision_node.m, + BNT/CPDs/@tabular_decision_node/Old/tabular_decision_node.m, + BNT/CPDs/@tabular_kernel/convert_to_pot.m, + BNT/CPDs/@tabular_kernel/convert_to_table.m, + BNT/CPDs/@tabular_kernel/get_field.m, + BNT/CPDs/@tabular_kernel/set_fields.m, + BNT/CPDs/@tabular_kernel/tabular_kernel.m, + BNT/CPDs/@tabular_kernel/Old/tabular_kernel.m, + BNT/CPDs/@tabular_utility_node/convert_to_pot.m, + BNT/CPDs/@tabular_utility_node/display.m, + BNT/CPDs/@tabular_utility_node/tabular_utility_node.m, + BNT/CPDs/@tree_CPD/display.m, + BNT/CPDs/@tree_CPD/evaluate_tree_performance.m, + BNT/CPDs/@tree_CPD/get_field.m, + BNT/CPDs/@tree_CPD/learn_params.m, BNT/CPDs/@tree_CPD/readme.txt, + BNT/CPDs/@tree_CPD/set_fields.m, BNT/CPDs/@tree_CPD/tree_CPD.m, + BNT/CPDs/Old/@linear_gaussian_CPD/linear_gaussian_CPD.m, + BNT/CPDs/Old/@linear_gaussian_CPD/log_marg_prob_node.m, + BNT/CPDs/Old/@linear_gaussian_CPD/update_params_complete.m, + BNT/CPDs/Old/@root_gaussian_CPD/log_marg_prob_node.m, + BNT/CPDs/Old/@root_gaussian_CPD/root_gaussian_CPD.m, + BNT/CPDs/Old/@root_gaussian_CPD/update_params_complete.m, + BNT/CPDs/Old/@tabular_chance_node/CPD_to_upot.m, + BNT/CPDs/Old/@tabular_chance_node/tabular_chance_node.m, + BNT/examples/dynamic/bat1.m, BNT/examples/dynamic/bkff1.m, + BNT/examples/dynamic/chmm1.m, + BNT/examples/dynamic/cmp_inference_dbn.m, + BNT/examples/dynamic/cmp_learning_dbn.m, + BNT/examples/dynamic/cmp_online_inference.m, + BNT/examples/dynamic/fhmm_infer.m, + BNT/examples/dynamic/filter_test1.m, + BNT/examples/dynamic/kalman1.m, + BNT/examples/dynamic/kjaerulff1.m, + BNT/examples/dynamic/loopy_dbn1.m, + BNT/examples/dynamic/mk_collage_from_clqs.m, + BNT/examples/dynamic/mk_fhmm.m, BNT/examples/dynamic/reveal1.m, + BNT/examples/dynamic/scg_dbn.m, + BNT/examples/dynamic/skf_data_assoc_gmux.m, + BNT/examples/dynamic/HHMM/add_hhmm_end_state.m, + BNT/examples/dynamic/HHMM/hhmm_jtree_clqs.m, + BNT/examples/dynamic/HHMM/mk_hhmm_topo.m, + BNT/examples/dynamic/HHMM/mk_hhmm_topo_F1.m, + BNT/examples/dynamic/HHMM/pretty_print_hhmm_parse.m, + BNT/examples/dynamic/HHMM/Motif/fixed_args_mk_motif_hhmm.m, + BNT/examples/dynamic/HHMM/Motif/mk_motif_hhmm.m, + BNT/examples/dynamic/HHMM/Motif/sample_motif_hhmm.m, + BNT/examples/dynamic/HHMM/Old/mk_abcd_hhmm.m, + BNT/examples/dynamic/HHMM/Old/mk_arrow_alpha_hhmm3.m, + BNT/examples/dynamic/HHMM/Old/mk_hhmm2.m, + BNT/examples/dynamic/HHMM/Old/mk_hhmm3.m, + BNT/examples/dynamic/HHMM/Old/mk_hhmm3_args.m, + BNT/examples/dynamic/HHMM/Old/motif_hhmm.m, + BNT/examples/dynamic/HHMM/Old/remove_hhmm_end_state.m, + BNT/examples/dynamic/HHMM/Square/get_square_data.m, + BNT/examples/dynamic/HHMM/Square/hhmm_inference.m, + BNT/examples/dynamic/HHMM/Square/is_F2_true_D3.m, + BNT/examples/dynamic/HHMM/Square/learn_square_hhmm_discrete.m, + BNT/examples/dynamic/HHMM/Square/mk_square_hhmm.m, + BNT/examples/dynamic/HHMM/Square/plot_square_hhmm.m, + BNT/examples/dynamic/HHMM/Square/sample_square_hhmm_cts.m, + BNT/examples/dynamic/HHMM/Square/sample_square_hhmm_discrete.m, + BNT/examples/dynamic/HHMM/Square/square4.mat, + BNT/examples/dynamic/HHMM/Square/square4_cases.mat, + BNT/examples/dynamic/HHMM/Square/test_square_fig.m, + BNT/examples/dynamic/HHMM/Square/test_square_fig.mat, + BNT/examples/dynamic/HHMM/Square/Old/learn_square_hhmm.m, + BNT/examples/dynamic/HHMM/Square/Old/mk_square_hhmm.m, + BNT/examples/dynamic/HHMM/Square/Old/plot_square_hhmm.m, + BNT/examples/dynamic/HHMM/Square/Old/sample_square_hhmm.m, + BNT/examples/dynamic/Old/chmm1.m, + BNT/examples/dynamic/Old/cmp_inference.m, + BNT/examples/dynamic/Old/kalman1.m, + BNT/examples/dynamic/Old/old.water1.m, + BNT/examples/dynamic/Old/online1.m, + BNT/examples/dynamic/Old/online2.m, + BNT/examples/dynamic/Old/scg_dbn.m, + BNT/examples/dynamic/SLAM/mk_gmux_robot_dbn.m, + BNT/examples/dynamic/SLAM/mk_linear_slam.m, + BNT/examples/dynamic/SLAM/slam_kf.m, + BNT/examples/dynamic/SLAM/slam_offline_loopy.m, + BNT/examples/dynamic/SLAM/slam_partial_kf.m, + BNT/examples/dynamic/SLAM/slam_stationary_loopy.m, + BNT/examples/dynamic/SLAM/Old/offline_loopy_slam.m, + BNT/examples/dynamic/SLAM/Old/paskin1.m, + BNT/examples/dynamic/SLAM/Old/skf_data_assoc_gmux2.m, + BNT/examples/dynamic/SLAM/Old/slam_kf.m, + BNT/examples/limids/id1.m, BNT/examples/limids/pigs1.m, + BNT/examples/static/cg1.m, BNT/examples/static/cg2.m, + BNT/examples/static/discrete2.m, BNT/examples/static/discrete3.m, + BNT/examples/static/fa1.m, BNT/examples/static/gaussian1.m, + BNT/examples/static/gibbs_test1.m, BNT/examples/static/lw1.m, + BNT/examples/static/mfa1.m, BNT/examples/static/mixexp1.m, + BNT/examples/static/mixexp2.m, BNT/examples/static/mixexp3.m, + BNT/examples/static/mog1.m, BNT/examples/static/qmr1.m, + BNT/examples/static/sample1.m, BNT/examples/static/softmax1.m, + BNT/examples/static/Belprop/belprop_loop1_discrete.m, + BNT/examples/static/Belprop/belprop_loop1_gauss.m, + BNT/examples/static/Belprop/belprop_loopy_cg.m, + BNT/examples/static/Belprop/belprop_loopy_discrete.m, + BNT/examples/static/Belprop/belprop_loopy_gauss.m, + BNT/examples/static/Belprop/belprop_polytree_cg.m, + BNT/examples/static/Belprop/belprop_polytree_gauss.m, + BNT/examples/static/Belprop/bp1.m, + BNT/examples/static/Belprop/gmux1.m, + BNT/examples/static/Brutti/Belief_IOhmm.m, + BNT/examples/static/Brutti/Belief_hmdt.m, + BNT/examples/static/Brutti/Belief_hme.m, + BNT/examples/static/Brutti/Sigmoid_Belief.m, + BNT/examples/static/HME/HMEforMatlab.jpg, + BNT/examples/static/HME/README, BNT/examples/static/HME/fhme.m, + BNT/examples/static/HME/gen_data.m, + BNT/examples/static/HME/hme_class_plot.m, + BNT/examples/static/HME/hme_reg_plot.m, + BNT/examples/static/HME/hme_topobuilder.m, + BNT/examples/static/HME/test_data_class.mat, + BNT/examples/static/HME/test_data_class2.mat, + BNT/examples/static/HME/test_data_reg.mat, + BNT/examples/static/HME/train_data_class.mat, + BNT/examples/static/HME/train_data_reg.mat, + BNT/examples/static/Misc/mixexp_data.txt, + BNT/examples/static/Misc/mixexp_graddesc.m, + BNT/examples/static/Misc/mixexp_plot.m, + BNT/examples/static/Misc/sprinkler.bif, + BNT/examples/static/Models/mk_cancer_bnet.m, + BNT/examples/static/Models/mk_car_bnet.m, + BNT/examples/static/Models/mk_ideker_bnet.m, + BNT/examples/static/Models/mk_incinerator_bnet.m, + BNT/examples/static/Models/mk_markov_chain_bnet.m, + BNT/examples/static/Models/mk_minimal_qmr_bnet.m, + BNT/examples/static/Models/mk_qmr_bnet.m, + BNT/examples/static/Models/mk_vstruct_bnet.m, + BNT/examples/static/Models/Old/mk_hmm_bnet.m, + BNT/examples/static/SCG/scg1.m, BNT/examples/static/SCG/scg2.m, + BNT/examples/static/SCG/scg3.m, + BNT/examples/static/SCG/scg_3node.m, + BNT/examples/static/SCG/scg_unstable.m, + BNT/examples/static/StructLearn/bic1.m, + BNT/examples/static/StructLearn/cooper_yoo.m, + BNT/examples/static/StructLearn/k2demo1.m, + BNT/examples/static/StructLearn/mcmc1.m, + BNT/examples/static/StructLearn/pc1.m, + BNT/examples/static/StructLearn/pc2.m, + BNT/examples/static/Zoubin/README, + BNT/examples/static/Zoubin/csum.m, + BNT/examples/static/Zoubin/ffa.m, + BNT/examples/static/Zoubin/mfa.m, + BNT/examples/static/Zoubin/mfa_cl.m, + BNT/examples/static/Zoubin/mfademo.m, + BNT/examples/static/Zoubin/rdiv.m, + BNT/examples/static/Zoubin/rprod.m, + BNT/examples/static/Zoubin/rsum.m, + BNT/examples/static/dtree/test_housing.m, + BNT/examples/static/dtree/test_restaurants.m, + BNT/examples/static/dtree/test_zoo1.m, + BNT/examples/static/dtree/tmp.dot, + BNT/examples/static/dtree/transform_data_into_bnt_format.m, + BNT/examples/static/fgraph/fg2.m, + BNT/examples/static/fgraph/fg3.m, + BNT/examples/static/fgraph/fg_mrf1.m, + BNT/examples/static/fgraph/fg_mrf2.m, + BNT/general/bnet_to_fgraph.m, + BNT/general/compute_fwd_interface.m, + BNT/general/compute_interface_nodes.m, + BNT/general/compute_minimal_interface.m, + BNT/general/dbn_to_bnet.m, + BNT/general/determine_elim_constraints.m, + BNT/general/do_intervention.m, BNT/general/dsep.m, + BNT/general/enumerate_scenarios.m, BNT/general/fgraph_to_bnet.m, + BNT/general/log_lik_complete.m, + BNT/general/log_marg_lik_complete.m, BNT/general/mk_bnet.m, + BNT/general/mk_fgraph.m, BNT/general/mk_limid.m, + BNT/general/mk_mutilated_samples.m, + BNT/general/mk_slice_and_half_dbn.m, + BNT/general/partition_dbn_nodes.m, + BNT/general/sample_bnet_nocell.m, BNT/general/sample_dbn.m, + BNT/general/score_bnet_complete.m, + BNT/general/unroll_dbn_topology.m, + BNT/general/Old/bnet_to_gdl_graph.m, + BNT/general/Old/calc_mpe_bucket.m, + BNT/general/Old/calc_mpe_dbn.m, + BNT/general/Old/calc_mpe_given_inf_engine.m, + BNT/general/Old/calc_mpe_global.m, + BNT/general/Old/compute_interface_nodes.m, + BNT/general/Old/mk_gdl_graph.m, GraphViz/draw_dbn.m, + GraphViz/make_layout.m, BNT/license.gpl.txt, + BNT/general/add_evidence_to_gmarginal.m, + BNT/inference/@inf_engine/bnet_from_engine.m, + BNT/inference/@inf_engine/get_field.m, + BNT/inference/@inf_engine/inf_engine.m, + BNT/inference/@inf_engine/marginal_family.m, + BNT/inference/@inf_engine/set_fields.m, + BNT/inference/@inf_engine/update_engine.m, + BNT/inference/@inf_engine/Old/marginal_family_pot.m, + BNT/inference/@inf_engine/Old/observed_nodes.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/bk_ff_hmm_inf_engine.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/dbn_init_bel.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/dbn_marginal_from_bel.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/dbn_predict_bel.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/dbn_update_bel.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/dbn_update_bel1.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/enter_evidence.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/marginal_family.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/marginal_nodes.m, 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BNT/inference/dynamic/@bk_inf_engine/update_engine.m, + BNT/inference/dynamic/@ff_inf_engine/enter_evidence.m, + BNT/inference/dynamic/@ff_inf_engine/enter_soft_evidence.m, + BNT/inference/dynamic/@ff_inf_engine/ff_inf_engine.m, + BNT/inference/dynamic/@ff_inf_engine/filter_evidence.m, + BNT/inference/dynamic/@ff_inf_engine/marginal_family.m, + BNT/inference/dynamic/@ff_inf_engine/marginal_nodes.m, + BNT/inference/dynamic/@ff_inf_engine/smooth_evidence.m, + BNT/inference/dynamic/@ff_inf_engine/Old/enter_soft_evidence.m, + BNT/inference/dynamic/@ff_inf_engine/Old/enter_soft_evidence1.m, + BNT/inference/dynamic/@ff_inf_engine/Old/marginal_family.m, + BNT/inference/dynamic/@frontier_inf_engine/enter_evidence.m, + BNT/inference/dynamic/@frontier_inf_engine/enter_soft_evidence.m, + BNT/inference/dynamic/@frontier_inf_engine/frontier_inf_engine.m, + BNT/inference/dynamic/@frontier_inf_engine/marginal_family.m, + BNT/inference/dynamic/@frontier_inf_engine/marginal_nodes.m, + 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BNT/inference/static/@gibbs_sampling_inf_engine/private/CPT.m, + BNT/inference/static/@gibbs_sampling_inf_engine/private/compute_children.m, + BNT/inference/static/@gibbs_sampling_inf_engine/private/compute_families.m, + BNT/inference/static/@gibbs_sampling_inf_engine/private/compute_families_dbn.m, + BNT/inference/static/@gibbs_sampling_inf_engine/private/compute_posterior.c, + BNT/inference/static/@gibbs_sampling_inf_engine/private/compute_posterior_dbn.m, + BNT/inference/static/@gibbs_sampling_inf_engine/private/compute_strides.m, + BNT/inference/static/@gibbs_sampling_inf_engine/private/get_cpts.m, + BNT/inference/static/@gibbs_sampling_inf_engine/private/get_slice_dbn.c, + BNT/inference/static/@gibbs_sampling_inf_engine/private/get_slice_dbn.m, + BNT/inference/static/@gibbs_sampling_inf_engine/private/my_sample_discrete.m, + BNT/inference/static/@gibbs_sampling_inf_engine/private/sample_single_discrete.c, + BNT/inference/static/@global_joint_inf_engine/global_joint_inf_engine.m, + BNT/inference/static/@global_joint_inf_engine/marginal_family.m, + BNT/inference/static/@jtree_inf_engine/cliques_from_engine.m, + BNT/inference/static/@jtree_inf_engine/clq_containing_nodes.m, + BNT/inference/static/@jtree_inf_engine/collect_evidence.m, + BNT/inference/static/@jtree_inf_engine/enter_soft_evidence.m, + BNT/inference/static/@jtree_inf_engine/marginal_family.m, + BNT/inference/static/@jtree_inf_engine/marginal_nodes.m, + BNT/inference/static/@jtree_inf_engine/set_fields.m, + BNT/inference/static/@jtree_inf_engine/Old/collect_evidence.m, + BNT/inference/static/@jtree_inf_engine/Old/distribute_evidence.m, + BNT/inference/static/@jtree_inf_engine/Old/enter_evidence.m, + BNT/inference/static/@jtree_inf_engine/Old/enter_soft_evidence.m, + BNT/inference/static/@jtree_limid_inf_engine/enter_evidence.m, + BNT/inference/static/@jtree_limid_inf_engine/jtree_limid_inf_engine.m, + BNT/inference/static/@jtree_limid_inf_engine/marginal_family.m, + BNT/inference/static/@jtree_limid_inf_engine/marginal_nodes.m, + BNT/inference/static/@jtree_limid_inf_engine/Old/marginal_family.m, + BNT/inference/static/@jtree_limid_inf_engine/Old/marginal_nodes_SS.m, + BNT/inference/static/@jtree_sparse_inf_engine/cliques_from_engine.m, + BNT/inference/static/@jtree_sparse_inf_engine/clq_containing_nodes.m, + BNT/inference/static/@jtree_sparse_inf_engine/collect_evidence.c, + BNT/inference/static/@jtree_sparse_inf_engine/distribute_evidence.c, + BNT/inference/static/@jtree_sparse_inf_engine/enter_evidence.m, + BNT/inference/static/@jtree_sparse_inf_engine/enter_soft_evidence.m, + BNT/inference/static/@jtree_sparse_inf_engine/init_pot.c, + BNT/inference/static/@jtree_sparse_inf_engine/marginal_family.m, + BNT/inference/static/@jtree_sparse_inf_engine/marginal_nodes.m, + BNT/inference/static/@jtree_sparse_inf_engine/set_fields.m, + BNT/inference/static/@jtree_sparse_inf_engine/old/collect_evidence.c, + BNT/inference/static/@jtree_sparse_inf_engine/old/distribute_evidence.c, + BNT/inference/static/@jtree_sparse_inf_engine/old/init_pot.c, + BNT/inference/static/@jtree_sparse_inf_engine/old/init_pot1.c, + BNT/inference/static/@jtree_sparse_inf_engine/old/init_pot1.m, + BNT/inference/static/@likelihood_weighting_inf_engine/enter_evidence.m, + BNT/inference/static/@likelihood_weighting_inf_engine/likelihood_weighting_inf_engine.m, + BNT/inference/static/@likelihood_weighting_inf_engine/marginal_nodes.m, + BNT/inference/static/@pearl_inf_engine/enter_evidence.m, + BNT/inference/static/@pearl_inf_engine/loopy_converged.m, + BNT/inference/static/@pearl_inf_engine/marginal_nodes.m, + BNT/inference/static/@pearl_inf_engine/private/compute_bel.m, + BNT/inference/static/@pearl_inf_engine/private/prod_lambda_msgs.m, + BNT/inference/static/@pearl_inf_engine/private/tree_protocol.m, + BNT/inference/static/@quickscore_inf_engine/enter_evidence.m, + BNT/inference/static/@quickscore_inf_engine/marginal_nodes.m, + BNT/inference/static/@quickscore_inf_engine/quickscore_inf_engine.m, + BNT/inference/static/@quickscore_inf_engine/private/C_quickscore.c, + BNT/inference/static/@quickscore_inf_engine/private/nr.h, + BNT/inference/static/@quickscore_inf_engine/private/nrutil.c, + BNT/inference/static/@quickscore_inf_engine/private/nrutil.h, + BNT/inference/static/@quickscore_inf_engine/private/quickscore.m, + BNT/learning/bayes_update_params.m, + BNT/learning/bic_score_family.m, + BNT/learning/compute_cooling_schedule.m, + BNT/learning/dirichlet_score_family.m, + BNT/learning/kpm_learn_struct_mcmc.m, + BNT/learning/learn_params_em.m, + BNT/learning/learn_struct_dbn_reveal.m, + BNT/learning/learn_struct_pdag_ic_star.m, + BNT/learning/mcmc_sample_to_hist.m, BNT/learning/mk_schedule.m, + BNT/learning/mk_tetrad_data_file.m, + BNT/learning/score_dags_old.m, HMM/dhmm_logprob_brute_force.m, + HMM/dhmm_logprob_path.m, HMM/mdp_sample.m, Kalman/AR_to_SS.m, + Kalman/SS_to_AR.m, Kalman/convert_to_lagged_form.m, + Kalman/ensure_AR.m, Kalman/eval_AR_perf.m, + Kalman/kalman_filter.m, Kalman/kalman_smoother.m, + Kalman/kalman_update.m, Kalman/learn_AR.m, + Kalman/learn_AR_diagonal.m, Kalman/learn_kalman.m, + Kalman/smooth_update.m, + BNT/general/convert_dbn_CPDs_to_tables_slow.m, + BNT/general/dispcpt.m, BNT/general/linear_gaussian_to_cpot.m, + BNT/general/partition_matrix_vec_3.m, + BNT/general/shrink_obs_dims_in_gaussian.m, + BNT/general/shrink_obs_dims_in_table.m, + BNT/potentials/CPD_to_pot.m, BNT/potentials/README, + BNT/potentials/check_for_cd_arcs.m, + BNT/potentials/determine_pot_type.m, + BNT/potentials/mk_initial_pot.m, + BNT/potentials/@cgpot/cg_can_to_mom.m, + BNT/potentials/@cgpot/cg_mom_to_can.m, + BNT/potentials/@cgpot/cgpot.m, BNT/potentials/@cgpot/display.m, + BNT/potentials/@cgpot/divide_by_pot.m, + BNT/potentials/@cgpot/domain_pot.m, + BNT/potentials/@cgpot/enter_cts_evidence_pot.m, + BNT/potentials/@cgpot/enter_discrete_evidence_pot.m, + BNT/potentials/@cgpot/marginalize_pot.m, + BNT/potentials/@cgpot/multiply_by_pot.m, + BNT/potentials/@cgpot/multiply_pots.m, + BNT/potentials/@cgpot/normalize_pot.m, + BNT/potentials/@cgpot/pot_to_marginal.m, + BNT/potentials/@cgpot/Old/normalize_pot.m, + BNT/potentials/@cgpot/Old/simple_marginalize_pot.m, + BNT/potentials/@cpot/cpot.m, BNT/potentials/@cpot/cpot_to_mpot.m, + BNT/potentials/@cpot/display.m, + BNT/potentials/@cpot/divide_by_pot.m, + BNT/potentials/@cpot/domain_pot.m, + BNT/potentials/@cpot/enter_cts_evidence_pot.m, + BNT/potentials/@cpot/marginalize_pot.m, + BNT/potentials/@cpot/multiply_by_pot.m, + BNT/potentials/@cpot/multiply_pots.m, + BNT/potentials/@cpot/normalize_pot.m, + BNT/potentials/@cpot/pot_to_marginal.m, + BNT/potentials/@cpot/rescale_pot.m, + BNT/potentials/@cpot/set_domain_pot.m, + BNT/potentials/@cpot/Old/cpot_to_mpot.m, + BNT/potentials/@cpot/Old/normalize_pot.convert.m, + BNT/potentials/@dpot/approxeq_pot.m, + BNT/potentials/@dpot/display.m, + BNT/potentials/@dpot/domain_pot.m, + BNT/potentials/@dpot/dpot_to_table.m, + BNT/potentials/@dpot/get_fields.m, + BNT/potentials/@dpot/multiply_pots.m, + BNT/potentials/@dpot/pot_to_marginal.m, + BNT/potentials/@dpot/set_domain_pot.m, + BNT/potentials/@mpot/display.m, + BNT/potentials/@mpot/marginalize_pot.m, + BNT/potentials/@mpot/mpot.m, BNT/potentials/@mpot/mpot_to_cpot.m, + BNT/potentials/@mpot/normalize_pot.m, + BNT/potentials/@mpot/pot_to_marginal.m, + BNT/potentials/@mpot/rescale_pot.m, + BNT/potentials/@upot/approxeq_pot.m, + BNT/potentials/@upot/display.m, + BNT/potentials/@upot/divide_by_pot.m, + BNT/potentials/@upot/marginalize_pot.m, + BNT/potentials/@upot/multiply_by_pot.m, + BNT/potentials/@upot/normalize_pot.m, + BNT/potentials/@upot/pot_to_marginal.m, + BNT/potentials/@upot/upot.m, + BNT/potentials/@upot/upot_to_opt_policy.m, + BNT/potentials/Old/comp_eff_node_sizes.m, + BNT/potentials/Tables/divide_by_sparse_table.c, + BNT/potentials/Tables/divide_by_table.c, + BNT/potentials/Tables/marg_sparse_table.c, + BNT/potentials/Tables/marg_table.c, + BNT/potentials/Tables/mult_by_sparse_table.c, + BNT/potentials/Tables/rep_mult.c, HMM/mk_leftright_transmat.m: + Initial import of code base from Kevin Murphy. + +2002-05-29 08:59 yozhik + + * BNT/@assocarray/assocarray.m, + BNT/CPDs/@boolean_CPD/boolean_CPD.m, + BNT/CPDs/@discrete_CPD/CPD_to_lambda_msg.m, + BNT/CPDs/@discrete_CPD/CPD_to_pi.m, + BNT/CPDs/@discrete_CPD/CPD_to_scgpot.m, + BNT/CPDs/@discrete_CPD/README, + BNT/CPDs/@discrete_CPD/convert_CPD_to_table_hidden_ps.m, + BNT/CPDs/@discrete_CPD/convert_obs_CPD_to_table.m, + BNT/CPDs/@discrete_CPD/convert_to_sparse_table.c, + BNT/CPDs/@discrete_CPD/convert_to_table.m, + BNT/CPDs/@discrete_CPD/discrete_CPD.m, + BNT/CPDs/@discrete_CPD/dom_sizes.m, + BNT/CPDs/@discrete_CPD/log_prob_node.m, + BNT/CPDs/@discrete_CPD/prob_node.m, + BNT/CPDs/@discrete_CPD/sample_node.m, + BNT/CPDs/@discrete_CPD/Old/convert_to_pot.m, + BNT/CPDs/@discrete_CPD/Old/convert_to_table.m, + BNT/CPDs/@discrete_CPD/Old/prob_CPD.m, + BNT/CPDs/@discrete_CPD/Old/prob_node.m, + BNT/CPDs/@discrete_CPD/private/prod_CPT_and_pi_msgs.m, + BNT/CPDs/@gaussian_CPD/CPD_to_lambda_msg.m, + BNT/CPDs/@gaussian_CPD/CPD_to_pi.m, + BNT/CPDs/@gaussian_CPD/CPD_to_scgpot.m, + BNT/CPDs/@gaussian_CPD/adjustable_CPD.m, + BNT/CPDs/@gaussian_CPD/convert_CPD_to_table_hidden_ps.m, + BNT/CPDs/@gaussian_CPD/display.m, + BNT/CPDs/@gaussian_CPD/get_field.m, + BNT/CPDs/@gaussian_CPD/reset_ess.m, + BNT/CPDs/@gaussian_CPD/sample_node.m, + BNT/CPDs/@gaussian_CPD/set_fields.m, + BNT/CPDs/@gaussian_CPD/Old/CPD_to_lambda_msg.m, + BNT/CPDs/@gaussian_CPD/Old/gaussian_CPD.m, + BNT/CPDs/@gaussian_CPD/Old/log_prob_node.m, + BNT/CPDs/@gaussian_CPD/Old/update_ess.m, + BNT/CPDs/@gaussian_CPD/Old/update_tied_ess.m, + BNT/CPDs/@gaussian_CPD/private/CPD_to_linear_gaussian.m, + BNT/CPDs/@generic_CPD/README, + BNT/CPDs/@generic_CPD/adjustable_CPD.m, + BNT/CPDs/@generic_CPD/display.m, + BNT/CPDs/@generic_CPD/generic_CPD.m, + BNT/CPDs/@generic_CPD/log_prior.m, + BNT/CPDs/@generic_CPD/set_clamped.m, + BNT/CPDs/@generic_CPD/Old/BIC_score_CPD.m, + BNT/CPDs/@generic_CPD/Old/CPD_to_dpots.m, + BNT/CPDs/@gmux_CPD/CPD_to_lambda_msg.m, + BNT/CPDs/@gmux_CPD/convert_to_pot.m, + BNT/CPDs/@gmux_CPD/CPD_to_pi.m, BNT/CPDs/@gmux_CPD/display.m, + BNT/CPDs/@gmux_CPD/gmux_CPD.m, BNT/CPDs/@gmux_CPD/sample_node.m, + BNT/CPDs/@gmux_CPD/Old/gmux_CPD.m, + BNT/CPDs/@hhmmF_CPD/log_prior.m, + BNT/CPDs/@hhmmF_CPD/maximize_params.m, + BNT/CPDs/@hhmmF_CPD/reset_ess.m, BNT/CPDs/@hhmmQ_CPD/log_prior.m, + BNT/CPDs/@hhmmQ_CPD/reset_ess.m, + BNT/CPDs/@mlp_CPD/convert_to_table.m, + BNT/CPDs/@mlp_CPD/maximize_params.m, BNT/CPDs/@mlp_CPD/mlp_CPD.m, + BNT/CPDs/@mlp_CPD/reset_ess.m, BNT/CPDs/@mlp_CPD/update_ess.m, + BNT/CPDs/@noisyor_CPD/CPD_to_lambda_msg.m, + BNT/CPDs/@noisyor_CPD/CPD_to_pi.m, + BNT/CPDs/@noisyor_CPD/noisyor_CPD.m, + BNT/CPDs/@noisyor_CPD/private/sum_prod_CPD_and_pi_msgs.m, + BNT/CPDs/@root_CPD/CPD_to_pi.m, + BNT/CPDs/@root_CPD/convert_to_pot.m, + BNT/CPDs/@root_CPD/log_marg_prob_node.m, + BNT/CPDs/@root_CPD/log_prob_node.m, + BNT/CPDs/@root_CPD/root_CPD.m, BNT/CPDs/@root_CPD/sample_node.m, + BNT/CPDs/@root_CPD/Old/CPD_to_CPT.m, + BNT/CPDs/@softmax_CPD/convert_to_pot.m, + BNT/CPDs/@softmax_CPD/display.m, + BNT/CPDs/@softmax_CPD/get_field.m, + BNT/CPDs/@softmax_CPD/maximize_params.m, + BNT/CPDs/@softmax_CPD/reset_ess.m, + BNT/CPDs/@softmax_CPD/sample_node.m, + BNT/CPDs/@softmax_CPD/set_fields.m, + BNT/CPDs/@softmax_CPD/update_ess.m, + BNT/CPDs/@softmax_CPD/private/extract_params.m, + BNT/CPDs/@tabular_CPD/CPD_to_CPT.m, + BNT/CPDs/@tabular_CPD/bayes_update_params.m, + BNT/CPDs/@tabular_CPD/log_nextcase_prob_node.m, + BNT/CPDs/@tabular_CPD/log_prior.m, + BNT/CPDs/@tabular_CPD/reset_ess.m, + BNT/CPDs/@tabular_CPD/update_ess.m, + BNT/CPDs/@tabular_CPD/update_ess_simple.m, + BNT/CPDs/@tabular_CPD/Old/BIC_score_CPD.m, + BNT/CPDs/@tabular_CPD/Old/bayesian_score_CPD.m, + BNT/CPDs/@tabular_CPD/Old/log_marg_prob_node_case.m, + BNT/CPDs/@tabular_CPD/Old/mult_CPD_and_pi_msgs.m, + BNT/CPDs/@tabular_CPD/Old/prob_CPT.m, + BNT/CPDs/@tabular_CPD/Old/prob_node.m, + BNT/CPDs/@tabular_CPD/Old/sample_node.m, + BNT/CPDs/@tabular_CPD/Old/sample_node_single_case.m, + BNT/CPDs/@tabular_CPD/Old/tabular_CPD.m, + BNT/CPDs/@tabular_CPD/Old/update_params.m, + BNT/CPDs/@tabular_decision_node/CPD_to_CPT.m, + BNT/CPDs/@tabular_decision_node/display.m, + BNT/CPDs/@tabular_decision_node/get_field.m, + BNT/CPDs/@tabular_decision_node/set_fields.m, + BNT/CPDs/@tabular_decision_node/tabular_decision_node.m, + BNT/CPDs/@tabular_decision_node/Old/tabular_decision_node.m, + BNT/CPDs/@tabular_kernel/convert_to_pot.m, + BNT/CPDs/@tabular_kernel/convert_to_table.m, + BNT/CPDs/@tabular_kernel/get_field.m, + BNT/CPDs/@tabular_kernel/set_fields.m, + BNT/CPDs/@tabular_kernel/tabular_kernel.m, + BNT/CPDs/@tabular_kernel/Old/tabular_kernel.m, + BNT/CPDs/@tabular_utility_node/convert_to_pot.m, + BNT/CPDs/@tabular_utility_node/display.m, + BNT/CPDs/@tabular_utility_node/tabular_utility_node.m, + BNT/CPDs/@tree_CPD/display.m, + BNT/CPDs/@tree_CPD/evaluate_tree_performance.m, + BNT/CPDs/@tree_CPD/get_field.m, + BNT/CPDs/@tree_CPD/learn_params.m, BNT/CPDs/@tree_CPD/readme.txt, + BNT/CPDs/@tree_CPD/set_fields.m, BNT/CPDs/@tree_CPD/tree_CPD.m, + BNT/CPDs/Old/@linear_gaussian_CPD/linear_gaussian_CPD.m, + BNT/CPDs/Old/@linear_gaussian_CPD/log_marg_prob_node.m, + BNT/CPDs/Old/@linear_gaussian_CPD/update_params_complete.m, + BNT/CPDs/Old/@root_gaussian_CPD/log_marg_prob_node.m, + BNT/CPDs/Old/@root_gaussian_CPD/root_gaussian_CPD.m, + BNT/CPDs/Old/@root_gaussian_CPD/update_params_complete.m, + BNT/CPDs/Old/@tabular_chance_node/CPD_to_upot.m, + BNT/CPDs/Old/@tabular_chance_node/tabular_chance_node.m, + BNT/examples/dynamic/bat1.m, BNT/examples/dynamic/bkff1.m, + BNT/examples/dynamic/chmm1.m, + BNT/examples/dynamic/cmp_inference_dbn.m, + BNT/examples/dynamic/cmp_learning_dbn.m, + BNT/examples/dynamic/cmp_online_inference.m, + BNT/examples/dynamic/fhmm_infer.m, + BNT/examples/dynamic/filter_test1.m, + BNT/examples/dynamic/kalman1.m, + BNT/examples/dynamic/kjaerulff1.m, + BNT/examples/dynamic/loopy_dbn1.m, + BNT/examples/dynamic/mk_collage_from_clqs.m, + BNT/examples/dynamic/mk_fhmm.m, BNT/examples/dynamic/reveal1.m, + BNT/examples/dynamic/scg_dbn.m, + BNT/examples/dynamic/skf_data_assoc_gmux.m, + BNT/examples/dynamic/HHMM/add_hhmm_end_state.m, + BNT/examples/dynamic/HHMM/hhmm_jtree_clqs.m, + BNT/examples/dynamic/HHMM/mk_hhmm_topo.m, + BNT/examples/dynamic/HHMM/mk_hhmm_topo_F1.m, + BNT/examples/dynamic/HHMM/pretty_print_hhmm_parse.m, + BNT/examples/dynamic/HHMM/Motif/fixed_args_mk_motif_hhmm.m, + BNT/examples/dynamic/HHMM/Motif/mk_motif_hhmm.m, + BNT/examples/dynamic/HHMM/Motif/sample_motif_hhmm.m, + BNT/examples/dynamic/HHMM/Old/mk_abcd_hhmm.m, + BNT/examples/dynamic/HHMM/Old/mk_arrow_alpha_hhmm3.m, + BNT/examples/dynamic/HHMM/Old/mk_hhmm2.m, + BNT/examples/dynamic/HHMM/Old/mk_hhmm3.m, + BNT/examples/dynamic/HHMM/Old/mk_hhmm3_args.m, + BNT/examples/dynamic/HHMM/Old/motif_hhmm.m, + BNT/examples/dynamic/HHMM/Old/remove_hhmm_end_state.m, + BNT/examples/dynamic/HHMM/Square/get_square_data.m, + BNT/examples/dynamic/HHMM/Square/hhmm_inference.m, + BNT/examples/dynamic/HHMM/Square/is_F2_true_D3.m, + BNT/examples/dynamic/HHMM/Square/learn_square_hhmm_discrete.m, + BNT/examples/dynamic/HHMM/Square/mk_square_hhmm.m, + BNT/examples/dynamic/HHMM/Square/plot_square_hhmm.m, + BNT/examples/dynamic/HHMM/Square/sample_square_hhmm_cts.m, + BNT/examples/dynamic/HHMM/Square/sample_square_hhmm_discrete.m, + BNT/examples/dynamic/HHMM/Square/square4.mat, + BNT/examples/dynamic/HHMM/Square/square4_cases.mat, + BNT/examples/dynamic/HHMM/Square/test_square_fig.m, + BNT/examples/dynamic/HHMM/Square/test_square_fig.mat, + BNT/examples/dynamic/HHMM/Square/Old/learn_square_hhmm.m, + BNT/examples/dynamic/HHMM/Square/Old/mk_square_hhmm.m, + BNT/examples/dynamic/HHMM/Square/Old/plot_square_hhmm.m, + BNT/examples/dynamic/HHMM/Square/Old/sample_square_hhmm.m, + BNT/examples/dynamic/Old/chmm1.m, + BNT/examples/dynamic/Old/cmp_inference.m, + BNT/examples/dynamic/Old/kalman1.m, + BNT/examples/dynamic/Old/old.water1.m, + BNT/examples/dynamic/Old/online1.m, + BNT/examples/dynamic/Old/online2.m, + BNT/examples/dynamic/Old/scg_dbn.m, + BNT/examples/dynamic/SLAM/mk_gmux_robot_dbn.m, + BNT/examples/dynamic/SLAM/mk_linear_slam.m, + BNT/examples/dynamic/SLAM/slam_kf.m, + BNT/examples/dynamic/SLAM/slam_offline_loopy.m, + BNT/examples/dynamic/SLAM/slam_partial_kf.m, + BNT/examples/dynamic/SLAM/slam_stationary_loopy.m, + BNT/examples/dynamic/SLAM/Old/offline_loopy_slam.m, + BNT/examples/dynamic/SLAM/Old/paskin1.m, + BNT/examples/dynamic/SLAM/Old/skf_data_assoc_gmux2.m, + BNT/examples/dynamic/SLAM/Old/slam_kf.m, + BNT/examples/limids/id1.m, BNT/examples/limids/pigs1.m, + BNT/examples/static/cg1.m, BNT/examples/static/cg2.m, + BNT/examples/static/discrete2.m, BNT/examples/static/discrete3.m, + BNT/examples/static/fa1.m, BNT/examples/static/gaussian1.m, + BNT/examples/static/gibbs_test1.m, BNT/examples/static/lw1.m, + BNT/examples/static/mfa1.m, BNT/examples/static/mixexp1.m, + BNT/examples/static/mixexp2.m, BNT/examples/static/mixexp3.m, + BNT/examples/static/mog1.m, BNT/examples/static/qmr1.m, + BNT/examples/static/sample1.m, BNT/examples/static/softmax1.m, + BNT/examples/static/Belprop/belprop_loop1_discrete.m, + BNT/examples/static/Belprop/belprop_loop1_gauss.m, + BNT/examples/static/Belprop/belprop_loopy_cg.m, + BNT/examples/static/Belprop/belprop_loopy_discrete.m, + BNT/examples/static/Belprop/belprop_loopy_gauss.m, + BNT/examples/static/Belprop/belprop_polytree_cg.m, + BNT/examples/static/Belprop/belprop_polytree_gauss.m, + BNT/examples/static/Belprop/bp1.m, + BNT/examples/static/Belprop/gmux1.m, + BNT/examples/static/Brutti/Belief_IOhmm.m, + BNT/examples/static/Brutti/Belief_hmdt.m, + BNT/examples/static/Brutti/Belief_hme.m, + BNT/examples/static/Brutti/Sigmoid_Belief.m, + BNT/examples/static/HME/HMEforMatlab.jpg, + BNT/examples/static/HME/README, BNT/examples/static/HME/fhme.m, + BNT/examples/static/HME/gen_data.m, + BNT/examples/static/HME/hme_class_plot.m, + BNT/examples/static/HME/hme_reg_plot.m, + BNT/examples/static/HME/hme_topobuilder.m, + BNT/examples/static/HME/test_data_class.mat, + BNT/examples/static/HME/test_data_class2.mat, + BNT/examples/static/HME/test_data_reg.mat, + BNT/examples/static/HME/train_data_class.mat, + BNT/examples/static/HME/train_data_reg.mat, + BNT/examples/static/Misc/mixexp_data.txt, + BNT/examples/static/Misc/mixexp_graddesc.m, + BNT/examples/static/Misc/mixexp_plot.m, + BNT/examples/static/Misc/sprinkler.bif, + BNT/examples/static/Models/mk_cancer_bnet.m, + BNT/examples/static/Models/mk_car_bnet.m, + BNT/examples/static/Models/mk_ideker_bnet.m, + BNT/examples/static/Models/mk_incinerator_bnet.m, + BNT/examples/static/Models/mk_markov_chain_bnet.m, + BNT/examples/static/Models/mk_minimal_qmr_bnet.m, + BNT/examples/static/Models/mk_qmr_bnet.m, + BNT/examples/static/Models/mk_vstruct_bnet.m, + BNT/examples/static/Models/Old/mk_hmm_bnet.m, + BNT/examples/static/SCG/scg1.m, BNT/examples/static/SCG/scg2.m, + BNT/examples/static/SCG/scg3.m, + BNT/examples/static/SCG/scg_3node.m, + BNT/examples/static/SCG/scg_unstable.m, + BNT/examples/static/StructLearn/bic1.m, + BNT/examples/static/StructLearn/cooper_yoo.m, + BNT/examples/static/StructLearn/k2demo1.m, + BNT/examples/static/StructLearn/mcmc1.m, + BNT/examples/static/StructLearn/pc1.m, + BNT/examples/static/StructLearn/pc2.m, + BNT/examples/static/Zoubin/README, + BNT/examples/static/Zoubin/csum.m, + BNT/examples/static/Zoubin/ffa.m, + BNT/examples/static/Zoubin/mfa.m, + BNT/examples/static/Zoubin/mfa_cl.m, + BNT/examples/static/Zoubin/mfademo.m, + BNT/examples/static/Zoubin/rdiv.m, + BNT/examples/static/Zoubin/rprod.m, + BNT/examples/static/Zoubin/rsum.m, + BNT/examples/static/dtree/test_housing.m, + BNT/examples/static/dtree/test_restaurants.m, + BNT/examples/static/dtree/test_zoo1.m, + BNT/examples/static/dtree/tmp.dot, + BNT/examples/static/dtree/transform_data_into_bnt_format.m, + BNT/examples/static/fgraph/fg2.m, + BNT/examples/static/fgraph/fg3.m, + BNT/examples/static/fgraph/fg_mrf1.m, + BNT/examples/static/fgraph/fg_mrf2.m, + BNT/general/bnet_to_fgraph.m, + BNT/general/compute_fwd_interface.m, + BNT/general/compute_interface_nodes.m, + BNT/general/compute_minimal_interface.m, + BNT/general/dbn_to_bnet.m, + BNT/general/determine_elim_constraints.m, + BNT/general/do_intervention.m, BNT/general/dsep.m, + BNT/general/enumerate_scenarios.m, BNT/general/fgraph_to_bnet.m, + BNT/general/log_lik_complete.m, + BNT/general/log_marg_lik_complete.m, BNT/general/mk_bnet.m, + BNT/general/mk_fgraph.m, BNT/general/mk_limid.m, + BNT/general/mk_mutilated_samples.m, + BNT/general/mk_slice_and_half_dbn.m, + BNT/general/partition_dbn_nodes.m, + BNT/general/sample_bnet_nocell.m, BNT/general/sample_dbn.m, + BNT/general/score_bnet_complete.m, + BNT/general/unroll_dbn_topology.m, + BNT/general/Old/bnet_to_gdl_graph.m, + BNT/general/Old/calc_mpe_bucket.m, + BNT/general/Old/calc_mpe_dbn.m, + BNT/general/Old/calc_mpe_given_inf_engine.m, + BNT/general/Old/calc_mpe_global.m, + BNT/general/Old/compute_interface_nodes.m, + BNT/general/Old/mk_gdl_graph.m, GraphViz/draw_dbn.m, + GraphViz/make_layout.m, BNT/license.gpl.txt, + BNT/general/add_evidence_to_gmarginal.m, + BNT/inference/@inf_engine/bnet_from_engine.m, + BNT/inference/@inf_engine/get_field.m, + BNT/inference/@inf_engine/inf_engine.m, + BNT/inference/@inf_engine/marginal_family.m, + BNT/inference/@inf_engine/set_fields.m, + BNT/inference/@inf_engine/update_engine.m, + BNT/inference/@inf_engine/Old/marginal_family_pot.m, + BNT/inference/@inf_engine/Old/observed_nodes.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/bk_ff_hmm_inf_engine.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/dbn_init_bel.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/dbn_marginal_from_bel.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/dbn_predict_bel.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/dbn_update_bel.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/dbn_update_bel1.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/enter_evidence.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/marginal_family.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/marginal_nodes.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/private/bk_ff_fb.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/private/combine_marginals_into_joint.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/private/dbn_to_hmm.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/private/mk_hmm_obs_lik_mat.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/private/mk_hmm_obs_lik_vec.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/private/mk_hmm_obs_lik_vec1.m, + BNT/inference/dynamic/@bk_ff_hmm_inf_engine/private/project_joint_onto_marginals.m, + BNT/inference/dynamic/@bk_inf_engine/bk_inf_engine.m, + BNT/inference/dynamic/@bk_inf_engine/dbn_init_bel.m, + BNT/inference/dynamic/@bk_inf_engine/dbn_marginal_from_bel.m, + BNT/inference/dynamic/@bk_inf_engine/dbn_update_bel.m, + BNT/inference/dynamic/@bk_inf_engine/dbn_update_bel1.m, + BNT/inference/dynamic/@bk_inf_engine/enter_evidence.m, + BNT/inference/dynamic/@bk_inf_engine/marginal_family.m, + BNT/inference/dynamic/@bk_inf_engine/marginal_nodes.m, + BNT/inference/dynamic/@bk_inf_engine/update_engine.m, + BNT/inference/dynamic/@ff_inf_engine/enter_evidence.m, + BNT/inference/dynamic/@ff_inf_engine/enter_soft_evidence.m, + BNT/inference/dynamic/@ff_inf_engine/ff_inf_engine.m, + BNT/inference/dynamic/@ff_inf_engine/filter_evidence.m, + BNT/inference/dynamic/@ff_inf_engine/marginal_family.m, + BNT/inference/dynamic/@ff_inf_engine/marginal_nodes.m, + BNT/inference/dynamic/@ff_inf_engine/smooth_evidence.m, + BNT/inference/dynamic/@ff_inf_engine/Old/enter_soft_evidence.m, + BNT/inference/dynamic/@ff_inf_engine/Old/enter_soft_evidence1.m, + BNT/inference/dynamic/@ff_inf_engine/Old/marginal_family.m, + BNT/inference/dynamic/@frontier_inf_engine/enter_evidence.m, + BNT/inference/dynamic/@frontier_inf_engine/enter_soft_evidence.m, + BNT/inference/dynamic/@frontier_inf_engine/frontier_inf_engine.m, + BNT/inference/dynamic/@frontier_inf_engine/marginal_family.m, + BNT/inference/dynamic/@frontier_inf_engine/marginal_nodes.m, + BNT/inference/dynamic/@frontier_inf_engine/set_fwdback.m, + BNT/inference/dynamic/@hmm_inf_engine/update_engine.m, + BNT/inference/dynamic/@hmm_inf_engine/Old/dhmm_inf_engine.m, + BNT/inference/dynamic/@hmm_inf_engine/Old/marginal_family.m, + BNT/inference/dynamic/@hmm_inf_engine/Old/marginal_nodes.m, + BNT/inference/dynamic/@jtree_dbn_inf_engine/marginal_family.m, + BNT/inference/dynamic/@jtree_dbn_inf_engine/Broken/enter_soft_evidence1.m, + BNT/inference/dynamic/@jtree_dbn_inf_engine/Broken/enter_soft_evidence2.m, + BNT/inference/dynamic/@jtree_dbn_inf_engine/Broken/enter_soft_evidence3.m, + BNT/inference/dynamic/@jtree_dbn_inf_engine/Broken/enter_soft_evidence4.m, + BNT/inference/dynamic/@jtree_dbn_inf_engine/Broken/marginal_nodes.m, + BNT/inference/dynamic/@jtree_dbn_inf_engine/Old/enter_soft_evidence_nonint.m, + BNT/inference/dynamic/@jtree_dbn_inf_engine/Old/enter_soft_evidence_trans.m, + BNT/inference/dynamic/@jtree_dbn_inf_engine/Old/jtree_dbn_inf_engine.m, + BNT/inference/dynamic/@jtree_dbn_inf_engine/Old/jtree_dbn_inf_engine1.m, + BNT/inference/dynamic/@jtree_dbn_inf_engine/Old/jtree_dbn_inf_engine2.m, + BNT/inference/dynamic/@jtree_unrolled_dbn_inf_engine/enter_evidence.m, + BNT/inference/dynamic/@jtree_unrolled_dbn_inf_engine/jtree_unrolled_dbn_inf_engine.m, + BNT/inference/dynamic/@jtree_unrolled_dbn_inf_engine/marginal_family.m, + BNT/inference/dynamic/@jtree_unrolled_dbn_inf_engine/marginal_nodes.m, + BNT/inference/dynamic/@jtree_unrolled_dbn_inf_engine/update_engine.m, + BNT/inference/dynamic/@jtree_unrolled_dbn_inf_engine/Old/marginal_family.m, + BNT/inference/dynamic/@jtree_unrolled_dbn_inf_engine/Old/marginal_nodes.m, + BNT/inference/dynamic/@kalman_inf_engine/enter_evidence.m, + BNT/inference/dynamic/@kalman_inf_engine/kalman_inf_engine.m, + BNT/inference/dynamic/@kalman_inf_engine/marginal_nodes.m, + BNT/inference/dynamic/@kalman_inf_engine/update_engine.m, + BNT/inference/dynamic/@kalman_inf_engine/private/dbn_to_lds.m, + BNT/inference/dynamic/@kalman_inf_engine/private/extract_params_from_gbn.m, + BNT/inference/dynamic/@pearl_dbn_inf_engine/enter_evidence.m, + BNT/inference/dynamic/@pearl_dbn_inf_engine/enter_soft_ev.m, + BNT/inference/dynamic/@pearl_dbn_inf_engine/marginal_nodes.m, + 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BNT/inference/static/@jtree_sparse_inf_engine/cliques_from_engine.m, + BNT/inference/static/@jtree_sparse_inf_engine/clq_containing_nodes.m, + BNT/inference/static/@jtree_sparse_inf_engine/collect_evidence.c, + BNT/inference/static/@jtree_sparse_inf_engine/distribute_evidence.c, + BNT/inference/static/@jtree_sparse_inf_engine/enter_evidence.m, + BNT/inference/static/@jtree_sparse_inf_engine/enter_soft_evidence.m, + BNT/inference/static/@jtree_sparse_inf_engine/init_pot.c, + BNT/inference/static/@jtree_sparse_inf_engine/marginal_family.m, + BNT/inference/static/@jtree_sparse_inf_engine/marginal_nodes.m, + BNT/inference/static/@jtree_sparse_inf_engine/set_fields.m, + BNT/inference/static/@jtree_sparse_inf_engine/old/collect_evidence.c, + BNT/inference/static/@jtree_sparse_inf_engine/old/distribute_evidence.c, + BNT/inference/static/@jtree_sparse_inf_engine/old/init_pot.c, + BNT/inference/static/@jtree_sparse_inf_engine/old/init_pot1.c, + BNT/inference/static/@jtree_sparse_inf_engine/old/init_pot1.m, + BNT/inference/static/@likelihood_weighting_inf_engine/enter_evidence.m, + BNT/inference/static/@likelihood_weighting_inf_engine/likelihood_weighting_inf_engine.m, + BNT/inference/static/@likelihood_weighting_inf_engine/marginal_nodes.m, + BNT/inference/static/@pearl_inf_engine/enter_evidence.m, + BNT/inference/static/@pearl_inf_engine/loopy_converged.m, + BNT/inference/static/@pearl_inf_engine/marginal_nodes.m, + BNT/inference/static/@pearl_inf_engine/private/compute_bel.m, + BNT/inference/static/@pearl_inf_engine/private/prod_lambda_msgs.m, + BNT/inference/static/@pearl_inf_engine/private/tree_protocol.m, + BNT/inference/static/@quickscore_inf_engine/enter_evidence.m, + BNT/inference/static/@quickscore_inf_engine/marginal_nodes.m, + BNT/inference/static/@quickscore_inf_engine/quickscore_inf_engine.m, + BNT/inference/static/@quickscore_inf_engine/private/C_quickscore.c, + BNT/inference/static/@quickscore_inf_engine/private/nr.h, + BNT/inference/static/@quickscore_inf_engine/private/nrutil.c, + BNT/inference/static/@quickscore_inf_engine/private/nrutil.h, + BNT/inference/static/@quickscore_inf_engine/private/quickscore.m, + BNT/learning/bayes_update_params.m, + BNT/learning/bic_score_family.m, + BNT/learning/compute_cooling_schedule.m, + BNT/learning/dirichlet_score_family.m, + BNT/learning/kpm_learn_struct_mcmc.m, + BNT/learning/learn_params_em.m, + BNT/learning/learn_struct_dbn_reveal.m, + BNT/learning/learn_struct_pdag_ic_star.m, + BNT/learning/mcmc_sample_to_hist.m, BNT/learning/mk_schedule.m, + BNT/learning/mk_tetrad_data_file.m, + BNT/learning/score_dags_old.m, HMM/dhmm_logprob_brute_force.m, + HMM/dhmm_logprob_path.m, HMM/mdp_sample.m, Kalman/AR_to_SS.m, + Kalman/SS_to_AR.m, Kalman/convert_to_lagged_form.m, + Kalman/ensure_AR.m, Kalman/eval_AR_perf.m, + Kalman/kalman_filter.m, Kalman/kalman_smoother.m, + Kalman/kalman_update.m, Kalman/learn_AR.m, + Kalman/learn_AR_diagonal.m, Kalman/learn_kalman.m, + Kalman/smooth_update.m, + BNT/general/convert_dbn_CPDs_to_tables_slow.m, + BNT/general/dispcpt.m, BNT/general/linear_gaussian_to_cpot.m, + BNT/general/partition_matrix_vec_3.m, + BNT/general/shrink_obs_dims_in_gaussian.m, + BNT/general/shrink_obs_dims_in_table.m, + BNT/potentials/CPD_to_pot.m, BNT/potentials/README, + BNT/potentials/check_for_cd_arcs.m, + BNT/potentials/determine_pot_type.m, + BNT/potentials/mk_initial_pot.m, + BNT/potentials/@cgpot/cg_can_to_mom.m, + BNT/potentials/@cgpot/cg_mom_to_can.m, + BNT/potentials/@cgpot/cgpot.m, BNT/potentials/@cgpot/display.m, + BNT/potentials/@cgpot/divide_by_pot.m, + BNT/potentials/@cgpot/domain_pot.m, + BNT/potentials/@cgpot/enter_cts_evidence_pot.m, + BNT/potentials/@cgpot/enter_discrete_evidence_pot.m, + BNT/potentials/@cgpot/marginalize_pot.m, + BNT/potentials/@cgpot/multiply_by_pot.m, + BNT/potentials/@cgpot/multiply_pots.m, + BNT/potentials/@cgpot/normalize_pot.m, + BNT/potentials/@cgpot/pot_to_marginal.m, + BNT/potentials/@cgpot/Old/normalize_pot.m, + BNT/potentials/@cgpot/Old/simple_marginalize_pot.m, + BNT/potentials/@cpot/cpot.m, BNT/potentials/@cpot/cpot_to_mpot.m, + BNT/potentials/@cpot/display.m, + BNT/potentials/@cpot/divide_by_pot.m, + BNT/potentials/@cpot/domain_pot.m, + BNT/potentials/@cpot/enter_cts_evidence_pot.m, + BNT/potentials/@cpot/marginalize_pot.m, + BNT/potentials/@cpot/multiply_by_pot.m, + BNT/potentials/@cpot/multiply_pots.m, + BNT/potentials/@cpot/normalize_pot.m, + BNT/potentials/@cpot/pot_to_marginal.m, + BNT/potentials/@cpot/rescale_pot.m, + BNT/potentials/@cpot/set_domain_pot.m, + BNT/potentials/@cpot/Old/cpot_to_mpot.m, + BNT/potentials/@cpot/Old/normalize_pot.convert.m, + BNT/potentials/@dpot/approxeq_pot.m, + BNT/potentials/@dpot/display.m, + BNT/potentials/@dpot/domain_pot.m, + BNT/potentials/@dpot/dpot_to_table.m, + BNT/potentials/@dpot/get_fields.m, + BNT/potentials/@dpot/multiply_pots.m, + BNT/potentials/@dpot/pot_to_marginal.m, + BNT/potentials/@dpot/set_domain_pot.m, + BNT/potentials/@mpot/display.m, + BNT/potentials/@mpot/marginalize_pot.m, + BNT/potentials/@mpot/mpot.m, BNT/potentials/@mpot/mpot_to_cpot.m, + BNT/potentials/@mpot/normalize_pot.m, + BNT/potentials/@mpot/pot_to_marginal.m, + BNT/potentials/@mpot/rescale_pot.m, + BNT/potentials/@upot/approxeq_pot.m, + BNT/potentials/@upot/display.m, + BNT/potentials/@upot/divide_by_pot.m, + BNT/potentials/@upot/marginalize_pot.m, + BNT/potentials/@upot/multiply_by_pot.m, + BNT/potentials/@upot/normalize_pot.m, + BNT/potentials/@upot/pot_to_marginal.m, + BNT/potentials/@upot/upot.m, + BNT/potentials/@upot/upot_to_opt_policy.m, + BNT/potentials/Old/comp_eff_node_sizes.m, + BNT/potentials/Tables/divide_by_sparse_table.c, + BNT/potentials/Tables/divide_by_table.c, + BNT/potentials/Tables/marg_sparse_table.c, + BNT/potentials/Tables/marg_table.c, + BNT/potentials/Tables/mult_by_sparse_table.c, + BNT/potentials/Tables/rep_mult.c, HMM/mk_leftright_transmat.m: + Initial revision + +2002-05-29 04:59 yozhik + + * BNT/inference/static/@stab_cond_gauss_inf_engine/: + clq_containing_nodes.m, problems.txt, push_pot_toclique.m, + Old/initialize_engine.m: Initial import of code base from Kevin + Murphy. + +2002-05-29 04:59 yozhik + + * BNT/inference/static/@stab_cond_gauss_inf_engine/: + clq_containing_nodes.m, problems.txt, push_pot_toclique.m, + Old/initialize_engine.m: Initial revision + +2002-05-19 15:11 yozhik + + * BNT/potentials/: @scgcpot/marginalize_pot.m, + @scgcpot/normalize_pot.m, @scgcpot/rescale_pot.m, + @scgcpot/scgcpot.m, @scgpot/direct_combine_pots.m, + @scgpot/pot_to_marginal.m: Initial import of code base from Kevin + Murphy. + +2002-05-19 15:11 yozhik + + * BNT/potentials/: @scgcpot/marginalize_pot.m, + @scgcpot/normalize_pot.m, @scgcpot/rescale_pot.m, + @scgcpot/scgcpot.m, @scgpot/direct_combine_pots.m, + @scgpot/pot_to_marginal.m: Initial revision + +2001-07-28 08:43 yozhik + + * BNT/potentials/genops.c: Initial import of code base from Kevin + Murphy. + +2001-07-28 08:43 yozhik + + * BNT/potentials/genops.c: Initial revision + diff --git a/sourcecodes/bnt-master/docs/Eqns/lin_reg_eqn.gif b/sourcecodes/bnt-master/docs/Eqns/lin_reg_eqn.gif new file mode 100644 index 00000000..6ba3fda5 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Eqns/lin_reg_eqn.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/CPTgrass.gif b/sourcecodes/bnt-master/docs/Figures/CPTgrass.gif new file mode 100644 index 00000000..a5ecccd1 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/CPTgrass.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/HMEforMatlab.jpg b/sourcecodes/bnt-master/docs/Figures/HMEforMatlab.jpg new file mode 100644 index 00000000..16682678 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/HMEforMatlab.jpg Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/ar1.fig 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1.50 90.00 180.00 + 6300 7500 6600 7200 +2 1 0 3 0 0 100 0 5 0.000 0 0 7 1 0 2 + 0 0 1.50 90.00 180.00 + 3600 5400 3900 5100 diff --git a/sourcecodes/bnt-master/docs/Figures/fa.eps b/sourcecodes/bnt-master/docs/Figures/fa.eps new file mode 100644 index 00000000..03102fbf --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/fa.eps @@ -0,0 +1,356 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: (ImageMagick) +%%Title: (fa.eps) +%%CreationDate: (Tue Nov 16 19:51:50 2004) +%%BoundingBox: 0 0 131 161 +%%DocumentData: Clean7Bit +%%LanguageLevel: 1 +%%Pages: 1 +%%EndComments + +%%BeginDefaults +%%EndDefaults + +%%BeginProlog +% +% Display a color image. The image is displayed in color on +% Postscript viewers or printers that support color, otherwise +% it is displayed as grayscale. +% +/DirectClassPacket +{ + % + % Get a DirectClass packet. + % + % Parameters: + % red. + % green. + % blue. + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + compression 0 eq + { + /number_pixels 3 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add 3 mul def + } ifelse + 0 3 number_pixels 1 sub + { + pixels exch color_packet putinterval + } for + pixels 0 number_pixels getinterval +} bind def + +/DirectClassImage +{ + % + % Display a DirectClass image. + % + systemdict /colorimage known + { + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { DirectClassPacket } false 3 colorimage + } + { + % + % No colorimage operator; convert to grayscale. + % + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { GrayDirectClassPacket } image + } ifelse +} bind def + +/GrayDirectClassPacket +{ + % + % Get a DirectClass packet; convert to grayscale. + % + % Parameters: + % red + % green + % blue + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/GrayPseudoClassPacket +{ + % + % Get a PseudoClass packet; convert to grayscale. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/PseudoClassPacket +{ + % + % Get a PseudoClass packet. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + compression 0 eq + { + /number_pixels 3 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add 3 mul def + } ifelse + 0 3 number_pixels 1 sub + { + pixels exch color_packet putinterval + } for + pixels 0 number_pixels getinterval +} bind def + +/PseudoClassImage +{ + % + % Display a PseudoClass image. + % + % Parameters: + % class: 0-PseudoClass or 1-Grayscale. + % + currentfile buffer readline pop + token pop /class exch def pop + class 0 gt + { + currentfile buffer readline pop + token pop /depth exch def pop + /grays columns 8 add depth sub depth mul 8 idiv string def + columns rows depth + [ + columns 0 0 + rows neg 0 rows + ] + { currentfile grays readhexstring pop } image + } + { + % + % Parameters: + % colors: number of colors in the colormap. + % colormap: red, green, blue color packets. + % + currentfile buffer readline pop + token pop /colors exch def pop + /colors colors 3 mul def + /colormap colors string def + currentfile colormap readhexstring pop pop + systemdict /colorimage known + { + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { PseudoClassPacket } false 3 colorimage + } + { + % + % No colorimage operator; convert to grayscale. + % + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { GrayPseudoClassPacket } image + } ifelse + } ifelse +} bind def + +/DisplayImage +{ + % + % Display a DirectClass or PseudoClass image. + % + % Parameters: + % x & y translation. + % x & y scale. + % label pointsize. + % image label. + % image columns & rows. + % class: 0-DirectClass or 1-PseudoClass. + % compression: 0-none or 1-RunlengthEncoded. + % hex color packets. + % + gsave + /buffer 512 string def + /byte 1 string def + /color_packet 3 string def + /pixels 768 string def + + currentfile buffer readline pop + token pop /x exch def + token pop /y exch def pop + x y translate + currentfile buffer readline pop + token pop /x exch def + token pop /y exch def pop + currentfile buffer readline pop + token pop /pointsize exch def pop + /Times-Roman findfont pointsize scalefont setfont + x y scale + currentfile buffer readline pop + token pop /columns exch def + token pop /rows exch def pop + currentfile buffer readline pop + token pop /class exch def pop + currentfile buffer readline pop + token pop /compression exch def pop + class 0 gt 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+ffffe0ffffffffffffffffffffffffffffffffe0ffffffffffffffffffffffffffffffff +e0 +end +%%PageTrailer +%%Trailer +%%EOF diff --git a/sourcecodes/bnt-master/docs/Figures/fa.fig b/sourcecodes/bnt-master/docs/Figures/fa.fig new file mode 100644 index 00000000..db4bec4a --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/fa.fig @@ -0,0 +1,16 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 900 1950 300 225 900 1950 1200 2175 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 900 825 300 225 900 825 1200 1050 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 900 1125 900 1650 +4 0 -1 0 0 0 12 0.0000 4 135 135 825 900 X\001 +4 0 -1 0 0 0 12 0.0000 4 135 135 825 2025 Y\001 diff --git a/sourcecodes/bnt-master/docs/Figures/fa.gif b/sourcecodes/bnt-master/docs/Figures/fa.gif new file mode 100644 index 00000000..420c1517 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/fa.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/fa_caption.fig b/sourcecodes/bnt-master/docs/Figures/fa_caption.fig new file mode 100644 index 00000000..71cd37f2 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/fa_caption.fig @@ -0,0 +1,13 @@ +#FIG 3.1 +Landscape +Center +Inches +1200 2 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 900 1950 300 225 900 1950 1200 2175 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 900 825 300 225 900 825 1200 1050 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 900 1125 900 1650 +4 0 -1 0 0 0 12 0.0000 4 135 135 750 2025 Y\001 +4 0 -1 0 0 0 12 0.0000 4 135 120 750 900 X\001 +4 0 -1 0 0 0 12 0.0000 4 180 1620 300 2625 Factor Analysis/PCA\001 diff --git a/sourcecodes/bnt-master/docs/Figures/fa_discrete.fig b/sourcecodes/bnt-master/docs/Figures/fa_discrete.fig new file mode 100644 index 00000000..b5be4ca5 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/fa_discrete.fig @@ -0,0 +1,43 @@ +#FIG 3.2 +Portrait +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 975 600 300 225 975 600 1275 825 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 2100 600 300 225 2100 600 2400 825 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 900 825 525 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 900 825 1425 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2025 825 600 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2025 825 1425 1425 +2 2 0 1 -1 0 0 0 2 0.000 0 0 7 0 0 5 + 300 1425 750 1425 750 1875 300 1875 300 1425 +2 2 0 1 -1 0 0 0 2 0.000 0 0 7 0 0 5 + 1275 1425 1725 1425 1725 1875 1275 1875 1275 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 900 825 3000 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2025 825 3000 1425 +2 2 0 1 0 0 100 0 2 3.000 0 0 7 0 0 5 + 2775 1425 3225 1425 3225 1875 2775 1875 2775 1425 +4 0 -1 0 0 0 24 0.0000 4 30 270 1350 750 ...\001 +4 0 -1 0 0 0 12 0.0000 4 135 210 1350 1725 R2\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 825 675 X1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1950 675 Xn\001 +4 0 -1 0 0 0 12 0.0000 4 135 210 375 1725 R1\001 +4 0 -1 0 0 0 12 0.0000 4 180 1920 600 2325 Discrete Factor Analysis\001 +4 0 -1 0 0 0 24 0.0000 4 30 270 2100 1650 ...\001 +4 0 0 100 0 -1 12 0.0000 4 135 210 2850 1725 Rn\001 diff --git a/sourcecodes/bnt-master/docs/Figures/fa_discrete.gif b/sourcecodes/bnt-master/docs/Figures/fa_discrete.gif new file mode 100644 index 00000000..54a8bbfb --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/fa_discrete.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/fa_discrete_single.fig b/sourcecodes/bnt-master/docs/Figures/fa_discrete_single.fig new file mode 100644 index 00000000..225a741c --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/fa_discrete_single.fig @@ -0,0 +1,18 @@ +#FIG 3.2 +Portrait +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +1 1 0 1 0 7 100 0 -1 0.000 1 0.0000 825 375 300 225 825 375 1125 600 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 600 825 1275 +2 2 0 1 0 0 100 0 2 0.000 0 0 7 0 0 5 + 600 1275 975 1275 975 1725 600 1725 600 1275 +4 0 -1 0 0 0 12 0.0000 4 135 120 750 1575 R\001 +4 0 -1 0 0 0 12 0.0000 4 135 135 750 450 X\001 +4 0 0 100 0 0 12 0.0000 4 180 1785 300 2025 discrete factor analysis\001 diff --git a/sourcecodes/bnt-master/docs/Figures/fa_regular.fig b/sourcecodes/bnt-master/docs/Figures/fa_regular.fig new file mode 100644 index 00000000..db30e0a0 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/fa_regular.fig @@ -0,0 +1,36 @@ +#FIG 3.1 +Landscape +Center +Inches +1200 2 +6 225 300 3450 3150 +1 3 0 1 -1 7 0 0 -1 0.000 1 0.0000 999 1082 335 335 999 1082 1149 1382 +1 3 0 1 -1 7 0 0 -1 0.000 1 0.0000 2124 1082 335 335 2124 1082 2274 1382 +1 3 0 1 -1 0 0 0 2 0.000 1 0.0000 624 2807 335 335 624 2807 774 3107 +1 3 0 1 -1 0 0 0 2 0.000 1 0.0000 1674 2807 335 335 1674 2807 1824 3107 +1 3 0 1 -1 0 0 0 2 0.000 1 0.0000 2874 2807 335 335 2874 2807 3024 3107 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 900 1425 675 2475 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 975 1425 1575 2400 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 975 1425 2775 2475 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1950 1425 750 2475 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1950 1425 1650 2325 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1950 1425 2775 2400 +4 0 -1 0 0 0 24 0.0000 4 330 3225 225 600 regular factor analysis\001 +-6 +4 0 -1 0 0 0 12 0.0000 4 135 210 825 1200 X1\001 +4 0 -1 0 0 0 12 0.0000 4 135 210 1950 1200 X2\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 525 2925 Y1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1500 2850 Y2\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 2700 2850 Y3\001 diff --git a/sourcecodes/bnt-master/docs/Figures/fa_regular.gif b/sourcecodes/bnt-master/docs/Figures/fa_regular.gif new file mode 100644 index 00000000..bb506a59 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/fa_regular.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/fa_scalar.eps b/sourcecodes/bnt-master/docs/Figures/fa_scalar.eps new file mode 100644 index 00000000..ccdc01cc --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/fa_scalar.eps @@ -0,0 +1,414 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: (ImageMagick) +%%Title: (fa_scalar.eps) +%%CreationDate: (Tue Nov 16 19:52:00 2004) +%%BoundingBox: 0 0 246 156 +%%DocumentData: Clean7Bit +%%LanguageLevel: 1 +%%Pages: 1 +%%EndComments + +%%BeginDefaults +%%EndDefaults + +%%BeginProlog +% +% Display a color image. 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1 -1 -1 0 0 -1 0.000 1 0.0000 2475 600 300 225 2475 600 2775 825 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 600 1650 300 225 600 1650 900 1875 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 3315 1654 300 225 3315 1654 3615 1879 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1575 1650 300 225 1575 1650 1875 1875 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 825 600 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1200 750 3075 1500 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2400 825 750 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2625 825 3225 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1125 825 1425 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2400 825 1650 1350 +4 0 -1 0 0 0 12 0.0000 4 135 225 900 675 X1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 2325 675 Xn\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 450 1725 Y1\001 +4 0 -1 0 0 0 12 0.0000 4 135 270 3150 1725 Ym\001 +4 0 -1 0 0 0 24 0.0000 4 30 270 1575 750 ...\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1425 1725 Y2\001 +4 0 -1 0 0 0 24 0.0000 4 30 270 2175 1650 ...\001 +-6 diff --git a/sourcecodes/bnt-master/docs/Figures/fa_scalar.gif b/sourcecodes/bnt-master/docs/Figures/fa_scalar.gif new file mode 100644 index 00000000..5e5a2598 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/fa_scalar.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/fa_scalar_caption.fig b/sourcecodes/bnt-master/docs/Figures/fa_scalar_caption.fig new file mode 100644 index 00000000..0862b150 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/fa_scalar_caption.fig @@ -0,0 +1,38 @@ +#FIG 3.1 +Landscape +Center +Inches +1200 2 +6 300 375 3675 1950 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 975 600 300 225 975 600 1275 825 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 2475 600 300 225 2475 600 2775 825 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 600 1650 300 225 600 1650 900 1875 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 3315 1654 300 225 3315 1654 3615 1879 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1575 1650 300 225 1575 1650 1875 1875 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 825 600 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1200 750 3075 1500 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2400 825 750 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2625 825 3225 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1125 825 1425 1425 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2400 825 1650 1350 +4 0 -1 0 0 0 12 0.0000 4 135 210 900 675 X1\001 +4 0 -1 0 0 0 12 0.0000 4 135 210 2325 675 Xn\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 450 1725 Y1\001 +4 0 -1 0 0 0 12 0.0000 4 135 270 3150 1725 Ym\001 +4 0 -1 0 0 0 24 0.0000 4 30 270 1575 750 ...\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1425 1725 Y2\001 +4 0 -1 0 0 0 24 0.0000 4 30 270 2175 1650 ...\001 +-6 +4 0 -1 0 0 0 12 0.0000 4 180 1170 1200 2325 Factor analysis\001 diff --git a/sourcecodes/bnt-master/docs/Figures/factorial_hmm3.fig b/sourcecodes/bnt-master/docs/Figures/factorial_hmm3.fig new file mode 100644 index 00000000..a42ed6bb --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/factorial_hmm3.fig @@ -0,0 +1,81 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +5 1 0 1 0 0 100 0 -1 0.000 0 1 1 0 2587.500 3637.500 1500 2775 1200 3675 1500 4500 + 0 0 1.00 60.00 120.00 +5 1 0 1 0 0 100 0 -1 0.000 0 1 1 0 2772.051 3261.376 1500 1875 900 3450 1425 4575 + 0 0 1.00 60.00 120.00 +5 1 0 1 0 0 100 0 -1 0.000 0 1 1 0 5962.500 3637.500 4875 2775 4575 3675 4875 4500 + 0 0 1.00 60.00 120.00 +5 1 0 1 0 0 100 0 -1 0.000 0 1 1 0 4162.500 3637.500 3075 2775 2775 3675 3075 4500 + 0 0 1.00 60.00 120.00 +5 1 0 1 0 0 100 0 -1 0.000 0 1 1 0 4347.051 3261.376 3075 1875 2475 3450 3000 4575 + 0 0 1.00 60.00 120.00 +5 1 0 1 0 0 100 0 -1 0.000 0 1 1 0 6147.051 3261.376 4875 1875 4275 3450 4800 4575 + 0 0 1.00 60.00 120.00 +6 1425 4350 2100 4875 +1 1 0 1 0 0 100 0 5 0.000 1 0.0000 1749 4596 300 225 1749 4596 2049 4821 +4 0 0 100 0 0 20 0.0000 4 195 345 1575 4725 Y1\001 +-6 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 1770 3645 300 225 1770 3645 2070 3870 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 3420 3645 300 225 3420 3645 3720 3870 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 5220 3645 300 225 5220 3645 5520 3870 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 1770 2745 300 225 1770 2745 2070 2970 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 3420 2745 300 225 3420 2745 3720 2970 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 5220 2745 300 225 5220 2745 5520 2970 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 5220 1845 300 225 5220 1845 5520 2070 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 3420 1845 300 225 3420 1845 3720 2070 +1 1 0 1 0 7 100 0 -1 0.000 1 0.0000 1770 1845 300 225 1770 1845 2070 2070 +1 1 0 1 0 0 100 0 5 0.000 1 0.0000 3399 4596 300 225 3399 4596 3699 4821 +1 1 0 1 0 0 100 0 5 0.000 1 0.0000 5199 4521 300 225 5199 4521 5499 4746 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 0 0 1 + 4200 4200 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2100 3600 3150 3600 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3750 3600 4950 3600 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 0 0 1 + 4200 3300 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2100 2700 3150 2700 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3750 2700 4950 2700 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 0 0 1 + 4200 2400 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2100 1800 3150 1800 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3750 1800 4950 1800 +2 1 0 1 0 0 100 0 5 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1725 3900 1725 4350 +2 1 0 1 0 0 100 0 5 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 5175 3825 5175 4275 +2 1 0 1 0 0 100 0 5 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3450 3900 3450 4350 +4 0 0 100 0 0 20 0.0000 4 195 345 1575 1950 A1\001 +4 0 0 100 0 0 20 0.0000 4 195 330 1575 2850 B1\001 +4 0 0 100 0 0 20 0.0000 4 195 330 1575 3750 C1\001 +4 0 0 100 0 0 20 0.0000 4 195 345 3225 1950 A2\001 +4 0 0 100 0 0 20 0.0000 4 195 330 3225 2850 B2\001 +4 0 0 100 0 0 20 0.0000 4 195 330 3225 3750 C2\001 +4 0 0 100 0 0 20 0.0000 4 195 345 5025 1950 A3\001 +4 0 0 100 0 0 20 0.0000 4 195 330 5025 2850 B3\001 +4 0 0 100 0 0 20 0.0000 4 195 330 5025 3750 C3\001 +4 0 0 100 0 0 30 0.0000 4 30 525 6375 3000 . . .\001 +4 0 0 100 0 0 20 0.0000 4 195 345 3225 4725 Y2\001 +4 0 0 100 0 0 20 0.0000 4 195 345 5025 4650 Y3\001 diff --git a/sourcecodes/bnt-master/docs/Figures/filter.eps b/sourcecodes/bnt-master/docs/Figures/filter.eps new file mode 100644 index 00000000..27fd6ced --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/filter.eps @@ -0,0 +1,9861 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: (ImageMagick) +%%Title: (filter.eps) +%%CreationDate: (Tue Nov 16 20:04:33 2004) +%%BoundingBox: 0 0 657 525 +%%DocumentData: Clean7Bit +%%LanguageLevel: 1 +%%Pages: 1 +%%EndComments + +%%BeginDefaults +%%EndDefaults + +%%BeginProlog +% +% Display a color image. The image is displayed in color on +% Postscript viewers or printers that support color, otherwise +% it is displayed as grayscale. +% +/DirectClassPacket +{ + % + % Get a DirectClass packet. + % + % Parameters: + % red. + % green. + % blue. + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + compression 0 eq + { + /number_pixels 3 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add 3 mul def + } ifelse + 0 3 number_pixels 1 sub + { + pixels exch color_packet putinterval + } for + pixels 0 number_pixels getinterval +} bind def + +/DirectClassImage +{ + % + % Display a DirectClass image. + % + systemdict /colorimage known + { + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { DirectClassPacket } false 3 colorimage + } + { + % + % No colorimage operator; convert to grayscale. + % + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { GrayDirectClassPacket } image + } ifelse +} bind def + +/GrayDirectClassPacket +{ + % + % Get a DirectClass packet; convert to grayscale. + % + % Parameters: + % red + % green + % blue + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/GrayPseudoClassPacket +{ + % + % Get a PseudoClass packet; convert to grayscale. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/PseudoClassPacket +{ + % + % Get a PseudoClass packet. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + compression 0 eq + { + /number_pixels 3 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add 3 mul def + } ifelse + 0 3 number_pixels 1 sub + { + pixels exch color_packet putinterval + } for + pixels 0 number_pixels getinterval +} bind def + +/PseudoClassImage +{ + % + % Display a PseudoClass image. + % + % Parameters: + % class: 0-PseudoClass or 1-Grayscale. + % + currentfile buffer readline pop + token pop /class exch def pop + class 0 gt + { + currentfile buffer readline pop + token pop /depth exch def pop + /grays columns 8 add depth sub depth mul 8 idiv string def + columns rows depth + [ + columns 0 0 + rows neg 0 rows + ] + { currentfile grays readhexstring pop } image + } + { + % + % Parameters: + % colors: number of colors in the colormap. + % colormap: red, green, blue color packets. + % + currentfile buffer readline pop + token pop /colors exch def pop + /colors colors 3 mul def + /colormap colors string def + currentfile colormap readhexstring pop pop + systemdict /colorimage known + { + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { PseudoClassPacket } false 3 colorimage + } + { + % + % No colorimage operator; convert to grayscale. + % + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { GrayPseudoClassPacket } image + } ifelse + } ifelse +} bind def + +/DisplayImage +{ + % + % Display a DirectClass or PseudoClass image. + % + % Parameters: + % x & y translation. + % x & y scale. + % label pointsize. + % image label. + % image columns & rows. + % class: 0-DirectClass or 1-PseudoClass. + % compression: 0-none or 1-RunlengthEncoded. + % hex color packets. + % + gsave + /buffer 512 string def + /byte 1 string def + /color_packet 3 string def + /pixels 768 string def + + currentfile buffer readline pop + token pop /x exch def + token pop /y exch def pop + x y translate + currentfile buffer readline pop + token pop /x exch def + token pop /y exch def pop + currentfile buffer readline pop + token pop /pointsize exch def pop + /Times-Roman findfont pointsize scalefont setfont + x y scale + currentfile buffer readline pop + token pop /columns exch def + token pop /rows exch def pop + currentfile buffer readline pop + token pop /class exch def pop + currentfile buffer readline pop + token pop /compression exch def pop + class 0 gt { PseudoClassImage } { DirectClassImage } ifelse + grestore +} bind def +%%EndProlog +%%Page: 1 1 +%%PageBoundingBox: 0 0 657 525 +userdict begin +DisplayImage +0 0 +657 525 +12.000000 +657 525 +1 +1 +1 +8 +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 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+ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffff +end +%%PageTrailer +%%Trailer +%%EOF diff --git a/sourcecodes/bnt-master/docs/Figures/filter.fig b/sourcecodes/bnt-master/docs/Figures/filter.fig new file mode 100644 index 00000000..4e125ccf --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/filter.fig @@ -0,0 +1,74 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +6 2550 5100 6300 6375 +2 2 0 1 0 0 100 0 10 0.000 0 0 -1 0 0 5 + 2550 5475 4050 5475 4050 5775 2550 5775 2550 5475 +2 1 0 1 0 0 100 0 10 0.000 0 0 -1 0 0 2 + 4050 5475 6300 5475 +2 1 0 1 0 0 100 0 10 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3300 6375 3300 5925 +2 1 0 1 0 0 100 0 10 0.000 0 0 -1 1 1 2 + 0 0 1.00 60.00 120.00 + 0 0 1.00 60.00 120.00 + 3450 6225 4050 6225 +4 0 0 100 0 0 18 0.0000 4 165 90 4050 5325 t\001 +4 0 0 100 0 0 18 0.0000 4 165 345 3600 6075 tau\001 +-6 +2 2 0 1 0 0 100 0 10 0.000 0 0 -1 0 0 5 + 2550 2025 4050 2025 4050 2325 2550 2325 2550 2025 +2 1 0 1 0 0 100 0 10 0.000 0 0 -1 0 0 2 + 4050 2025 6300 2025 +2 1 0 1 0 0 100 0 10 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 4050 2925 4050 2475 +2 2 0 1 0 0 100 0 10 0.000 0 0 -1 0 0 5 + 2550 7200 6375 7200 6375 7500 2550 7500 2550 7200 +2 1 0 1 0 0 100 0 10 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3975 8025 3975 7575 +2 2 0 1 0 0 100 0 10 0.000 0 0 -1 0 0 5 + 2475 3525 3975 3525 3975 3825 2475 3825 2475 3525 +2 1 0 1 0 0 100 0 10 0.000 0 0 -1 0 0 2 + 3975 3525 6225 3525 +2 1 0 1 0 0 100 0 10 0.000 0 0 -1 1 1 2 + 0 0 1.00 60.00 120.00 + 0 0 1.00 60.00 120.00 + 3975 4275 4575 4275 +2 1 0 1 0 0 100 0 10 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 4650 4425 4650 3975 +2 2 0 1 0 0 100 0 10 0.000 0 0 -1 0 0 5 + 2550 600 4050 600 4050 900 2550 900 2550 600 +2 1 0 1 0 0 100 0 10 0.000 0 0 -1 0 0 2 + 4050 600 6300 600 +2 1 0 1 0 0 100 0 10 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 4050 1500 4050 1050 +4 0 0 100 0 0 18 0.0000 4 165 90 4050 1875 t\001 +4 0 0 50 0 0 24 0.0000 4 330 3540 6675 2175 argmax P(x(1:t) | y(1:t))\001 +4 0 0 50 0 0 24 0.0000 4 330 810 6975 2550 x(1:t)\001 +4 0 0 100 0 0 18 0.0000 4 165 90 3975 7125 t\001 +4 0 0 100 0 0 18 0.0000 4 195 180 6225 7125 T\001 +4 0 0 100 0 0 18 0.0000 4 165 90 3975 3375 t\001 +4 0 0 100 0 0 18 0.0000 4 195 540 3975 4125 delta\001 +4 0 0 100 0 0 18 0.0000 4 165 90 4050 450 t\001 +4 0 0 50 0 0 24 0.0000 4 330 1950 6675 750 P(X(t)|y(1:t))\001 +4 0 0 50 0 0 24 0.0000 4 330 2865 6450 3600 P(X(t+delta)|y(1:t))\001 +4 0 0 50 0 0 24 0.0000 4 330 2550 6600 5550 P(X(t-tau)|y(1:t))\001 +4 0 0 50 0 0 24 0.0000 4 330 2085 6750 7425 P(X(t)|y(1:T))\001 +4 0 0 50 0 0 24 0.0000 4 330 1140 600 900 filtering\001 +4 0 0 50 0 0 24 0.0000 4 330 1470 450 3750 prediction\001 +4 0 0 50 0 0 24 0.0000 4 255 1005 600 2175 Viterbi\001 +4 0 0 50 0 0 24 0.0000 4 330 1305 600 5775 fixed-lag\001 +4 0 0 50 0 0 24 0.0000 4 330 1575 525 7740 smoothing\001 +4 0 0 50 0 0 24 0.0000 4 330 1170 525 8055 (offline)\001 +4 0 0 50 0 0 24 0.0000 4 330 1575 600 6090 smoothing\001 +4 0 0 50 0 0 24 0.0000 4 255 1920 525 7425 fixed interval\001 diff --git a/sourcecodes/bnt-master/docs/Figures/filter.gif b/sourcecodes/bnt-master/docs/Figures/filter.gif new file mode 100644 index 00000000..bda3de61 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/filter.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/filter.tex b/sourcecodes/bnt-master/docs/Figures/filter.tex new file mode 100644 index 00000000..871f8dfd --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/filter.tex @@ -0,0 +1,51 @@ +%Latex +\documentstyle[fleqn,psfig,12pt,pstricks,pst-node,pst-tree]{article} + +%\documentclass[fleqn,12pt]{article} +%\usepackage{pstricks,pst-node,pst-tree} + +%latex2e +%\documentclass{article} +%\usepackage{epsfig,alltt,fancybox} + +\setlength{\textwidth}{6.5in} +\setlength{\oddsidemargin}{0in} +\setlength{\textheight}{8.5in} +\setlength{\headheight}{0in} +\setlength{\headsep}{-0.5in} +\setlength{\parindent}{0in} % block style +\setlength{\parskip}{0.3cm} + +\newcommand{\mytitle}[1]{\newpage \huge \begin{center}#1\vspace{0.8cm}\end{center} \LARGE} + + +\begin{document} + +\mytitle{Hello world} + +Hello world + +\centering +$ +\pstree[treemode=R]{\Tcircle{b}}{% + \pstree{\TC*^{a_1}}{% + \Tr{b_{11}}^{x_1} + \Tr{b_{12}}_{x_2} + } + \pstree{\TC*_{a_2}}{% + \Tr{b_{21}}^{x_1} + \Tr{b_{22}}_{x_2} + } +} +$ + + + +\mytitle{Hello world 2} + +\psline[linewidth=0.5cm](0,0)(2,0) +\psline[linewidth=0.05cm](2,-1)(6,-1) + + + +\end{document} diff --git a/sourcecodes/bnt-master/docs/Figures/gaussplot.png b/sourcecodes/bnt-master/docs/Figures/gaussplot.png new file mode 100644 index 00000000..29cb66ec --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/gaussplot.png Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hme.fig b/sourcecodes/bnt-master/docs/Figures/hme.fig new file mode 100644 index 00000000..dffda256 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hme.fig @@ -0,0 +1,35 @@ +#FIG 3.1 +Landscape +Center +Inches +1200 2 +6 225 150 2625 3825 +6 300 150 1800 3450 +5 1 0 1 -1 -1 0 0 -1 0.000 0 1 1 0 2212.500 2250.000 825 1425 600 2325 825 3075 + 0 0 1.00 60.00 120.00 +5 1 0 1 -1 -1 0 0 -1 0.000 0 1 1 0 3083.468 1905.242 750 525 375 2025 675 3150 + 0 0 1.00 60.00 120.00 +5 1 0 1 -1 -1 0 0 -1 0.000 0 0 1 0 679.747 1409.810 1200 450 1725 1725 1275 2325 + 0 0 1.00 60.00 120.00 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 975 375 300 225 975 375 1275 600 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 1050 3225 300 225 1050 3225 1350 3450 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 825 1050 1200 1050 1200 1500 825 1500 825 1050 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 825 2100 1200 2100 1200 2550 825 2550 825 2100 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1050 600 1050 1050 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1050 1575 1050 2100 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1050 2625 1050 3000 +-6 +4 0 -1 0 0 0 12 0.0000 4 180 2370 225 3750 Hierarchical Mixture of Experts\001 +4 0 -1 0 0 0 12 0.0000 4 135 120 825 450 X\001 +4 0 -1 0 0 0 12 0.0000 4 165 225 900 1350 Q1\001 +4 0 -1 0 0 0 12 0.0000 4 165 225 900 2400 Q2\001 +4 0 -1 0 0 0 12 0.0000 4 135 135 900 3300 Y\001 +-6 diff --git a/sourcecodes/bnt-master/docs/Figures/hme.gif b/sourcecodes/bnt-master/docs/Figures/hme.gif new file mode 100644 index 00000000..85150fbb --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hme.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hme_dec_boundary.gif b/sourcecodes/bnt-master/docs/Figures/hme_dec_boundary.gif new file mode 100644 index 00000000..15cd6b1d --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hme_dec_boundary.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hme_dec_boundary.png b/sourcecodes/bnt-master/docs/Figures/hme_dec_boundary.png new file mode 100644 index 00000000..a12cb885 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hme_dec_boundary.png Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm3.fig b/sourcecodes/bnt-master/docs/Figures/hmm3.fig new file mode 100644 index 00000000..61578c4e --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm3.fig @@ -0,0 +1,39 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +75.00 +Single +-2 +1200 2 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 525 1650 270 270 525 1650 675 1875 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 1575 1650 270 270 1575 1650 1725 1875 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 2625 1650 270 270 2625 1650 2775 1875 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 375 600 750 600 750 1050 375 1050 375 600 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 525 1050 525 1425 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 825 1350 825 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1425 600 1800 600 1800 1050 1425 1050 1425 600 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1575 1050 1575 1425 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1800 825 2325 825 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 2400 600 2775 600 2775 1050 2400 1050 2400 600 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2550 1050 2550 1425 +4 0 0 100 0 2 12 0.0000 4 135 90 525 975 1\001 +4 0 0 100 0 2 12 0.0000 4 135 90 1575 975 3\001 +4 0 0 100 0 2 12 0.0000 4 135 90 2475 900 5\001 +4 0 0 100 0 2 12 0.0000 4 135 90 450 1725 2\001 +4 0 0 100 0 2 12 0.0000 4 135 90 1425 1725 4\001 +4 0 0 100 0 2 12 0.0000 4 135 90 2550 1725 6\001 diff --git a/sourcecodes/bnt-master/docs/Figures/hmm3.gif b/sourcecodes/bnt-master/docs/Figures/hmm3.gif new file mode 100644 index 00000000..05a6f592 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm3.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm3letter.fig b/sourcecodes/bnt-master/docs/Figures/hmm3letter.fig new file mode 100644 index 00000000..3a190045 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm3letter.fig @@ -0,0 +1,39 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +75.00 +Single +-2 +1200 2 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 525 1650 270 270 525 1650 675 1875 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 1575 1650 270 270 1575 1650 1725 1875 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 2625 1650 270 270 2625 1650 2775 1875 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 375 600 750 600 750 1050 375 1050 375 600 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 525 1050 525 1425 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 825 1350 825 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1425 600 1800 600 1800 1050 1425 1050 1425 600 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1575 1050 1575 1425 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1800 825 2325 825 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 2400 600 2775 600 2775 1050 2400 1050 2400 600 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2550 1050 2550 1425 +4 0 0 100 0 0 12 0.0000 4 135 225 450 1725 Y1\001 +4 0 0 100 0 0 12 0.0000 4 135 225 1425 1725 Y2\001 +4 0 0 100 0 0 12 0.0000 4 135 225 2475 1725 Y3\001 +4 0 0 100 0 0 12 0.0000 4 135 225 450 900 X1\001 +4 0 0 100 0 0 12 0.0000 4 135 225 2475 900 X3\001 +4 0 0 100 0 0 12 0.0000 4 135 225 1500 900 X2\001 diff --git a/sourcecodes/bnt-master/docs/Figures/hmm3letter.gif b/sourcecodes/bnt-master/docs/Figures/hmm3letter.gif new file mode 100644 index 00000000..540fb75b --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm3letter.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm3letter.jpg b/sourcecodes/bnt-master/docs/Figures/hmm3letter.jpg new file mode 100644 index 00000000..29695dfd --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm3letter.jpg Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm4.fig b/sourcecodes/bnt-master/docs/Figures/hmm4.fig new file mode 100644 index 00000000..92066fb9 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm4.fig @@ -0,0 +1,55 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +75.00 +Single +-2 +1200 2 +1 3 0 2 0 7 50 0 -1 0.000 1 0.0000 600 1200 300 300 600 1200 900 1200 +1 3 0 2 0 7 50 0 -1 0.000 1 0.0000 1800 1200 300 300 1800 1200 2100 1200 +1 3 0 2 0 7 50 0 -1 0.000 1 0.0000 3000 1200 300 300 3000 1200 3300 1200 +1 3 0 2 0 7 50 0 -1 0.000 1 0.0000 4200 1200 300 300 4200 1200 4500 1200 +1 3 0 2 0 0 50 0 7 0.000 1 0.0000 600 2400 300 300 600 2400 900 2400 +1 3 0 2 0 0 50 0 7 0.000 1 0.0000 3000 2400 300 300 3000 2400 3300 2400 +1 3 0 2 0 0 50 0 7 0.000 1 0.0000 4200 2400 300 300 4200 2400 4500 2400 +1 3 0 2 0 0 50 0 7 0.000 1 0.0000 1800 2400 300 300 1800 2400 2100 2400 +2 1 0 2 0 0 50 0 7 0.000 0 0 -1 1 0 2 + 1 1 2.00 120.00 240.00 + 600 1500 600 2100 +2 1 0 2 0 0 50 0 7 0.000 0 0 -1 1 0 2 + 1 1 2.00 120.00 240.00 + 900 1200 1500 1200 +2 1 0 2 0 0 50 0 7 0.000 0 0 -1 1 0 2 + 1 1 2.00 120.00 240.00 + 2100 1200 2700 1200 +2 1 0 2 0 0 50 0 7 0.000 0 0 -1 1 0 2 + 1 1 2.00 120.00 240.00 + 3375 1200 3900 1200 +2 1 0 2 0 0 50 0 7 0.000 0 0 -1 1 0 2 + 1 1 2.00 120.00 240.00 + 1800 1500 1800 2100 +2 1 0 2 0 0 50 0 7 0.000 0 0 -1 1 0 2 + 1 1 2.00 120.00 240.00 + 3000 1500 3000 2100 +2 1 0 2 0 0 50 0 7 0.000 0 0 -1 1 0 2 + 1 1 2.00 120.00 240.00 + 4200 1500 4200 2100 +4 0 0 100 0 0 30 0.0000 4 60 600 5025 1200 . . .\001 +4 0 0 50 0 0 24 0.0000 4 255 270 450 1275 X\001 +4 0 0 50 0 0 24 0.0000 4 255 270 1650 1275 X\001 +4 0 0 50 0 0 24 0.0000 4 255 270 2850 1275 X\001 +4 0 0 50 0 0 24 0.0000 4 255 240 450 2475 Y\001 +4 0 0 50 0 0 24 0.0000 4 255 240 2850 2475 Y\001 +4 0 0 50 0 0 24 0.0000 4 255 240 4050 2475 Y\001 +4 0 0 50 0 0 24 0.0000 4 255 180 675 1425 1\001 +4 0 0 50 0 0 24 0.0000 4 255 180 1875 1350 2\001 +4 0 0 50 0 0 24 0.0000 4 255 180 3075 1350 3\001 +4 0 0 50 0 0 24 0.0000 4 255 180 4275 1425 4\001 +4 0 0 50 0 0 24 0.0000 4 255 180 600 2550 1\001 +4 0 0 50 0 0 24 0.0000 4 255 180 3000 2550 3\001 +4 0 0 50 0 0 24 0.0000 4 255 180 4200 2550 4\001 +4 0 0 50 0 0 24 0.0000 4 255 270 4050 1275 X\001 +4 0 0 50 0 0 24 0.0000 4 255 180 1800 2625 2\001 +4 0 0 50 0 0 24 0.0000 4 255 240 1650 2475 Y\001 diff --git a/sourcecodes/bnt-master/docs/Figures/hmm4.gif b/sourcecodes/bnt-master/docs/Figures/hmm4.gif new file mode 100644 index 00000000..336bc5d6 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm4.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm4_params.fig b/sourcecodes/bnt-master/docs/Figures/hmm4_params.fig new file mode 100644 index 00000000..b254e1e5 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm4_params.fig @@ -0,0 +1,81 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +75.00 +Single +-2 +1200 2 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 3150 2550 270 270 3150 2550 3300 2775 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 3975 2550 270 270 3975 2550 4125 2775 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 1350 2550 270 270 1350 2550 1500 2775 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 2250 2550 270 270 2250 2550 2400 2775 +1 3 1 1 0 7 100 0 -1 4.000 1 0.0000 2625 3825 270 270 2625 3825 2775 4050 +1 3 1 1 0 7 100 0 -1 4.000 1 0.0000 1425 600 270 270 1425 600 1575 825 +1 3 1 1 0 7 100 0 -1 4.000 1 0.0000 3000 600 270 270 3000 600 3150 825 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 2925 1500 3300 1500 3300 1950 2925 1950 2925 1500 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3075 1950 3075 2325 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 3750 1500 4125 1500 4125 1950 3750 1950 3750 1500 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3300 1725 3750 1725 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3975 1950 3975 2250 +2 1 0 1 0 0 100 0 20 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 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0 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3075 900 3075 1425 +2 1 1 1 0 7 100 0 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3225 750 3900 1500 +4 0 0 100 0 0 12 0.0000 4 135 225 3000 2625 Y3\001 +4 0 0 100 0 0 12 0.0000 4 165 225 3825 1800 Q4\001 +4 0 0 100 0 0 12 0.0000 4 165 225 3000 1800 Q3\001 +4 0 0 100 0 0 12 0.0000 4 135 225 3825 2625 Y4\001 +4 0 0 100 0 0 12 0.0000 4 135 225 1275 2625 Y1\001 +4 0 0 100 0 0 12 0.0000 4 165 225 1275 1800 Q1\001 +4 0 0 100 0 0 12 0.0000 4 135 225 2100 2625 Y2\001 +4 0 0 100 0 0 12 0.0000 4 165 225 2175 1800 Q2\001 +4 0 0 100 0 0 12 0.0000 4 135 195 1275 675 P1\001 +4 0 0 100 0 0 12 0.0000 4 135 195 2850 675 P2\001 +4 0 0 100 0 0 12 0.0000 4 135 195 2550 3900 P3\001 +4 0 0 100 0 0 30 0.0000 4 30 525 4350 1800 . . .\001 diff --git a/sourcecodes/bnt-master/docs/Figures/hmm4_params.gif b/sourcecodes/bnt-master/docs/Figures/hmm4_params.gif new file mode 100644 index 00000000..656bc13f --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm4_params.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm4_paramsX.fig b/sourcecodes/bnt-master/docs/Figures/hmm4_paramsX.fig new file mode 100644 index 00000000..5deec7e1 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm4_paramsX.fig @@ -0,0 +1,81 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +75.00 +Single +-2 +1200 2 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 3150 2550 270 270 3150 2550 3300 2775 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 3975 2550 270 270 3975 2550 4125 2775 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 1350 2550 270 270 1350 2550 1500 2775 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 2250 2550 270 270 2250 2550 2400 2775 +1 3 1 1 0 7 100 0 -1 4.000 1 0.0000 1425 600 270 270 1425 600 1575 825 +1 3 1 1 0 7 100 0 -1 4.000 1 0.0000 3000 600 270 270 3000 600 3150 825 +1 3 1 1 0 7 100 0 -1 4.000 1 0.0000 2615 3863 270 270 2615 3863 2765 4088 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 2925 1500 3300 1500 3300 1950 2925 1950 2925 1500 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3075 1950 3075 2325 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 3750 1500 4125 1500 4125 1950 3750 1950 3750 1500 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3300 1725 3750 1725 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3975 1950 3975 2250 +2 1 0 1 0 0 100 0 20 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1575 1725 2100 1725 +2 1 0 1 0 0 100 0 20 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2475 1725 2925 1725 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1200 1500 1575 1500 1575 1950 1200 1950 1200 1500 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1350 1950 1350 2325 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 2100 1500 2475 1500 2475 1950 2100 1950 2100 1500 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2250 1950 2250 2325 +2 1 1 1 0 7 100 0 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2475 3600 1500 2775 +2 1 1 1 0 7 100 0 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2550 3600 2400 2850 +2 1 1 1 0 7 100 0 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2700 3525 3075 2775 +2 1 1 1 0 7 100 0 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2850 3675 3825 2775 +2 1 1 1 0 7 100 0 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1425 900 1425 1500 +2 1 1 1 0 7 100 0 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2925 900 2400 1500 +2 1 1 1 0 7 100 0 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3075 900 3075 1425 +2 1 1 1 0 7 100 0 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3225 750 3900 1500 +4 0 0 100 0 0 12 0.0000 4 135 225 3000 2625 Y3\001 +4 0 0 100 0 0 12 0.0000 4 135 225 3825 2625 Y4\001 +4 0 0 100 0 0 12 0.0000 4 135 225 1275 2625 Y1\001 +4 0 0 100 0 0 12 0.0000 4 135 225 2100 2625 Y2\001 +4 0 0 100 0 0 30 0.0000 4 60 600 4350 1800 . . .\001 +4 0 0 50 0 0 12 0.0000 4 135 225 1275 1800 X1\001 +4 0 0 50 0 0 12 0.0000 4 135 225 2175 1800 X2\001 +4 0 0 50 0 0 12 0.0000 4 135 225 3000 1800 X3\001 +4 0 0 50 0 0 12 0.0000 4 135 225 3825 1800 X4\001 +4 0 0 50 0 0 18 0.0000 4 195 180 2550 3975 B\001 +4 0 0 50 0 0 18 0.0000 4 255 210 1275 750 pi\001 +4 0 0 50 0 0 18 0.0000 4 195 225 2850 750 A\001 diff --git a/sourcecodes/bnt-master/docs/Figures/hmm4_square.fig b/sourcecodes/bnt-master/docs/Figures/hmm4_square.fig new file mode 100644 index 00000000..9f0226f0 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm4_square.fig @@ -0,0 +1,55 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +75.00 +Single +-2 +1200 2 +6 1425 600 2025 1950 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 1725 1650 270 270 1725 1650 1875 1875 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1575 600 1950 600 1950 1050 1575 1050 1575 600 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1725 1050 1725 1425 +4 0 0 100 0 0 12 0.0000 4 135 225 1575 1725 Y2\001 +4 0 0 100 0 0 12 0.0000 4 165 225 1650 900 Q2\001 +-6 +6 525 600 1125 1950 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 825 1650 270 270 825 1650 975 1875 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 675 600 1050 600 1050 1050 675 1050 675 600 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 1050 825 1425 +4 0 0 100 0 0 12 0.0000 4 135 225 750 1725 Y1\001 +4 0 0 100 0 0 12 0.0000 4 165 225 750 900 Q1\001 +-6 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 2625 1650 270 270 2625 1650 2775 1875 +1 3 0 1 0 0 100 0 2 0.000 1 0.0000 3450 1650 270 270 3450 1650 3600 1875 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 2400 600 2775 600 2775 1050 2400 1050 2400 600 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2550 1050 2550 1425 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 3225 600 3600 600 3600 1050 3225 1050 3225 600 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2775 825 3225 825 +2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3450 1050 3450 1350 +2 1 0 1 0 0 100 0 20 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1050 825 1575 825 +2 1 0 1 0 0 100 0 20 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1950 825 2400 825 +4 0 0 100 0 0 12 0.0000 4 135 225 2475 1725 Y3\001 +4 0 0 100 0 0 12 0.0000 4 165 225 3300 900 Q4\001 +4 0 0 100 0 0 12 0.0000 4 165 225 2475 900 Q3\001 +4 0 0 100 0 0 12 0.0000 4 135 225 3300 1725 Y4\001 +4 0 0 100 0 0 30 0.0000 4 30 525 3825 975 . . .\001 diff --git a/sourcecodes/bnt-master/docs/Figures/hmm_ar.fig b/sourcecodes/bnt-master/docs/Figures/hmm_ar.fig new file mode 100644 index 00000000..8aefe002 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_ar.fig @@ -0,0 +1,30 @@ +#FIG 3.1 +Landscape +Center +Inches +1200 2 +6 225 225 2025 1875 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 525 1275 300 225 525 1275 825 1500 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1575 1275 300 225 1575 1275 1875 1500 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 450 1350 450 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 375 225 750 225 750 675 375 675 375 225 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 525 675 525 1050 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1425 225 1800 225 1800 675 1425 675 1425 225 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1575 675 1575 1050 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 1275 1275 1275 +4 0 -1 0 0 0 12 0.0000 4 165 225 450 525 Q1\001 +4 0 -1 0 0 0 12 0.0000 4 165 225 1500 525 Q2\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 375 1350 Y1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1425 1350 Y2\001 +4 0 -1 0 0 0 12 0.0000 4 180 1785 225 1800 Auto Regressive HMM\001 +-6 diff --git a/sourcecodes/bnt-master/docs/Figures/hmm_ar.gif b/sourcecodes/bnt-master/docs/Figures/hmm_ar.gif new file mode 100644 index 00000000..57d57c4d --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_ar.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm_coupled.fig b/sourcecodes/bnt-master/docs/Figures/hmm_coupled.fig new file mode 100644 index 00000000..6eab0030 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_coupled.fig @@ -0,0 +1,65 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +75.00 +Single +-2 +1200 2 +6 300 225 1950 675 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 600 450 300 225 600 450 900 675 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1650 450 300 225 1650 450 1950 675 +-6 +6 450 1875 1875 2325 +6 450 1875 1875 2325 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 900 2100 1425 2100 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 450 1875 825 1875 825 2325 450 2325 450 1875 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1500 1875 1875 1875 1875 2325 1500 2325 1500 1875 +-6 +-6 +6 450 1050 1875 1500 +6 450 1050 1875 1500 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 900 1275 1425 1275 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 450 1050 825 1050 825 1500 450 1500 450 1050 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1500 1050 1875 1050 1875 1500 1500 1500 1500 1050 +-6 +-6 +6 300 2700 1950 3150 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 600 2925 300 225 600 2925 900 3150 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1650 2925 300 225 1650 2925 1950 3150 +-6 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 600 2325 600 2700 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1650 2325 1650 2700 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 600 1050 600 675 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1650 1050 1650 675 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 900 2100 1425 1350 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 900 1275 1350 2025 +4 0 -1 0 0 0 12 0.0000 4 135 210 450 525 C1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 525 1350 A1\001 +4 0 -1 0 0 0 12 0.0000 4 135 210 525 2175 B1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 450 3000 D1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1500 3075 D2\001 +4 0 -1 0 0 0 12 0.0000 4 135 210 1500 525 C2\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1575 1350 A2\001 +4 0 -1 0 0 0 12 0.0000 4 135 210 1575 2175 B2\001 diff --git a/sourcecodes/bnt-master/docs/Figures/hmm_coupled.gif b/sourcecodes/bnt-master/docs/Figures/hmm_coupled.gif new file mode 100644 index 00000000..1dd2859c --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_coupled.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm_factorial.fig b/sourcecodes/bnt-master/docs/Figures/hmm_factorial.fig new file mode 100644 index 00000000..94e6010c --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_factorial.fig @@ -0,0 +1,50 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +75.00 +Single +-2 +1200 2 +5 1 0 1 -1 7 0 0 -1 0.000 0 1 1 0 1501.355 1323.343 675 450 300 1275 525 2025 + 0 0 1.00 60.00 120.00 +5 1 0 1 -1 7 0 0 -1 0.000 0 1 1 0 2263.600 1307.555 1650 525 1275 1200 1575 2025 + 0 0 1.00 60.00 120.00 +6 675 1050 2100 1500 +6 675 1050 2100 1500 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1125 1275 1650 1275 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 675 1050 1050 1050 1050 1500 675 1500 675 1050 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1725 1050 2100 1050 2100 1500 1725 1500 1725 1050 +-6 +-6 +6 675 225 2100 675 +6 675 225 2100 675 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1125 450 1650 450 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 675 225 1050 225 1050 675 675 675 675 225 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1725 225 2100 225 2100 675 1725 675 1725 225 +-6 +-6 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 825 2100 300 225 825 2100 1125 2325 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1875 2100 300 225 1875 2100 2175 2325 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 1500 825 1875 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1875 1500 1875 1875 +4 0 -1 0 0 0 12 0.0000 4 135 225 675 2175 Y1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1800 2250 Y2\001 +4 0 -1 0 0 0 12 0.0000 4 135 1185 675 2850 Factorial HMM\001 +4 0 0 100 0 0 12 0.0000 4 135 225 750 525 A1\001 +4 0 0 100 0 0 12 0.0000 4 135 225 1800 525 A2\001 +4 0 0 100 0 0 12 0.0000 4 135 210 750 1350 B1\001 +4 0 0 100 0 0 12 0.0000 4 135 210 1800 1350 B2\001 diff --git a/sourcecodes/bnt-master/docs/Figures/hmm_factorial.gif b/sourcecodes/bnt-master/docs/Figures/hmm_factorial.gif new file mode 100644 index 00000000..6ee4d113 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_factorial.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm_gauss.fig b/sourcecodes/bnt-master/docs/Figures/hmm_gauss.fig new file mode 100644 index 00000000..bb26e360 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_gauss.fig @@ -0,0 +1,27 @@ +#FIG 3.1 +Landscape +Center +Inches +1200 2 +6 225 600 1875 1875 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 525 1650 300 225 525 1650 825 1875 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1575 1650 300 225 1575 1650 1875 1875 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 825 1350 825 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 375 600 750 600 750 1050 375 1050 375 600 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 525 1050 525 1425 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1425 600 1800 600 1800 1050 1425 1050 1425 600 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1575 1050 1575 1425 +-6 +4 0 -1 0 0 0 12 0.0000 4 165 225 450 900 Q1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 375 1725 Y1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1500 1725 Y2\001 +4 0 -1 0 0 0 12 0.0000 4 180 2130 150 2250 HMM with Gaussian output\001 +4 0 -1 0 0 0 12 0.0000 4 165 225 1500 900 Q2\001 diff --git a/sourcecodes/bnt-master/docs/Figures/hmm_gauss.gif b/sourcecodes/bnt-master/docs/Figures/hmm_gauss.gif new file mode 100644 index 00000000..d0aa117b --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_gauss.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm_io.fig b/sourcecodes/bnt-master/docs/Figures/hmm_io.fig new file mode 100644 index 00000000..1f84aafd --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_io.fig @@ -0,0 +1,44 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +75.00 +Single +-2 +1200 2 +5 1 0 1 -1 7 0 0 -1 0.000 0 1 1 0 1725.000 1200.000 600 450 375 1275 600 1950 + 0 0 1.00 60.00 120.00 +5 1 0 1 -1 7 0 0 -1 0.000 0 1 1 0 2859.375 1303.125 1650 525 1425 1200 1575 1950 + 0 0 1.00 60.00 120.00 +6 525 150 2175 2325 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1875 2100 300 225 1875 2100 2175 2325 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 825 2100 300 225 825 2100 1125 2325 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 900 375 300 225 900 375 1200 600 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1875 375 300 225 1875 375 2175 600 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1125 1275 1650 1275 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 675 1050 1050 1050 1050 1500 675 1500 675 1050 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 1500 825 1875 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1725 1050 2100 1050 2100 1500 1725 1500 1725 1050 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1875 1500 1875 1875 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 675 825 1050 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1875 675 1875 1050 +-6 +4 0 -1 0 0 0 12 0.0000 4 165 225 750 1350 Q1\001 +4 0 -1 0 0 0 12 0.0000 4 165 225 1800 1350 Q2\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 750 450 U1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 750 2175 Y1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1725 450 U2\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1725 2175 Y2\001 diff --git a/sourcecodes/bnt-master/docs/Figures/hmm_io.gif b/sourcecodes/bnt-master/docs/Figures/hmm_io.gif new file mode 100644 index 00000000..24dfc2da --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_io.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm_mixgauss.fig b/sourcecodes/bnt-master/docs/Figures/hmm_mixgauss.fig new file mode 100644 index 00000000..cbfb1c63 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_mixgauss.fig @@ -0,0 +1,52 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +75.00 +Single +-2 +1200 2 +6 675 1650 1275 2100 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 975 1875 300 225 975 1875 1275 2100 +4 0 -1 0 0 0 12 0.0000 4 135 225 825 1950 Y1\001 +-6 +6 1725 1725 2325 2175 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 2025 1950 300 225 2025 1950 2325 2175 +4 0 -1 0 0 0 12 0.0000 4 135 225 1950 2025 Y2\001 +-6 +6 1500 900 1875 1350 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1500 900 1875 900 1875 1350 1500 1350 1500 900 +4 0 -1 0 0 0 12 0.0000 4 135 255 1575 1200 M2\001 +-6 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1275 450 1800 450 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 825 225 1200 225 1200 675 825 675 825 225 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1875 225 2250 225 2250 675 1875 675 1875 225 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 225 900 600 900 600 1350 225 1350 225 900 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 675 450 900 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 525 1350 750 1650 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1050 675 1050 1650 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1875 675 1725 900 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1800 1350 1950 1725 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2100 675 2100 1725 +4 0 -1 0 0 0 12 0.0000 4 165 225 900 525 Q1\001 +4 0 -1 0 0 0 12 0.0000 4 135 255 300 1200 M1\001 +4 0 -1 0 0 0 12 0.0000 4 165 225 1950 525 Q2\001 diff --git a/sourcecodes/bnt-master/docs/Figures/hmm_mixgauss.gif b/sourcecodes/bnt-master/docs/Figures/hmm_mixgauss.gif new file mode 100644 index 00000000..33d758b8 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_mixgauss.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm_zoo.fig b/sourcecodes/bnt-master/docs/Figures/hmm_zoo.fig new file mode 100644 index 00000000..05d34578 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_zoo.fig @@ -0,0 +1,162 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +75.00 +Single +-2 +1200 2 +5 1 0 2 0 0 50 0 -1 0.000 0 1 1 0 2784.375 4153.125 1575 3375 1350 4050 1500 4800 + 1 1 1.00 60.00 120.00 +5 1 0 2 0 0 50 0 -1 0.000 0 1 1 0 1650.000 4050.000 525 3300 300 4125 525 4800 + 1 1 1.00 60.00 120.00 +5 1 0 2 0 0 50 0 -1 0.000 0 1 1 0 6376.355 4323.343 5550 3450 5175 4275 5400 5025 + 1 1 1.00 60.00 120.00 +5 1 0 2 0 0 50 0 -1 0.000 0 1 1 0 7138.600 4307.555 6525 3525 6150 4200 6450 5025 + 1 1 1.00 60.00 120.00 +1 1 0 2 7 0 50 0 20 0.000 1 0.0000 750 4950 300 225 750 4950 1050 5175 +1 1 0 2 7 0 50 0 20 0.000 1 0.0000 1800 4950 300 225 1800 4950 2100 5175 +1 1 0 2 7 0 50 0 20 0.000 1 0.0000 825 3225 300 225 825 3225 1125 3450 +1 1 0 2 7 0 50 0 20 0.000 1 0.0000 1800 3225 300 225 1800 3225 2100 3450 +1 1 0 2 7 0 50 0 20 0.000 1 0.0000 3150 3150 300 225 3150 3150 3450 3375 +1 1 0 2 7 0 50 0 20 0.000 1 0.0000 4200 3150 300 225 4200 3150 4500 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a/sourcecodes/bnt-master/docs/Figures/hmm_zoo.gif b/sourcecodes/bnt-master/docs/Figures/hmm_zoo.gif new file mode 100644 index 00000000..c16ddffa --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_zoo.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/hmm_zoo_names.fig b/sourcecodes/bnt-master/docs/Figures/hmm_zoo_names.fig new file mode 100644 index 00000000..b53a7143 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/hmm_zoo_names.fig @@ -0,0 +1,169 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +75.00 +Single +-2 +1200 2 +5 1 0 1 -1 7 0 0 -1 0.000 0 1 1 0 1650.000 4050.000 525 3300 300 4125 525 4800 + 0 0 1.00 60.00 120.00 +5 1 0 1 -1 7 0 0 -1 0.000 0 1 1 0 2784.375 4153.125 1575 3375 1350 4050 1500 4800 + 0 0 1.00 60.00 120.00 +5 1 0 1 -1 7 0 0 -1 0.000 0 1 1 0 6376.355 4323.343 5550 3450 5175 4275 5400 5025 + 0 0 1.00 60.00 120.00 +5 1 0 1 -1 7 0 0 -1 0.000 0 1 1 0 7138.600 4307.555 6525 3525 6150 4200 6450 5025 + 0 0 1.00 60.00 120.00 +1 1 0 1 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The image is displayed in color on +% Postscript viewers or printers that support color, otherwise +% it is displayed as grayscale. +% +/DirectClassPacket +{ + % + % Get a DirectClass packet. + % + % Parameters: + % red. + % green. + % blue. + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + compression 0 eq + { + /number_pixels 3 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add 3 mul def + } ifelse + 0 3 number_pixels 1 sub + { + pixels exch color_packet putinterval + } for + pixels 0 number_pixels getinterval +} bind def + +/DirectClassImage +{ + % + % Display a DirectClass image. + % + systemdict /colorimage known + { + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { DirectClassPacket } false 3 colorimage + } + { + % + % No colorimage operator; convert to grayscale. + % + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { GrayDirectClassPacket } image + } ifelse +} bind def + +/GrayDirectClassPacket +{ + % + % Get a DirectClass packet; convert to grayscale. + % + % Parameters: + % red + % green + % blue + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/GrayPseudoClassPacket +{ + % + % Get a PseudoClass packet; convert to grayscale. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + color_packet 0 get 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index 00000000..864d5112 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/ifa.fig @@ -0,0 +1,59 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +5 1 0 1 -1 7 0 0 -1 0.000 0 1 0 0 2362.500 1537.500 1725 2325 2400 2550 3000 2325 +5 1 0 1 -1 7 0 0 -1 0.000 0 1 0 0 1950.000 818.750 675 2400 1950 2850 3225 2400 +6 225 900 3600 2475 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 900 1125 300 225 900 1125 1200 1350 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 2400 1125 300 225 2400 1125 2700 1350 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 525 2175 300 225 525 2175 825 2400 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 3240 2179 300 225 3240 2179 3540 2404 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1500 2175 300 225 1500 2175 1800 2400 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 750 1350 525 1950 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1125 1275 3000 2025 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2325 1350 675 1950 +2 1 0 1 -1 7 0 0 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1 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 2 + 2625 2175 2925 2175 +4 0 -1 0 0 0 12 0.0000 4 165 225 675 525 Q1\001 +4 0 -1 0 0 0 12 0.0000 4 165 225 2325 525 Qn\001 diff --git a/sourcecodes/bnt-master/docs/Figures/ifa.gif b/sourcecodes/bnt-master/docs/Figures/ifa.gif new file mode 100644 index 00000000..d77cd76a --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/ifa.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/kf.fig b/sourcecodes/bnt-master/docs/Figures/kf.fig new file mode 100644 index 00000000..a2e24c82 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/kf.fig @@ -0,0 +1,27 @@ +#FIG 3.1 +Landscape +Center +Inches +1200 2 +6 225 225 1950 2250 +6 225 225 1950 675 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 525 450 300 225 525 450 825 675 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 1650 450 300 225 1650 450 1950 675 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 450 1350 450 +-6 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 525 1575 300 225 525 1575 825 1800 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1650 1575 300 225 1650 1575 1950 1800 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 525 750 525 1275 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1650 750 1650 1275 +4 0 -1 0 0 0 12 0.0000 4 135 1455 225 2250 Kalman filter model\001 +-6 +4 0 -1 0 0 0 12 0.0000 4 135 210 375 525 X1\001 +4 0 -1 0 0 0 12 0.0000 4 135 210 1500 525 X2\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 375 1650 Y1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1500 1650 Y2\001 diff --git a/sourcecodes/bnt-master/docs/Figures/kf.gif b/sourcecodes/bnt-master/docs/Figures/kf.gif new file mode 100644 index 00000000..9d0ae34e --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/kf.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/kf_input.fig b/sourcecodes/bnt-master/docs/Figures/kf_input.fig new file mode 100644 index 00000000..e51d73fa --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/kf_input.fig @@ -0,0 +1,40 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +5 1 0 1 0 7 100 0 -1 0.000 0 1 1 0 3131.250 1462.500 1575 450 1275 1425 1575 2475 + 0 0 1.00 60.00 120.00 +5 1 0 1 0 7 100 0 -1 0.000 0 1 1 0 4272.606 1559.043 2775 525 2475 1275 2700 2475 + 0 0 1.00 60.00 120.00 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1875 2550 300 225 1875 2550 2175 2775 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 3000 2550 300 225 3000 2550 3300 2775 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 1875 1425 300 225 1875 1425 2175 1650 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 3000 1425 300 225 3000 1425 3300 1650 +1 1 0 1 0 0 100 0 2 0.000 1 0.0000 1875 375 300 225 1875 375 2175 600 +1 1 0 1 0 0 100 0 2 0.000 1 0.0000 3000 375 300 225 3000 375 3300 600 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1875 1725 1875 2250 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3000 1725 3000 2250 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2175 1425 2700 1425 +2 1 0 1 0 7 100 0 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b/sourcecodes/bnt-master/docs/Figures/kf_obs_track.fig @@ -0,0 +1,43 @@ +#FIG 3.2 +Portrait +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +1 4 0 2 0 7 100 0 -1 0.000 1 0.0000 1350 1500 335 335 1200 1200 1500 1800 +1 4 0 2 0 7 100 0 -1 0.000 1 0.0000 1350 2700 335 335 1200 2400 1500 3000 +1 4 0 2 0 7 100 0 -1 0.000 1 0.0000 1350 3900 335 335 1200 3600 1500 4200 +1 4 0 2 0 7 100 0 -1 0.000 1 0.0000 1350 5100 335 335 1200 4800 1500 5400 +1 4 0 2 0 7 100 0 -1 0.000 1 0.0000 2850 5100 335 335 2700 4800 3000 5400 +1 4 0 2 0 7 100 0 -1 0.000 1 0.0000 2850 3900 335 335 2700 3600 3000 4200 +1 4 0 2 0 7 100 0 -1 0.000 1 0.0000 2850 2700 335 335 2700 2400 3000 3000 +1 4 0 2 0 7 100 0 -1 0.000 1 0.0000 2850 1500 335 335 2700 1200 3000 1800 +2 1 0 2 0 0 100 0 2 0.000 0 0 -1 1 0 2 + 0 0 4.00 120.00 240.00 + 1725 1500 2550 1500 +2 1 0 2 0 0 100 0 2 0.000 0 0 -1 1 0 2 + 0 0 4.00 120.00 240.00 + 1725 2700 2550 1650 +2 1 0 2 0 0 100 0 2 0.000 0 0 -1 1 0 2 + 0 0 4.00 120.00 240.00 + 1725 3900 2475 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files differdiff --git a/sourcecodes/bnt-master/docs/Figures/mcmc_accept.jpg b/sourcecodes/bnt-master/docs/Figures/mcmc_accept.jpg new file mode 100644 index 00000000..f383627e --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/mcmc_accept.jpg Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/mcmc_post.jpg b/sourcecodes/bnt-master/docs/Figures/mcmc_post.jpg new file mode 100644 index 00000000..bbe172b5 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/mcmc_post.jpg Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/mfa.eps b/sourcecodes/bnt-master/docs/Figures/mfa.eps new file mode 100644 index 00000000..9b40d66c --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/mfa.eps @@ -0,0 +1,347 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: (ImageMagick) +%%Title: (mfa.eps) +%%CreationDate: (Tue Nov 16 19:52:06 2004) +%%BoundingBox: 0 0 126 151 +%%DocumentData: Clean7Bit +%%LanguageLevel: 1 +%%Pages: 1 +%%EndComments + +%%BeginDefaults +%%EndDefaults + +%%BeginProlog +% +% Display a color image. The image is displayed in color on +% Postscript viewers or printers that support color, otherwise +% it is displayed as grayscale. +% +/DirectClassPacket +{ + % + % Get a DirectClass packet. + % + % Parameters: + % red. + % green. + % blue. + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + compression 0 eq + { + /number_pixels 3 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add 3 mul def + } ifelse + 0 3 number_pixels 1 sub + { + pixels exch color_packet putinterval + } for + pixels 0 number_pixels getinterval +} bind def + +/DirectClassImage +{ + % + % Display a DirectClass image. + % + systemdict /colorimage known + { + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { DirectClassPacket } false 3 colorimage + } + { + % + % No colorimage operator; convert to grayscale. + % + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { GrayDirectClassPacket } image + } ifelse +} bind def + +/GrayDirectClassPacket +{ + % + % Get a DirectClass packet; convert to grayscale. + % + % Parameters: + % red + % green + % blue + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/GrayPseudoClassPacket +{ + % + % Get a PseudoClass packet; convert to grayscale. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/PseudoClassPacket +{ + % + % Get a PseudoClass packet. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + compression 0 eq + { + /number_pixels 3 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add 3 mul def + } ifelse + 0 3 number_pixels 1 sub + { + pixels exch color_packet putinterval + } for + pixels 0 number_pixels getinterval +} bind def + +/PseudoClassImage +{ + % + % Display a PseudoClass image. + % + % Parameters: + % class: 0-PseudoClass or 1-Grayscale. + % + currentfile buffer readline pop + token pop /class exch def pop + class 0 gt + { + currentfile buffer readline pop + token pop /depth exch def pop + /grays columns 8 add depth sub depth mul 8 idiv string def + columns rows depth + [ + columns 0 0 + rows neg 0 rows + ] + { currentfile grays readhexstring pop } image + } + { + % + % Parameters: + % colors: number of colors in the colormap. + % colormap: red, green, blue color packets. + % + currentfile buffer readline pop + token pop /colors exch def pop + /colors colors 3 mul def + /colormap colors string def + currentfile colormap readhexstring pop pop + systemdict /colorimage known + { + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { PseudoClassPacket } false 3 colorimage + } + { + % + % No colorimage operator; convert to grayscale. + % + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { GrayPseudoClassPacket } image + } ifelse + } ifelse +} bind def + +/DisplayImage +{ + % + % Display a DirectClass or PseudoClass image. + % + % Parameters: + % x & y translation. + % x & y scale. + % label pointsize. + % image label. + % image columns & rows. + % class: 0-DirectClass or 1-PseudoClass. + % compression: 0-none or 1-RunlengthEncoded. + % hex color packets. + % + gsave + /buffer 512 string def + /byte 1 string def + /color_packet 3 string def + /pixels 768 string def + + currentfile buffer readline pop + token pop /x exch def + token pop /y exch def pop + x y translate + currentfile buffer readline pop + token pop /x exch def + token pop /y exch def pop + currentfile buffer readline pop + token pop /pointsize exch def pop + /Times-Roman findfont pointsize scalefont setfont + x y scale + currentfile buffer readline pop + token pop /columns exch def + token pop /rows exch def pop + currentfile buffer readline pop + token pop /class exch def pop + currentfile buffer readline pop + token pop /compression exch def pop + class 0 gt { PseudoClassImage } { DirectClassImage } ifelse + grestore +} bind def +%%EndProlog +%%Page: 1 1 +%%PageBoundingBox: 0 0 126 151 +userdict begin +DisplayImage +0 0 +126 151 +12.000000 +126 151 +1 +1 +1 +1 +fffffffffffffffffffffffffffffffcfffffffffffffffffffffffffffffffcffffffff +fffffffffffffffffffffffcfffffffffffffffffffffffffffffffcffffffffffffffff +fffffffffffffffcfffffffffffffffffffffffffffffffcffffffffffffffffffffffff +fffffffcfffffffffffffffffffffffffffffffcfffffffffffffffffffffffffffffffc +ff8000001fffffffffffff800ffffffcffbfffffdffffffffffffc7ff1fffffcffbfffff +dfffffffffffe3fffe3ffffcffbfffffdfffffffffffdfffffdffffcffbfffffdfffffff +ffff3fffffe7fffcffbfffffdffffffffffefffffffbfffcffbfffffdffffffffffdffff +fffdfffcffbfffffdffffffffffbfffffffefffcffbfffffdffffffffff7ffffffff7ffc +ffbfffffdffffffffff7ffffffff7ffcffbfffffdfffffffffefffffffffbffcffbf0fff +dfffffffffeff18fffffbffcffbe67ffdfffffffffdffbdfffffdffcffbef7ffdfffffff 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-1 1 0 2 + 0 0 1.00 60.00 120.00 + 525 675 900 1200 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1425 675 1050 1200 +-6 +4 0 -1 0 0 0 12 0.0000 4 135 135 825 1500 Y\001 +4 0 -1 0 0 0 12 0.0000 4 165 135 300 525 Q\001 +4 0 -1 0 0 0 12 0.0000 4 135 135 1350 525 X\001 diff --git a/sourcecodes/bnt-master/docs/Figures/mfa.gif b/sourcecodes/bnt-master/docs/Figures/mfa.gif new file mode 100644 index 00000000..3325b3a4 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/mfa.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/mixexp.fig b/sourcecodes/bnt-master/docs/Figures/mixexp.fig new file mode 100644 index 00000000..6999ae55 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/mixexp.fig @@ -0,0 +1,23 @@ +#FIG 3.1 +Landscape +Center +Inches +1200 2 +6 525 300 1575 2925 +5 1 0 1 -1 7 0 0 -1 0.000 0 1 1 0 2055.000 1650.000 975 675 600 1650 975 2625 + 0 0 1.00 60.00 120.00 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1275 525 300 225 1275 525 1575 750 +1 1 0 1 -1 -1 0 0 -1 0.000 1 0.0000 1275 2700 300 225 1275 2700 1575 2925 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1050 1350 1425 1350 1425 1800 1050 1800 1050 1350 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1275 825 1275 1350 +2 1 0 1 -1 7 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1275 1800 1275 2475 +-6 +4 0 -1 0 0 0 12 0.0000 4 180 1425 450 3300 Mixture of Experts\001 +4 0 -1 0 0 0 12 0.0000 4 135 120 1200 600 X\001 +4 0 -1 0 0 0 12 0.0000 4 165 135 1125 1650 Q\001 +4 0 -1 0 0 0 12 0.0000 4 135 135 1200 2775 Y\001 diff --git a/sourcecodes/bnt-master/docs/Figures/mixexp.gif b/sourcecodes/bnt-master/docs/Figures/mixexp.gif new file mode 100644 index 00000000..1b76e58c --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/mixexp.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/mixexp_after.gif b/sourcecodes/bnt-master/docs/Figures/mixexp_after.gif new file mode 100644 index 00000000..06525bf4 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/mixexp_after.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/mixexp_before.gif b/sourcecodes/bnt-master/docs/Figures/mixexp_before.gif new file mode 100644 index 00000000..6eef2add --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/mixexp_before.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/mixexp_data.gif b/sourcecodes/bnt-master/docs/Figures/mixexp_data.gif new file mode 100644 index 00000000..7a770216 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/mixexp_data.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/model_select.png b/sourcecodes/bnt-master/docs/Figures/model_select.png new file mode 100644 index 00000000..25ee5da4 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/model_select.png Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/qmr.fig b/sourcecodes/bnt-master/docs/Figures/qmr.fig new file mode 100644 index 00000000..0d852876 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/qmr.fig @@ -0,0 +1,50 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +1 3 0 3 0 0 100 0 5 0.000 1 0.0000 750 2700 309 309 750 2700 1050 2775 +1 3 0 3 0 0 100 0 5 0.000 1 0.0000 3225 2700 309 309 3225 2700 3525 2775 +1 3 0 3 0 0 100 0 5 0.000 1 0.0000 4500 2700 309 309 4500 2700 4800 2775 +1 3 0 3 0 0 100 0 -1 0.000 1 0.0000 1950 2700 309 309 1950 2700 2025 3000 +1 3 0 3 0 0 100 0 -1 0.000 1 0.0000 5775 2700 309 309 5775 2700 5850 3000 +1 3 0 3 0 0 100 0 -1 0.000 1 0.0000 4125 1200 309 309 4125 1200 4200 1500 +1 3 0 3 0 0 100 0 -1 0.000 1 0.0000 3000 1200 309 309 3000 1200 3075 1500 +1 3 0 3 0 0 100 0 -1 0.000 1 0.0000 1875 1200 309 309 1875 1200 1950 1500 +2 1 0 3 0 7 100 0 -1 0.000 0 0 7 1 1 2 + 0 0 3.00 180.00 360.00 + 0 0 3.00 180.00 360.00 + 450 3750 6150 3750 +2 1 0 3 0 7 100 0 -1 0.000 0 0 7 1 1 2 + 0 0 3.00 180.00 360.00 + 0 0 3.00 180.00 360.00 + 1350 525 4650 525 +2 1 0 3 0 7 100 0 -1 0.000 0 0 7 1 0 2 + 0 0 3.00 180.00 360.00 + 1725 1500 900 2475 +2 1 0 3 0 7 100 0 -1 0.000 0 0 7 1 0 2 + 0 0 3.00 180.00 360.00 + 2850 1500 1050 2475 +2 1 0 3 0 7 100 0 -1 0.000 0 0 7 1 0 2 + 0 0 3.00 180.00 360.00 + 2925 1500 2100 2400 +2 1 0 3 0 7 100 0 -1 0.000 0 0 7 1 0 2 + 0 0 3.00 180.00 360.00 + 3975 1500 2100 2400 +2 1 0 3 0 7 100 0 -1 0.000 0 0 7 1 0 2 + 0 0 3.00 180.00 360.00 + 3075 1575 3150 2325 +2 1 0 3 0 7 100 0 -1 0.000 0 0 7 1 0 2 + 0 0 3.00 180.00 360.00 + 3150 1575 4200 2400 +2 1 0 3 0 7 100 0 -1 0.000 0 0 7 1 0 2 + 0 0 3.00 180.00 360.00 + 4200 1500 5475 2475 +4 0 0 100 0 0 24 0.0000 4 255 720 1950 4200 4000\001 +4 0 0 100 0 0 24 0.0000 4 330 1575 2850 4200 Symptoms\001 +4 0 0 100 0 0 24 0.0000 4 255 540 2100 375 600\001 +4 0 0 100 0 0 24 0.0000 4 255 1290 2775 375 Diseases\001 diff --git a/sourcecodes/bnt-master/docs/Figures/qmr.gif b/sourcecodes/bnt-master/docs/Figures/qmr.gif new file mode 100644 index 00000000..f32294fe --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/qmr.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/qmr.jpg b/sourcecodes/bnt-master/docs/Figures/qmr.jpg new file mode 100644 index 00000000..40a6c57c --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/qmr.jpg Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/qmr.rnd.jpg b/sourcecodes/bnt-master/docs/Figures/qmr.rnd.jpg new file mode 100644 index 00000000..57026cd7 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/qmr.rnd.jpg Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/rainer_dbn.jpg b/sourcecodes/bnt-master/docs/Figures/rainer_dbn.jpg new file mode 100644 index 00000000..a546ff4f --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/rainer_dbn.jpg Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/rainer_tied.fig b/sourcecodes/bnt-master/docs/Figures/rainer_tied.fig new file mode 100644 index 00000000..49cf565c --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/rainer_tied.fig @@ -0,0 +1,88 @@ +#FIG 3.2 +Portrait +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +5 1 0 2 0 7 100 0 -1 0.000 0 1 1 0 3196.875 3000.000 1275 1875 975 3150 1275 4125 + 1 1 1.00 60.00 120.00 +5 1 0 2 0 7 100 0 -1 0.000 0 1 1 0 5596.875 3000.000 3675 1875 3375 3150 3675 4125 + 1 1 1.00 60.00 120.00 +5 1 0 2 0 7 100 0 -1 0.000 0 0 1 0 3849.948 3549.006 2475 3375 3075 2400 4500 2325 + 1 1 1.00 60.00 120.00 +1 3 0 2 0 7 100 0 -1 0.000 1 0.0000 1500 1875 237 237 1500 1875 1575 2100 +1 3 0 2 0 7 100 0 -1 0.000 1 0.0000 1500 2775 237 237 1500 2775 1575 3000 +1 3 0 2 0 7 100 0 -1 0.000 1 0.0000 2400 2325 237 237 2400 2325 2475 2550 +1 3 0 2 0 7 100 0 -1 0.000 1 0.0000 2325 3600 237 237 2325 3600 2400 3825 +1 3 0 2 0 7 100 0 -1 0.000 1 0.0000 1500 4275 237 237 1500 4275 1575 4500 +1 3 0 2 0 7 100 0 -1 0.000 1 0.0000 3900 1875 237 237 3900 1875 3975 2100 +1 3 0 2 0 7 100 0 -1 0.000 1 0.0000 3900 2775 237 237 3900 2775 3975 3000 +1 3 0 2 0 7 100 0 -1 0.000 1 0.0000 4800 2325 237 237 4800 2325 4875 2550 +1 3 0 2 0 7 100 0 -1 0.000 1 0.0000 4725 3600 237 237 4725 3600 4800 3825 +1 3 0 2 0 7 100 0 -1 0.000 1 0.0000 3900 4275 237 237 3900 4275 3975 4500 +1 1 1 2 0 7 100 0 -1 6.000 1 0.0000 2625 1875 1725 300 2625 1875 4350 2175 +1 1 1 2 0 7 100 0 -1 6.000 1 0.0000 2625 4275 1800 300 2625 4275 4425 4575 +1 1 1 2 0 7 100 0 -1 6.000 1 0.0000 3375 3600 1725 300 3375 3600 5100 3900 +1 1 1 2 0 7 100 0 -1 6.000 1 0.0000 1500 2775 375 300 1500 2775 1875 3075 +1 1 1 2 0 7 100 0 -1 6.000 1 0.0000 3900 2775 375 300 3900 2775 4275 3075 +1 1 1 2 0 7 100 0 -1 6.000 1 0.0000 4800 2325 375 300 4800 2325 5175 2625 +1 1 1 2 0 7 100 0 -1 6.000 1 0.0000 2400 2325 375 225 2400 2325 2775 2550 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 1725 1950 2175 2250 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 2175 2400 1725 2700 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 2400 2550 2400 3375 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 1575 3000 2100 3450 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 3 + 1 1 1.00 60.00 120.00 + 1725 1950 2175 3375 2175 3375 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 2175 3750 1650 4125 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 4125 1950 4575 2250 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 4575 2400 4125 2700 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 4800 2550 4800 3375 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 3975 3000 4500 3450 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 3 + 1 1 1.00 60.00 120.00 + 4125 1950 4575 3375 4575 3375 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 4575 3750 4050 4125 +2 1 0 2 0 7 100 0 -1 0.000 0 0 -1 1 0 2 + 1 1 1.00 60.00 120.00 + 2700 2325 3675 2700 +4 0 0 100 0 2 12 0.0000 4 135 90 1425 1950 1\001 +4 0 0 100 0 2 12 0.0000 4 135 90 2325 2400 2\001 +4 0 0 100 0 2 12 0.0000 4 135 90 1425 2850 3\001 +4 0 0 100 0 2 12 0.0000 4 135 90 2250 3675 4\001 +4 0 0 100 0 2 12 0.0000 4 135 90 1425 4350 5\001 +4 0 0 100 0 2 12 0.0000 4 135 90 3750 1950 6\001 +4 0 0 100 0 2 12 0.0000 4 135 90 4650 2400 7\001 +4 0 0 100 0 2 12 0.0000 4 135 90 3750 2850 8\001 +4 0 0 100 0 2 12 0.0000 4 135 90 4650 3675 9\001 +4 0 0 100 0 2 12 0.0000 4 135 180 3750 4350 10\001 +4 0 0 100 0 2 12 0.0000 4 135 210 2475 2700 E2\001 +4 0 0 100 0 2 12 0.0000 4 135 210 1350 4725 E5\001 +4 0 0 100 0 2 12 0.0000 4 135 210 1725 3900 E4\001 +4 0 0 100 0 2 12 0.0000 4 135 210 1350 3300 E3\001 +4 0 0 100 0 2 12 0.0000 4 135 210 1350 2325 E1\001 +4 0 0 100 0 2 12 0.0000 4 135 210 3750 3225 E7\001 +4 0 0 100 0 2 12 0.0000 4 135 210 4800 2850 E6\001 diff --git a/sourcecodes/bnt-master/docs/Figures/rainer_tied.gif b/sourcecodes/bnt-master/docs/Figures/rainer_tied.gif new file mode 100644 index 00000000..19cb8279 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/rainer_tied.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/sar.fig b/sourcecodes/bnt-master/docs/Figures/sar.fig new file mode 100644 index 00000000..f6f54d1d --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/sar.fig @@ -0,0 +1,37 @@ +#FIG 3.1 +Landscape +Center +Inches +1200 2 +6 375 225 2175 2325 +6 375 1350 2100 1800 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 675 1575 300 225 675 1575 975 1800 +1 1 0 1 -1 0 0 0 2 0.000 1 0.0000 1800 1575 300 225 1800 1575 2100 1800 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 975 1575 1500 1575 +-6 +6 450 225 1950 675 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 450 225 825 225 825 675 450 675 450 225 +2 2 0 1 -1 7 0 0 -1 0.000 0 0 -1 0 0 5 + 1575 225 1950 225 1950 675 1575 675 1575 225 +2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 825 450 1575 450 +-6 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 675 675 675 1350 +2 1 0 1 -1 0 0 0 2 0.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1800 675 1800 1350 +2 1 1 1 -1 0 0 0 2 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 975 1425 1500 525 +4 0 -1 0 0 0 12 0.0000 4 180 1530 600 2250 Switching AR model\001 +-6 +4 0 -1 0 0 0 12 0.0000 4 165 225 525 525 Q1\001 +4 0 -1 0 0 0 12 0.0000 4 165 225 1650 525 Q2\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 525 1650 Y1\001 +4 0 -1 0 0 0 12 0.0000 4 135 225 1650 1650 Y2\001 diff --git a/sourcecodes/bnt-master/docs/Figures/sar.gif b/sourcecodes/bnt-master/docs/Figures/sar.gif new file mode 100644 index 00000000..d22e3e2d --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/sar.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/Figures/skf.fig b/sourcecodes/bnt-master/docs/Figures/skf.fig new file mode 100644 index 00000000..85ff267f --- /dev/null +++ b/sourcecodes/bnt-master/docs/Figures/skf.fig @@ -0,0 +1,48 @@ +#FIG 3.1 +Landscape +Center +Inches +1200 2 +6 300 300 2550 3375 +5 1 0 1 -1 -1 0 0 -1 0.000 0 1 1 0 3718.581 1734.122 1725 600 1425 1725 1575 2550 + 0 0 1.00 60.00 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b/sourcecodes/bnt-master/docs/Talks/BNT_mathworks.ppt new file mode 100644 index 00000000..fb41110b --- /dev/null +++ b/sourcecodes/bnt-master/docs/Talks/BNT_mathworks.ppt Binary files differdiff --git a/sourcecodes/bnt-master/docs/Talks/gR03.ppt b/sourcecodes/bnt-master/docs/Talks/gR03.ppt new file mode 100644 index 00000000..fe7b6f74 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Talks/gR03.ppt Binary files differdiff --git a/sourcecodes/bnt-master/docs/Talks/stair_BNT_mathworks.ppt b/sourcecodes/bnt-master/docs/Talks/stair_BNT_mathworks.ppt new file mode 100644 index 00000000..a400eba0 --- /dev/null +++ b/sourcecodes/bnt-master/docs/Talks/stair_BNT_mathworks.ppt Binary files differdiff --git a/sourcecodes/bnt-master/docs/adj2pajek2.m b/sourcecodes/bnt-master/docs/adj2pajek2.m new file mode 100644 index 00000000..5276a4e3 --- /dev/null +++ b/sourcecodes/bnt-master/docs/adj2pajek2.m @@ -0,0 +1,86 @@ +% ADJ2PAJEK2 Converts an adjacency matrix representation to a Pajek .net read format +% adj2pajek2(adj, filename-stem, 'argname1', argval1, ...) +% +% Set A(i,j)=-1 to get a dotted line +% +% Optional arguments +% +% nodeNames - cell array, defaults to {'v1','v2,...} +% shapes - cell array, defaults to {'ellipse','ellipse',...} +% Choices are 'ellipse', 'box', 'diamond', 'triangle', 'cross', 'empty' +% partition - vector of integers, defaults to [1 1 ... 1] +% This will automatically color-code the vertices by their partition +% +% Run pajek (available from http://vlado.fmf.uni-lj.si/pub/networks/pajek/) +% Choose File->Network->Read from the menu +% Then press ctrl-G (Draw->Draw) +% Optional: additionally load the partition file then press ctrl-P (Draw->partition) +% +% Examples +% A=zeros(5,5);A(1,2)=-1;A(2,1)=-1;A(1,[3 4])=1;A(2,5)=1; +% adj2pajek2(A,'foo') % makes foo.net +% +% adj2pajek2(A,'foo','partition',[1 1 2 2 2]) % makes foo.net and foo.clu +% +% adj2pajek2(A,'foo',... +% 'nodeNames',{'TF1','TF2','G1','G2','G3'},... +% 'shapes',{'box','box','ellipse','ellipse','ellipse'}); +% +% +% The file format is documented on p68 of the pajek manual +% and good examples are on p58, p72 +% +% Written by Kevin Murphy, 30 May 2007 +% Based on adj2pajek by Gergana Bounova +% http://stuff.mit.edu/people/gerganaa/www/matlab/routines.html +% Fixes a small bug (opens files as 'wt' instead of 'w' so it works in windows) +% Also, simplified her code and added some features. + +function []=adj2pajek2(adj,filename, varargin) + +N = length(adj); +for i=1:N + nodeNames{i} = strcat('"v',num2str(i),'"'); + shapes{i} = 'ellipse'; +end + +[nodeNames, shapes, partition] = process_options(varargin, ... + 'nodeNames', nodeNames, 'shapes', shapes, 'partition', []); + +if ~isempty(partition) + fid = fopen(sprintf('%s.clu', filename),'wt','native'); + fprintf(fid,'*Vertices %6i\n',N); + for i=1:N + fprintf(fid, '%d\n', partition(i)); + end + fclose(fid); +end + +fid = fopen(sprintf('%s.net', filename),'wt','native'); + +fprintf(fid,'*Vertices %6i\n',N); +for i=1:N + fprintf(fid,'%3i %s %s\n', i, nodeNames{i}, shapes{i}); +end + +%fprintf(fid,'*Edges\n'); +fprintf(fid,'*Arcs\n'); % directed +for i=1:N + for j=1:N + if adj(i,j) ~= 0 + fprintf(fid,' %4i %4i %2i\n',i,j,adj(i,j)); + end + end +end +fclose(fid) + + +if 0 +adj2pajek2(A,'foo',... + 'nodeNames',{'TF1','TF2','G1','G2','G3'},... + 'shapes',{'box','box','ellipse','ellipse','ellipse'}); + +N = 100; part = ones(1,N); part(intersect(reg.tfidxTest,1:N))=2; +G = reg.Atest(1:N, 1:N)'; +adj2pajek2(G, 'Ecoli100', 'partition', part) +end diff --git a/sourcecodes/bnt-master/docs/bnsoftOld.html b/sourcecodes/bnt-master/docs/bnsoftOld.html new file mode 100644 index 00000000..9da92fa1 --- /dev/null +++ b/sourcecodes/bnt-master/docs/bnsoftOld.html @@ -0,0 +1,1078 @@ +<head> +<TITLE>Software Packages for Graphical Models / Bayesian Networks</TITLE> +</head> + +<body> + +<h1> +Software Packages for Graphical Models / Bayesian Networks +</h1> +<p> +Written by Kevin Murphy. +<br> +Last updated 31 October 2005. + +<h3>Remarks</h3> +<ul> + +<li> +A much more detailed comparison of some of these software packages is +available from Appendix B of +<a href="http://www.csse.monash.edu.au/bai/">Bayesian AI</a>, by +Ann Nicholson and Kevin Korb. +This appendix is +available +<a +href="http://www.csse.monash.edu.au/bai/book/appendix_b.pdf">here</a>, +and is based on the online comparison below. + +<li> +An online <a +href="http://servasi.insa-rouen.fr/enseignement/siteUV/rna/BNT/software.html">French +version</a> of this page is also available (not necessarily up-to-date). + +<!-- +<li> +Matt Turk has made a +<a href="http://ilab.cs.ucsb.edu/BayesNets/Bayesia_Software.pdf">pdf +version of this page</a> (not necessarily up-to-date) with some extra +comments on the packages. +--> +<!-- +<br> +(He also has a +<a href="http://ilab.cs.ucsb.edu/BayesNets">cached copy</a> of several of +the packages listed below, which you can look at if links are down.) +--> + +</ul> + +<h3>What do the headers in the table mean?</h3> + +<ul> +<li> Src = source code included? (N=no) If so, what language? + +<li> API = application program interface included? +(N means the program cannot be integrated into your code, i.e., it +must be run as a standalone executable.) + +<li> Exec = Executable runs on W = Windows (95/98/NT), U = Unix, M = +Mac, or - = any machine with a compiler. + +<li> Cts = are continuous (latent) nodes supported? +G = (conditionally) Gaussians nodes supported analytically, +Cs = continuous nodes supported by sampling, +Cd = continuous nodes supported by discretization, +Cx = continuous nodes supported by some unspecified method, +D = only discrete nodes supported. + + +<li> GUI = Graphical User Interface included? + +<li> Learns parameters? + +<li> Learns structure? CI = means uses conditional independency tests + +<!-- +<li> Sample = sampling methods (e.g., likelihood weighting, MCMC) supported? +--> + +<li> Utility = utility and decision nodes (i.e., influence diagrams) +supported? + +<li> Free? +0 = free (although possibly only for academic use). +$ = commercial software (although most have free versions +which are restricted in +various ways, e.g., the model size is limited, or models cannot be +saved, or there is no API.) + + +<li> Undir? +What kind of graphs are supported? +U = only undirected graphs, +D = only directed graphs, +UD = both undirected and directed, +CG = chain graphs (mixed directed/undirected). + + +<li> Inference = which inference algorithm is used? +jtree = junction tree, +varelim = variable (bucket) elimination, +MH = Metropols Hastings, +G = Gibbs sampling, +IS = importance sampling, +sampling = some other Monte Carlo method, +polytree = Pearl's algorithm restricted to a graph with no cycles, +none = no inference supported (hence the program is only designed for +structure learning from completely observed data) + +<li> Comments. +If in "quotes", I am quoting the authors at their request. + +</ul> + + + +<!-- +The following table is automatically generated as follows +bnsoft_to_html.pl < bnsoft.txt > ! soft.html +Then soft.html is inserted and the last line is tidied up. +--> + +<table border> +<tr> +<td> Name +<td> Authors +<td> Src +<td> API +<td> Exec +<td> Cts +<td> GUI +<td> Params +<td> Struct +<td> Utility +<td> Free +<td> Undir +<td> Inference +<td> Comments + + +<tr> +<td> <!--Name--> <a href=http://www.agenarisk.com > AgenaRisk</a> +<td> <!--Authors-->Agena +<td> <!--Src-->N +<td> <!--API-->Y +<td> <!--Exec-->W,U +<td> <!--Cts-->Cx +<td> <!--GUI-->Y +<td> <!--Params-->Y +<td> <!--Struct-->N +<td> <!--Utility-->N +<td> <!--Free-->$ +<td> <!--Undir-->D +<td> <!--Inference-->JTree +<td> <!--Comments-->Simulation by Dynamic discretisation + +<tr> +<td> <!--Name--> <a href=http://www.lumina.com > Analytica</a> +<td> <!--Authors-->Lumina +<td> <!--Src-->N +<td> <!--API-->Y +<td> <!--Exec-->W,M +<td> <!--Cts-->G +<td> <!--GUI-->Y +<td> <!--Params-->N +<td> <!--Struct-->N +<td> <!--Utility-->Y +<td> <!--Free-->$ +<td> <!--Undir-->D +<td> <!--Inference-->sampling +<td> <!--Comments-->spread sheet compatible + +<tr> +<td> <!--Name--> <a href=http://www.cs.duke.edu/~amink/software/banjo/> Banjo</a> +<td> <!--Authors-->Hartemink +<td> <!--Src-->Java +<td> <!--API-->Y +<td> <!--Exec-->W,U,M +<td> <!--Cts-->Cd +<td> <!--GUI-->N +<td> <!--Params-->N +<td> <!--Struct-->Y +<td> <!--Utility-->N +<td> <!--Free-->0 +<td> <!--Undir-->D +<td> <!--Inference-->none +<td> <!--Comments--> structure learning of +static or dynamic networks of discrete variables + + +<tr> +<td><a href=http://www.cs.Helsinki.FI/research/fdk/bassist/> Bassist</a> +<td> U. Helsinki +<td> C++ +<td> Y +<td> U +<td> G +<td> N +<td> Y +<td> N +<td> N +<td> 0 +<td> D +<td> MH +<td> Generates C++ for MCMC. + +<tr> +<td><a href=http://www.cs.Helsinki.FI/research/cosco/Projects/NONE/SW/ > Bayda</a> +<td> U. Helsinki +<td> Java +<td> Y +<td> WUM +<td> G +<td> Y +<td> Y +<td> N +<td> N +<td> 0 +<td> D +<td> ? +<td> Bayesian Naive Bayes classifier. + +<tr> +<td><a href=http://www.mbfys.kun.nl/snn/Research/bayesbuilder/ > BayesBuilder</a> +<td> Nijman (U. Nijmegen) +<td> N +<td> N +<td> W +<td> D +<td> Y +<td> N +<td> N +<td> N +<td> 0 +<td> D +<td> ? +<td> - + +<tr> +<td> <!--Name--> <a +href="http://www.bayesia.com">BayesiaLab</a> +<td> <!--Authors-->Bayesia Ltd +<td> <!--Src-->N +<td> <!--API-->N +<td> <!--Exec-->- +<td> <!--Cts-->Cd +<td> <!--GUI-->Y +<td> <!--Params-->Y +<td> <!--Struct-->Y +<td> <!--Utility-->N +<td> <!--Free-->$ +<td> <!--Undir-->CG +<td> <!--Inference-->jtree,G +<td> <!--Comments--> +<!-- +Structural learning (association discovery, +classification), Bayesian clustering with analysis report generation, +missing values +processing, essential graphs, analysis toolbox, adaptive +questionnaires, dynamic models +--> +Structural learning, adaptive +questionnaires, dynamic models + + + + +<tr> + +<td><a href=http://www.bayesware.com>Bayesware Discoverer</a> +<td> Bayesware +<td> N +<td> N +<td> WUM +<td> Cd +<td> Y +<td> Y +<td> Y +<td> N +<td> $ +<td> D +<td> ? +<td> Uses bound and collapse for learning with missing data. + +<tr> +<td><a href=http://B-Course.hiit.fi/> B-course</a> +<td> U. Helsinki +<td> N +<td> N +<td> WUM +<td> Cd +<td> Y +<td> Y +<td> Y +<td> N +<td> 0 +<td> D +<td> ? +<td> Runs on their server: view results using a web browser. + +<!-- +<tr> +<td><a href="http://www.aist.go.jp/ETL/etl/suri/motomura/BN/bn-java.html">Bayonnet</a> +<td> Motomura (ETL) +<td> Java +<td> Y +<td> WUM +<td> NN +<td> Cx +<td> Y +<td> N +<td> N +<td> 0 +<td> D +<td> ? +<td> For learning, represents BN as a neural net. +--> + +<tr> +<td><a href=http://www.cs.ualberta.ca/~jcheng/bnpc.htm > Belief net power constructor</a> +<td> Cheng (U.Alberta) +<td> N +<td> W +<td> W +<td> D +<td> Y +<td> Y +<td> CI +<td> N +<td> 0 +<td> D +<td> ? +<td> - + +<tr> +<td><a href=http://www.cs.berkeley.edu/~murphyk/Bayes/bnt.html > BNT</a> +<td> Murphy (U.C.Berkeley) +<td> Matlab/C +<td> Y +<td> WUM +<td> G +<td> N +<td> Y +<td> Y +<td> Y +<td> 0 +<td> D,U +<td> Many +<td> Also handles dynamic models, like HMMs and Kalman filters. + + +<tr> +<td> <!--Name--> <a href="http://bndev.sourceforge.net/">BNJ</a> +<td> <!--Authors-->Hsu (Kansas) +<td> <!--Src-->Java +<td> <!--API-->- +<td> <!--Exec-->- +<td> <!--Cts-->D +<td> <!--GUI-->Y +<td> <!--Params-->N +<td> <!--Struct-->Y +<td> <!--Utility-->N +<td> <!--Free-->0 +<td> <!--Undir-->D +<td> <!--Inference-->jtree, IS +<td> <!--Comments-->- + +<!-- +<tr> +<td><a href=http://94.parkview-court.net/project/ > BN Toolkit</a> +<td> Gowans (Imperial) +<td> Visual Basic +<td> Y +<td> W +<td> D +<td> Y +<td> N +<td> Y +<td> N +<td> 0 +<td> D +<td> Polytree +<td> Parser and GUI for the XML-BIF format. +--> + +<tr> +<td><a href=http://www.ics.uci.edu/~irinar > BucketElim</a> +<td> Rish (U.C.Irvine) +<td> C++ +<td> Y +<td> WU +<td> D +<td> N +<td> N +<td> N +<td> N +<td> 0 +<td> D +<td> Varelim +<td> - + +<tr> +<td><a href=http://www.mrc-bsu.cam.ac.uk/bugs > BUGS</a> +<td> MRC/Imperial College +<td> N +<td> N +<td> WU +<td> Cs +<td> W +<td> Y +<td> N +<td> N +<td> 0 +<td> D +<td> Gibbs +<td> - + +<tr> +<td><a href=http://www.data-digest.com > Business Navigator 5</a> +<td> Data Digest Corp +<td> N +<td> N +<td> W +<td> Cd +<td> Y +<td> Y +<td> Y +<td> N +<td> $ +<td> D +<td> Jtree +<td> - + +<tr> +<td><a href=http://www-pcd.stanford.edu/cousins/caben-1.1.tar.gz > CABeN</a> +<td> Cousins et al. (Wash. U.) +<td> C +<td> Y +<td> WU +<td> D +<td> N +<td> N +<td> N +<td> N +<td> 0 +<td> D +<td> 5 Sampling methods +<td> - + +<tr> +<td> <!--Name--> <a href= +http://discover1.mc.vanderbilt.edu/discover/public/>Causal discoverer</a> +<td> <!--Authors-->Vanderbilt +<td> <!--Src-->N +<td> <!--API-->N +<td> <!--Exec-->W +<td> <!--Cts-->- +<td> <!--GUI-->- +<td> <!--Params-->N +<td> <!--Struct-->Y +<td> <!--Utility-->N +<td> <!--Free-->0 +<td> <!--Undir-->D +<td> <!--Inference-->- +<td> <!--Comments-->structure learning only + + +<tr> +<td><a href=http://www.math.auc.dk/~jhb/CoCo/information.html > CoCo+Xlisp</a> +<td> Badsberg (U. Aalborg) +<td> C/lisp +<td> Y +<td> U +<td> D +<td> Y +<td> Y +<td> CI +<td> N +<td> 0 +<td> U +<td> Jtree +<td> Designed for contingency tables. + +<tr> +<td><a href=http://www.cs.ubc.ca/labs/lci/CIspace/ > CIspace</a> +<td> Poole et al. (UBC) +<td> Java +<td> N +<td> WU +<td> D +<td> Y +<td> N +<td> N +<td> N +<td> 0 +<td> D +<td> Varelim +<td> - + +<tr> +<td> <!--Name--> <a href="http://www.robots.ox.ac.uk/~parg/software.html">DBNbox</a> +<td> <!--Authors-->Roberts et al +<td> <!--Src-->Matlab +<td> <!--API-->- +<td> <!--Exec-->- +<td> <!--Cts-->Y +<td> <!--GUI-->N +<td> <!--Params-->Y +<td> <!--Struct-->N +<td> <!--Utility-->N + +<td> <!--Free-->Y +<td> <!--Undir-->D +<td> <!--Inference-->Various +<td> <!--Comments-->DBNs + +<tr> +<td> <!--Name--><a href=http://www.math.auc.dk/novo/deal>Deal</a> +<td> <!--Authors-->Bottcher et al +<td> <!--Src-->R +<td> <!--API-->- +<td> <!--Exec-->- +<td> <!--Cts-->G +<td> <!--GUI-->Y +<td> <!--Params-->Y +<td> <!--Struct-->Y +<td> <!--Utility-->N +<td> <!--Free-->0 +<td> <!--Undir-->D +<td> <!--Inference-->None +<td> <!--Comments--> +Structure learning. + +<tr> +<td><a href=http://www.deriveit.com >DeriveIt</a> +<td> <!--Authors-->DeriveIt LLC +<td> <!--Src-->N +<td> <!--API-->- +<td> <!--Exec-->- +<td> <!--Cts-->? +<td> <!--GUI-->? +<td> <!--Params-->Y +<td> <!--Struct-->Y +<td> <!--Utility-->? +<td> <!--Free-->$ +<td> <!--Undir-->D +<td> <!--Inference-->Jtree +<td> <!--Comments--> +Exploits local structure in CPDs. + + +<tr> +<td><a href=http://www.noeticsystems.com>Ergo</> +<td> <!--Authors-->Noetic systems +<td> <!--Src-->N +<td> <!--API-->Y +<td> <!--Exec-->W,M +<td> <!--Cts-->D +<td> <!--GUI-->Y +<td> <!--Params-->N +<td> <!--Struct-->N +<td> <!--Utility-->N +<td> <!--Free-->$ +<td> <!--Undir-->D +<td> <!--Inference-->jtree +<td> <!--Comments-->- + +<!-- +<tr> +<td><a href=http://www.ens-lyon.fr/~jnarboux/Floue/index.html>FLoUE/BIFtoN</a> +<td> ENS Lyon +<td> Java +<td> Y +<td> WUM +<td> D +<td> N +<td> N +<td> N +<td> N +<td> 0 +<td> D +<td> Jtree +<td> - +--> + +<tr> +<td><a href=http://www.staff.ncl.ac.uk/d.j.wilkinson/software/gdagsim/ > GDAGsim</a> +<td> Wilkinson (U. Newcastle) +<td> C +<td> Y +<td> WUM +<td> G +<td> N +<td> N +<td> N +<td> N +<td> 0 +<td> D +<td> Exact +<td> Bayesian analysis of large linear Gaussian directed models. + + +<tr> +<td><a href=http://www2.sis.pitt.edu/~genie > Genie</a> +<td> U. Pittsburgh +<td> N +<td> WU +<td> WU +<td> D +<td> W +<td> N +<td> N +<td> Y +<td> 0 +<td> D +<td> Jtree +<td> - + +<tr> +<td><a href=http://www.math.ntnu.no/~hrue/GMRFsim/ > GMRFsim</a> +<td> Rue (U. Trondheim) +<td> C +<td> Y +<td> WUM +<td> G +<td> N +<td> N +<td> N +<td> N +<td> 0 +<td> U +<td> MCMC +<td> Bayesian analysis of large linear Gaussian undirected models. + + +<tr> +<td> <!--Name--> <a href=http://ssli.ee.washington.edu/~bilmes/gmtk/>GMTk</a> +<td> <!--Authors-->Bilmes (UW), Zweig (IBM) +<td> <!--Src-->N +<td> <!--API-->Y +<td> <!--Exec-->U +<td> <!--Cts-->D +<td> <!--GUI-->N +<td> <!--Params-->Y +<td> <!--Struct-->Y +<td> <!--Utility-->N +<td> <!--Free-->0 +<td> <!--Undir-->D +<td> <!--Inference-->Jtree +<td> <!--Comments--> +Designed for speech recognition. + +<tr> +<td> <!--Name--> <a href=http://www.r-project.org/gR>gR</a> +<td> <!--Authors-->Lauritzen et al. +<td> <!--Src-->R +<td> <!--API-->- +<td> <!--Exec-->- +<td> <!--Cts-->- +<td> <!--GUI-->- +<td> <!--Params-->- +<td> <!--Struct-->- +<td> <!--Utility-->- +<td> <!--Free-->0 +<td> <!--Undir-->- +<td> <!--Inference-->- +<td> <!--Comments-->Currently vaporware + + +<tr> +<td> <!--Name--> <a href=http://www.stats.bris.ac.uk/~peter/Grappa/>Grappa</a> +<td> <!--Authors-->Green (Bristol) +<td> <!--Src-->R +<td> <!--API-->- +<td> <!--Exec-->- +<td> <!--Cts-->D +<td> <!--GUI-->N +<td> <!--Params-->N) +<td> <!--Struct-->N +<td> <!--Utility-->N +<td> <!--Free-->0 +<td> <!--Undir-->D +<td> <!--Inference-->Jtree +<td> <!--Comments-->- + + +<tr> +<td><a href=http://www.hugin.com > Hugin Expert</a> +<td> Hugin +<td> N +<td> Y +<td> W +<td> G +<td> W +<td> Y +<td> CI +<td> Y +<td> $ +<td> CG +<td> Jtree +<td> - + + +<tr> +<td> <!--Name--> <a href=http://www.warnes.net/GregsSoftwareLinks/index_html?Tab=MCMC>Hydra</a> +<td> <!--Authors-->Warnes (U.Wash.) +<td> <!--Src-->Java +<td> <!--API-->- +<td> <!--Exec-->- +<td> <!--Cts-->Cs +<td> <!--GUI-->Y +<td> <!--Params-->Y +<td> <!--Struct-->N +<td> <!--Utility-->N +<td> <!--Free-->0 +<td> <!--Undir-->U,D +<td> <!--Inference-->MCMC +<td> <!--Comments-->- + + +<tr> +<!-- http://www.rpal.rockwell.com/ideal.html --> +<td><a href=http://yoda.cis.temple.edu:8080/ideal>Ideal</a> +<td> Rockwell +<td> Lisp +<td> Y +<td> WUM +<td> D +<td> Y +<td> N +<td> N +<td> Y +<td> 0 +<td> D +<td> Jtree +<td> GUI requires Allegro Lisp. + +<tr> +<td><a href=http://www.cs.cmu.edu/~javabayes/Home/ > Java Bayes</a> +<td> Cozman (CMU) +<td> Java +<td> Y +<td> WUM +<td> D +<td> Y +<td> N +<td> N +<td> Y +<td> 0 +<td> D +<td> Varelim, jtree +<td> - + + +<tr> +<td> <!--Name--><a href="http://www.codeas.com/kbaseai.php" >KBaseAI</a> +<td> <!--Authors-->Codeas +<td> <!--Src-->N +<td> <!--API-->Y +<td> <!--Exec-->W,U +<td> <!--Cts-->D +<td> <!--GUI-->N +<td> <!--Params-->N +<td> <!--Struct-->N +<td> <!--Utility-->N +<td> <!--Free-->$ +<td> <!--Undir-->D +<td> <!--Inference-->varelim +<td> <!--Comments-->client/server architecture, multiple users, access +control, query language + + +<tr> +<td> <!--Name--> <a href=http://www.cs.huji.ac.il/labs/compbio/LibB>LibB</a> +<td> <!--Authors-->Friedman (Hebrew U) +<td> <!--Src-->N +<td> <!--API-->Y +<td> <!--Exec-->W +<td> <!--Cts-->D +<td> <!--GUI-->N +<td> <!--Params-->Y +<td> <!--Struct-->Y +<td> <!--Utility-->N +<td> <!--Free-->0 +<td> <!--Undir-->D +<td> <!--Inference-->none +<td> <!--Comments--> +Structure learning + +<tr> +<td><a href=http://www.hypergraph.dk/ > MIM</a> +<td> HyperGraph Software +<td> N +<td> N +<td> W +<td> G +<td> Y +<td> Y +<td> Y +<td> N +<td> $ +<td> CG +<td> Jtree +<td> Up to 52 variables. + +<tr> +<td><a href=http://research.microsoft.com/adapt/MSBNx/ > MSBNx</a> +<td> Microsoft +<td> N +<td> Y +<td> W +<td> D +<td> W +<td> N +<td> N +<td> Y +<td> 0 +<td> D +<td> Jtree +<td> - + +<tr> +<td><a href=http://www.norsys.com > Netica</a> +<td> Norsys +<td> N +<td> WUM +<td> W +<td> G +<td> W +<td> Y +<td> N +<td> Y +<td> $ +<td> D +<td> jtree +<td> - + +<tr> +<td> <!--Name--> <a href="http://www.autonlab.org/autonweb/showSoftware/149/">Optimal +Reinsertion</a> +<td> <!--Authors-->Moore, Wong (CMU) +<td> <!--Src-->N +<td> <!--API-->N +<td> <!--Exec-->W,U +<td> <!--Cts-->D +<td> <!--GUI-->N +<td> <!--Params-->Y +<td> <!--Struct-->Y +<td> <!--Utility-->N +<td> <!--Free-->0 +<td> <!--Undir-->D +<td> <!--Inference-->none +<td> <!--Comments-->structure learning + + +<tr> +<td> <!--Name--> <a href="http://people.bu.edu/vladimir/pmt/index.html">PMT</a> +<td> <!--Authors-->Pavlovic (BU) +<td> <!--Src-->Matlab/C +<td> <!--API-->- +<td> <!--Exec-->- +<td> <!--Cts-->D +<td> <!--GUI-->N +<td> <!--Params-->Y +<td> <!--Struct-->N +<td> <!--Utility-->N +<td> <!--Free-->0 +<td> <!--Undir-->D +<td> <!--Inference-->special purpose +<td> <!--Comments-->- + + + +<tr> +<td> <!--Name--> <a href=http://www.intel.com/research/mrl/pnl/>PNL</a> +<td> <!--Authors-->Eruhimov (Intel) +<td> <!--Src-->C++ +<td> <!--API-->- +<td> <!--Exec-->- +<td> <!--Cts-->D +<td> <!--GUI-->N +<td> <!--Params-->Y +<td> <!--Struct-->Y +<td> <!--Utility-->N +<td> <!--Free-->0 +<td> <!--Undir-->U,D +<td> <!--Inference-->Jtree +<td> <!--Comments--> +A C++ version of BNT; will be released 12/03. + + +<tr> +<td><a href=http://iridia.ulb.ac.be/pulcinella/Welcome.html > Pulcinella</a> +<td> IRIDIA +<td> Lisp +<td> Y +<td> WUM +<td> D +<td> Y +<td> N +<td> N +<td> N +<td> 0 +<td> D +<td> ? +<td> Uses valuation systems for non-probabilistic calculi. + +<tr> +<!-- http://civil.colorado.edu/~dodier/sonero-mirror/index.html --> +<td><a href=http://sourceforge.net/projects/riso>RISO</a> +<td> Dodier (U.Colorado) +<td> Java +<td> Y +<td> WUM +<td> G +<td> Y +<td> N +<td> N +<td> N +<td> 0 +<td> D +<td> Polytree +<td> Distributed implementation. + + + +<tr> +<td> <!--Name--> <a href=http://reasoning.cs.ucla.edu/samiam/> Sam Iam</a> +<td> <!--Authors-->Darwiche (UCLA) +<td> <!--Src-->N +<td> <!--API-->N ? +<td> <!--Exec-->WU ? (Java executable) +<td> <!--Cts-->G ? +<td> <!--GUI-->Y +<td> <!--Params-->Y +<td> <!--Struct-->N ? +<td> <!--Utility-->Y +<td> <!--Free-->0 +<td> <!--Undir-->D +<td> <!--Inference-->Recursive conditioning +<td> <!--Comments-->Also does sensitivity Analysis + + +<tr> +<td><a href=http://www.phil.cmu.edu/tetrad/>Tetrad</a> +<td> CMU +<td> N +<td> N +<td> WU +<td> G +<td> N +<td> Y +<td> CI +<td> N +<td> 0 +<td> U,D +<td> None +<td> - + + +<tr> +<td> <!--Name--> <a href="https://sourceforge.net/projects/unbbayes/">UnBBayes</a> +<td> <!--Authors-->? +<td> <!--Src-->Java +<td> <!--API-->- +<td> <!--Exec-->- +<td> <!--Cts-->D +<td> <!--GUI-->Y +<td> <!--Params-->N +<td> <!--Struct-->Y +<td> <!--Utility-->N +<td> <!--Free-->0 +<td> <!--Undir-->D +<td> <!--Inference-->jtree +<td> <!--Comments-->K2 for struct learning + + +<tr> +<td><a href=http://www.inference.phy.cam.ac.uk/jmw39/>Vibes</a> +<td> Winn & Bishop (U. Cambridge) +<td> Java +<td> Y +<td> WU +<td> Cx +<td> Y +<td> Y +<td> N +<td> N +<td> 0 +<td> D +<td> Variational +<td> +Not yet available. + + +<tr> +<td><a href=http://snowhite.cis.uoguelph.ca/faculty_info/yxiang/ww3/> Web Weaver</a> +<td> Xiang (U.Regina) +<td> Java +<td> Y +<td> WUM +<td> D +<td> Y +<td> N +<td> N +<td> Y +<td> 0 +<td> D +<td> ? +<td> - + +<tr> +<td><a href=http://research.microsoft.com/~dmax/WinMine/tooldoc.htm > WinMine</a> +<td> Microsoft +<td> N +<td> N +<td> W +<td> Cx +<td> Y +<td> Y +<td> Y +<td> N +<td> 0 +<td> U,D +<td> None +<td> Learns BN or dependency net structure. + +<tr> +<td><a href=http://www.staff.city.ac.uk/~rgc/webpages/xbpage.html > XBAIES 2.0</a> +<td> Cowell (City U.) +<td> N +<td> N +<td> W +<td> G +<td> Y +<td> Y +<td> N +<td> Y +<td> 0 +<td> CG +<td> Jtree +<td> - + +<tr> + +</table> + + +<p> +<h2>Other sites related to (software for) graphical models</h2> +<ul> + + +<li> +<a +href="http://directory.google.com/Top/Computers/Artificial_Intelligence/Belief_Networks/Software/"> +Google's list</a> of Bayes net software. + +<!-- +<li> <a href="http://www.sis.pitt.edu/~dsl/da-software.html"> +Decision analysis programs</a>, list maintained by +Marek Druzdzel. +--> + +<li> <a href="http://www.cs.cmu.edu/~lorens/papers/mscthesis.html"> +Comparison of decision analysis software packages</a> by +Håkan L. Younes. The emphasis is on real-time decision making. + +<li> <a href="http://bayes.stat.washington.edu/almond/belief.html"> +Older belief net programs</a> (c 1996), a list created (but no longer maintained) +by Russ Almond. + +<li> +<a +href="http://www.cs.huji.ac.il/labs/compbio/Repository"> +Repository of Bayes nets</a>. + +</document> diff --git a/sourcecodes/bnt-master/docs/bnt.html b/sourcecodes/bnt-master/docs/bnt.html new file mode 100644 index 00000000..d43e9fda --- /dev/null +++ b/sourcecodes/bnt-master/docs/bnt.html @@ -0,0 +1,359 @@ +<html> <head> +<title>Bayes Net Toolbox for Matlab</title> +</head> + +<body> +<!--<body bgcolor="#FFFFFF"> --> + +<h1>Bayes Net Toolbox for Matlab</h1> +Written by Kevin Murphy, 1997--2002. +Last updated: 19 October 2007. + +<P><P> +<table> +<tr> +<td> +<img align=left src="Figures/mathbymatlab.gif" alt="Matlab logo"> +<!-- <img align=left src="toolbox.gif" alt="Toolbox logo">--> +<td> +<!--<center>--> +<a href="http://groups.yahoo.com/group/BayesNetToolbox/join"> +<img src="http://groups.yahoo.com/img/ui/join.gif" border=0><br> +Click to subscribe to the BNT email list</a> +<br> +(<a href="http://groups.yahoo.com/group/BayesNetToolbox"> +http://groups.yahoo.com/group/BayesNetToolbox</a>) +<!--</center>--> +</table> + + +<p> +<ul> +<li> <a href="changelog.html">Changelog</a> + +<li> <a +href="http://www.cs.ubc.ca/~murphyk/Software/BNT/FullBNT-1.0.4.zip">Download +zip file</a>. + +<li> <a href="install.html">Installation</a> + +<li> <a href="license.gpl">Terms and conditions of use (GNU Library GPL)</a> + + +<li> <a href="usage.html">How to use the toolbox</a> + +<li> <a href="whyNotSourceforge.html">Why I closed the sourceforge +site</a>. + +<!-- +<li> <a href="Talks/BNT_mathworks.ppt">Powerpoint slides on graphical models +and BNT</a>, presented to the Mathworks, June 2003 + + +<li> <a href="Talks/gR03.ppt">Powerpoint slides on BNT and object +recognition</a>, presented at the <a +href="http://www.math.auc.dk/gr/gr2003.html">gR</a> workshop, +September 2003. +--> + +<!-- +<li> <a href="gR03.pdf">Proposed design for gR, a graphical models +toolkit in R</a>, September 2003. +(For more information on the gR project, +click <a href="http://www.r-project.org/gR/">here</a>.) +--> + +<li> +<!-- +<img src = "../new.gif" alt="new"> +--> + +<a href="../../Papers/bnt.pdf">Invited paper on BNT</a>, +published in +Computing Science and Statistics, 2001. + +<li> <a href="../bnsoft.html">Other Bayes net software</a> + +<!--<li> <a href="software.html">Other Matlab software</a>--> + +<li> <a href="../../Bayes/bnintro.html">A brief introduction to +Bayesian Networks</a> + + +<li> <a href="#features">Major features</a> +<li> <a href="#models">Supported models</a> +<!--<li> <a href="#future">Future work</a>--> +<li> <a href="#give_away">Why do I give the code away?</a> +<li> <a href="#why_matlab">Why Matlab?</a> +<li> <a href="#ack">Acknowledgments</a> +</ul> +<p> + + + +<h2><a name="features">Major features</h2> +<ul> + +<li> BNT supports many types of +<b>conditional probability distributions</b> (nodes), +and it is easy to add more. +<ul> +<li>Tabular (multinomial) +<li>Gaussian +<li>Softmax (logistic/ sigmoid) +<li>Multi-layer perceptron (neural network) +<li>Noisy-or +<li>Deterministic +</ul> +<p> + +<li> BNT supports <b>decision and utility nodes</b>, as well as chance +nodes, +i.e., influence diagrams as well as Bayes nets. +<p> + +<li> BNT supports static and dynamic BNs (useful for modelling dynamical systems +and sequence data). +<p> + +<li> BNT supports many different <b>inference algorithms</b>, +and it is easy to add more. + +<ul> +<li> Exact inference for static BNs: +<ul> +<li>junction tree +<li>variable elimination +<li>brute force enumeration (for discrete nets) +<li>linear algebra (for Gaussian nets) +<li>Pearl's algorithm (for polytrees) +<li>quickscore (for QMR) +</ul> + +<p> +<li> Approximate inference for static BNs: +<ul> +<li>likelihood weighting +<li> Gibbs sampling +<li>loopy belief propagation +</ul> + +<p> +<li> Exact inference for DBNs: +<ul> +<li>junction tree +<li>frontier algorithm +<li>forwards-backwards (for HMMs) +<li>Kalman-RTS (for LDSs) +</ul> + +<p> +<li> Approximate inference for DBNs: +<ul> +<li>Boyen-Koller +<li>factored-frontier/loopy belief propagation +</ul> + +</ul> +<p> + +<li> +BNT supports several methods for <b>parameter learning</b>, +and it is easy to add more. +<ul> + +<li> Batch MLE/MAP parameter learning using EM. +(Each node type has its own M method, e.g. softmax nodes use IRLS,<br> +and each inference engine has its own E method, so the code is fully modular.) + +<li> Sequential/batch Bayesian parameter learning (for fully observed tabular nodes only). +</ul> + + +<p> +<li> +BNT supports several methods for <b>regularization</b>, +and it is easy to add more. +<ul> +<li> Any node can have its parameters clamped (made non-adjustable). +<li> Any set of compatible nodes can have their parameters tied (c.f., +weight sharing in a neural net). +<li> Some node types (e.g., tabular) supports priors for MAP estimation. +<li> Gaussian covariance matrices can be declared full or diagonal, and can +be tied across states of their discrete parents (if any). +</ul> + +<p> +<li> +BNT supports several methods for <b>structure learning</b>, +and it is easy to add more. +<ul> + +<li> Bayesian structure learning, +using MCMC or local search (for fully observed tabular nodes only). + +<li> Constraint-based structure learning (IC/PC and IC*/FCI). +</ul> + + +<p> +<li> The source code is extensively documented, object-oriented, and free, making it +an excellent tool for teaching, research and rapid prototyping. + +</ul> + + + +<h2><a name="models">Supported probabilistic models</h2> +<p> +It is trivial to implement all of +the following probabilistic models using the toolbox. +<ul> +<li>Static +<ul> +<li> Linear regression, logistic regression, hierarchical mixtures of experts + +<li> Naive Bayes classifiers, mixtures of Gaussians, +sigmoid belief nets + +<li> Factor analysis, probabilistic +PCA, probabilistic ICA, mixtures of these models + +</ul> + +<li>Dynamic +<ul> + +<li> HMMs, Factorial HMMs, coupled HMMs, input-output HMMs, DBNs + +<li> Kalman filters, ARMAX models, switching Kalman filters, +tree-structured Kalman filters, multiscale AR models + +</ul> + +<li> Many other combinations, for which there are (as yet) no names! + +</ul> + + +<!-- +<h2><a name="future">Future work</h2> + +I have a long <a href="wish.txt">wish list</a> +of features I would like to add to BNT +at some point in the future. +Please email me (<a +href="mailto:murphyk@cs.berkeley.edu">murphyk@cs.berkeley.edu</a>) +if you are interested in contributing! +--> + + + +<h2><a name="give_away">Why do I give the code away?</h2> + +<ul> + +<li> +I was hoping for a Linux-style effect, whereby people would contribute +their own Matlab code so that the package would grow. With a few +exceptions, this has not happened, +although several people have provided bug-fixes (see the <a +href="#ack">acknowledgements</a>). +Perhaps the <a +href="http://www.cs.berkeley.edu/~murphyk/OpenBayes/index.html">Open +Bayes Project</a> will be more +succesful in this regard, although the evidence to date is not promising. + +<p> +<li> +Knowing that someone else might read your code forces one to +document it properly, a good practice in any case, as anyone knows who +has revisited old code. +In addition, by having many "eye balls", it is easier to spot bugs. + + +<p> +<li> +I believe in the concept of +<a href="http://www-stat.stanford.edu/~donoho/Reports/1995/wavelab.pdf"> +reproducible research</a>. +Good science requires that other people be able +to replicate your experiments. +Often a paper does not give enough details about how exactly an +algorithm was implemented (e.g., how were the parameters chosen? what +initial conditions were used?), and these can make a big difference in +practice. +Hence one should release the code that +was actually used to generate the results in one's paper. +This also prevents re-inventing the wheel. + +<p> +<li> +I was fed up with reading papers where all people do is figure out how +to do exact inference and/or learning +in a model which is just a trivial special case of a general Bayes net, e.g., +input-output HMMs, coupled-HMMs, auto-regressive HMMs. +My hope is that, by releasing general purpose software, the field can +move on to more interesting questions. +As Alfred North Whitehead said in 1911, +"Civilization advances by extending the number of important operations +that we can do without thinking about them." + +</ul> + + + + + +<h2><a name="why_matlab">Why Matlab?</h2> + +Matlab is an interactive, matrix-oriented programming language that +enables one to express one's (mathematical) ideas very concisely and directly, +without having to worry about annoying details like memory allocation +or type checking. This considerably reduces development time and +keeps code short, readable and fully portable. +Matlab has excellent built-in support for many data analysis and +visualization routines. In addition, there are many useful toolboxes, e.g., for +neural networks, signal and image processing. +The main disadvantages of Matlab are that it can be slow (which is why +we are currently rewriting parts of BNT in C), and that the commercial +license is expensive (although the student version is only $100 in the US). +<p> +Many people ask me why I did not use +<a href="http://www.octave.org/">Octave</a>, +an open-source Matlab clone. +The reason is that +Octave does not support multi-dimensional arrays, +cell arrays, objects, etc. +<p> +Click <a href="../which_language.html">here</a> for a more detailed +comparison of matlab and other languages. + + + +<h2><a name="ack">Acknowledgments</h2> + +I would like to thank numerous people for bug fixes, including: +Rainer Deventer, Michael Robert James, Philippe Leray, Pedrito Maynard-Reid II, Andrew Ng, +Ron Parr, Ilya Shpitser, Xuejing Sun, Ursula Sondhauss. +<p> +I would like to thank the following people for contributing code: +Pierpaolo Brutti, Ali Taylan Cemgil, Tamar Kushnir, +Tom Murray, +Nicholas Saunier, +Ken Shan, +Yair Weiss, +Bob Welch, +Ron Zohar. +<p> +The following Intel employees have also contributed code: +Qian Diao, Shan Huang, Yimin Zhang and especially Wei Hu. + +<p> +I would like to thank Stuart Russell for funding me over the years as +I developed BNT, and Gary Bradksi for hiring me as an intern at Intel, +which has supported much of the recent developments of BNT. + + +</body> + diff --git a/sourcecodes/bnt-master/docs/bnt_download.html b/sourcecodes/bnt-master/docs/bnt_download.html new file mode 100644 index 00000000..83d32bf4 --- /dev/null +++ b/sourcecodes/bnt-master/docs/bnt_download.html @@ -0,0 +1,16 @@ +<title>Download BNT</title> + +Download BNT from <a href="http://bnt.sourceforge.net/">BNT sourceforge site</a> + +<!-- +Number of hits since 13 November 2002: +<!-- Begin of WebCounter.com code. Do not modify --> +<applet codebase="http://vc.webcounter.com/" code="vc5.class" width=88 height=31> +<param name=a value=37322><a href="http://www.webcounter.com/"> +<img src="http://vc.webcounter.com/0/t?a=37322&s=&g=" width=88 height=31></a></applet> +<!-- End of WebCounter.com code. Do not modify --> + +<p> +<a href="../FullBNT.zip">Download BNT</a> +<p> +--> diff --git a/sourcecodes/bnt-master/docs/bnt_pre_sf.html b/sourcecodes/bnt-master/docs/bnt_pre_sf.html new file mode 100644 index 00000000..9b1fe2cb --- /dev/null +++ b/sourcecodes/bnt-master/docs/bnt_pre_sf.html @@ -0,0 +1,350 @@ +<html> <head> +<title>Bayes Net Toolbox for Matlab</title> +</head> + +<body> +<!--<body bgcolor="#FFFFFF"> --> + +<h1>Bayes Net Toolbox for Matlab</h1> +Written by Kevin Murphy. +<br> +<b>BNT is now available from <a href="http://bnt.sourceforge.net/">sourceforge</a>!</b> +<!-- +Last updated on 9 June 2004 (<a href="changelog.html">Detailed +changelog</a>) +--> +<p> + +<P><P> +<table> +<tr> +<td> +<img align=left src="Figures/mathbymatlab.gif" alt="Matlab logo"> +<!-- <img align=left src="toolbox.gif" alt="Toolbox logo">--> +<td> +<!--<center>--> +<a href="http://groups.yahoo.com/group/BayesNetToolbox/join"> +<img src="http://groups.yahoo.com/img/ui/join.gif" border=0><br> +Click to subscribe to the BNT email list</a> +<br> +(<a href="http://groups.yahoo.com/group/BayesNetToolbox"> +http://groups.yahoo.com/group/BayesNetToolbox</a>) +<!--</center>--> +</table> + + +<p> +<ul> +<!--<li> <a href="bnt_download.html">Download toolbox</a>--> +<li> Download BNT from <a href="http://bnt.sourceforge.net/">BNT sourceforge site</a> + +<li> <a href="license.gpl">Terms and conditions of use (GNU Library GPL)</a> + +<li> <a href="usage.html">How to use the toolbox</a> + + +<li> <a href="Talks/BNT_mathworks.ppt">Powerpoint slides on graphical models +and BNT</a>, presented to the Mathworks, June 2003 + +<!-- +<li> <a href="Talks/gR03.ppt">Powerpoint slides on BNT and object +recognition</a>, presented at the <a +href="http://www.math.auc.dk/gr/gr2003.html">gR</a> workshop, +September 2003. +--> + +<li> <a href="gR03.pdf">Proposed design for gR, a graphical models +toolkit in R</a>, September 2003. +<!-- +(For more information on the gR project, +click <a href="http://www.r-project.org/gR/">here</a>.) +--> + +<li> +<!-- +<img src = "../new.gif" alt="new"> +--> +<a href="../../Papers/bnt.pdf">Invited paper on BNT</a>, +published in +Computing Science and Statistics, 2001. + +<li> <a href="bnsoft.html">Other Bayes net software</a> + +<!--<li> <a href="software.html">Other Matlab software</a>--> + +<li> <a href="../../Bayes/bnintro.html">A brief introduction to +Bayesian Networks</a> + +<li> <a href="#features">Major features</a> +<li> <a href="#models">Supported models</a> +<!--<li> <a href="#future">Future work</a>--> +<li> <a href="#give_away">Why do I give the code away?</a> +<li> <a href="#why_matlab">Why Matlab?</a> +<li> <a href="#ack">Acknowledgments</a> +</ul> +<p> + + + +<h2><a name="features">Major features</h2> +<ul> + +<li> BNT supports many types of +<b>conditional probability distributions</b> (nodes), +and it is easy to add more. +<ul> +<li>Tabular (multinomial) +<li>Gaussian +<li>Softmax (logistic/ sigmoid) +<li>Multi-layer perceptron (neural network) +<li>Noisy-or +<li>Deterministic +</ul> +<p> + +<li> BNT supports <b>decision and utility nodes</b>, as well as chance +nodes, +i.e., influence diagrams as well as Bayes nets. +<p> + +<li> BNT supports static and dynamic BNs (useful for modelling dynamical systems +and sequence data). +<p> + +<li> BNT supports many different <b>inference algorithms</b>, +and it is easy to add more. + +<ul> +<li> Exact inference for static BNs: +<ul> +<li>junction tree +<li>variable elimination +<li>brute force enumeration (for discrete nets) +<li>linear algebra (for Gaussian nets) +<li>Pearl's algorithm (for polytrees) +<li>quickscore (for QMR) +</ul> + +<p> +<li> Approximate inference for static BNs: +<ul> +<li>likelihood weighting +<li> Gibbs sampling +<li>loopy belief propagation +</ul> + +<p> +<li> Exact inference for DBNs: +<ul> +<li>junction tree +<li>frontier algorithm +<li>forwards-backwards (for HMMs) +<li>Kalman-RTS (for LDSs) +</ul> + +<p> +<li> Approximate inference for DBNs: +<ul> +<li>Boyen-Koller +<li>factored-frontier/loopy belief propagation +</ul> + +</ul> +<p> + +<li> +BNT supports several methods for <b>parameter learning</b>, +and it is easy to add more. +<ul> + +<li> Batch MLE/MAP parameter learning using EM. +(Each node type has its own M method, e.g. softmax nodes use IRLS,<br> +and each inference engine has its own E method, so the code is fully modular.) + +<li> Sequential/batch Bayesian parameter learning (for fully observed tabular nodes only). +</ul> + + +<p> +<li> +BNT supports several methods for <b>regularization</b>, +and it is easy to add more. +<ul> +<li> Any node can have its parameters clamped (made non-adjustable). +<li> Any set of compatible nodes can have their parameters tied (c.f., +weight sharing in a neural net). +<li> Some node types (e.g., tabular) supports priors for MAP estimation. +<li> Gaussian covariance matrices can be declared full or diagonal, and can +be tied across states of their discrete parents (if any). +</ul> + +<p> +<li> +BNT supports several methods for <b>structure learning</b>, +and it is easy to add more. +<ul> + +<li> Bayesian structure learning, +using MCMC or local search (for fully observed tabular nodes only). + +<li> Constraint-based structure learning (IC/PC and IC*/FCI). +</ul> + + +<p> +<li> The source code is extensively documented, object-oriented, and free, making it +an excellent tool for teaching, research and rapid prototyping. + +</ul> + + + +<h2><a name="models">Supported probabilistic models</h2> +<p> +It is trivial to implement all of +the following probabilistic models using the toolbox. +<ul> +<li>Static +<ul> +<li> Linear regression, logistic regression, hierarchical mixtures of experts + +<li> Naive Bayes classifiers, mixtures of Gaussians, +sigmoid belief nets + +<li> Factor analysis, probabilistic +PCA, probabilistic ICA, mixtures of these models + +</ul> + +<li>Dynamic +<ul> + +<li> HMMs, Factorial HMMs, coupled HMMs, input-output HMMs, DBNs + +<li> Kalman filters, ARMAX models, switching Kalman filters, +tree-structured Kalman filters, multiscale AR models + +</ul> + +<li> Many other combinations, for which there are (as yet) no names! + +</ul> + + +<!-- +<h2><a name="future">Future work</h2> + +I have a long <a href="wish.txt">wish list</a> +of features I would like to add to BNT +at some point in the future. +Please email me (<a +href="mailto:murphyk@cs.berkeley.edu">murphyk@cs.berkeley.edu</a>) +if you are interested in contributing! +--> + + + +<h2><a name="give_away">Why do I give the code away?</h2> + +<ul> + +<li> +I was hoping for a Linux-style effect, whereby people would contribute +their own Matlab code so that the package would grow. With a few +exceptions, this has not happened, +although several people have provided bug-fixes (see the <a +href="#ack">acknowledgements</a>). +Perhaps the <a +href="http://www.cs.berkeley.edu/~murphyk/OpenBayes/index.html">Open +Bayes Project</a> will be more +succesful in this regard, although the evidence to date is not promising. + +<p> +<li> +Knowing that someone else might read your code forces one to +document it properly, a good practice in any case, as anyone knows who +has revisited old code. +In addition, by having many "eye balls", it is easier to spot bugs. + + +<p> +<li> +I believe in the concept of +<a href="http://www-stat.stanford.edu/~donoho/Reports/1995/wavelab.pdf"> +reproducible research</a>. +Good science requires that other people be able +to replicate your experiments. +Often a paper does not give enough details about how exactly an +algorithm was implemented (e.g., how were the parameters chosen? what +initial conditions were used?), and these can make a big difference in +practice. +Hence one should release the code that +was actually used to generate the results in one's paper. +This also prevents re-inventing the wheel. + +<p> +<li> +I was fed up with reading papers where all people do is figure out how +to do exact inference and/or learning +in a model which is just a trivial special case of a general Bayes net, e.g., +input-output HMMs, coupled-HMMs, auto-regressive HMMs. +My hope is that, by releasing general purpose software, the field can +move on to more interesting questions. +As Alfred North Whitehead said in 1911, +"Civilization advances by extending the number of important operations +that we can do without thinking about them." + +</ul> + + + + + +<h2><a name="why_matlab">Why Matlab?</h2> + +Matlab is an interactive, matrix-oriented programming language that +enables one to express one's (mathematical) ideas very concisely and directly, +without having to worry about annoying details like memory allocation +or type checking. This considerably reduces development time and +keeps code short, readable and fully portable. +Matlab has excellent built-in support for many data analysis and +visualization routines. In addition, there are many useful toolboxes, e.g., for +neural networks, signal and image processing. +The main disadvantages of Matlab are that it can be slow (which is why +we are currently rewriting parts of BNT in C), and that the commercial +license is expensive (although the student version is only $100 in the US). +<p> +Many people ask me why I did not use +<a href="http://www.octave.org/">Octave</a>, +an open-source Matlab clone. +The reason is that +Octave does not support multi-dimensional arrays, +cell arrays, objects, etc. +<p> +Click <a href="../which_language.html">here</a> for a more detailed +comparison of matlab and other languages. + + + +<h2><a name="ack">Acknowledgments</h2> + +I would like to thank numerous people for bug fixes, including: +Rainer Deventer, Michael Robert James, Philippe Leray, Pedrito Maynard-Reid II, Andrew Ng, +Ron Parr, Ilya Shpitser, Xuejing Sun, Ursula Sondhauss. +<p> +I would like to thank the following people for contributing code: +Pierpaolo Brutti, Ali Taylan Cemgil, Tamar Kushnir, Ken Shan, +<a href="http://www.cs.berkeley.edu/~yweiss">Yair Weiss</a>, +Ron Zohar. +<p> +The following Intel employees have also contributed code: +Qian Diao, Shan Huang, Yimin Zhang and especially Wei Hu. + +<p> +I would like to thank Stuart Russell for funding me over the years as +I developed BNT, and Gary Bradksi for hiring me as an intern at Intel, +which has supported much of the recent developments of BNT. + + +</body> + diff --git a/sourcecodes/bnt-master/docs/cellarray.html b/sourcecodes/bnt-master/docs/cellarray.html new file mode 100644 index 00000000..19a8da21 --- /dev/null +++ b/sourcecodes/bnt-master/docs/cellarray.html @@ -0,0 +1,59 @@ +Cell arrays are a little tricky in Matlab. +Consider this example. + +C=num2cell(rand(2,3)) +C = + [0.4565] [0.8214] [0.6154] + [0.0185] [0.4447] [0.7919] + +C{1,2} % this is the contents of this cell (could be a vector or a string) +ans = + 0.8214 + +C{1:2,2} % this is the contents of these cells - returns multiple +answers! +ans = + 0.8214 +ans = + 0.4447 + +A = C(1:2,2) % this is a slice of the cell array +ans = + [0.8214] + [0.4447] + +A{1} % A is itself a cell array +ans = + 0.8214 + + + +>> C(1:2,2)=0 % can't assign a scalar to a cell array +C(1:2,2)=0 +??? Conversion to cell from double is not possible. + + +>> C(1:2,2)={0;0} % can assign a cell array to a cell array +C(1:2,2)={0;0} +C = + [0.4565] [0] [0.6154] + [0.0185] [0] [0.7919] + + +BTW, I use cell arrays for evidence for 2 reasons: + +1. [] indicates missing values +2. it can easily represent vector-valued nodes + +The following example makes this clear + +C{1,1} = [] +C{2,1} = rand(3,1) + +C = + [] [0] [0.6154] + [3x1 double] [0] [0.7919] + + +Hope this helps, +Kevin diff --git a/sourcecodes/bnt-master/docs/changelog.html b/sourcecodes/bnt-master/docs/changelog.html new file mode 100644 index 00000000..23f7a9a3 --- /dev/null +++ b/sourcecodes/bnt-master/docs/changelog.html @@ -0,0 +1,1617 @@ +<title>History of changes to BNT</title> +<h1>History of changes to BNT</h1> + + +<h2>Changes since 4 Oct 2007</h2> + +<pre> +- 19 Oct 07 murphyk + +* BNT\CPDs\@noisyor_CPD\CPD_to_CPT.m: 2nd half of the file is a repeat +of the first half and was deleted (thanks to Karl Kuschner) + +* KPMtools\myismember.m should return logical for use in "assert" so add line at end + p=logical(p); this prevents "assert" from failing on an integer input. +(thanks to Karl Kuschner) + + + +- 17 Oct 07 murphyk + +* Updated subv2ind and ind2subv in KPMtools to Tom Minka's implementation. +His ind2subv is faster (vectorized), but I had to modify it so it +matched the behavior of my version when called with siz=[]. +His subv2inv is slightly simpler than mine because he does not treat +the siz=[2 2 ... 2] case separately. +Note: there is now no need to ever use the C versions of these +functions (or any others, for that matter). + +* removed BNT/add_BNT_to_path since no longer needed. + + + +- 4 Oct 07 murphyk + +* moved code from sourceforge to UBC website, made version 1.0.4 + +* @pearl_inf_engine/pearl_inf_engine line 24, default +argument for protocol changed from [] to 'parallel'. +Also, changed private/parallel_protocol so it doesn't write to an +empty file id (Matlab 7 issue) + +* added foptions (Matlab 7 issue) + +* changed genpathKPM to exclude svn. Put it in toplevel directory to +massively simplify the installation process. + +</pre> + + +<h2>Sourceforge changelog</h2> + +BNT was first ported to sourceforge on 28 July 2001 by yozhik. +BNT was removed from sourceforge on 4 October 2007 by Kevin Murphy; +that version is cached as <a +href="FullBNT-1.0.3.zip">FullBNT-1.0.3.zip</a>. +See <a href="ChangeLog.Sourceforge.txt">Changelog from +sourceforge</a> for a history of that version of the code, +which formed the basis of the branch currently on Murphy's web page. + + +<h2> Changes from August 1998 -- July 2004</h2> + +Kevin Murphy made the following changes to his own private copy. +(Other small changes were made between July 2004 and October 2007, but were +not documented.) +These may or may not be reflected in the sourceforge version of the +code (which was independently maintained). + + +<ul> +<li> 9 June 2004 +<ul> +<li> Changed tabular_CPD/learn_params back to old syntax, to make it +compatible with gaussian_CPD/learn_params (and re-enabled +generic_CPD/learn_params). +Modified learning/learn_params.m and learning/score_family +appropriately. +(In particular, I undid the change Sonia Leach had to make to +score_family to handle this asymmetry.) +Added examples/static/gaussian2 to test this new functionality. + +<li> Added bp_mrf2 (for generic pairwise MRFs) to +inference/static/@bp_belprop_mrf2_inf_engine. [MRFs are not +"officially" supported in BNT, so this code is just for expert +hackers.] + +<li> Added examples/static/nodeorderExample.m to illustrate importance +of using topological ordering. + +<li> Ran dos2unix on all *.c files within BNT to eliminate compiler +warnings. + +</ul> + +<li> 7 June 2004 +<ul> +<li> Replaced normaliseC with normalise in HMM/fwdback, for maximum +portability (and negligible loss in speed). +<li> Ensured FullBNT versions of HMM, KPMstats etc were as up-to-date +as stand-alone versions. +<li> Changed add_BNT_to_path so it no longer uses addpath(genpath()), +which caused Old versions of files to mask new ones. +</ul> + +<li> 18 February 2004 +<ul> +<li> A few small bug fixes to BNT, as posted to the Yahoo group. +<li> Several new functions added to KPMtools, KPMstats and Graphviz +(none needed by BNT). +<li> Added CVS to some of my toolboxes. +</ul> + +<li> 30 July 2003 +<ul> +<li> qian.diao fixed @mpot/set_domain_pot and @cgpot/set_domain_pot +<li> Marco Grzegorczyk found, and Sonia Leach fixed, a bug in +do_removal inside learn_struct_mcmc +</ul> + + +<li> 28 July 2003 +<ul> +<li> Sebastian Luehr provided 2 minor bug fixes, to HMM/fwdback (if any(scale==0)) +and FullBNT\HMM\CPDs\@hhmmQ_CPD\update_ess.m (wrong transpose). +</ul> + +<li> 8 July 2003 +<ul> +<li> Removed buggy BNT/examples/static/MRF2/Old/mk_2D_lattice.m which was +masking correct graph/mk_2D_lattice. +<li> Fixed bug in graph/mk_2D_lattice_slow in the non-wrap-around case +(line 78) +</ul> + + +<li> 2 July 2003 +<ul> +<li> Sped up normalize(., 1) in KPMtools by avoiding general repmat +<li> Added assign_cols and marginalize_table to KPMtools +</ul> + + +<li> 29 May 2003 +<ul> +<li> Modified KPMstats/mixgauss_Mstep so it repmats Sigma in the tied +covariance case (bug found by galt@media.mit.edu). + +<li> Bob Welch found bug in gaussian_CPDs/maximize_params in the way +cpsz was computed. + +<li> Added KPMstats/mixgauss_em, because my code is easier to +understand/modify than netlab's (at least for me!). + +<li> Modified BNT/examples/dynamic/viterbi1 to call multinomial_prob +instead of mk_dhmm_obs_lik. + +<li> Moved parzen window and partitioned models code to KPMstats. + +<li> Rainer Deventer fixed some bugs in his scgpot code, as follows: +1. complement_pot.m +Problems occured for probabilities equal to zero. The result is an +division by zero error. +<br> +2. normalize_pot.m +This function is used during the calculation of the log-likelihood. +For a probability of zero a warning "log of zero" occurs. I have not +realy fixed the bug. As a workaround I suggest to calculate the +likelihhod based on realmin (the smallest real number) instead of +zero. +<br> +3. recursive_combine_pots +At the beginning of the function there was no test for the trivial case, +which defines the combination of two potentials as equal to the direct +combination. The result might be an infinite recursion which leads to +a stack overflow in matlab. +</ul> + + + +<li> 11 May 2003 +<ul> +<li> Fixed bug in gaussian_CPD/maximize_params so it is compatible +with the new clg_Mstep routine +<li> Modified KPMstats/cwr_em to handle single cluster case +separately. +<li> Fixed bug in netlab/gmminit. +<li> Added hash tables to KPMtools. +</ul> + + +<li> 4 May 2003 +<ul> +<li> +Renamed many functions in KPMstats so the name of the +distribution/model type comes first, +Mstep_clg -> clg_Mstep, +Mstep_cond_gauss -> mixgauss_Mstep. +Also, renamed eval_pdf_xxx functions to xxx_prob, e.g. +eval_pdf_cond_mixgauss -> mixgauss_prob. +This is simpler and shorter. + +<li> +Renamed many functions in HMM toolbox so the name of the +distribution/model type comes first, +log_lik_mhmm -> mhmm_logprob, etc. +mk_arhmm_obs_lik has finally been re-implemented in terms of clg_prob +and mixgauss_prob (for slice 1). +Removed the Demos directory, and put them in the main directory. +This code is not backwards compatible. + +<li> Removed some of the my_xxx functions from KPMstats (these were +mostly copies of functions from the Mathworks stats toolbox). + + +<li> Modified BNT to take into account changes to KPMstats and +HMM toolboxes. + +<li> Fixed KPMstats/Mstep_clg (now called clg_Mstep) for spherical Gaussian case. +(Trace was wrongly parenthesised, and I used YY instead of YTY. +The spherical case now gives the same result as the full case +for cwr_demo.) +Also, mixgauss_Mstep now adds 0.01 to the ML estimate of Sigma, +to act as a regularizer (it used to add 0.01 to E[YY'], but this was +ignored in the spherical case). + +<li> Added cluster weighted regression to KPMstats. + +<li> Added KPMtools/strmatch_substr. +</ul> + + + +<li> 28 Mar 03 +<ul> +<li> Added mc_stat_distrib and eval_pdf_cond_prod_parzen to KPMstats +<li> Fixed GraphViz/arrow.m incompatibility with matlab 6.5 +(replace all NaN's with 0). +Modified GraphViz/graph_to_dot so it also works on windows. +<li> I removed dag_to_jtree and added graph_to_jtree to the graph +toolbox; the latter expects an undirected graph as input. +<li> I added triangulate_2Dlattice_demo.m to graph. +<li> Rainer Deventer fixed the stable conditional Gaussian potential +classes (scgpot and scgcpot) and inference engine +(stab_cond_gauss_inf_engine). +<li> Rainer Deventer added (stable) higher-order Markov models (see +inference/dynamic/@stable_ho_inf_engine). +</ul> + + +<li> 14 Feb 03 +<ul> +<li> Simplified learning/learn_params so it no longer returns BIC +score. Also, simplified @tabular_CPD/learn_params so it only takes +local evidence. +Added learn_params_dbn, which does ML estimation of fully observed +DBNs. +<li> Vectorized KPMstats/eval_pdf_cond_mixgauss for tied Sigma +case (much faster!). +Also, now works in log-domain to prevent underflow. +eval_pdf_mixgauss now calls eval_pdf_cond_mixgauss and inherits these benefits. +<li> add_BNT_to_path now calls genpath with 2 arguments if using +matlab version 5. +</ul> + + +<li> 30 Jan 03 +<ul> +<li> Vectorized KPMstats/eval_pdf_cond_mixgauss for scalar Sigma +case (much faster!) +<li> Renamed mk_dotfile_from_hmm to draw_hmm and moved it to the +GraphViz library. +<li> Rewrote @gaussian_CPD/maximize_params.m so it calls +KPMstats/Mstep_clg. +This fixes bug when using clamped means (found by Rainer Deventer +and Victor Eruhimov) +and a bug when using a Wishart prior (no gamma term in the denominator). +It is also easier to read. +I rewrote the technical report re-deriving all the equations in a +clearer notation, making the solution to the bugs more obvious. +(See www.ai.mit.edu/~murphyk/Papers/learncg.pdf) +Modified Mstep_cond_gauss to handle priors. +<li> Fixed bug reported by Ramgopal Mettu in which add_BNT_to_path +calls genpath with only 1 argument, whereas version 5 requires 2. +<li> Fixed installC and uninstallC to search in FullBNT/BNT. +</ul> + + +<li> 24 Jan 03 +<ul> +<li> Major simplification of HMM code. +The API is not backwards compatible. +No new functionality has been added, however. +There is now only one fwdback function, instead of 7; +different behaviors are controlled through optional arguments. +I renamed 'evaluate observation likelihood' (local evidence) +to 'evaluate conditional pdf', since this is more general. +i.e., renamed +mk_dhmm_obs_lik to eval_pdf_cond_multinomial, +mk_ghmm_obs_lik to eval_pdf_cond_gauss, +mk_mhmm_obs_lik to eval_pdf_cond_mog. +These functions have been moved to KPMstats, +so they can be used by other toolboxes. +ghmm's have been eliminated, since they are just a special case of +mhmm's with M=1 mixture component. +mixgauss HMMs can now handle a different number of +mixture components per state. +init_mhmm has been eliminated, and replaced with init_cond_mixgauss +(in KPMstats) and mk_leftright/rightleft_transmat. +learn_dhmm can no longer handle inputs (although this is easy to add back). +</ul> + + + + + +<li> 20 Jan 03 +<ul> +<li> Added arrow.m to GraphViz directory, and commented out line 922, +in response to a bug report. +</ul> + +<li> 18 Jan 03 +<ul> +<li> Major restructuring of BNT file structure: +all code that is not specific to Bayes nets has been removed; +these packages must be downloaded separately. (Or just download FullBNT.) +This makes it easier to ensure different toolboxes are consistent. +misc has been slimmed down and renamed KPMtools, so it can be shared by other toolboxes, +such as HMM and Kalman; some of the code has been moved to BNT/general. +The Graphics directory has been slimmed down and renamed GraphViz. +The graph directory now has no dependence on BNT (dag_to_jtree has +been renamed graph_to_jtree and has a new API). +netlab2 no longer contains any netlab files, only netlab extensions. +None of the functionality has changed. +</ul> + + + +<li> 11 Jan 03 +<ul> +<li> jtree_dbn_inf_engine can now support soft evidence. + +<li> Rewrote graph/dfs to make it clearer. +Return arguments have changed, as has mk_rooted_tree. +The acyclicity check for large undirected graphs can cause a stack overflow. +It turns out that this was not a bug, but is because Matlab's stack depth +bound is very low by default. + +<li> Renamed examples/dynamic/filter2 to filter_test1, so it does not +conflict with the filter2 function in the image processing toolbox. + +<li> Ran test_BNT on various versions of matlab to check compatibility. +On matlab 6.5 (r13), elapsed time = 211s, cpu time = 204s. +On matlab 6.1 (r12), elapsed time = 173s, cpu time = 164s. +On matlab 5.3 (r11), elapsed time = 116s, cpu time = 114s. +So matlab is apparently getting slower with time!! +(All results were with a linux PIII machine.) +</ul> + + +<li> 14 Nov 02 +<ul> +<li> Removed all ndx inference routines, since they are only +marginally faster on toy problems, +and are slower on large problems due to having to store and lookup +the indices (causes cache misses). +In particular, I removed jtree_ndx_inf_eng and jtree_ndx_dbn_inf_eng, all the *ndx* +routines from potentials/Tables, and all the UID stuff from +add_BNT_to_path, +thus simplifying the code. +This required fixing hmm_(2TBN)_inf_engine/marginal_nodes\family, +and updating installC. + + +<li> Removed jtree_C_inf_engine and jtree_C_dbn_inf_engine. +The former is basically the same as using jtree_inf_engine with +mutliply_by_table.c and marginalize_table.c. +The latter benefited slightly by assuming potentials were tables +(arrays not objects), but these negligible savings don't justify the +complexity and code duplication. + +<li> Removed stab_cond_gauss_inf_engine and +scg_unrolled_dbn_inf_engine, +written by shan.huang@intel.com, since the code was buggy. + +<li> Removed potential_engine, which was only experimental anyway. + +</ul> + + + +<li> 13 Nov 02 +<ul> +<li> <b>Released version 5</b>. +The previous version, released on 7/28/02, is available +<a href="BNT4.zip">here</a>. + +<li> Moved code and documentation to MIT. + +<li> Added repmat.c from Thomas Minka's lightspeed library. +Modified it so it can return an empty matrix. + +<li> Tomas Kocka fixed bug in the BDeu option for tabular_CPD, +and contributed graph/dag_to_eg, to convert to essential graphs. + +<!--<li> Wrote a <a href="../Papers/fastmult.pdf">paper</a> which explains +the ndx methods and the ndx cache BNT uses for fast +multiplication/ marginalization of multi-dimensional arrays. +--> + +<li> Modified definition of hhmmQ_CPD, so that Qps can now accept +parents in either the current or previous slice. + +<li> Added hhmm2Q_CPD class, which is simpler than hhmmQ (no embedded +sub CPDs, etc), and which allows the conditioning parents, Qps, to +be before (in the topological ordering) the F or Q(t-1) nodes. +See BNT/examples/dynamic/HHMM/Map/mk_map_hhmm for an example. +</ul> + + +<li> 7/28/02 +<ul> +<li> Changed graph/best_first_elim_order from min-fill to min-weight. +<li> Ernest Chan fixed bug in Kalman/sample_lds (G{i} becomes G{m} in +line 61). +<li> Tal Blum <bloom@cs.huji.ac.il> fixed bug in HMM/init_ghmm (Q +becomes K, the number of states). +<li> Fixed jtree_2tbn_inf_engine/set_fields so it correctly sets the +maximize flag to 1 even in subengines. +<li> Gary Bradksi did a simple mod to the PC struct learn alg so you can pass it an +adjacency matrix as a constraint. Also, CovMat.m reads a file and +produces a covariance matrix. +<li> KNOWN BUG in CPDs/@hhmmQ_CPD/update_ess.m at line 72 caused by +examples/dynamic/HHMM/Square/learn_square_hhmm_cts.m at line 57. +<li> +The old version is available from www.cs.berkeley.edu/~murphyk/BNT.24june02.zip +</ul> + + +<li> 6/24/02 +<ul> +<li> Renamed dag_to_dot as graph_to_dot and added support for +undirected graphs. +<li> Changed syntax for HHMM CPD constructors: no need to specify d/D +anymore,so they can be used for more complex models. +<li> Removed redundant first argument to mk_isolated_tabular_CPD. +</ul> + + +<li> 6/19/02 +<ul> +<li> +Fixed most probable explanation code. +Replaced calc_mpe with find_mpe, which is now a method of certain +inference engines, e.g., jtree, belprop. +calc_mpe_global has become the find_mpe method of global_joint. +calc_mpe_bucket has become the find_mpe method of var_elim. +calc_mpe_dbn has become the find_mpe method of smoother. +These routines now correctly find the jointly most probable +explanation, instead of the marginally most probable assignments. +See examples/static/mpe1\mpe2 and examples/dynamic/viterbi1 +for examples. +Removed maximize flag from constructor and enter_evidence +methods, since this no longer needs to be specified by the user. + +<li> Rainer Deventer fixed in a bug in +CPDs/@gaussian_CPD/udpate_ess.m: +now, hidden_cps = any(hidden_bitv(cps)), whereas it used to be +hidden_cps = all(hidden_bitv(cps)). + +</ul> + + +<li> 5/29/02 +<ul> +<li> CPDs/@gaussian_CPD/udpate_ess.m fixed WX,WXX,WXY (thanks to Rainer Deventer and +Yohsuke Minowa for spotting the bug). Does the C version work?? +<li> potentials/@cpot/mpot_to_cpot fixed K==0 case (thanks to Rainer Deventer). +<li> CPDs/@gaussian_CPD/log_prob_node now accepts non-cell array data +on self (thanks to rishi <rishi@capsl.udel.edu> for catching this). +</ul> + + +<li> 5/19/02 +<ul> + +<!-- +<li> Finally added <a href="../Papers/wei_ndx.ps.gz">paper</a> by Wei Hu (written +November 2001) +describing ndxB, ndxD, and ndxSD. +--> + +<li> Wei Hu made the following changes. +<ul> +<li> Memory leak repair: + a. distribute_evidence.c in static/@jtree_C directory + b. distribute_evidence.c in static/@jtree_ndx directory + c. marg_tablec. in Tables dir + +<li> Add "@jtree_ndx_2TBN_inf_engine" in inference/online dir + +<li> Add "@jtree_sparse_inf_engine" in inference/static dir + +<li> Add "@jtree_sparse_2TBN_inf_engine" in inference/online dir + +<li> Modify "tabular_CPD.m" in CPDs/@tabular_CPD dir , used for sparse + +<li> In "@discrete_CPD" dir: + a. modify "convert_to_pot.m", used for sparse + b. add "convert_to_sparse_table.c" + +<li> In "potentials/@dpot" dir: + a. remove "divide_by_pot.c" and "multiply_by_pot.c" + b. add "divide_by_pot.m" and "multiply_by_pot.m" + c. modify "dpot.m", "marginalize_pot.m" and "normalize_pot.m" + +<li> In "potentials/Tables" dir: + a. modify mk_ndxB.c;(for speedup) + b. add "mult_by_table.m", + "divide_by_table.m", + "divide_by_table.c", + "marg_sparse_table.c", + "mult_by_sparse_table.c", + "divide_by_sparse_table.c". + +<li> Modify "normalise.c" in misc dir, used for sparse. + +<li>And, add discrete2, discrete3, filter2 and filter3 as test applications in test_BNT.m +Modify installC.m +</ul> + +<li> Kevin made the following changes related to strong junction +trees: +<ul> +<li> jtree_inf_engin line 75: +engine.root_clq = length(engine.cliques); +the last clq is guaranteed to be a strong root + +<li> dag_to_jtree line 38: [jtree, root, B, w] = +cliques_to_jtree(cliques, ns); +never call cliques_to_strong_jtree + +<li> strong_elim_order: use Ilya's code instead of topological sorting. +</ul> + +<li> Kevin fixed CPDs/@generic_CPD/learn_params, so it always passes +in the correct hidden_bitv field to update_params. + +</ul>. + + +<li> 5/8/02 +<ul> + +<li> Jerod Weinman helped fix some bugs in HHMMQ_CPD/maximize_params. + +<li> Removed broken online inference from hmm_inf_engine. +It has been replaced filter_inf_engine, which can take hmm_inf_engine +as an argument. + +<li> Changed graph visualization function names. +'draw_layout' is now 'draw_graph', +'draw_layout_dbn' is now 'draw_dbn', +'plotgraph' is now 'dag_to_dot', +'plothmm' is now 'hmm_to_dot', +added 'dbn_to_dot', +'mkdot' no longer exists': its functioality has been subsumed by dag_to_dot. +The dot functions now all take optional args in string/value format. +</ul> + + +<li> 4/1/02 +<ul> +<li> Added online inference classes. +See BNT/inference/online and BNT/examples/dynamic/filter1. +This is work in progress. +<li> Renamed cmp_inference to cmp_inference_dbn, and made its + interface and behavior more similar to cmp_inference_static. +<li> Added field rep_of_eclass to bnet and dbn, to simplify +parameter tying (see ~murphyk/Bayes/param_tieing.html). +<li> Added gmux_CPD (Gaussian mulitplexers). +See BNT/examples/dynamic/SLAM/skf_data_assoc_gmux for an example. +<li> Modified the forwards sampling routines. +general/sample_dbn and sample_bnet now take optional arguments as +strings, and can sample with pre-specified evidence. +sample_bnet can only generate a single sample, and it is always a cell +array. +sample_node can only generate a single sample, and it is always a +scalar or vector. +This eliminates the false impression that the function was +ever vectorized (which was only true for tabular_CPDs). +(Calling sample_bnet inside a for-loop is unlikely to be a bottleneck.) +<li> Updated usage.html's description of CPDs (gmux) and inference +(added gibbs_sampling and modified the description of pearl). +<li> Modified BNT/Kalman/kalman_filter\smoother so they now optionally +take an observed input (control) sequence. +Also, optional arguments are now passed as strings. +<li> Removed BNT/examples/static/uci_data to save space. +</ul> + +<li> 3/14/02 +<ul> +<li> pearl_inf_engine now works for (vector) Gaussian nodes, as well +as discrete. compute_pi has been renamed CPD_to_pi. compute_lambda_msg + has been renamed CPD_to_lambda_msg. These are now implemented for + the discrete_CPD class instead of tabular_CPD. noisyor and + Gaussian have their own private implemenations. +Created examples/static/Belprop subdirectory. +<li> Added examples/dynamic/HHMM/Motif. +<li> Added Matt Brand's entropic prior code. +<li> cmp_inference_static has changed. It no longer returns err. It + can check for convergence. It can accept 'observed'. +</ul> + + +<li> 3/4/02 +<ul> +<li> Fixed HHMM code. Now BNT/examples/dynamic/HHMM/mk_abcd_hhmm +implements the example in the NIPS paper. See also +Square/sample_square_hhmm_discrete and other files. + +<li> Included Bhaskara Marthi's gibbs_sampling_inf_engine. Currently +this only works if all CPDs are tabular and if you call installC. + +<li> Modified Kalman/tracking_demo so it calls plotgauss2d instead of + gaussplot. + +<li> Included Sonia Leach's speedup of mk_rnd_dag. +My version created all NchooseK subsets, and then picked among them. Sonia +reorders the possible parents randomly and choose +the first k. This saves on having to enumerate the large number of +possible subsets before picking from one. + +<li> Eliminated BNT/inference/static/Old, which contained some old +.mexglx files which wasted space. +</ul> + + + +<li> 2/15/02 +<ul> +<li> Removed the netlab directory, since most of it was not being +used, and it took up too much space (the goal is to have BNT.zip be +less than 1.4MB, so if fits on a floppy). +The required files have been copied into netlab2. +</ul> + +<li> 2/14/02 +<ul> +<li> Shan Huang fixed most (all?) of the bugs in his stable CG code. +scg1-3 now work, but scg_3node and scg_unstable give different + behavior than that reported in the Cowell book. + +<li> I changed gaussplot so it plots an ellipse representing the + eigenvectors of the covariance matrix, rather than numerically + evaluating the density and using a contour plot; this + is much faster and gives better pictures. The new function is + called plotgauss2d in BNT/Graphics. + +<li> Joni Alon <jalon@cs.bu.edu> fixed some small bugs: + mk_dhmm_obs_lik called forwards with the wrong args, and + add_BNT_to_path should quote filenames with spaces. + +<li> I added BNT/stats2/myunidrnd which is called by learn_struct_mcmc. + +<li> I changed BNT/potentials/@dpot/multiply_by_dpot so it now says +Tbig.T(:) = Tbig.T(:) .* Ts(:); +</ul> + + +<li> 2/6/02 +<ul> +<li> Added hierarchical HMMs. See BNT/examples/dynamic/HHMM and +CPDs/@hhmmQ_CPD and @hhmmF_CPD. +<li> sample_dbn can now sample until a certain condition is true. +<li> Sonia Leach fixed learn_struct_mcmc and changed mk_nbrs_of_digraph +so it only returns DAGs. +Click <a href="sonia_mcmc.txt">here</a> for details of her changes. +</ul> + + +<li> 2/4/02 +<ul> +<li> Wei Hu fixed a bug in +jtree_ndx_inf_engine/collect\distribute_evidence.c which failed when +maximize=1. +<li> +I fixed various bugs to do with conditional Gaussians, +so mixexp3 now works (thansk to Gerry Fung <gerry.fung@utoronto.ca> + for spotting the error). Specifically: +Changed softmax_CPD/convert_to_pot so it now puts cts nodes in cdom, and no longer inherits + this function from discrete_CPD. + Changed root_CPD/convert_to_put so it puts self in cdom. +</ul> + + +<li> 1/31/02 +<ul> +<li> Fixed log_lik_mhmm (thanks to ling chen <real_lingchen@yahoo.com> +for spotting the typo) +<li> Now many scripts in examples/static call cmp_inference_static. +Also, SCG scripts have been simplified (but still don't work!). +<li> belprop and belprop_fg enter_evidence now returns [engine, ll, + niter], with ll=0, so the order of the arguments is compatible with other engines. +<li> Ensured that all enter_evidence methods support optional + arguments such as 'maximize', even if they ignore them. +<li> Added Wei Hu's potentials/Tables/rep_mult.c, which is used to + totally eliminate all repmats from gaussian_CPD/update_ess. +</ul> + + +<li> 1/30/02 +<ul> +<li> update_ess now takes hidden_bitv instead of hidden_self and +hidden_ps. This allows gaussian_CPD to distinguish hidden discrete and +cts parents. Now learn_params_em, as well as learn_params_dbn_em, + passes in this info, for speed. + +<li> gaussian_CPD update_ess is now vectorized for any case where all + the continuous nodes are observed (eg., Gaussian HMMs, AR-HMMs). + +<li> mk_dbn now automatically detects autoregressive nodes. + +<li> hmm_inf_engine now uses indexes in marginal_nodes/family for + speed. Marginal_ndoes can now only handle single nodes. + (SDndx is hard-coded, to avoid the overhead of using marg_ndx, + which is slow because of the case and global statements.) + +<li> add_ev_to_dmarginal now retains the domain field. + +<li> Wei Hu wrote potentials/Tables/repmat_and_mult.c, which is used to + avoid some of the repmat's in gaussian_CPD/update_ess. + +<li> installC now longer sets the global USEC, since USEC is set to 0 + by add_BNT_to_path, even if the C files have already been compiled + in a previous session. Instead, gaussian_CPD checks to + see if repmat_and_mult exists, and (bat1, chmm1, water1, water2) + check to see if jtree_C_inf_engine/collect_evidence exists. + Note that checking if a file exists is slow, so we do the check + inside the gaussian_CPD constructor, not inside update_ess. + +<li> uninstallC now deletes both .mex and .dll files, just in case I + accidently ship a .zip file with binaries. It also deletes mex + files from jtree_C_inf_engine. + +<li> Now marginal_family for both jtree_limid_inf_engine and + global_joint_inf_engine returns a marginal structure and + potential, as required by solve_limid. + Other engines (eg. jtree_ndx, hmm) are not required to return a potential. +</ul> + + + +<li> 1/22/02 +<ul> +<li> Added an optional argument to mk_bnet and mk_dbn which lets you +add names to nodes. This uses the new assoc_array class. + +<li> Added Yimin Zhang's (unfinished) classification/regression tree +code to CPDs/tree_CPD. + +</ul> + + + +<li> 1/14/02 +<ul> +<li> Incorporated some of Shan Huang's (still broken) stable CG code. +</ul> + + +<li> 1/9/02 +<ul> +<li> Yimin Zhang vectorized @discrete_CPD/prob_node, which speeds up +structure learning considerably. I fixed this to handle softmax CPDs. + +<li> Shan Huang changed the stable conditional Gaussian code to handle +vector-valued nodes, but it is buggy. + +<li> I vectorized @gaussian_CPD/update_ess for a special case. + +<li> Removed denom=min(1, ... Z) from gaussian_CPD/maximize_params +(added to cope with negative temperature for entropic prior), which +gives wrong results on mhmm1. +</ul> + + +<li> 1/7/02 + +<ul> +<li> Removed the 'xo' typo from mk_qmr_bnet. + +<li> convert_dbn_CPDs_to_tables has been vectorized; it is now +substantially faster to compute the conditional likelihood for long sequences. + +<li> Simplified constructors for tabular_CPD and gaussian_CPD, so they +now both only take the form CPD(bnet, i, ...) for named arguments - +the CPD('self', i, ...) format is gone. Modified mk_fgraph_given_ev +to use mk_isolated_tabular_CPD instead. + +<li> Added entropic prior to tabular and Gaussian nodes. +For tabular_CPD, changed name of arguments to the constructor to +distinguish Dirichlet and entropic priors. In particular, +tabular_CPD(bnet, i, 'prior', 2) is now +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_weight', 2). + +<li> Added deterministic annealing to learn_params_dbn_em for use with +entropic priors. The old format learn(engine, cases, max_iter) has +been replaced by learn(engine, cases, 'max_iter', max_iter). + +<li> Changed examples/dynamic/bat1 and kjaerulff1, since default +equivalence classes have changed from untied to tied. +</ul> + +<li> 12/30/01 +<ul> +<li> DBN default equivalence classes for slice 2 has changed, so that +now parameters are tied for nodes with 'equivalent' parents in slices +1 and 2 (e.g., observed leaf nodes). This essentially makes passing in +the eclass arguments redundant (hooray!). +</ul> + + +<li> 12/20/01 +<ul> +<li> <b>Released version 4</b>. +Version 4 is considered a major new release +since it is not completely backwards compatible with V3. +Observed nodes are now specified when the bnet/dbn is created, +not when the engine is created. This changes the interface to many of +the engines, making the code no longer backwards compatible. +Hence support for non-named optional arguments (BNT2 style) has also +been removed; hence mk_dbn etc. requires arguments to be passed by name. + +<li> Ilya Shpitser's C code for triangulation now compiles under +Windows as well as Unix, thanks to Wei Hu. + +<li> All the ndx engines have been combined, and now take an optional +argument specifying what kind of index to use. + +<li> learn_params_dbn_em is now more efficient: +@tabular_CPD/update_ess for nodes whose families +are hidden does not need need to call add_evidence_to_dmarginal, which +is slow. + +<li> Wei Hu fixed bug in jtree_ndxD, so now the matlab and C versions +both work. + +<li> dhmm_inf_engine replaces hmm_inf_engine, since the former can +handle any kind of topology and is slightly more efficient. dhmm is +extended to handle Gaussian, as well as discrete, +observed nodes. The new hmm_inf_engine no longer supports online +inference (which was broken anyway). + +<li> Added autoregressive HMM special case to hmm_inf_engine for +speed. + +<li> jtree_ndxSD_dbn_inf_engine now computes likelihood of the +evidence in a vectorized manner, where possible, just like +hmm_inf_engine. + +<li> Added mk_limid, and hence simplified mk_bnet and mk_dbn. + + +<li> Gaussian_CPD now uses 0.01*I prior on covariance matrix by +default. To do ML estimation, set 'cov_prior_weight' to 0. + +<li> Gaussian_CPD and tabular_CPD +optional binary arguments are now set using 0/1 rather no 'no'/'yes'. + +<li> Removed Shan Huang's PDAG and decomposable graph code, which will +be put in a separate structure learning library. +</ul> + + +<li> 12/11/01 +<ul> +<li> Wei Hu fixed jtree_ndx*_dbn_inf_engine and marg_table.c. + +<li> Shan Huang contributed his implementation of stable conditional +Gaussian code (Lauritzen 1999), and methods to search through the +space of PDAGs (Markov equivalent DAGs) and undirected decomposable +graphs. The latter is still under development. +</ul> + + +<li> 12/10/01 +<ul> +<li> Included Wei Hu's new versions of the ndx* routines, which use +integers instead of doubles. The new versions are about 5 times faster +in C. In general, ndxSD is the best choice. + +<li> Fixed misc/add_ev_to_dmarginal so it works with the ndx routines +in bat1. + +<li> Added calc_mpe_dbn to do Viterbi parsing. + +<li> Updated dhmm_inf_engine so it computes marginals. +</ul> + + + +<li> 11/23/01 +<ul> +<li> learn_params now does MAP estimation (i.e., uses Dirichlet prior, +if define). Thanks to Simon Keizer skeizer@cs.utwente.nl for spotting +this. +<li> Changed plotgraph so it calls ghostview with the output of dotty, +instead of converting from .ps to .tif. The resulting image is much +easier to read. +<li> Fixed cgpot/multiply_by_pots.m. +<li> Wei Hu fixed ind2subv.c. +<li> Changed arguments to compute_joint_pot. +</ul> + + +<li> 11/1/01 +<ul> +<li> Changed sparse to dense in @dpot/multiply_pots, because sparse +arrays apparently cause a bug in the NT version of Matlab. + +<li> Fixed the bug in gaussian_CPD/log_prob_node.m which +incorrectly called the vectorized gaussian_prob with different means +when there were continuous parents and more than one case. +(Thanks to Dave Andre for finding this.) + +<li> Fixed the bug in root_CPD/convert_to_pot which did not check for +pot_type='g'. +(Thanks to Dave Andre for finding this.) + +<li> Changed calc_mpe and calc_mpe_global so they now return a cell array. + +<li> Combine pearl and loopy_pearl into a single inference engine +called 'pearl_inf_engine', which now takes optional arguments passed +in using the name/value pair syntax. +marginal_nodes/family now takes the optional add_ev argument (same as +jtree), which is the opposite of the previous shrink argument. + +<li> Created pearl_unrolled_dbn_inf_engine and "resurrected" +pearl_dbn_inf_engine in a simplified (but still broken!) form. + +<li> Wei Hi fixed the bug in ind2subv.c, so now ndxSD works. +He also made C versions of ndxSD and ndxB, and added (the unfinished) ndxD. + +</ul> + + +<li> 10/20/01 + +<ul> +<li> Removed the use_ndx option from jtree_inf, +and created 2 new inference engines: jtree_ndxSD_inf_engine and +jtree_ndxB_inf_engine. +The former stores 2 sets of indices for the small and difference +domains; the latter stores 1 set of indices for the big domain. +In Matlab, the ndxB version is often significantly faster than ndxSD +and regular jree, except when the clique size is large. +When compiled to C, the difference between ndxB and ndxSD (in terms of +speed) vanishes; again, both are faster than compiled jtree, except +when the clique size is large. +Note: ndxSD currently has a bug in it, so it gives the wrong results! +(The DBN analogs are jtree_dbn_ndxSD_inf_engine and +jtree_dbn_ndxB_inf_engine.) + +<li> Removed duplicate files from the HMM and Kalman subdirectories. +e.g., normalise is now only in BNT/misc, so when compiled to C, it +masks the unique copy of the Matlab version. +</ul> + + + +<li> 10/17/01 +<ul> +<li> Fixed bugs introduced on 10/15: +Renamed extract_gaussian_CPD_params_given_ev_on_dps.m to +gaussian_CPD_params_given_dps.m since Matlab can't cope with such long +names (this caused cg1 to fail). Fixed bug in +gaussian_CPD/convert_to_pot, which now calls convert_to_table in the +discrete case. + +<li> Fixed bug in bk_inf_engine/marginal_nodes. +The test 'if nodes < ss' is now +'if nodes <= ss' (bug fix due to Stephen seg_ma@hotmail.com) + +<li> Simplified uninstallC. +</ul> + + +<li> 10/15/01 +<ul> + +<li> Added use_ndx option to jtree_inf and jtree_dbn_inf. +This pre-computes indices for multiplying, dividing and marginalizing +discrete potentials. +This is like the old jtree_fast_inf_engine, but we use an extra level +of indirection to reduce the number of indices needed (see +uid_generator object). +Sometimes this is faster than the original way... +This is work in progress. + +<li> The constructor for dpot no longer calls myreshape, which is very +slow. +But new dpots still must call myones. +Hence discrete potentials are only sometimes 1D vectors (but should +always be thought of as multi-D arrays). This is work in progress. +</ul> + + +<li> 10/6/01 +<ul> +<li> Fixed jtree_dbn_inf_engine, and added kjaerulff1 to test this. +<li> Added option to jtree_inf_engine/marginal_nodes to return "full +sized" marginals, even on observed nodes. +<li> Clustered BK in examples/dynamic/bat1 seems to be broken, +so it has been commented out. +BK will be re-implemented on top of jtree_dbn, which should much more +efficient. +</ul> + +<li> 9/25/01 +<ul> +<li> jtree_dbn_inf_engine is now more efficient than calling BK with +clusters = exact, since it only uses the interface nodes, instead of +all of them, to maintain the belief state. +<li> Uninstalled the broken C version of strong_elim_order. +<li> Changed order of arguments to unroll_dbn_topology, so that intra1 +is no longer required. +<li> Eliminated jtree_onepass, which can be simulated by calling +collect_evidence on jtree. +<li> online1 is no longer in the test_BNT suite, since there is some +problem with online prediction with mixtures of Gaussians using BK. +This functionality is no longer supported, since doing it properly is +too much work. +</ul> +</li> + +<li> 9/7/01 +<ul> +<li> Added Ilya Shpitser's C triangulation code (43x faster!). +Currently this only compiles under linux; windows support is being added. +</ul> + + +<li> 9/5/01 +<ul> +<li> Fixed typo in CPDs/@tabular_kernel/convert_to_table (thanks, +Philippe!) +<li> Fixed problems with clamping nodes in tabular_CPD, learn_params, +learn_params_tabular, and bayes_update_params. See +BNT/examples/static/learn1 for a demo. +</ul> + + +<li> 9/3/01 +<ul> +<li> Fixed typo on line 87 of gaussian_CPD which caused error in cg1.m +<li> Installed Wei Hu's latest version of jtree_C_inf_engine, which +can now compute marginals on any clique/cluster. +<li> Added Yair Weiss's code to compute the Bethe free energy +approximation to the log likelihood in loopy_pearl (still need to add +this to belprop). The return arguments are now: engine, loglik and +niter, which is different than before. +</ul> + + + +<li> 8/30/01 +<ul> +<li> Fixed bug in BNT/examples/static/id1 which passed hard-coded +directory name to belprop_inf_engine. + +<li> Changed tabular_CPD and gaussian_CPD so they can now be created +without having to pass in a bnet. + +<li> Finished mk_fgraph_given_ev. See the fg* files in examples/static +for demos of factor graphs (work in progress). +</ul> + + + +<li> 8/22/01 +<ul> + +<li> Removed jtree_compiled_inf_engine, +since the C code it generated was so big that it would barf on large +models. + +<li> Tidied up the potentials/Tables directory. +Removed mk_marg/mult_ndx.c, +which have been superceded by the much faster mk_marg/mult_index.c +(written by Wei Hu). +Renamed the Matlab versions mk_marginalise/multiply_table_ndx.m +to be mk_marg/mult_index.m to be compatible with the C versions. +Note: nobody calls these routines anymore! +(jtree_C_inf_engine/enter_softev.c has them built-in.) +Removed mk_ndx.c, which was only used by jtree_compiled. +Removed mk_cluster_clq_ndx.m, mk_CPD_clq_ndx, and marginalise_table.m +which were not used. +Moved shrink_obs_dims_in_table.m to misc. + +<li> In potentials/@dpot directory: removed multiply_by_pot_C_old.c. +Now marginalize_pot.c can handle maximization, +and divide_by_pot.c has been implmented. +marginalize/multiply/divide_by_pot.m no longer have useC or genops options. +(To get the C versions, use installC.m) + +<li> Removed useC and genops options from jtree_inf_engine.m +To use the C versions, install the C code. + +<li> Updated BNT/installC.m. + +<li> Added fclose to @loopy_pearl_inf/enter_evidence. + +<li> Changes to MPE routines in BNT/general. +The maximize parameter is now specified inside enter_evidence +instead of when the engine is created. +Renamed calc_mpe_given_inf_engine to just calc_mpe. +Added Ron Zohar's optional fix to handle the case of ties. +Now returns log-likelihood instead of likelihood. +Added calc_mpe_global. +Removed references to genops in calc_mpe_bucket.m +Test file is now called mpe1.m + +<li> For DBN inference, filter argument is now passed by name, +as is maximize. This is NOT BACKWARDS COMPATIBLE. + +<li> Removed @loopy_dbn_inf_engine, which will was too complicated. +In the future, a new version, which applies static loopy to the +unrolled DBN, will be provided. + +<li> discrete_CPD class now contains the family sizes and supports the +method dom_sizes. This is because it could not access the child field +CPD.sizes, and mysize(CPT) may give the wrong answer. + +<li> Removed all functions of the form CPD_to_xxx, where xxx = dpot, cpot, +cgpot, table, tables. These have been replaced by convert_to_pot, +which takes a pot_type argument. +@discrete_CPD calls convert_to_table to implement a default +convert_to_pot. +@discrete_CPD calls CPD_to_CPT to implement a default +convert_to_table. +The convert_to_xxx routines take fewer arguments (no need to pass in +the globals node_sizes and cnodes!). +Eventually, convert_to_xxx will be vectorized, so it will operate on +all nodes in the same equivalence class "simultaneously", which should +be significantly quicker, at least for Gaussians. + +<li> Changed discrete_CPD/sample_node and prob_node to use +convert_to_table, instead of CPD_to_CPT, so mlp/softmax nodes can +benefit. + +<li> Removed @tabular_CPD/compute_lambda_msg_fast and +private/prod_CPD_and_pi_msgs_fast, since no one called them. + +<li> Renamed compute_MLE to learn_params, +by analogy with bayes_update_params (also because it may compute an +MAP estimate). + +<li> Renamed set_params to set_fields +and get_params to get_field for CPD and dpot objects, to +avoid confusion with the parameters of the CPD. + +<li> Removed inference/doc, which has been superceded +by the web page. + +<li> Removed inference/static/@stab_cond_gauss_inf_engine, which is +broken, and all references to stable CG. + +</ul> + + + + + +<li> 8/12/01 +<ul> +<li> I removed potentials/@dpot/marginalize_pot_max. +Now marginalize_pot for all potential classes take an optional third +argument, specifying whether to sum out or max out. +The dpot class also takes in optional arguments specifying whether to +use C or genops (the global variable USE_GENOPS has been eliminated). + +<li> potentials/@dpot/marginalize_pot has been simplified by assuming +that 'onto' is always in ascending order (i.e., we remove +Maynard-Reid's patch). This is to keep the code identical to the C +version and the other class implementations. + +<li> Added Ron Zohar's general/calc_mpe_bucket function, +and my general/calc_mpe_given_inf_engine, for calculating the most +probable explanation. + + +<li> Added Wei Hu's jtree_C_inf_engine. +enter_softev.c is about 2 times faster than enter_soft_evidence.m. + +<li> Added the latest version of jtree_compiled_inf_engine by Wei Hu. +The 'C' ndx_method now calls potentials/Tables/mk_marg/mult_index, +and the 'oldC' ndx_method calls potentials/Tables/mk_marg/mult_ndx. + +<li> Added potentials/@dpot/marginalize_pot_C.c and +multiply_by_pot_C.c by Wei Hu. +These can be called by setting the 'useC' argument in +jtree_inf_engine. + +<li> Added BNT/installC.m to compile all the mex files. + +<li> Renamed prob_fully_instantiated_bnet to log_lik_complete. + +<li> Added Shan Huang's unfinished stable conditional Gaussian +inference routines. +</ul> + + + +<li> 7/13/01 +<ul> +<li> Added the latest version of jtree_compiled_inf_engine by Wei Hu. +<li> Added the genops class by Doug Schwarz (see +BNT/genopsfun/README). This provides a 1-2x speed-up of +potentials/@dpot/multiply_by_pot and divide_by_pot. +<li> The function BNT/examples/static/qmr_compiled compares the +performance gains of these new functions. +</ul> + +<li> 7/6/01 +<ul> +<li> Made bk_inf_engine use the name/value argument syntax. This can +now do max-product (Viterbi) as well as sum-product +(forward-backward). +<li> Changed examples/static/mfa1 to use the new name/value argument +syntax. +</ul> + + +<li> 6/28/01 + +<ul> + +<li> <b>Released version 3</b>. +Version 3 is considered a major new release +since it is not completely backwards compatible with V2. +V3 supports decision and utility nodes, loopy belief propagation on +general graphs (including undirected), structure learning for non-tabular nodes, +a simplified way of handling optional +arguments to functions, +and many other features which are described below. +In addition, the documentation has been substantially rewritten. + +<li> The following functions can now take optional arguments specified +as name/value pairs, instead of passing arguments in a fixed order: +mk_bnet, jtree_inf_engine, tabular_CPD, gaussian_CPD, softmax_CPD, mlp_CPD, +enter_evidence. +This is very helpful if you want to use default values for most parameters. +The functions remain backwards compatible with BNT2. + +<li> dsoftmax_CPD has been renamed softmax_CPD, replacing the older +version of softmax. The directory netlab2 has been updated, and +contains weighted versions of some of the learning routines in netlab. +(This code is still being developed by P. Brutti.) + +<li> The "fast" versions of the inference engines, which generated +matlab code, have been removed. +@jtree_compiled_inf_engine now generates C code. +(This feature is currently being developed by Wei Hu of Intel (China), +and is not yet ready for public use.) + +<li> CPD_to_dpot, CPD_to_cpot, CPD_to_cgpot and CPD_to_upot +are in the process of being replaced by convert_to_pot. + +<li> determine_pot_type now takes as arguments (bnet, onodes) +instead of (onodes, cnodes, dag), +so it can detect the presence of utility nodes as well as continuous +nodes. +Hence this function is not backwards compatible with BNT2. + +<li> The structure learning code (K2, mcmc) now works with any node +type, not just tabular. +mk_bnets_tabular has been eliminated. +bic_score_family and dirichlet_score_family will be replaced by score_family. +Note: learn_struct_mcmc has a new interface that is not backwards +compatible with BNT2. + +<li> update_params_complete has been renamed bayes_update_params. +Also, learn_params_tabular has been replaced by learn_params, which +works for any CPD type. + +<li> Added decision/utility nodes. +</ul> + + +<li> 6/6/01 +<ul> +<li> Added soft evidence to jtree_inf_engine. +<li> Changed the documentation slightly (added soft evidence and +parameter tying, and separated parameter and structure learning). +<li> Changed the parameters of determine_pot_type, so it no longer +needs to be passed a DAG argument. +<li> Fixed parameter tying in mk_bnet (num. CPDs now equals num. equiv +classes). +<li> Made learn_struct_mcmc work in matlab version 5.2 (thanks to +Nimrod Megiddo for finding this bug). +<li> Made 'acyclic.m' work for undirected graphs. +</ul> + + +<li> 5/23/01 +<ul> +<li> Added Tamar Kushnir's code for the IC* algorithm +(learn_struct_pdag_ic_star). This learns the +structure of a PDAG, and can identify the presence of latent +variables. + +<li> Added Yair Weiss's code for computing the MAP assignment using +junction tree (i.e., a new method called @dpot/marginalize_pot_max +instead of marginalize_pot.) + +<li> Added @discrete_CPD/prob_node in addition to log_prob_node to handle +deterministic CPDs. +</ul> + + +<li> 5/12/01 +<ul> +<li> Pierpaolo Brutti updated his mlp and dsoftmax CPD classes, +and improved the HME code. + +<li> HME example now added to web page. (The previous example was non-hierarchical.) + +<li> Philippe Leray (author of the French documentation for BNT) +pointed out that I was including netlab.tar unnecessarily. +</ul> + + +<li> 5/4/01 +<ul> +<li> Added mlp_CPD which defines a CPD as a (conditional) multi-layer perceptron. +This class was written by Pierpaolo Brutti. + +<li> Added hierarchical mixtures of experts demo (due to Pierpaolo Brutti). + +<li> Fixed some bugs in dsoftmax_CPD. + +<li> Now the BNT distribution includes the whole +<a href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a> library in a +subdirectory. +It also includes my HMM and Kalman filter toolboxes, instead of just +fragments of them. +</ul> + + +<li> 5/2/01 +<ul> +<li> gaussian_inf_engine/enter_evidence now correctly returns the +loglik, even if all nodes are instantiated (bug fix due to +Michael Robert James). + +<li> Added dsoftmax_CPD which allows softmax nodes to have discrete +and continuous parents; the discrete parents act as indices into the +parameters for the continuous node, by analogy with conditional +Gaussian nodes. This class was written by Pierpaolo Brutti. +</ul> + + +<li> 3/27/01 +<ul> +<li> learn_struct_mcmc no longer returns sampled_bitv. +<li> Added mcmc_sample_to_hist to post-process the set of samples. +</ul> + +<li> 3/21/01 +<ul> +<li> Changed license from UC to GNU Library GPL. + +<li> Made all CPD constructors accept 0 arguments, so now bnets can be +saved to and loaded from files. + +<li> Improved the implementation of sequential and batch Bayesian +parameter learning for tabular CPDs with completely observed data (see +log_marg_lik_complete and update_params_complete). This code also +handles interventional data. + +<li> Added MCMC structure learning for completely observed, discrete, +static BNs. + +<li> Started implementing Bayesian estimation of linear Gaussian +nodes. See root_gaussian_CPD and +linear_gaussian_CPD. The old gaussian_CPD class has not been changed. + +<li> Renamed evaluate_CPD to log_prob_node, and simplified its +arguments. + +<li> Renamed sample_CPD to sample_node, simplified its +arguments, and vectorized it. + +<li> Renamed "learn_params_tabular" to "update_params_complete". +This does Bayesian updating, but no longer computes the BIC score. + +<li> Made routines for completely observed networks (sampling, +complete data likelihood, etc.) handle cell arrays or regular arrays, +which are faster. +If some nodes are not scalars, or are hidden, you must use cell arrays. +You must convert to a cell array before passing to an inference routine. + +<li> Fixed bug in gaussian_CPD constructor. When creating CPD with +more than 1 discrete parent with random parameters, the matrices were +the wrong shape (Bug fix due to Xuejing Sun). +</ul> + + + +<li> 11/24/00 +<ul> +<li> Renamed learn_params and learn_params_dbn to learn_params_em/ +learn_params_dbn_em. The return arguments are now [bnet, LLtrace, +engine] instead of [engine, LLtrace]. +<li> Added structure learning code for static nets (K2, PC). +<li> Renamed learn_struct_inter_full_obs as learn_struct_dbn_reveal, +and reimplemented it to make it simpler and faster. +<li> Added sequential Bayesian parameter learning (learn_params_tabular). +<li> Major rewrite of the documentation. +</ul> + +<!-- +<li> 6/1/00 +<ul> +<li> Subtracted 1911 off the counter, so now it counts hits from +5/22/00. (The initial value of 1911 was a conservative lower bound on the number of +hits from the time the page was created.) +</ul> +--> + +<li> 5/22/00 +<ul> +<li> Added online filtering and prediction. +<li> Added the factored frontier and loopy_dbn algorithms. +<li> Separated the online user manual into two, for static and dynamic +networks. +<!-- +<li> Added a counter to the BNT web page, and initialized it to 1911, +which is the number of people who have downloaded my software (BNT and +other toolboxes) since 8/24/98. +--> +<li> Added a counter to the BNT web page. +<!-- +Up to this point, 1911 people had downloaded my software (BNT and +other toolboxes) since 8/24/98. +--> +</ul> + + +<li> 4/27/00 +<ul> +<li> Fixed the typo in bat1.m +<li> Added preliminary code for online inference in DBNs +<li> Added coupled HMM example +</ul> + +<li> 4/23/00 +<ul> +<li> Fixed the bug in the fast inference routines where the indices +are empty (arises in bat1.m). +<li> Sped up marginal_family for the fast engines by precomputing indices. +</ul> + +<li> 4/17/00 +<ul> +<li> Simplified implementation of BK_inf_engine by using soft evidence. +<li> Added jtree_onepass_inf_engine (which computes a single marginal) +and modified jtree_dbn_fast to use it. +</ul> + +<li> 4/14/00 +<ul> +<li> Added fast versions of jtree and BK, which are +designed for models where the division into hidden/observed is fixed, +and all hidden variables are discrete. These routines are 2-3 times +faster than their non-fast counterparts. + +<li> Added graph drawing code +contributed by Ali Taylan Cemgil from the University of Nijmegen. +</ul> + +<li> 4/10/00 +<ul> +<li> Distinguished cnodes and cnodes_slice in DBNs so that kalman1 +works with BK. +<li> Removed dependence on cellfun (which only exists in matlab 5.3) +by adding isemptycell. Now the code works in 5.2. +<li> Changed the UC copyright notice. +</ul> + + + +<li> 3/29/00 +<ul> +<li><b>Released BNT 2.0</b>, now with objects! +Here are the major changes. + +<li> There are now 3 classes of objects in BNT: +Conditional Probability Distributions, potentials (for junction tree), +and inference engines. +Making an inference algorithm (junction tree, sampling, loopy belief +propagation, etc.) an object might seem counter-intuitive, but in +fact turns out to be a good idea, since the code and documentation +can be made modular. +(In Java, each algorithm would be a class that implements the +inferenceEngine interface. Since Matlab doesn't support interfaces, +inferenceEngine is an abstract (virtual) base class.) + +<p> +<li> +In version 1, instead of Matlab's built-in objects, +I used structs and a + simulated dispatch mechanism based on the type-tag system in the + classic textbook by Abelson + and Sussman ("Structure and Interpretation of Computer Programs", + MIT Press, 1985). This required editing the dispatcher every time a + new object type was added. It also required unique (and hence long) + names for each method, and allowed the user unrestricted access to + the internal state of objects. + +<p> +<li> The Bayes net itself is now a lightweight struct, and can be +used to specify a model independently of the inference algorithm used +to process it. +In version 1, the inference engine was stored inside the Bayes net. + +<!-- +See the list of <a href="differences2.html">changes from version +1</a>. +--> +</ul> + + + +<li> 11/24/99 +<ul> +<li> Added fixed lag smoothing, online EM and the ability to learn +switching HMMs (POMDPs) to the HMM toolbox. +<li> Renamed the HMM toolbox function 'mk_dhmm_obs_mat' to +'mk_dhmm_obs_lik', and similarly for ghmm and mhmm. Updated references +to these functions in BNT. +<li> Changed the order of return params from kalman_filter to make it +more natural. Updated references to this function in BNT. +</ul> + + + +<li>10/27/99 +<ul> +<li>Fixed line 42 of potential/cg/marginalize_cgpot and lines 32-39 of bnet/add_evidence_to_marginal +(thanks to Rainer Deventer for spotting these bugs!) +</ul> + + +<li>10/21/99 +<ul> +<li>Completely changed the blockmatrix class to make its semantics +more sensible. The constructor is not backwards compatible! +</ul> + +<li>10/6/99 +<ul> +<li>Fixed all_vals = cat(1, vals{:}) in user/enter_evidence +<li>Vectorized ind2subv and sub2indv and removed the C versions. +<li>Made mk_CPT_from_mux_node much faster by having it call vectorized +ind2subv +<li>Added Sondhauss's bug fix to line 68 of bnet/add_evidence_to_marginal +<li>In dbn/update_belief_state, instead of adding eps to likelihood if 0, +we leave it at 0, and set the scale factor to 0 instead of dividing. +</ul> + +<li>8/19/99 +<ul> +<li>Added Ghahramani's mfa code to examples directory to compare with +fa1, which uses BNT +<li>Changed all references of assoc to stringmatch (e.g., in +examples/mk_bat_topology) +</ul> + +<li>June 1999 +<ul> +<li><b>Released BNT 1.0</b> on the web. +</ul> + + +<li>August 1998 +<ul> +<li><b>Released BNT 0.0</b> via email. +</ul> + + +<li>October 1997 +<ul> +<li>First started working on Matlab version of BNT. +</ul> + +<li>Summer 1997 +<ul> +<li> First started working on C++ version of BNT while working at DEC (now Compaq) CRL. +</ul> + +<!-- +<li>Fall 1996 +<ul> +<li>Made a C++ program that generates DBN-specific C++ code +for inference using the frontier algorithm. +</ul> + +<li>Fall 1995 +<ul> +<li>Arrive in Berkeley, and first learn about Bayes Nets. Start using +Geoff Zweig's C++ code. +</ul> +--> + +</ul> diff --git a/sourcecodes/bnt-master/docs/dbn_hmm_demo.m b/sourcecodes/bnt-master/docs/dbn_hmm_demo.m new file mode 100644 index 00000000..30933b8f --- /dev/null +++ b/sourcecodes/bnt-master/docs/dbn_hmm_demo.m @@ -0,0 +1,37 @@ + Example due to Wang Hee Lin" <engp1622@nus.edu.sg + + +intra = zeros(2); +intra(1,2) = 1; +inter = zeros(2); +inter(1,1) = 1; + +Q = 2; % num hidden states +O = 2; % num observable symbols +ns = [Q O];%number of states +dnodes = 1:2; +%onodes = [1:2]; % only possible with jtree, not hmm +onodes = [2]; +bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes, 'observed', onodes); +for i=1:4 + bnet.CPD{i} = tabular_CPD(bnet, i); +end + +prior0 = normalise(rand(Q,1)); +transmat0 = mk_stochastic(rand(Q,Q)); +obsmat0 = mk_stochastic(rand(Q,O)); + +%engine = smoother_engine(hmm_2TBN_inf_engine(bnet)); +engine = smoother_engine(jtree_2TBN_inf_engine(bnet)); + +ss = 2;%slice size(ss) +ncases = 10;%number of examples +T=10; +max_iter=2;%iterations for EM +cases = cell(1, ncases); +for i=1:ncases + ev = sample_dbn(bnet, T); + cases{i} = cell(ss,T); + cases{i}(onodes,:) = ev(onodes, :); +end +[bnet2, LLtrace] = learn_params_dbn_em(engine, cases, 'max_iter', 4); diff --git a/sourcecodes/bnt-master/docs/gr03.pdf b/sourcecodes/bnt-master/docs/gr03.pdf new file mode 100644 index 00000000..8ad8cf49 --- /dev/null +++ b/sourcecodes/bnt-master/docs/gr03.pdf Binary files differdiff --git a/sourcecodes/bnt-master/docs/graph_to_dot.m b/sourcecodes/bnt-master/docs/graph_to_dot.m new file mode 100644 index 00000000..c9f692d1 --- /dev/null +++ b/sourcecodes/bnt-master/docs/graph_to_dot.m @@ -0,0 +1,86 @@ +function graph_to_dot(adj, varargin) +%GRAPH_TO_DOT Makes a GraphViz (AT&T) file representing an adjacency matrix +% graph_to_dot(adj, ...) writes to the specified filename. +% +% Optional arguments can be passed as name/value pairs: [default] +% +% 'filename' - if omitted, writes to 'tmp.dot' +% 'arc_label' - arc_label{i,j} is a string attached to the i-j arc [""] +% 'node_label' - node_label{i} is a string attached to the node i ["i"] +% 'width' - width in inches [10] +% 'height' - height in inches [10] +% 'leftright' - 1 means layout left-to-right, 0 means top-to-bottom [0] +% 'directed' - 1 means use directed arcs, 0 means undirected [1] +% +% For details on graphviz, See http://www.research.att.com/sw/tools/graphviz +% +% See also dot_to_graph and draw_dot. + +% First version written by Kevin Murphy 2002. +% Modified by Leon Peshkin, Jan 2004. +% Bugfix by Tom Minka, Mar 2004. + +node_label = []; arc_label = []; % set default args +width = 10; height = 10; +leftright = 0; directed = 1; filename = 'tmp.dot'; + +for i = 1:2:nargin-1 % get optional args + switch varargin{i} + case 'filename', filename = varargin{i+1}; + case 'node_label', node_label = varargin{i+1}; + case 'arc_label', arc_label = varargin{i+1}; + case 'width', width = varargin{i+1}; + case 'height', height = varargin{i+1}; + case 'leftright', leftright = varargin{i+1}; + case 'directed', directed = varargin{i+1}; + end +end +% minka +if ~directed + adj = triu(adj | adj'); +end + +fid = fopen(filename, 'w'); +if directed + fprintf(fid, 'digraph G {\n'); + arctxt = '->'; + if isempty(arc_label) + labeltxt = ''; + else + labeltxt = '[label="%s"]'; + end +else + fprintf(fid, 'graph G {\n'); + arctxt = '--'; + if isempty(arc_label) + labeltxt = '[dir=none]'; + else + labeltext = '[label="%s",dir=none]'; + end +end +edgeformat = strcat(['%d ',arctxt,' %d ',labeltxt,';\n']); +fprintf(fid, 'center = 1;\n'); +fprintf(fid, 'size=\"%d,%d\";\n', width, height); +if leftright + fprintf(fid, 'rankdir=LR;\n'); +end +Nnds = length(adj); +for node = 1:Nnds % process nodes + if isempty(node_label) + fprintf(fid, '%d;\n', node); + else + fprintf(fid, '%d [ label = "%s" ];\n', node, node_label{node}); + end +end +for node1 = 1:Nnds % process edges + arcs = find(adj(node1,:)); % children(adj, node); + for node2 = arcs + if ~isempty(arc_label) + fprintf(fid, edgeformat,node1,node2,arc_label{node1,node2}); + else + fprintf(fid, edgeformat, node1, node2); + end + end +end +fprintf(fid, '}'); +fclose(fid); diff --git a/sourcecodes/bnt-master/docs/graphviz.html b/sourcecodes/bnt-master/docs/graphviz.html new file mode 100644 index 00000000..5ad3e777 --- /dev/null +++ b/sourcecodes/bnt-master/docs/graphviz.html @@ -0,0 +1,66 @@ +<h1>Visualizing graph structures in matlab</h1> + +We discuss some methods for visualizing graphs/ networks, including automatic +layout of the nodes. +We assume the graph is represented as an adjacency matrix. +If using BNT, you can access the DAG using +<pre> +G = bnet.dag; +</pre> + +<h2>Matlab's biograph function</h2> + +The Mathworks computational biology toolbox +has many useful graph related functions, including visualization. +<br> +Click +<a href="http://www.mathworks.com/products/bioinfo/demos.html?file=/products/demos/shipping/bioinfo/graphtheorydemo.html#4"> +here</a> +for a demo. + + + +<h2>Cemgil's draw_graph</h2> + +You can visualize an arbitrary graph (such as one learned using the +structure learning routines) with Matlab code written by +<a href="http://www-sigproc.eng.cam.ac.uk/~atc27/matlab/layout.html"> +Ali Taylan Cemgil</a> +from the University of Cambridge. +A modified version of this code +is <a href="GraphViz.zip">here</a> +(this is already bundled with BNT). +Just type +<pre> +draw_graph(G); +</pre> +For example, this is the output produced on a +<a href="http://www.cs.ubc.ca/~murphyk/Software/BNT/usage.html#qmr">random QMR-like model</a>: +<p> +<img src="Figures/qmr.rnd.jpg"> +<p> + +<h2>Pajek</h2> + +<a href="http://vlado.fmf.uni-lj.si/pub/networks/pajek">Pajek</a> +is an excellent, free Windows program for graph layout. +Use <a href="adj2pajek2.m">adj2pajek2.m</a> to convert a graph to the +Pajek file format. +<br> +Then Choose File->Network->Read from the menu. + +<h2>AT&T Graphviz</h2> + +<a href="http://www.research.att.com/sw/tools/graphviz">graphhviz</a> +is an +open-source graph visualization package from AT&T. +Use +<a href="graph_to_dot.m">graph_to_dot</a> +to convert an adjacency matrix to +the AT&T file format (the "dot" format). +You then use dot to convert it to postscript: +<pre> +graph_to_dot(G, 'filename', 'foo.dot'); +dot -Tps foo.dot -o foo.ps +ghostview foo.ps & +</pre> diff --git a/sourcecodes/bnt-master/docs/install.html b/sourcecodes/bnt-master/docs/install.html new file mode 100644 index 00000000..989dd8ff --- /dev/null +++ b/sourcecodes/bnt-master/docs/install.html @@ -0,0 +1,32 @@ +To install, just unzip FullBNT.zip, start Matlab, and proceed as +follows +<pre> +>> cd C:\kmurphy\FullBNT\FullBNT-1.0.4 +>> addpath(genpathKPM(pwd)) + +Warning: Function C:\kmurphy\FullBNT\FullBNT-1.0.4\KPMtools\isvector.m has the same name as a MATLAB builtin. We suggest you rename the function to avoid a potential name conflict. +> In path at 110 + In addpath at 89 +Warning: Function C:\kmurphy\FullBNT\FullBNT-1.0.4\KPMtools\isscalar.m has the same name as a MATLAB builtin. We suggest you rename the function to avoid a potential name conflict. +> In path at 110 + In addpath at 89 +Warning: Function C:\kmurphy\FullBNT\FullBNT-1.0.4\KPMtools\assert.m has the same name as a MATLAB builtin. We suggest you rename the function to avoid a potential name conflict. +> In path at 110 + In addpath at 89 + +>> test_BNT +</pre> +The genpathKPM function is like the builtin genpath function, but it +does not add directories called 'Old' to the path, thus preventing old +versions of functions accidently shadowing new ones. +The warnings occur because Matlab 7 added functions with the same +names as my functions. The BNT versions will shadow the built-in ones, +but this should be harmless. +<p> +Note: the functions installC_BNT etc. are not needed anymore: all C +code has either been removed or is unnecessary. +<p> Note: as mentioned above, with new versions of matlab you +will get lots of warning messages, but everything should still work. +<p> Note: Cory Reith tells me that, as of +15 Sep 2009, octave 3.2.2 can run most of BNT. +Please send email to crieth@ucsd.edu if you have questions. diff --git a/sourcecodes/bnt-master/docs/join.gif b/sourcecodes/bnt-master/docs/join.gif new file mode 100644 index 00000000..04692115 --- /dev/null +++ b/sourcecodes/bnt-master/docs/join.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/license.gpl b/sourcecodes/bnt-master/docs/license.gpl new file mode 100644 index 00000000..6a2d5712 --- /dev/null +++ b/sourcecodes/bnt-master/docs/license.gpl @@ -0,0 +1,450 @@ +This library is free software; you can redistribute it and/or +modify it under the terms of the GNU Library General Public +License version 2 as published by the Free Software Foundation. + +This library is distributed in the hope that it will be useful, +but WITHOUT ANY WARRANTY; without even the implied warranty of +MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. + +GNU Library General Public License + +---------------------------------------------------------------------------- + +Table of Contents + + * GNU LIBRARY GENERAL PUBLIC LICENSE + o Preamble + o TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION + +---------------------------------------------------------------------------- + +GNU LIBRARY GENERAL PUBLIC LICENSE + +Version 2, June 1991 + +Copyright (C) 1991 Free Software Foundation, Inc. +675 Mass Ave, Cambridge, MA 02139, USA +Everyone is permitted to copy and distribute verbatim copies +of this license document, but changing it is not allowed. + +[This is the first released version of the library GPL. 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FOR A PARTICULAR PURPOSE. + + diff --git a/sourcecodes/bnt-master/docs/majorFeatures.html b/sourcecodes/bnt-master/docs/majorFeatures.html new file mode 100644 index 00000000..11005a85 --- /dev/null +++ b/sourcecodes/bnt-master/docs/majorFeatures.html @@ -0,0 +1,113 @@ + +<h2><a name="features">Major features</h2> +<ul> + +<li> BNT supports many types of +<b>conditional probability distributions</b> (nodes), +and it is easy to add more. +<ul> +<li>Tabular (multinomial) +<li>Gaussian +<li>Softmax (logistic/ sigmoid) +<li>Multi-layer perceptron (neural network) +<li>Noisy-or +<li>Deterministic +</ul> +<p> + +<li> BNT supports <b>decision and utility nodes</b>, as well as chance +nodes, +i.e., influence diagrams as well as Bayes nets. +<p> + +<li> BNT supports static and dynamic BNs (useful for modelling dynamical systems +and sequence data). +<p> + +<li> BNT supports many different <b>inference algorithms</b>, +and it is easy to add more. + +<ul> +<li> Exact inference for static BNs: +<ul> +<li>junction tree +<li>variable elimination +<li>brute force enumeration (for discrete nets) +<li>linear algebra (for Gaussian nets) +<li>Pearl's algorithm (for polytrees) +<li>quickscore (for QMR) +</ul> + +<p> +<li> Approximate inference for static BNs: +<ul> +<li>likelihood weighting +<li> Gibbs sampling +<li>loopy belief propagation +</ul> + +<p> +<li> Exact inference for DBNs: +<ul> +<li>junction tree +<li>frontier algorithm +<li>forwards-backwards (for HMMs) +<li>Kalman-RTS (for LDSs) +</ul> + +<p> +<li> Approximate inference for DBNs: +<ul> +<li>Boyen-Koller +<li>factored-frontier/loopy belief propagation +</ul> + +</ul> +<p> + +<li> +BNT supports several methods for <b>parameter learning</b>, +and it is easy to add more. +<ul> + +<li> Batch MLE/MAP parameter learning using EM. +(Each node type has its own M method, e.g. softmax nodes use IRLS,<br> +and each inference engine has its own E method, so the code is fully modular.) + +<li> Sequential/batch Bayesian parameter learning (for fully observed tabular nodes only). +</ul> + + +<p> +<li> +BNT supports several methods for <b>regularization</b>, +and it is easy to add more. +<ul> +<li> Any node can have its parameters clamped (made non-adjustable). +<li> Any set of compatible nodes can have their parameters tied (c.f., +weight sharing in a neural net). +<li> Some node types (e.g., tabular) supports priors for MAP estimation. +<li> Gaussian covariance matrices can be declared full or diagonal, and can +be tied across states of their discrete parents (if any). +</ul> + +<p> +<li> +BNT supports several methods for <b>structure learning</b>, +and it is easy to add more. +<ul> + +<li> Bayesian structure learning, +using MCMC or local search (for fully observed tabular nodes only). + +<li> Constraint-based structure learning (IC/PC and IC*/FCI). +</ul> + + +<p> +<li> The source code is extensively documented, object-oriented, and free, making it +an excellent tool for teaching, research and rapid prototyping. + +</ul> + + \ No newline at end of file diff --git a/sourcecodes/bnt-master/docs/mathbymatlab.gif b/sourcecodes/bnt-master/docs/mathbymatlab.gif new file mode 100644 index 00000000..0de8d7a0 --- /dev/null +++ b/sourcecodes/bnt-master/docs/mathbymatlab.gif Binary files differdiff --git a/sourcecodes/bnt-master/docs/matlab_comparison.html b/sourcecodes/bnt-master/docs/matlab_comparison.html new file mode 100644 index 00000000..bdf75aae --- /dev/null +++ b/sourcecodes/bnt-master/docs/matlab_comparison.html @@ -0,0 +1,62 @@ +<html> <head> +<title>Comparison of Matlab, R/S/Splus, Gauss, etc.</title> +</head> + +<body> +<!--<body bgcolor="#FFFFFF"> --> + +<h1>Comparison of Matlab, R/S/Splus, Gauss, etc.</h1> + +<ul> +<li> <a href="http://www.scientificweb.com/ncrunch/">Comparison of +mathematical programs for data analysis</a>, +Stefan Steinhaus, tech report, 2000. +<br> +This is a very detailed comparison of features and speed of several +interactive scientific programming environments, e.g. Matlab, +Mathematica, Splus. + + +<!-- +<p> +<li> <a href="Papers/gauss.econ.review.ps">Econometric +programming environments: Gauss, Ox and S-PLUS</a>, +Francisco Cribari-Neto. +J. of Applied Econometrics, 12(1):77-89, 1997 +<br> +Ox can not be used interactively, and has a C-style syntax (it even +requires users to pre-declare variables!). Its only advantage is speed. +S-Plus has tons of features and good documentation, but is slow. +Gauss is somewhere in between. + + +<p> +<li> <a href="Papers/matlab.econ.review.ps">MATLAB as an econometric +programming environment</a>, +Francisco Cribari-Neto and Mark J. Jensen. +J. of Applied Econometrics, 12(6):735-432, 1997. +<br> +The basic conclusion is that Matlab has excellent graphics and +sparse-matrix facilities, but is slower than Gauss/Ox (especially on +code +with loops), and has few statistical routines built-in (one must buy the +stats toolbox). + + + +<p> +<li> <a href="Papers/R.econ.review.ps">R: Yet another econometric +programming environment</a>, +Francisco Cribari-Neto and S. Zarkos. +J. of Applied Econometrics, 14(3):319-329, 1999. +<br> +The basic conclusion is that R is much faster than Splus on +code with loops, but a little bit slower on vectorized code. (Gauss/ +Ox is much faster than both; in my experience, R and Matlab have about +the same speed.) +However, R has much better memory management than Splus, and R is free. Otherwise, R/S/Splus +are essentially the same. +--> + + +</ul> diff --git a/sourcecodes/bnt-master/docs/numDAGsEqn.png b/sourcecodes/bnt-master/docs/numDAGsEqn.png new file mode 100644 index 00000000..3524947b --- /dev/null +++ b/sourcecodes/bnt-master/docs/numDAGsEqn.png Binary files differdiff --git a/sourcecodes/bnt-master/docs/numDAGsEqn2.png b/sourcecodes/bnt-master/docs/numDAGsEqn2.png new file mode 100644 index 00000000..0898c969 --- /dev/null +++ b/sourcecodes/bnt-master/docs/numDAGsEqn2.png Binary files differdiff --git a/sourcecodes/bnt-master/docs/param_tieing.html b/sourcecodes/bnt-master/docs/param_tieing.html new file mode 100644 index 00000000..b426fe30 --- /dev/null +++ b/sourcecodes/bnt-master/docs/param_tieing.html @@ -0,0 +1,47 @@ + +<h2>Example of more complex parameter tieing</h2> + +We now give a more complex pattern of parameter tieing. +(This example is due to Rainer Deventer.) +The structure is as follows: +<p> +<img src="Figures/rainer_tied.gif" height="400"> +<!--<img src="rainer_dbn.jpg" height="600">--> +<p> +Since nodes 2 and 3 in slice 2 (N7 and N8) +have different parents than their counterparts in slice 1 (N2 and N3), +they must be put into different equivalence classes. +Hence we define +<pre> +eclass1 = [1 2 3 4 5]; +eclass2 = [1 6 7 4 5]; +</pre> +The dotted bubbles represent the equivalence classes. +Node 7 is the representative node for equivalence class +6, and node 8 is the rep. for class 7, so we need to write +<pre> +bnet.CPD{6} = xxx_CPD(bnet, 7, xxx); +bnet.CPD{7} = xxx_CPD(bnet, 8, xxx); +</pre> +In general, you can use the following code fragment: +<pre> +eclass = bnet.equiv_class(:); +for e=1:max(eclass) + i = bnet.rep_of_eclass(e); + bnet.CPD{e} = xxx_CPD(bnet,i); +end +</pre> +<!-- +which is equivalent to +<pre> +E = max(eclass); +rep = zeros(1,E); +for e=1:E + mems = find(eclass==e); + rep(e) = mems(1); +end +for e=1:E + bnet.CPD{e} = xxx_CPD(bnet, rep(e)); +end +</pre> +--> diff --git a/sourcecodes/bnt-master/docs/supportedModels.html b/sourcecodes/bnt-master/docs/supportedModels.html new file mode 100644 index 00000000..af29d18e --- /dev/null +++ b/sourcecodes/bnt-master/docs/supportedModels.html @@ -0,0 +1,32 @@ + <h2><a name="models">Supported probabilistic models</h2> +<p> +It is trivial to implement all of +the following probabilistic models using the toolbox. +<ul> +<li>Static +<ul> +<li> Linear regression, logistic regression, hierarchical mixtures of experts + +<li> Naive Bayes classifiers, mixtures of Gaussians, +sigmoid belief nets + +<li> Factor analysis, probabilistic +PCA, probabilistic ICA, mixtures of these models + +</ul> + +<li>Dynamic +<ul> + +<li> HMMs, Factorial HMMs, coupled HMMs, input-output HMMs, DBNs + +<li> Kalman filters, ARMAX models, switching Kalman filters, +tree-structured Kalman filters, multiscale AR models + +</ul> + +<li> Many other combinations, for which there are (as yet) no names! + +</ul> + + \ No newline at end of file diff --git a/sourcecodes/bnt-master/docs/usage.html b/sourcecodes/bnt-master/docs/usage.html new file mode 100644 index 00000000..018301e3 --- /dev/null +++ b/sourcecodes/bnt-master/docs/usage.html @@ -0,0 +1,3072 @@ +<HEAD> +<TITLE>How to use the Bayes Net Toolbox</TITLE> +</HEAD> + +<BODY BGCOLOR="#FFFFFF"> +<!-- white background is better for the pictures and equations --> + +<h1>How to use the Bayes Net Toolbox</h1> + +This documentation was last updated on 29 October 2007. +<br> +Click +<a href="http://bnt.insa-rouen.fr/">here</a> +for a French version of this documentation (last updated in 2005). +<p> + +<ul> +<li> <a href="install.html">Installation</a> + +<li> <a href="#basics">Creating your first Bayes net</a> + <ul> + <li> <a href="#basics">Creating a model by hand</a> + <li> <a href="#file">Loading a model from a file</a> + <li> <a href="#GUI">Creating a model using a GUI</a> + <li> <a href="graphviz.html">Graph visualization</a> + </ul> + +<li> <a href="#inference">Inference</a> + <ul> + <li> <a href="#marginal">Computing marginal distributions</a> + <li> <a href="#joint">Computing joint distributions</a> + <li> <a href="#soft">Soft/virtual evidence</a> + <li> <a href="#mpe">Most probable explanation</a> + </ul> + +<li> <a href="#cpd">Conditional Probability Distributions</a> + <ul> + <li> <a href="#tabular">Tabular (multinomial) nodes</a> + <li> <a href="#noisyor">Noisy-or nodes</a> + <li> <a href="#deterministic">Other (noisy) deterministic nodes</a> + <li> <a href="#softmax">Softmax (multinomial logit) nodes</a> + <li> <a href="#mlp">Neural network nodes</a> + <li> <a href="#root">Root nodes</a> + <li> <a href="#gaussian">Gaussian nodes</a> + <li> <a href="#glm">Generalized linear model nodes</a> + <li> <a href="#dtree">Classification/regression tree nodes</a> + <li> <a href="#nongauss">Other continuous distributions</a> + <li> <a href="#cpd_summary">Summary of CPD types</a> + </ul> + +<li> <a href="#examples">Example models</a> + <ul> + <li> <a + href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt"> +Gaussian mixture models</a> + <li> <a href="#pca">PCA, ICA, and all that</a> + <li> <a href="#mixep">Mixtures of experts</a> + <li> <a href="#hme">Hierarchical mixtures of experts</a> + <li> <a href="#qmr">QMR</a> + <li> <a href="#cg_model">Conditional Gaussian models</a> + <li> <a href="#hybrid">Other hybrid models</a> + </ul> + +<li> <a href="#param_learning">Parameter learning</a> + <ul> + <li> <a href="#load_data">Loading data from a file</a> + <li> <a href="#mle_complete">Maximum likelihood parameter estimation from complete data</a> + <li> <a href="#prior">Parameter priors</a> + <li> <a href="#bayes_learn">(Sequential) Bayesian parameter updating from complete data</a> + <li> <a href="#em">Maximum likelihood parameter estimation with missing values (EM)</a> + <li> <a href="#tying">Parameter tying</a> + </ul> + +<li> <a href="#structure_learning">Structure learning</a> + <ul> + <li> <a href="#enumerate">Exhaustive search</a> + <li> <a href="#K2">K2</a> + <li> <a href="#hill_climb">Hill-climbing</a> + <li> <a href="#mcmc">MCMC</a> + <li> <a href="#active">Active learning</a> + <li> <a href="#struct_em">Structural EM</a> + <li> <a href="#graphdraw">Visualizing the learned graph structure</a> + <li> <a href="#constraint">Constraint-based methods</a> + </ul> + + +<li> <a href="#engines">Inference engines</a> + <ul> + <li> <a href="#jtree">Junction tree</a> + <li> <a href="#varelim">Variable elimination</a> + <li> <a href="#global">Global inference methods</a> + <li> <a href="#quickscore">Quickscore</a> + <li> <a href="#belprop">Belief propagation</a> + <li> <a href="#sampling">Sampling (Monte Carlo)</a> + <li> <a href="#engine_summary">Summary of inference engines</a> + </ul> + + +<li> <a href="#influence">Influence diagrams/ decision making</a> + + +<li> <a href="usage_dbn.html">DBNs, HMMs, Kalman filters and all that</a> +</ul> + +</ul> + + + +<h1><a name="basics">Creating your first Bayes net</h1> + +To define a Bayes net, you must specify the graph structure and then +the parameters. We look at each in turn, using a simple example +(adapted from Russell and +Norvig, "Artificial Intelligence: a Modern Approach", Prentice Hall, +1995, p454). + + +<h2>Graph structure</h2> + + +Consider the following network. + +<p> +<center> +<IMG SRC="Figures/sprinkler.gif"> +</center> +<p> + +<P> +To specify this directed acyclic graph (dag), we create an adjacency matrix: +<PRE> +N = 4; +dag = zeros(N,N); +C = 1; S = 2; R = 3; W = 4; +dag(C,[R S]) = 1; +dag(R,W) = 1; +dag(S,W)=1; +</PRE> +<P> +We have numbered the nodes as follows: +Cloudy = 1, Sprinkler = 2, Rain = 3, WetGrass = 4. +<b>The nodes must always be numbered in topological order, i.e., +ancestors before descendants.</b> +For a more complicated graph, this is a little inconvenient: we will +see how to get around this <a href="usage_dbn.html#bat">below</a>. +<p> +In Matlab 6, you can use logical arrays instead of double arrays, +which are 4 times smaller: +<pre> +dag = false(N,N); +dag(C,[R S]) = true; +... +</pre> +However, <b>some graph functions (eg acyclic) do not work on +logical arrays</b>! +<p> +You can visualize the resulting graph structure using +the methods discussed <a href="#graphdraw">below</a>. +For details on GUIs, +click <a href="#GUI">here</a>. + +<h2>Creating the Bayes net shell</h2> + +In addition to specifying the graph structure, +we must specify the size and type of each node. +If a node is discrete, its size is the +number of possible values +each node can take on; if a node is continuous, +it can be a vector, and its size is the length of this vector. +In this case, we will assume all nodes are discrete and binary. +<PRE> +discrete_nodes = 1:N; +node_sizes = 2*ones(1,N); +</pre> +If the nodes were not binary, you could type e.g., +<pre> +node_sizes = [4 2 3 5]; +</pre> +meaning that Cloudy has 4 possible values, +Sprinkler has 2 possible values, etc. +Note that these are cardinal values, not ordinal, i.e., +they are not ordered in any way, like 'low', 'medium', 'high'. +<p> +We are now ready to make the Bayes net: +<pre> +bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes); +</PRE> +By default, all nodes are assumed to be discrete, so we can also just +write +<pre> +bnet = mk_bnet(dag, node_sizes); +</PRE> +You may also specify which nodes will be observed. +If you don't know, or if this not fixed in advance, +just use the empty list (the default). +<pre> +onodes = []; +bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes, 'observed', onodes); +</PRE> +Note that optional arguments are specified using a name/value syntax. +This is common for many BNT functions. +In general, to find out more about a function (e.g., which optional +arguments it takes), please see its +documentation string by typing +<pre> +help mk_bnet +</pre> +See also other <a href="matlab_tips.html">useful Matlab tips</a>. +<p> +It is possible to associate names with nodes, as follows: +<pre> +bnet = mk_bnet(dag, node_sizes, 'names', {'cloudy','S','R','W'}, 'discrete', 1:4); +</pre> +You can then refer to a node by its name: +<pre> +C = bnet.names('cloudy'); % bnet.names is an associative array +bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]); +</pre> +This feature uses my own associative array class. + + +<h2><a name="cpt">Parameters</h2> + +A model consists of the graph structure and the parameters. +The parameters are represented by CPD objects (CPD = Conditional +Probability Distribution), which define the probability distribution +of a node given its parents. +(We will use the terms "node" and "random variable" interchangeably.) +The simplest kind of CPD is a table (multi-dimensional array), which +is suitable when all the nodes are discrete-valued. Note that the discrete +values are not assumed to be ordered in any way; that is, they +represent categorical quantities, like male and female, rather than +ordinal quantities, like low, medium and high. +(We will discuss CPDs in more detail <a href="#cpd">below</a>.) +<p> +Tabular CPDs, also called CPTs (conditional probability tables), +are stored as multidimensional arrays, where the dimensions +are arranged in the same order as the nodes, e.g., the CPT for node 4 +(WetGrass) is indexed by Sprinkler (2), Rain (3) and then WetGrass (4) itself. +Hence the child is always the last dimension. +If a node has no parents, its CPT is a column vector representing its +prior. +Note that in Matlab (unlike C), arrays are indexed +from 1, and are layed out in memory such that the first index toggles +fastest, e.g., the CPT for node 4 (WetGrass) is as follows +<P> +<P><IMG ALIGN=BOTTOM SRC="Figures/CPTgrass.gif"><P> +<P> +where we have used the convention that false==1, true==2. +We can create this CPT in Matlab as follows +<PRE> +CPT = zeros(2,2,2); +CPT(1,1,1) = 1.0; +CPT(2,1,1) = 0.1; +... +</PRE> +Here is an easier way: +<PRE> +CPT = reshape([1 0.1 0.1 0.01 0 0.9 0.9 0.99], [2 2 2]); +</PRE> +In fact, we don't need to reshape the array, since the CPD constructor +will do that for us. So we can just write +<pre> +bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [1 0.1 0.1 0.01 0 0.9 0.9 0.99]); +</pre> +The other nodes are created similarly (using the old syntax for +optional parameters) +<PRE> +bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]); +bnet.CPD{R} = tabular_CPD(bnet, R, [0.8 0.2 0.2 0.8]); +bnet.CPD{S} = tabular_CPD(bnet, S, [0.5 0.9 0.5 0.1]); +bnet.CPD{W} = tabular_CPD(bnet, W, [1 0.1 0.1 0.01 0 0.9 0.9 0.99]); +</PRE> + + +<h2><a name="rnd_cpt">Random Parameters</h2> + +If we do not specify the CPT, random parameters will be +created, i.e., each "row" of the CPT will be drawn from the uniform distribution. +To ensure repeatable results, use +<pre> +rand('state', seed); +randn('state', seed); +</pre> +To control the degree of randomness (entropy), +you can sample each row of the CPT from a Dirichlet(p,p,...) distribution. +If p << 1, this encourages "deterministic" CPTs (one entry near 1, the rest near 0). +If p = 1, each entry is drawn from U[0,1]. +If p >> 1, the entries will all be near 1/k, where k is the arity of +this node, i.e., each row will be nearly uniform. +You can do this as follows, assuming this node +is number i, and ns is the node_sizes. +<pre> +k = ns(i); +ps = parents(dag, i); +psz = prod(ns(ps)); +CPT = sample_dirichlet(p*ones(1,k), psz); +bnet.CPD{i} = tabular_CPD(bnet, i, 'CPT', CPT); +</pre> + + +<h2><a name="file">Loading a network from a file</h2> + +If you already have a Bayes net represented in the XML-based +<a href="http://www.cs.cmu.edu/afs/cs/user/fgcozman/www/Research/InterchangeFormat/"> +Bayes Net Interchange Format (BNIF)</a> (e.g., downloaded from the +<a +href="http://www.cs.huji.ac.il/labs/compbio/Repository"> +Bayes Net repository</a>), +you can convert it to BNT format using +the +<a href="http://www.digitas.harvard.edu/~ken/bif2bnt/">BIF-BNT Java +program</a> written by Ken Shan. +(This is not necessarily up-to-date.) +<p> +<b>It is currently not possible to save/load a BNT matlab object to +file</b>, but this is easily fixed if you modify all the constructors +for all the classes (see matlab documentation). + +<h2><a name="GUI">Creating a model using a GUI</h2> + +<ul> +<li>Senthil Nachimuthu +has started (Oct 07) an open source +GUI for BNT called +<a href="http://projeny.sourceforge.net">projeny</a> +using Java. This is a successor to BNJ. + +<li> +Philippe LeRay has written (Sep 05) +a +<a href="http://banquiseasi.insa-rouen.fr/projects/bnt-editor/"> +BNT GUI</a> in matlab. + +<li> +<a +href="http://www.dataonstage.com/BNT/PACKAGES/LinkStrength/index.html"> +LinkStrength</a>, + a package by Imme Ebert-Uphoff for visualizing the strength of +dependencies between nodes. + +</ul> + +<h2>Graph visualization</h2> + +Click <a href="graphviz.html">here</a> for more information +on graph visualization. + + +<h1><a name="inference">Inference</h1> + +Having created the BN, we can now use it for inference. +There are many different algorithms for doing inference in Bayes nets, +that make different tradeoffs between speed, +complexity, generality, and accuracy. +BNT therefore offers a variety of different inference +"engines". We will discuss these +in more detail <a href="#engines">below</a>. +For now, we will use the junction tree +engine, which is the mother of all exact inference algorithms. +This can be created as follows. +<pre> +engine = jtree_inf_engine(bnet); +</pre> +The other engines have similar constructors, but might take +additional, algorithm-specific parameters. +All engines are used in the same way, once they have been created. +We illustrate this in the following sections. + + +<h2><a name="marginal">Computing marginal distributions</h2> + +Suppose we want to compute the probability that the sprinker was on +given that the grass is wet. +The evidence consists of the fact that W=2. All the other nodes +are hidden (unobserved). We can specify this as follows. +<pre> +evidence = cell(1,N); +evidence{W} = 2; +</pre> +We use a 1D cell array instead of a vector to +cope with the fact that nodes can be vectors of different lengths. +In addition, the value [] can be used +to denote 'no evidence', instead of having to specify the observation +pattern as a separate argument. +(Click <a href="cellarray.html">here</a> for a quick tutorial on cell +arrays in matlab.) +<p> +We are now ready to add the evidence to the engine. +<pre> +[engine, loglik] = enter_evidence(engine, evidence); +</pre> +The behavior of this function is algorithm-specific, and is discussed +in more detail <a href="#engines">below</a>. +In the case of the jtree engine, +enter_evidence implements a two-pass message-passing scheme. +The first return argument contains the modified engine, which +incorporates the evidence. The second return argument contains the +log-likelihood of the evidence. (Not all engines are capable of +computing the log-likelihood.) +<p> +Finally, we can compute p=P(S=2|W=2) as follows. +<PRE> +marg = marginal_nodes(engine, S); +marg.T +ans = + 0.57024 + 0.42976 +p = marg.T(2); +</PRE> +We see that p = 0.4298. +<p> +Now let us add the evidence that it was raining, and see what +difference it makes. +<PRE> +evidence{R} = 2; +[engine, loglik] = enter_evidence(engine, evidence); +marg = marginal_nodes(engine, S); +p = marg.T(2); +</PRE> +We find that p = P(S=2|W=2,R=2) = 0.1945, +which is lower than +before, because the rain can ``explain away'' the +fact that the grass is wet. +<p> +You can plot a marginal distribution over a discrete variable +as a barchart using the built 'bar' function: +<pre> +bar(marg.T) +</pre> +This is what it looks like + +<p> +<center> +<IMG SRC="Figures/sprinkler_bar.gif"> +</center> +<p> + +<h2><a name="observed">Observed nodes</h2> + +What happens if we ask for the marginal on an observed node, e.g. P(W|W=2)? +An observed discrete node effectively only has 1 value (the observed + one) --- all other values would result in 0 probability. +For efficiency, BNT treats observed (discrete) nodes as if they were + set to 1, as we see below: +<pre> +evidence = cell(1,N); +evidence{W} = 2; +engine = enter_evidence(engine, evidence); +m = marginal_nodes(engine, W); +m.T +ans = + 1 +</pre> +This can get a little confusing, since we assigned W=2. +So we can ask BNT to add the evidence back in by passing in an optional argument: +<pre> +m = marginal_nodes(engine, W, 1); +m.T +ans = + 0 + 1 +</pre> +This shows that P(W=1|W=2) = 0 and P(W=2|W=2) = 1. + + + +<h2><a name="joint">Computing joint distributions</h2> + +We can compute the joint probability on a set of nodes as in the +following example. +<pre> +evidence = cell(1,N); +[engine, ll] = enter_evidence(engine, evidence); +m = marginal_nodes(engine, [S R W]); +</pre> +m is a structure. The 'T' field is a multi-dimensional array (in +this case, 3-dimensional) that contains the joint probability +distribution on the specified nodes. +<pre> +>> m.T +ans(:,:,1) = + 0.2900 0.0410 + 0.0210 0.0009 +ans(:,:,2) = + 0 0.3690 + 0.1890 0.0891 +</pre> +We see that P(S=1,R=1,W=2) = 0, since it is impossible for the grass +to be wet if both the rain and sprinkler are off. +<p> +Let us now add some evidence to R. +<pre> +evidence{R} = 2; +[engine, ll] = enter_evidence(engine, evidence); +m = marginal_nodes(engine, [S R W]) +m = + domain: [2 3 4] + T: [2x1x2 double] +>> m.T +m.T +ans(:,:,1) = + 0.0820 + 0.0018 +ans(:,:,2) = + 0.7380 + 0.1782 +</pre> +The joint T(i,j,k) = P(S=i,R=j,W=k|evidence) +should have T(i,1,k) = 0 for all i,k, since R=1 is incompatible +with the evidence that R=2. +Instead of creating large tables with many 0s, BNT sets the effective +size of observed (discrete) nodes to 1, as explained above. +This is why m.T has size 2x1x2. +To get a 2x2x2 table, type +<pre> +m = marginal_nodes(engine, [S R W], 1) +m = + domain: [2 3 4] + T: [2x2x2 double] +>> m.T +m.T +ans(:,:,1) = + 0 0.082 + 0 0.0018 +ans(:,:,2) = + 0 0.738 + 0 0.1782 +</pre> + +<p> +Note: It is not always possible to compute the joint on arbitrary +sets of nodes: it depends on which inference engine you use, as discussed +in more detail <a href="#engines">below</a>. + + +<h2><a name="soft">Soft/virtual evidence</h2> + +Sometimes a node is not observed, but we have some distribution over +its possible values; this is often called "soft" or "virtual" +evidence. +One can use this as follows +<pre> +[engine, loglik] = enter_evidence(engine, evidence, 'soft', soft_evidence); +</pre> +where soft_evidence{i} is either [] (if node i has no soft evidence) +or is a vector representing the probability distribution over i's +possible values. +For example, if we don't know i's exact value, but we know its +likelihood ratio is 60/40, we can write evidence{i} = [] and +soft_evidence{i} = [0.6 0.4]. +<p> +Currently only jtree_inf_engine supports this option. +It assumes that all hidden nodes, and all nodes for +which we have soft evidence, are discrete. +For a longer example, see BNT/examples/static/softev1.m. + + +<h2><a name="mpe">Most probable explanation</h2> + +To compute the most probable explanation (MPE) of the evidence (i.e., +the most probable assignment, or a mode of the joint), use +<pre> +[mpe, ll] = calc_mpe(engine, evidence); +</pre> +mpe{i} is the most likely value of node i. +This calls enter_evidence with the 'maximize' flag set to 1, which +causes the engine to do max-product instead of sum-product. +The resulting max-marginals are then thresholded. +If there is more than one maximum probability assignment, we must take + care to break ties in a consistent manner (thresholding the + max-marginals may give the wrong result). To force this behavior, + type +<pre> +[mpe, ll] = calc_mpe(engine, evidence, 1); +</pre> +Note that computing the MPE is someties called abductive reasoning. + +<p> +You can also use <tt>calc_mpe_bucket</tt> written by Ron Zohar, +that does a forwards max-product pass, and then a backwards traceback +pass, which is how Viterbi is traditionally implemented. + + + +<h1><a name="cpd">Conditional Probability Distributions</h1> + +A Conditional Probability Distributions (CPD) +defines P(X(i) | X(Pa(i))), where X(i) is the i'th node, and X(Pa(i)) +are the parents of node i. There are many ways to represent this +distribution, which depend in part on whether X(i) and X(Pa(i)) are +discrete, continuous, or a combination. +We will discuss various representations below. + + +<h2><a name="tabular">Tabular nodes</h2> + +If the CPD is represented as a table (i.e., if it is a multinomial +distribution), it has a number of parameters that is exponential in +the number of parents. See the example <a href="#cpt">above</a>. + + +<h2><a name="noisyor">Noisy-or nodes</h2> + +A noisy-OR node is like a regular logical OR gate except that +sometimes the effects of parents that are on get inhibited. +Let the prob. that parent i gets inhibited be q(i). +Then a node, C, with 2 parents, A and B, has the following CPD, where +we use F and T to represent off and on (1 and 2 in BNT). +<pre> +A B P(C=off) P(C=on) +--------------------------- +F F 1.0 0.0 +T F q(A) 1-q(A) +F T q(B) 1-q(B) +T T q(A)q(B) 1-q(A)q(B) +</pre> +Thus we see that the causes get inhibited independently. +It is common to associate a "leak" node with a noisy-or CPD, which is +like a parent that is always on. This can account for all other unmodelled +causes which might turn the node on. +<p> +The noisy-or distribution is similar to the logistic distribution. +To see this, let the nodes, S(i), have values in {0,1}, and let q(i,j) +be the prob. that j inhibits i. Then +<pre> +Pr(S(i)=1 | parents(S(i))) = 1 - prod_{j} q(i,j)^S(j) +</pre> +Now define w(i,j) = -ln q(i,j) and rho(x) = 1-exp(-x). Then +<pre> +Pr(S(i)=1 | parents(S(i))) = rho(sum_j w(i,j) S(j)) +</pre> +For a sigmoid node, we have +<pre> +Pr(S(i)=1 | parents(S(i))) = sigma(-sum_j w(i,j) S(j)) +</pre> +where sigma(x) = 1/(1+exp(-x)). Hence they differ in the choice of +the activation function (although both are monotonically increasing). +In addition, in the case of a noisy-or, the weights are constrained to be +positive, since they derive from probabilities q(i,j). +In both cases, the number of parameters is <em>linear</em> in the +number of parents, unlike the case of a multinomial distribution, +where the number of parameters is exponential in the number of parents. +We will see an example of noisy-OR nodes <a href="#qmr">below</a>. + + +<h2><a name="deterministic">Other (noisy) deterministic nodes</h2> + +Deterministic CPDs for discrete random variables can be created using +the deterministic_CPD class. It is also possible to 'flip' the output +of the function with some probability, to simulate noise. +The boolean_CPD class is just a special case of a +deterministic CPD, where the parents and child are all binary. +<p> +Both of these classes are just "syntactic sugar" for the tabular_CPD +class. + + + +<h2><a name="softmax">Softmax nodes</h2> + +If we have a discrete node with a continuous parent, +we can define its CPD using a softmax function +(also known as the multinomial logit function). +This acts like a soft thresholding operator, and is defined as follows: +<pre> + exp(w(:,i)'*x + b(i)) +Pr(Q=i | X=x) = ----------------------------- + sum_j exp(w(:,j)'*x + b(j)) + +</pre> +The parameters of a softmax node, w(:,i) and b(i), i=1..|Q|, have the +following interpretation: w(:,i)-w(:,j) is the normal vector to the +decision boundary between classes i and j, +and b(i)-b(j) is its offset (bias). For example, suppose +X is a 2-vector, and Q is binary. Then +<pre> +w = [1 -1; + 0 0]; + +b = [0 0]; +</pre> +means class 1 are points in the 2D plane with positive x coordinate, +and class 2 are points in the 2D plane with negative x coordinate. +If w has large magnitude, the decision boundary is sharp, otherwise it +is soft. +In the special case that Q is binary (0/1), the softmax function reduces to the logistic +(sigmoid) function. +<p> +Fitting a softmax function can be done using the iteratively reweighted +least squares (IRLS) algorithm. +We use the implementation from +<a href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>. +Note that since +the softmax distribution is not in the exponential family, it does not +have finite sufficient statistics, and hence we must store all the +training data in uncompressed form. +If this takes too much space, one should use online (stochastic) gradient +descent (not implemented in BNT). +<p> +If a softmax node also has discrete parents, +we use a different set of w/b parameters for each combination of +parent values, as in the <a href="#gaussian">conditional linear +Gaussian CPD</a>. +This feature was implemented by Pierpaolo Brutti. +He is currently extending it so that discrete parents can be treated +as if they were continuous, by adding indicator variables to the X +vector. +<p> +We will see an example of softmax nodes <a href="#mixexp">below</a>. + + +<h2><a name="mlp">Neural network nodes</h2> + +Pierpaolo Brutti has implemented the mlp_CPD class, which uses a multi layer perceptron +to implement a mapping from continuous parents to discrete children, +similar to the softmax function. +(If there are also discrete parents, it creates a mixture of MLPs.) +It uses code from <a +href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>. +This is work in progress. + +<h2><a name="root">Root nodes</h2> + +A root node has no parents and no parameters; it can be used to model +an observed, exogeneous input variable, i.e., one which is "outside" +the model. +This is useful for conditional density models. +We will see an example of root nodes <a href="#mixexp">below</a>. + + +<h2><a name="gaussian">Gaussian nodes</h2> + +We now consider a distribution suitable for the continuous-valued nodes. +Suppose the node is called Y, its continuous parents (if any) are +called X, and its discrete parents (if any) are called Q. +The distribution on Y is defined as follows: +<pre> +- no parents: Y ~ N(mu, Sigma) +- cts parents : Y|X=x ~ N(mu + W x, Sigma) +- discrete parents: Y|Q=i ~ N(mu(:,i), Sigma(:,:,i)) +- cts and discrete parents: Y|X=x,Q=i ~ N(mu(:,i) + W(:,:,i) * x, Sigma(:,:,i)) +</pre> +where N(mu, Sigma) denotes a Normal distribution with mean mu and +covariance Sigma. Let |X|, |Y| and |Q| denote the sizes of X, Y and Q +respectively. +If there are no discrete parents, |Q|=1; if there is +more than one, then |Q| = a vector of the sizes of each discrete parent. +If there are no continuous parents, |X|=0; if there is more than one, +then |X| = the sum of their sizes. +Then mu is a |Y|*|Q| vector, Sigma is a |Y|*|Y|*|Q| positive +semi-definite matrix, and W is a |Y|*|X|*|Q| regression (weight) +matrix. +<p> +We can create a Gaussian node with random parameters as follows. +<pre> +bnet.CPD{i} = gaussian_CPD(bnet, i); +</pre> +We can specify the value of one or more of the parameters as in the +following example, in which |Y|=2, and |Q|=1. +<pre> +bnet.CPD{i} = gaussian_CPD(bnet, i, 'mean', [0; 0], 'weights', randn(Y,X), 'cov', eye(Y)); +</pre> +<p> +We will see an example of conditional linear Gaussian nodes <a +href="#cg_model">below</a>. +<p> +<b>When learning Gaussians from data</b>, it is helpful to ensure the +data has a small magnitde +(see e.g., KPMstats/standardize) to prevent numerical problems. +Unless you have a lot of data, it is also a very good idea to use +diagonal instead of full covariance matrices. +(BNT does not currently support spherical covariances, although it +would be easy to add, since KPMstats/clg_Mstep supports this option; +you would just need to modify gaussian_CPD/update_ess to accumulate +weighted inner products.) + + + +<h2><a name="nongauss">Other continuous distributions</h2> + +Currently BNT does not support any CPDs for continuous nodes other +than the Gaussian. +However, you can use a mixture of Gaussians to +approximate other continuous distributions. We will see some an example +of this with the IFA model <a href="#pca">below</a>. + + +<h2><a name="glm">Generalized linear model nodes</h2> + +In the future, we may incorporate some of the functionality of +<a href = +"http://www.sci.usq.edu.au/staff/dunn/glmlab/glmlab.html">glmlab</a> +into BNT. + + +<h2><a name="dtree">Classification/regression tree nodes</h2> + +We plan to add classification and regression trees to define CPDs for +discrete and continuous nodes, respectively. +Trees have many advantages: they are easy to interpret, they can do +feature selection, they can +handle discrete and continuous inputs, they do not make strong +assumptions about the form of the distribution, the number of +parameters can grow in a data-dependent way (i.e., they are +semi-parametric), they can handle missing data, etc. +However, they are not yet implemented. +<!-- +Yimin Zhang is currently (Feb '02) implementing this. +--> + + +<h2><a name="cpd_summary">Summary of CPD types</h2> + +We list all the different types of CPDs supported by BNT. +For each CPD, we specify if the child and parents can be discrete (D) or +continuous (C) (Binary (B) nodes are a special case). +We also specify which methods each class supports. +If a method is inherited, the name of the parent class is mentioned. +If a parent class calls a child method, this is mentioned. +<p> +The <tt>CPD_to_CPT</tt> method converts a CPD to a table; this +requires that the child and all parents are discrete. +The CPT might be exponentially big... +<tt>convert_to_table</tt> evaluates a CPD with evidence, and +represents the the resulting potential as an array. +This requires that the child is discrete, and any continuous parents +are observed. +<tt>convert_to_pot</tt> evaluates a CPD with evidence, and +represents the resulting potential as a dpot, gpot, cgpot or upot, as +requested. (d=discrete, g=Gaussian, cg = conditional Gaussian, u = +utility). + +<p> +When we sample a node, all the parents are observed. +When we compute the (log) probability of a node, all the parents and +the child are observed. +<p> +We also specify if the parameters are learnable. +For learning with EM, we require +the methods <tt>reset_ess</tt>, <tt>update_ess</tt> and +<tt>maximize_params</tt>. +For learning from fully observed data, we require +the method <tt>learn_params</tt>. +By default, all classes inherit this from generic_CPD, which simply +calls <tt>update_ess</tt> N times, once for each data case, followed +by <tt>maximize_params</tt>, i.e., it is like EM, without the E step. +Some classes implement a batch formula, which is quicker. +<p> +Bayesian learning means computing a posterior over the parameters +given fully observed data. +<p> +Pearl means we implement the methods <tt>compute_pi</tt> and +<tt>compute_lambda_msg</tt>, used by +<tt>pearl_inf_engine</tt>, which runs on directed graphs. +<tt>belprop_inf_engine</tt> only needs <tt>convert_to_pot</tt>.H +The pearl methods can exploit special properties of the CPDs for +computing the messages efficiently, whereas belprop does not. +<p> +The only method implemented by generic_CPD is <tt>adjustable_CPD</tt>, +which is not shown, since it is not very interesting. + + +<p> + + +<table> +<table border units = pixels><tr> +<td align=center>Name +<td align=center>Child +<td align=center>Parents +<td align=center>Comments +<td align=center>CPD_to_CPT +<td align=center>conv_to_table +<td align=center>conv_to_pot +<td align=center>sample +<td align=center>prob +<td align=center>learn +<td align=center>Bayes +<td align=center>Pearl + + +<tr> +<!-- Name--><td> +<!-- Child--><td> +<!-- Parents--><td> +<!-- Comments--><td> +<!-- CPD_to_CPT--><td> +<!-- conv_to_table--><td> +<!-- conv_to_pot--><td> +<!-- sample--><td> +<!-- prob--><td> +<!-- learn--><td> +<!-- Bayes--><td> +<!-- Pearl--><td> + +<tr> +<!-- Name--><td>boolean +<!-- Child--><td>B +<!-- Parents--><td>B +<!-- Comments--><td>Syntactic sugar for tabular +<!-- CPD_to_CPT--><td>- +<!-- conv_to_table--><td>- +<!-- conv_to_pot--><td>- +<!-- sample--><td>- +<!-- prob--><td>- +<!-- learn--><td>- +<!-- Bayes--><td>- +<!-- Pearl--><td>- + +<tr> +<!-- Name--><td>deterministic +<!-- Child--><td>D +<!-- Parents--><td>D +<!-- Comments--><td>Syntactic sugar for tabular +<!-- CPD_to_CPT--><td>- +<!-- conv_to_table--><td>- +<!-- conv_to_pot--><td>- +<!-- sample--><td>- +<!-- prob--><td>- +<!-- learn--><td>- +<!-- Bayes--><td>- +<!-- Pearl--><td>- + +<tr> +<!-- Name--><td>Discrete +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>Virtual class +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Calls CPD_to_CPT +<!-- conv_to_pot--><td>Calls conv_to_table +<!-- sample--><td>Calls conv_to_table +<!-- prob--><td>Calls conv_to_table +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + +<tr> +<!-- Name--><td>Gaussian +<!-- Child--><td>C +<!-- Parents--><td>C/D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>Y +<!-- prob--><td>Y +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + +<tr> +<!-- Name--><td>gmux +<!-- Child--><td>C +<!-- Parents--><td>C/D +<!-- Comments--><td>multiplexer +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>N +<!-- prob--><td>N +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>Y + + +<tr> +<!-- Name--><td>MLP +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>multi layer perceptron +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Y +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>noisy-or +<!-- Child--><td>B +<!-- Parents--><td>B +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>Y +<!-- conv_to_table--><td>Inherits from discrete +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>Y + + +<tr> +<!-- Name--><td>root +<!-- Child--><td>C/D +<!-- Parents--><td>none +<!-- Comments--><td>no params +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>Y +<!-- prob--><td>Y +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>softmax +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Y +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>generic +<!-- Child--><td>C/D +<!-- Parents--><td>C/D +<!-- Comments--><td>Virtual class +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>N +<!-- sample--><td>N +<!-- prob--><td>N +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>Tabular +<!-- Child--><td>D +<!-- Parents--><td>D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>Y +<!-- conv_to_table--><td>Inherits from discrete +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>Y +<!-- Pearl--><td>Y + +</table> + + + +<h1><a name="examples">Example models</h1> + + +<h2>Gaussian mixture models</h2> + +Richard W. DeVaul has made a detailed tutorial on how to fit mixtures +of Gaussians using BNT. Available +<a href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt">here</a>. + + +<h2><a name="pca">PCA, ICA, and all that </h2> + +In Figure (a) below, we show how Factor Analysis can be thought of as a +graphical model. Here, X has an N(0,I) prior, and +Y|X=x ~ N(mu + Wx, Psi), +where Psi is diagonal and W is called the "factor loading matrix". +Since the noise on both X and Y is diagonal, the components of these +vectors are uncorrelated, and hence can be represented as individual +scalar nodes, as we show in (b). +(This is useful if parts of the observations on the Y vector are occasionally missing.) +We usually take k=|X| << |Y|=D, so the model tries to explain +many observations using a low-dimensional subspace. + + +<center> +<table> +<tr> +<td><img src="Figures/fa.gif"> +<td><img src="Figures/fa_scalar.gif"> +<td><img src="Figures/mfa.gif"> +<td><img src="Figures/ifa.gif"> +<tr> +<td align=center> (a) +<td align=center> (b) +<td align=center> (c) +<td align=center> (d) +</table> +</center> + +<p> +We can create this model in BNT as follows. +<pre> +ns = [k D]; +dag = zeros(2,2); +dag(1,2) = 1; +bnet = mk_bnet(dag, ns, 'discrete', []); +bnet.CPD{1} = gaussian_CPD(bnet, 1, 'mean', zeros(k,1), 'cov', eye(k), ... + 'cov_type', 'diag', 'clamp_mean', 1, 'clamp_cov', 1); +bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(D,1), 'cov', diag(Psi0), 'weights', W0, ... + 'cov_type', 'diag', 'clamp_mean', 1); +</pre> + +The root node is clamped to the N(0,I) distribution, so that we will +not update these parameters during learning. +The mean of the leaf node is clamped to 0, +since we assume the data has been centered (had its mean subtracted +off); this is just for simplicity. +Finally, the covariance of the leaf node is constrained to be +diagonal. W0 and Psi0 are the initial parameter guesses. + +<p> +We can fit this model (i.e., estimate its parameters in a maximum +likelihood (ML) sense) using EM, as we +explain <a href="#em">below</a>. +Not surprisingly, the ML estimates for mu and Psi turn out to be +identical to the +sample mean and variance, which can be computed directly as +<pre> +mu_ML = mean(data); +Psi_ML = diag(cov(data)); +</pre> +Note that W can only be identified up to a rotation matrix, because of +the spherical symmetry of the source. + +<p> +If we restrict Psi to be spherical, i.e., Psi = sigma*I, +there is a closed-form solution for W as well, +i.e., we do not need to use EM. +In particular, W contains the first |X| eigenvectors of the sample covariance +matrix, with scalings determined by the eigenvalues and sigma. +Classical PCA can be obtained by taking the sigma->0 limit. +For details, see + +<ul> +<li> <a href="ftp://hope.caltech.edu/pub/roweis/Empca/empca.ps"> +"EM algorithms for PCA and SPCA"</a>, Sam Roweis, NIPS 97. +(<a href="ftp://hope.caltech.edu/pub/roweis/Code/empca.tar.gz"> +Matlab software</a>) + +<p> +<li> +<a +href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003> +"Mixtures of probabilistic principal component analyzers"</a>, +Tipping and Bishop, Neural Computation 11(2):443--482, 1999. +</ul> + +<p> +By adding a hidden discrete variable, we can create mixtures of FA +models, as shown in (c). +Now we can explain the data using a set of subspaces. +We can create this model in BNT as follows. +<pre> +ns = [M k D]; +dag = zeros(3); +dag(1,3) = 1; +dag(2,3) = 1; +bnet = mk_bnet(dag, ns, 'discrete', 1); +bnet.CPD{1} = tabular_CPD(bnet, 1, Pi0); +bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(k, 1), 'cov', eye(k), 'cov_type', 'diag', ... + 'clamp_mean', 1, 'clamp_cov', 1); +bnet.CPD{3} = gaussian_CPD(bnet, 3, 'mean', Mu0', 'cov', repmat(diag(Psi0), [1 1 M]), ... + 'weights', W0, 'cov_type', 'diag', 'tied_cov', 1); +</pre> +Notice how the covariance matrix for Y is the same for all values of +Q; that is, the noise level in each sub-space is assumed the same. +However, we allow the offset, mu, to vary. +For details, see +<ul> + +<LI> +<a HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/tr-96-1.ps.gz"> The EM +Algorithm for Mixtures of Factor Analyzers </A>, +Ghahramani, Z. and Hinton, G.E. (1996), +University of Toronto +Technical Report CRG-TR-96-1. +(<A HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/mfa.tar.gz">Matlab software</A>) + +<p> +<li> +<a +href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003> +"Mixtures of probabilistic principal component analyzers"</a>, +Tipping and Bishop, Neural Computation 11(2):443--482, 1999. +</ul> + +<p> +I have included Zoubin's specialized MFA code (with his permission) +with the toolbox, so you can check that BNT gives the same results: +see 'BNT/examples/static/mfa1.m'. + +<p> +Independent Factor Analysis (IFA) generalizes FA by allowing a +non-Gaussian prior on each component of X. +(Note that we can approximate a non-Gaussian prior using a mixture of +Gaussians.) +This means that the likelihood function is no longer rotationally +invariant, so we can uniquely identify W and the hidden +sources X. +IFA also allows a non-diagonal Psi (i.e. correlations between the components of Y). +We recover classical Independent Components Analysis (ICA) +in the Psi -> 0 limit, and by assuming that |X|=|Y|, so that the +weight matrix W is square and invertible. +For details, see +<ul> +<li> +<a href="http://www.gatsby.ucl.ac.uk/~hagai/ifa.ps">Independent Factor +Analysis</a>, H. Attias, Neural Computation 11: 803--851, 1998. +</ul> + + + +<h2><a name="mixexp">Mixtures of experts</h2> + +As an example of the use of the softmax function, +we introduce the Mixture of Experts model. +<!-- +We also show +the Hierarchical Mixture of Experts model, where the hierarchy has two +levels. +(This is essentially a probabilistic decision tree of height two.) +--> +As before, +circles denote continuous-valued nodes, +squares denote discrete nodes, clear +means hidden, and shaded means observed. +<p> +<center> +<table> +<tr> +<td><img src="Figures/mixexp.gif"> +<!-- +<td><img src="Figures/hme.gif"> +--> +</table> +</center> +<p> +X is the observed +input, Y is the output, and +the Q nodes are hidden "gating" nodes, which select the appropriate +set of parameters for Y. During training, Y is assumed observed, +but for testing, the goal is to predict Y given X. +Note that this is a <em>conditional</em> density model, so we don't +associate any parameters with X. +Hence X's CPD will be a root CPD, which is a way of modelling +exogenous nodes. +If the output is a continuous-valued quantity, +we assume the "experts" are linear-regression units, +and set Y's CPD to linear-Gaussian. +If the output is discrete, we set Y's CPD to a softmax function. +The Q CPDs will always be softmax functions. + +<p> +As a concrete example, consider the mixture of experts model where X and Y are +scalars, and Q is binary. +This is just piecewise linear regression, where +we have two line segments, i.e., +<P> +<IMG ALIGN=BOTTOM SRC="Eqns/lin_reg_eqn.gif"> +<P> +We can create this model with random parameters as follows. +(This code is bundled in BNT/examples/static/mixexp2.m.) +<PRE> +X = 1; +Q = 2; +Y = 3; +dag = zeros(3,3); +dag(X,[Q Y]) = 1 +dag(Q,Y) = 1; +ns = [1 2 1]; % make X and Y scalars, and have 2 experts +onodes = [1 3]; +bnet = mk_bnet(dag, ns, 'discrete', 2, 'observed', onodes); + +rand('state', 0); +randn('state', 0); +bnet.CPD{1} = root_CPD(bnet, 1); +bnet.CPD{2} = softmax_CPD(bnet, 2); +bnet.CPD{3} = gaussian_CPD(bnet, 3); +</PRE> +Now let us fit this model using <a href="#em">EM</a>. +First we <a href="#load_data">load the data</a> (1000 training cases) and plot them. +<P> +<PRE> +data = load('/examples/static/Misc/mixexp_data.txt', '-ascii'); +plot(data(:,1), data(:,2), '.'); +</PRE> +<p> +<center> +<IMG SRC="Figures/mixexp_data.gif"> +</center> +<p> +This is what the model looks like before training. +(Thanks to Thomas Hofman for writing this plotting routine.) +<p> +<center> +<IMG SRC="Figures/mixexp_before.gif"> +</center> +<p> +Now let's train the model, and plot the final performance. +(We will discuss how to train models in more detail <a href="#param_learning">below</a>.) +<P> +<PRE> +ncases = size(data, 1); % each row of data is a training case +cases = cell(3, ncases); +cases([1 3], :) = num2cell(data'); % each column of cases is a training case +engine = jtree_inf_engine(bnet); +max_iter = 20; +[bnet2, LLtrace] = learn_params_em(engine, cases, max_iter); +</PRE> +(We specify which nodes will be observed when we create the engine. +Hence BNT knows that the hidden nodes are all discrete. +For complex models, this can lead to a significant speedup.) +Below we show what the model looks like after 16 iterations of EM +(with 100 IRLS iterations per M step), when it converged +using the default convergence tolerance (that the +fractional change in the log-likelihood be less than 1e-3). +Before learning, the log-likelihood was +-322.927442; afterwards, it was -13.728778. +<p> +<center> +<IMG SRC="Figures/mixexp_after.gif"> +</center> +(See BNT/examples/static/mixexp2.m for details of the code.) + + + +<h2><a name="hme">Hierarchical mixtures of experts</h2> + +A hierarchical mixture of experts (HME) extends the mixture of experts +model by having more than one hidden node. A two-level example is shown below, along +with its more traditional representation as a neural network. +This is like a (balanced) probabilistic decision tree of height 2. +<p> +<center> +<IMG SRC="Figures/HMEforMatlab.jpg"> +</center> +<p> +<a href="mailto:pbrutti@stat.cmu.edu">Pierpaolo Brutti</a> +has written an extensive set of routines for HMEs, +which are bundled with BNT: see the examples/static/HME directory. +These routines allow you to choose the number of hidden (gating) +layers, and the form of the experts (softmax or MLP). +See the file hmemenu, which provides a demo. +For example, the figure below shows the decision boundaries learned +for a ternary classification problem, using a 2 level HME with softmax +gates and softmax experts; the training set is on the left, the +testing set on the right. +<p> +<center> +<!--<IMG SRC="Figures/hme_dec_boundary.gif">--> +<IMG SRC="Figures/hme_dec_boundary.png"> +</center> +<p> + + +<p> +For more details, see the following: +<ul> + +<li> <a href="http://www.cs.berkeley.edu/~jordan/papers/hierarchies.ps.Z"> +Hierarchical mixtures of experts and the EM algorithm</a> +M. I. Jordan and R. A. Jacobs. Neural Computation, 6, 181-214, 1994. + +<li> <a href = +"http://www.cs.berkeley.edu/~dmartin/software">David Martin's +matlab code for HME</a> + +<li> <a +href="http://www.cs.berkeley.edu/~jordan/papers/uai.ps.Z">Why the +logistic function? A tutorial discussion on +probabilities and neural networks.</a> M. I. Jordan. MIT Computational +Cognitive Science Report 9503, August 1995. + +<li> "Generalized Linear Models", McCullagh and Nelder, Chapman and +Halll, 1983. + +<li> +"Improved learning algorithms for mixtures of experts in multiclass +classification". +K. Chen, L. Xu, H. Chi. +Neural Networks (1999) 12: 1229-1252. + +<li> <a href="http://www.oigeeza.com/steve/"> +Classification Using Hierarchical Mixtures of Experts</a> +S.R. Waterhouse and A.J. Robinson. +In Proc. IEEE Workshop on Neural Network for Signal Processing IV (1994), pp. 177-186 + +<li> <a href="http://www.idiap.ch/~perry/"> +Localized mixtures of experts</a>, +P. Moerland, 1998. + +<li> "Nonlinear gated experts for time series", +A.S. Weigend and M. Mangeas, 1995. + +</ul> + + +<h2><a name="qmr">QMR</h2> + +Bayes nets originally arose out of an attempt to add probabilities to +expert systems, and this is still the most common use for BNs. +A famous example is +QMR-DT, a decision-theoretic reformulation of the Quick Medical +Reference (QMR) model. +<p> +<center> +<IMG ALIGN=BOTTOM SRC="Figures/qmr.gif"> +</center> +Here, the top layer represents hidden disease nodes, and the bottom +layer represents observed symptom nodes. +The goal is to infer the posterior probability of each disease given +all the symptoms (which can be present, absent or unknown). +Each node in the top layer has a Bernoulli prior (with a low prior +probability that the disease is present). +Since each node in the bottom layer has a high fan-in, we use a +noisy-OR parameterization; each disease has an independent chance of +causing each symptom. +The real QMR-DT model is copyright, but +we can create a random QMR-like model as follows. +<pre> +function bnet = mk_qmr_bnet(G, inhibit, leak, prior) +% MK_QMR_BNET Make a QMR model +% bnet = mk_qmr_bnet(G, inhibit, leak, prior) +% +% G(i,j) = 1 iff there is an arc from disease i to finding j +% inhibit(i,j) = inhibition probability on i->j arc +% leak(j) = inhibition prob. on leak->j arc +% prior(i) = prob. disease i is on + +[Ndiseases Nfindings] = size(inhibit); +N = Ndiseases + Nfindings; +finding_node = Ndiseases+1:N; +ns = 2*ones(1,N); +dag = zeros(N,N); +dag(1:Ndiseases, finding_node) = G; +bnet = mk_bnet(dag, ns, 'observed', finding_node); + +for d=1:Ndiseases + CPT = [1-prior(d) prior(d)]; + bnet.CPD{d} = tabular_CPD(bnet, d, CPT'); +end + +for i=1:Nfindings + fnode = finding_node(i); + ps = parents(G, i); + bnet.CPD{fnode} = noisyor_CPD(bnet, fnode, leak(i), inhibit(ps, i)); +end +</pre> +In the file BNT/examples/static/qmr1, we create a random bipartite +graph G, with 5 diseases and 10 findings, and random parameters. +(In general, to create a random dag, use 'mk_random_dag'.) +We can visualize the resulting graph structure using +the methods discussed <a href="#graphdraw">below</a>, with the +following results: +<p> +<img src="Figures/qmr.rnd.jpg"> + +<p> +Now let us put some random evidence on all the leaves except the very +first and very last, and compute the disease posteriors. +<pre> +pos = 2:floor(Nfindings/2); +neg = (pos(end)+1):(Nfindings-1); +onodes = myunion(pos, neg); +evidence = cell(1, N); +evidence(findings(pos)) = num2cell(repmat(2, 1, length(pos))); +evidence(findings(neg)) = num2cell(repmat(1, 1, length(neg))); + +engine = jtree_inf_engine(bnet); +[engine, ll] = enter_evidence(engine, evidence); +post = zeros(1, Ndiseases); +for i=diseases(:)' + m = marginal_nodes(engine, i); + post(i) = m.T(2); +end +</pre> +Junction tree can be quite slow on large QMR models. +Fortunately, it is possible to exploit properties of the noisy-OR +function to speed up exact inference using an algorithm called +<a href="#quickscore">quickscore</a>, discussed below. + + + + + +<h2><a name="cg_model">Conditional Gaussian models</h2> + +A conditional Gaussian model is one in which, conditioned on all the discrete +nodes, the distribution over the remaining (continuous) nodes is +multivariate Gaussian. This means we can have arcs from discrete (D) +to continuous (C) nodes, but not vice versa. +(We <em>are</em> allowed C->D arcs if the continuous nodes are observed, +as in the <a href="#mixexp">mixture of experts</a> model, +since this distribution can be represented with a discrete potential.) +<p> +We now give an example of a CG model, from +the paper "Propagation of Probabilities, Means amd +Variances in Mixed Graphical Association Models", Steffen Lauritzen, +JASA 87(420):1098--1108, 1992 (reprinted in the book "Probabilistic Networks and Expert +Systems", R. G. Cowell, A. P. Dawid, S. L. Lauritzen and +D. J. Spiegelhalter, Springer, 1999.) + +<h3>Specifying the graph</h3> + +Consider the model of waste emissions from an incinerator plant shown below. +We follow the standard convention that shaded nodes are observed, +clear nodes are hidden. +We also use the non-standard convention that +square nodes are discrete (tabular) and round nodes are +Gaussian. + +<p> +<center> +<IMG SRC="Figures/cg1.gif"> +</center> +<p> + +We can create this model as follows. +<pre> +F = 1; W = 2; E = 3; B = 4; C = 5; D = 6; Min = 7; Mout = 8; L = 9; +n = 9; + +dag = zeros(n); +dag(F,E)=1; +dag(W,[E Min D]) = 1; +dag(E,D)=1; +dag(B,[C D])=1; +dag(D,[L Mout])=1; +dag(Min,Mout)=1; + +% node sizes - all cts nodes are scalar, all discrete nodes are binary +ns = ones(1, n); +dnodes = [F W B]; +cnodes = mysetdiff(1:n, dnodes); +ns(dnodes) = 2; + +bnet = mk_bnet(dag, ns, 'discrete', dnodes); +</pre> +'dnodes' is a list of the discrete nodes; 'cnodes' is the continuous +nodes. 'mysetdiff' is a faster version of the built-in 'setdiff'. +<p> + + +<h3>Specifying the parameters</h3> + +The parameters of the discrete nodes can be specified as follows. +<pre> +bnet.CPD{B} = tabular_CPD(bnet, B, 'CPT', [0.85 0.15]); % 1=stable, 2=unstable +bnet.CPD{F} = tabular_CPD(bnet, F, 'CPT', [0.95 0.05]); % 1=intact, 2=defect +bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [2/7 5/7]); % 1=industrial, 2=household +</pre> + +<p> +The parameters of the continuous nodes can be specified as follows. +<pre> +bnet.CPD{E} = gaussian_CPD(bnet, E, 'mean', [-3.9 -0.4 -3.2 -0.5], ... + 'cov', [0.00002 0.0001 0.00002 0.0001]); +bnet.CPD{D} = gaussian_CPD(bnet, D, 'mean', [6.5 6.0 7.5 7.0], ... + 'cov', [0.03 0.04 0.1 0.1], 'weights', [1 1 1 1]); +bnet.CPD{C} = gaussian_CPD(bnet, C, 'mean', [-2 -1], 'cov', [0.1 0.3]); +bnet.CPD{L} = gaussian_CPD(bnet, L, 'mean', 3, 'cov', 0.25, 'weights', -0.5); +bnet.CPD{Min} = gaussian_CPD(bnet, Min, 'mean', [0.5 -0.5], 'cov', [0.01 0.005]); +bnet.CPD{Mout} = gaussian_CPD(bnet, Mout, 'mean', 0, 'cov', 0.002, 'weights', [1 1]); +</pre> + + +<h3><a name="cg_infer">Inference</h3> + +<!--Let us perform inference in the <a href="#cg_model">waste incinerator example</a>.--> +First we compute the unconditional marginals. +<pre> +engine = jtree_inf_engine(bnet); +evidence = cell(1,n); +[engine, ll] = enter_evidence(engine, evidence); +marg = marginal_nodes(engine, E); +</pre> +<!--(Of course, we could use <tt>cond_gauss_inf_engine</tt> instead of jtree.)--> +'marg' is a structure that contains the fields 'mu' and 'Sigma', which +contain the mean and (co)variance of the marginal on E. +In this case, they are both scalars. +Let us check they match the published figures (to 2 decimal places). +<!--(We can't expect +more precision than this in general because I have implemented the algorithm of +Lauritzen (1992), which can be numerically unstable.)--> +<pre> +tol = 1e-2; +assert(approxeq(marg.mu, -3.25, tol)); +assert(approxeq(sqrt(marg.Sigma), 0.709, tol)); +</pre> +We can compute the other posteriors similarly. +Now let us add some evidence. +<pre> +evidence = cell(1,n); +evidence{W} = 1; % industrial +evidence{L} = 1.1; +evidence{C} = -0.9; +[engine, ll] = enter_evidence(engine, evidence); +</pre> +Now we find +<pre> +marg = marginal_nodes(engine, E); +assert(approxeq(marg.mu, -3.8983, tol)); +assert(approxeq(sqrt(marg.Sigma), 0.0763, tol)); +</pre> + + +We can also compute the joint probability on a set of nodes. +For example, P(D, Mout | evidence) is a 2D Gaussian: +<pre> +marg = marginal_nodes(engine, [D Mout]) +marg = + domain: [6 8] + mu: [2x1 double] + Sigma: [2x2 double] + T: 1.0000 +</pre> +The mean is +<pre> +marg.mu +ans = + 3.6077 + 4.1077 +</pre> +and the covariance matrix is +<pre> +marg.Sigma +ans = + 0.1062 0.1062 + 0.1062 0.1182 +</pre> +It is easy to visualize this posterior using standard Matlab plotting +functions, e.g., +<pre> +gaussplot2d(marg.mu, marg.Sigma); +</pre> +produces the following picture. + +<p> +<center> +<IMG SRC="Figures/gaussplot.png"> +</center> +<p> + + +The T field indicates that the mixing weight of this Gaussian +component is 1.0. +If the joint contains discrete and continuous variables, the result +will be a mixture of Gaussians, e.g., +<pre> +marg = marginal_nodes(engine, [F E]) + domain: [1 3] + mu: [-3.9000 -0.4003] + Sigma: [1x1x2 double] + T: [0.9995 4.7373e-04] +</pre> +The interpretation is +Sigma(i,j,k) = Cov[ E(i) E(j) | F=k ]. +In this case, E is a scalar, so i=j=1; k specifies the mixture component. +<p> +We saw in the sprinkler network that BNT sets the effective size of +observed discrete nodes to 1, since they only have one legal value. +For continuous nodes, BNT sets their length to 0, +since they have been reduced to a point. +For example, +<pre> +marg = marginal_nodes(engine, [B C]) + domain: [4 5] + mu: [] + Sigma: [] + T: [0.0123 0.9877] +</pre> +It is simple to post-process the output of marginal_nodes. +For example, the file BNT/examples/static/cg1 sets the mu term of +observed nodes to their observed value, and the Sigma term to 0 (since +observed nodes have no variance). + +<p> +Note that the implemented version of the junction tree is numerically +unstable when using CG potentials +(which is why, in the example above, we only required our answers to agree with +the published ones to 2dp.) +This is why you might want to use <tt>stab_cond_gauss_inf_engine</tt>, +implemented by Shan Huang. This is described in + +<ul> +<li> "Stable Local Computation with Conditional Gaussian Distributions", +S. Lauritzen and F. Jensen, Tech Report R-99-2014, +Dept. Math. Sciences, Allborg Univ., 1999. +</ul> + +However, even the numerically stable version +can be computationally intractable if there are many hidden discrete +nodes, because the number of mixture components grows exponentially e.g., in a +<a href="usage_dbn.html#lds">switching linear dynamical system</a>. +In general, one must resort to approximate inference techniques: see +the discussion on <a href="#engines">inference engines</a> below. + + +<h2><a name="hybrid">Other hybrid models</h2> + +When we have C->D arcs, where C is hidden, we need to use +approximate inference. +One approach (not implemented in BNT) is described in +<ul> +<li> <a +href="http://www.cs.berkeley.edu/~murphyk/Papers/hybrid_uai99.ps.gz">A +Variational Approximation for Bayesian Networks with +Discrete and Continuous Latent Variables</a>, +K. Murphy, UAI 99. +</ul> +Of course, one can always use <a href="#sampling">sampling</a> methods +for approximate inference in such models. + + + +<h1><a name="param_learning">Parameter Learning</h1> + +The parameter estimation routines in BNT can be classified into 4 +types, depending on whether the goal is to compute +a full (Bayesian) posterior over the parameters or just a point +estimate (e.g., Maximum Likelihood or Maximum A Posteriori), +and whether all the variables are fully observed or there is missing +data/ hidden variables (partial observability). +<p> + +<TABLE BORDER> +<tr> + <TH></TH> + <th>Full obs</th> + <th>Partial obs</th> +</tr> +<tr> + <th>Point</th> + <td><tt>learn_params</tt></td> + <td><tt>learn_params_em</tt></td> +</tr> +<tr> + <th>Bayes</th> + <td><tt>bayes_update_params</tt></td> + <td>not yet supported</td> +</tr> +</table> + + +<h2><a name="load_data">Loading data from a file</h2> + +To load numeric data from an ASCII text file called 'dat.txt', where each row is a +case and columns are separated by white-space, such as +<pre> +011979 1626.5 0.0 +021979 1367.0 0.0 +... +</pre> +you can use +<pre> +data = load('dat.txt'); +</pre> +or +<pre> +load dat.txt -ascii +</pre> +In the latter case, the data is stored in a variable called 'dat' (the +filename minus the extension). +Alternatively, suppose the data is stored in a .csv file (has commas +separating the columns, and contains a header line), such as +<pre> +header info goes here +ORD,011979,1626.5,0.0 +DSM,021979,1367.0,0.0 +... +</pre> +You can load this using +<pre> +[a,b,c,d] = textread('dat.txt', '%s %d %f %f', 'delimiter', ',', 'headerlines', 1); +</pre> +If your file is not in either of these formats, you can either use Perl to convert +it to this format, or use the Matlab scanf command. +Type +<tt> +help iofun +</tt> +for more information on Matlab's file functions. +<!-- +<p> +To load data directly from Excel, +you should buy the +<a href="http://www.mathworks.com/products/excellink/">Excel Link</a>. +To load data directly from a relational database, +you should buy the +<a href="http://www.mathworks.com/products/database">Database +toolbox</a>. +--> +<p> +BNT learning routines require data to be stored in a cell array. +data{i,m} is the value of node i in case (example) m, i.e., each +<em>column</em> is a case. +If node i is not observed in case m (missing value), set +data{i,m} = []. +(Not all the learning routines can cope with such missing values, however.) +In the special case that all the nodes are observed and are +scalar-valued (as opposed to vector-valued), the data can be +stored in a matrix (as opposed to a cell-array). +<p> +Suppose, as in the <a href="#mixexp">mixture of experts example</a>, +that we have 3 nodes in the graph: X(1) is the observed input, X(3) is +the observed output, and X(2) is a hidden (gating) node. We can +create the dataset as follows. +<pre> +data = load('dat.txt'); +ncases = size(data, 1); +cases = cell(3, ncases); +cases([1 3], :) = num2cell(data'); +</pre> +Notice how we transposed the data, to convert rows into columns. +Also, cases{2,m} = [] for all m, since X(2) is always hidden. + + +<h2><a name="mle_complete">Maximum likelihood parameter estimation from complete data</h2> + +As an example, let's generate some data from the sprinkler network, randomize the parameters, +and then try to recover the original model. +First we create some training data using forwards sampling. +<pre> +samples = cell(N, nsamples); +for i=1:nsamples + samples(:,i) = sample_bnet(bnet); +end +</pre> +samples{j,i} contains the value of the j'th node in case i. +sample_bnet returns a cell array because, in general, each node might +be a vector of different length. +In this case, all nodes are discrete (and hence scalars), so we +could have used a regular array instead (which can be quicker): +<pre> +data = cell2num(samples); +</pre +So now data(j,i) = samples{j,i}. +<p> +Now we create a network with random parameters. +(The initial values of bnet2 don't matter in this case, since we can find the +globally optimal MLE independent of where we start.) +<pre> +% Make a tabula rasa +bnet2 = mk_bnet(dag, node_sizes); +seed = 0; +rand('state', seed); +bnet2.CPD{C} = tabular_CPD(bnet2, C); +bnet2.CPD{R} = tabular_CPD(bnet2, R); +bnet2.CPD{S} = tabular_CPD(bnet2, S); +bnet2.CPD{W} = tabular_CPD(bnet2, W); +</pre> +Finally, we find the maximum likelihood estimates of the parameters. +<pre> +bnet3 = learn_params(bnet2, samples); +</pre> +To view the learned parameters, we use a little Matlab hackery. +<pre> +CPT3 = cell(1,N); +for i=1:N + s=struct(bnet3.CPD{i}); % violate object privacy + CPT3{i}=s.CPT; +end +</pre> +Here are the parameters learned for node 4. +<pre> +dispcpt(CPT3{4}) +1 1 : 1.0000 0.0000 +2 1 : 0.2000 0.8000 +1 2 : 0.2273 0.7727 +2 2 : 0.0000 1.0000 +</pre> +So we see that the learned parameters are fairly close to the "true" +ones, which we display below. +<pre> +dispcpt(CPT{4}) +1 1 : 1.0000 0.0000 +2 1 : 0.1000 0.9000 +1 2 : 0.1000 0.9000 +2 2 : 0.0100 0.9900 +</pre> +We can get better results by using a larger training set, or using +informative priors (see <a href="#prior">below</a>). + + + +<h2><a name="prior">Parameter priors</h2> + +Currently, only tabular CPDs can have priors on their parameters. +The conjugate prior for a multinomial is the Dirichlet. +(For binary random variables, the multinomial is the same as the +Bernoulli, and the Dirichlet is the same as the Beta.) +<p> +The Dirichlet has a simple interpretation in terms of pseudo counts. +If we let N_ijk = the num. times X_i=k and Pa_i=j occurs in the +training set, where Pa_i are the parents of X_i, +then the maximum likelihood (ML) estimate is +T_ijk = N_ijk / N_ij (where N_ij = sum_k' N_ijk'), which will be 0 if N_ijk=0. +To prevent us from declaring that (X_i=k, Pa_i=j) is impossible just because this +event was not seen in the training set, +we can pretend we saw value k of X_i, for each value j of Pa_i some number (alpha_ijk) +of times in the past. +The MAP (maximum a posterior) estimate is then +<pre> +T_ijk = (N_ijk + alpha_ijk) / (N_ij + alpha_ij) +</pre> +and is never 0 if all alpha_ijk > 0. +For example, consider the network A->B, where A is binary and B has 3 +values. +A uniform prior for B has the form +<pre> + B=1 B=2 B=3 +A=1 1 1 1 +A=2 1 1 1 +</pre> +which can be created using +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'unif'); +</pre> +This prior does not satisfy the likelihood equivalence principle, +which says that <a href="#markov_equiv">Markov equivalent</a> models +should have the same marginal likelihood. +A prior that does satisfy this principle is shown below. +Heckerman (1995) calls this the +BDeu prior (likelihood equivalent uniform Bayesian Dirichlet). +<pre> + B=1 B=2 B=3 +A=1 1/6 1/6 1/6 +A=2 1/6 1/6 1/6 +</pre> +where we put N/(q*r) in each bin; N is the equivalent sample size, +r=|A|, q = |B|. +This can be created as follows +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'BDeu'); +</pre> +Here, 1 is the equivalent sample size, and is the strength of the +prior. +You can change this using +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', ... + 'BDeu', 'dirichlet_weight', 10); +</pre> +<!--where counts is an array of pseudo-counts of the same size as the +CPT.--> +<!-- +<p> +When you specify a prior, you should set row i of the CPT to the +normalized version of row i of the pseudo-count matrix, i.e., to the +expected values of the parameters. This will ensure that computing the +marginal likelihood sequentially (see <a +href="#bayes_learn">below</a>) and in batch form gives the same +results. +To do this, proceed as follows. +<pre> +tabular_CPD(bnet, i, 'prior', counts, 'CPT', mk_stochastic(counts)); +</pre> +For a non-informative prior, you can just write +<pre> +tabular_CPD(bnet, i, 'prior', 'unif', 'CPT', 'unif'); +</pre> +--> + + +<h2><a name="bayes_learn">(Sequential) Bayesian parameter updating from complete data</h2> + +If we use conjugate priors and have fully observed data, we can +compute the posterior over the parameters in batch form as follows. +<pre> +cases = sample_bnet(bnet, nsamples); +bnet = bayes_update_params(bnet, cases); +LL = log_marg_lik_complete(bnet, cases); +</pre> +bnet.CPD{i}.prior contains the new Dirichlet pseudocounts, +and bnet.CPD{i}.CPT is set to the mean of the posterior (the +normalized counts). +(Hence if the initial pseudo counts are 0, +<tt>bayes_update_params</tt> and <tt>learn_params</tt> will give the +same result.) + + + + +<p> +We can compute the same result sequentially (on-line) as follows. +<pre> +LL = 0; +for m=1:nsamples + LL = LL + log_marg_lik_complete(bnet, cases(:,m)); + bnet = bayes_update_params(bnet, cases(:,m)); +end +</pre> + +The file <tt>BNT/examples/static/StructLearn/model_select1</tt> has an example of +sequential model selection which uses the same idea. +We generate data from the model A->B +and compute the posterior prob of all 3 dags on 2 nodes: + (1) A B, (2) A <- B , (3) A -> B +Models 2 and 3 are <a href="#markov_equiv">Markov equivalent</a>, and therefore indistinguishable from +observational data alone, so we expect their posteriors to be the same +(assuming a prior which satisfies likelihood equivalence). +If we use random parameters, the "true" model only gets a higher posterior after 2000 trials! +However, if we make B a noisy NOT gate, the true model "wins" after 12 +trials, as shown below (red = model 1, blue/green (superimposed) +represents models 2/3). +<p> +<img src="Figures/model_select.png"> +<p> +The use of marginal likelihood for model selection is discussed in +greater detail in the +section on <a href="structure_learning">structure learning</a>. + + + + +<h2><a name="em">Maximum likelihood parameter estimation with missing values</h2> + +Now we consider learning when some values are not observed. +Let us randomly hide half the values generated from the water +sprinkler example. +<pre> +samples2 = samples; +hide = rand(N, nsamples) > 0.5; +[I,J]=find(hide); +for k=1:length(I) + samples2{I(k), J(k)} = []; +end +</pre> +samples2{i,l} is the value of node i in training case l, or [] if unobserved. +<p> +Now we will compute the MLEs using the EM algorithm. +We need to use an inference algorithm to compute the expected +sufficient statistics in the E step; the M (maximization) step is as +above. +<pre> +engine2 = jtree_inf_engine(bnet2); +max_iter = 10; +[bnet4, LLtrace] = learn_params_em(engine2, samples2, max_iter); +</pre> +LLtrace(i) is the log-likelihood at iteration i. We can plot this as +follows: +<pre> +plot(LLtrace, 'x-') +</pre> +Let's display the results after 10 iterations of EM. +<pre> +celldisp(CPT4) +CPT4{1} = + 0.6616 + 0.3384 +CPT4{2} = + 0.6510 0.3490 + 0.8751 0.1249 +CPT4{3} = + 0.8366 0.1634 + 0.0197 0.9803 +CPT4{4} = +(:,:,1) = + 0.8276 0.0546 + 0.5452 0.1658 +(:,:,2) = + 0.1724 0.9454 + 0.4548 0.8342 +</pre> +We can get improved performance by using one or more of the following +methods: +<ul> +<li> Increasing the size of the training set. +<li> Decreasing the amount of hidden data. +<li> Running EM for longer. +<li> Using informative priors. +<li> Initialising EM from multiple starting points. +</ul> + +Click <a href="#gaussian">here</a> for a discussion of learning +Gaussians, which can cause numerical problems. +<p> +For a more complete example of learning with EM, +see the script BNT/examples/static/learn1.m. + +<h2><a name="tying">Parameter tying</h2> + +In networks with repeated structure (e.g., chains and grids), it is +common to assume that the parameters are the same at every node. This +is called parameter tying, and reduces the amount of data needed for +learning. +<p> +When we have tied parameters, there is no longer a one-to-one +correspondence between nodes and CPDs. +Rather, each CPD species the parameters for a whole equivalence class +of nodes. +It is easiest to see this by example. +Consider the following <a href="usage_dbn.html#hmm">hidden Markov +model (HMM)</a> +<p> +<img src="Figures/hmm3.gif"> +<p> +<!-- +We can create this graph structure, assuming we have T time-slices, +as follows. +(We number the nodes as shown in the figure, but we could equally well +number the hidden nodes 1:T, and the observed nodes T+1:2T.) +<pre> +N = 2*T; +dag = zeros(N); +hnodes = 1:2:2*T; +for i=1:T-1 + dag(hnodes(i), hnodes(i+1))=1; +end +onodes = 2:2:2*T; +for i=1:T + dag(hnodes(i), onodes(i)) = 1; +end +</pre> +<p> +The hidden nodes are always discrete, and have Q possible values each, +but the observed nodes can be discrete or continuous, and have O possible values/length. +<pre> +if cts_obs + dnodes = hnodes; +else + dnodes = 1:N; +end +ns = ones(1,N); +ns(hnodes) = Q; +ns(onodes) = O; +</pre> +--> +When HMMs are used for semi-infinite processes like speech recognition, +we assume the transition matrix +P(H(t+1)|H(t)) is the same for all t; this is called a time-invariant +or homogenous Markov chain. +Hence hidden nodes 2, 3, ..., T +are all in the same equivalence class, say class Hclass. +Similarly, the observation matrix P(O(t)|H(t)) is assumed to be the +same for all t, so the observed nodes are all in the same equivalence +class, say class Oclass. +Finally, the prior term P(H(1)) is in a class all by itself, say class +H1class. +This is illustrated below, where we explicitly represent the +parameters as random variables (dotted nodes). +<p> +<img src="Figures/hmm4_params.gif"> +<p> +In BNT, we cannot represent parameters as random variables (nodes). +Instead, we "hide" the +parameters inside one CPD for each equivalence class, +and then specify that the other CPDs should share these parameters, as +follows. +<pre> +hnodes = 1:2:2*T; +onodes = 2:2:2*T; +H1class = 1; Hclass = 2; Oclass = 3; +eclass = ones(1,N); +eclass(hnodes(2:end)) = Hclass; +eclass(hnodes(1)) = H1class; +eclass(onodes) = Oclass; +% create dag and ns in the usual way +bnet = mk_bnet(dag, ns, 'discrete', dnodes, 'equiv_class', eclass); +</pre> +Finally, we define the parameters for each equivalence class: +<pre> +bnet.CPD{H1class} = tabular_CPD(bnet, hnodes(1)); % prior +bnet.CPD{Hclass} = tabular_CPD(bnet, hnodes(2)); % transition matrix +if cts_obs + bnet.CPD{Oclass} = gaussian_CPD(bnet, onodes(1)); +else + bnet.CPD{Oclass} = tabular_CPD(bnet, onodes(1)); +end +</pre> +In general, if bnet.CPD{e} = xxx_CPD(bnet, j), then j should be a +member of e's equivalence class; that is, it is not always the case +that e == j. You can use bnet.rep_of_eclass(e) to return the +representative of equivalence class e. +BNT will look up the parents of j to determine the size +of the CPT to use. It assumes that this is the same for all members of +the equivalence class. +Click <a href="param_tieing.html">here</a> for +a more complex example of parameter tying. +<p> +Note: +Normally one would define an HMM as a +<a href = "usage_dbn.html">Dynamic Bayes Net</a> +(see the function BNT/examples/dynamic/mk_chmm.m). +However, one can define an HMM as a static BN using the function +BNT/examples/static/Models/mk_hmm_bnet.m. + + + +<h1><a name="structure_learning">Structure learning</h1> + +Update (9/29/03): +Phillipe LeRay is developing some additional structure learning code +on top of BNT. Click +<a href="http://banquiseasi.insa-rouen.fr/projects/bnt-slp/"> +here</a> +for details. + +<p> + +There are two very different approaches to structure learning: +constraint-based and search-and-score. +In the <a href="#constraint">constraint-based approach</a>, +we start with a fully connected graph, and remove edges if certain +conditional independencies are measured in the data. +This has the disadvantage that repeated independence tests lose +statistical power. +<p> +In the more popular search-and-score approach, +we perform a search through the space of possible DAGs, and either +return the best one found (a point estimate), or return a sample of the +models found (an approximation to the Bayesian posterior). +<p> +The number of DAGs as a function of the number of +nodes, G(n), is super-exponential in n, +and is given by the following recurrence +<!--(where R(i)=G(n)):--> +<p> +<center> +<IMG SRC="numDAGsEqn2.png"> +</center> +<p> +The first few values +are shown below. + +<table> +<tr> <th>n</th> <th align=left>G(n)</th> </tr> +<tr> <td>1</td> <td>1</td> </tr> +<tr> <td>2</td> <td>3</td> </tr> +<tr> <td>3</td> <td>25</td> </tr> +<tr> <td>4</td> <td>543</td> </tr> +<tr> <td>5</td> <td>29,281</td> </tr> +<tr> <td>6</td> <td>3,781,503</td> </tr> +<tr> <td>7</td> <td>1.1 x 10^9</td> </tr> +<tr> <td>8</td> <td>7.8 x 10^11</td> </tr> +<tr> <td>9</td> <td>1.2 x 10^15</td> </tr> +<tr> <td>10</td> <td>4.2 x 10^18</td> </tr> +</table> + +Since the number of DAGs is super-exponential in the number of nodes, +we cannot exhaustively search the space, so we either use a local +search algorithm (e.g., greedy hill climbining, perhaps with multiple +restarts) or a global search algorithm (e.g., Markov Chain Monte +Carlo). +<p> +If we know a total ordering on the nodes, +finding the best structure amounts to picking the best set of parents +for each node independently. +This is what the K2 algorithm does. +If the ordering is unknown, we can search over orderings, +which is more efficient than searching over DAGs (Koller and Friedman, 2000). +<p> +In addition to the search procedure, we must specify the scoring +function. There are two popular choices. The Bayesian score integrates +out the parameters, i.e., it is the marginal likelihood of the model. +The BIC (Bayesian Information Criterion) is defined as +log P(D|theta_hat) - 0.5*d*log(N), where D is the data, theta_hat is +the ML estimate of the parameters, d is the number of parameters, and +N is the number of data cases. +The BIC method has the advantage of not requiring a prior. +<p> +BIC can be derived as a large sample +approximation to the marginal likelihood. +(It is also equal to the Minimum Description Length of a model.) +However, in practice, the sample size does not need to be very large +for the approximation to be good. +For example, in the figure below, we plot the ratio between the log marginal likelihood +and the BIC score against data-set size; we see that the ratio rapidly +approaches 1, especially for non-informative priors. +(This plot was generated by the file BNT/examples/static/bic1.m. It +uses the water sprinkler BN with BDeu Dirichlet priors with different +equivalent sample sizes.) + +<p> +<center> +<IMG SRC="Figures/bic.png"> +</center> +<p> + +<p> +As with parameter learning, handling missing data/ hidden variables is +much harder than the fully observed case. +The structure learning routines in BNT can therefore be classified into 4 +types, analogously to the parameter learning case. +<p> + +<TABLE BORDER> +<tr> + <TH></TH> + <th>Full obs</th> + <th>Partial obs</th> +</tr> +<tr> + <th>Point</th> + <td><tt>learn_struct_K2</tt> <br> +<!-- <tt>learn_struct_hill_climb</tt></td> --> + <td><tt>not yet supported</tt></td> +</tr> +<tr> + <th>Bayes</th> + <td><tt>learn_struct_mcmc</tt></td> + <td>not yet supported</td> +</tr> +</table> + + +<h2><a name="markov_equiv">Markov equivalence</h2> + +If two DAGs encode the same conditional independencies, they are +called Markov equivalent. The set of all DAGs can be paritioned into +Markov equivalence classes. Graphs within the same class can +have +the direction of some of their arcs reversed without changing any of +the CI relationships. +Each class can be represented by a PDAG +(partially directed acyclic graph) called an essential graph or +pattern. This specifies which edges must be oriented in a certain +direction, and which may be reversed. + +<p> +When learning graph structure from observational data, +the best one can hope to do is to identify the model up to Markov +equivalence. To distinguish amongst graphs within the same equivalence +class, one needs interventional data: see the discussion on <a +href="#active">active learning</a> below. + + + +<h2><a name="enumerate">Exhaustive search</h2> + +The brute-force approach to structure learning is to enumerate all +possible DAGs, and score each one. This provides a "gold standard" +with which to compare other algorithms. We can do this as follows. +<pre> +dags = mk_all_dags(N); +score = score_dags(data, ns, dags); +</pre> +where data(i,m) is the value of node i in case m, +and ns(i) is the size of node i. +If the DAGs have a lot of families in common, we can cache the sufficient statistics, +making this potentially more efficient than scoring the DAGs one at a time. +(Caching is not currently implemented, however.) +<p> +By default, we use the Bayesian scoring metric, and assume CPDs are +represented by tables with BDeu(1) priors. +We can override these defaults as follows. +If we want to use uniform priors, we can say +<pre> +params = cell(1,N); +for i=1:N + params{i} = {'prior', 'unif'}; +end +score = score_dags(data, ns, dags, 'params', params); +</pre> +params{i} is a cell-array, containing optional arguments that are +passed to the constructor for CPD i. +<p> +Now suppose we want to use different node types, e.g., +Suppose nodes 1 and 2 are Gaussian, and nodes 3 and 4 softmax (both +these CPDs can support discrete and continuous parents, which is +necessary since all other nodes will be considered as parents). +The Bayesian scoring metric currently only works for tabular CPDs, so +we will use BIC: +<pre> +score = score_dags(data, ns, dags, 'discrete', [3 4], 'params', [], + 'type', {'gaussian', 'gaussian', 'softmax', softmax'}, 'scoring_fn', 'bic') +</pre> +In practice, one can't enumerate all possible DAGs for N > 5, +but one can evaluate any reasonably-sized set of hypotheses in this +way (e.g., nearest neighbors of your current best guess). +Think of this as "computer assisted model refinement" as opposed to de +novo learning. + + +<h2><a name="K2">K2</h2> + +The K2 algorithm (Cooper and Herskovits, 1992) is a greedy search algorithm that works as follows. +Initially each node has no parents. It then adds incrementally that parent whose addition most +increases the score of the resulting structure. When the addition of no single +parent can increase the score, it stops adding parents to the node. +Since we are using a fixed ordering, we do not need to check for +cycles, and can choose the parents for each node independently. +<p> +The original paper used the Bayesian scoring +metric with tabular CPDs and Dirichlet priors. +BNT generalizes this to allow any kind of CPD, and either the Bayesian +scoring metric or BIC, as in the example <a href="#enumerate">above</a>. +In addition, you can specify +an optional upper bound on the number of parents for each node. +The file BNT/examples/static/k2demo1.m gives an example of how to use K2. +We use the water sprinkler network and sample 100 cases from it as before. +Then we see how much data it takes to recover the generating structure: +<pre> +order = [C S R W]; +max_fan_in = 2; +sz = 5:5:100; +for i=1:length(sz) + dag2 = learn_struct_K2(data(:,1:sz(i)), node_sizes, order, 'max_fan_in', max_fan_in); + correct(i) = isequal(dag, dag2); +end +</pre> +Here are the results. +<pre> +correct = + Columns 1 through 12 + 0 0 0 0 0 0 0 1 0 1 1 1 + Columns 13 through 20 + 1 1 1 1 1 1 1 1 +</pre> +So we see it takes about sz(10)=50 cases. (BIC behaves similarly, +showing that the prior doesn't matter too much.) +In general, we cannot hope to recover the "true" generating structure, +only one that is in its <a href="#markov_equiv">Markov equivalence +class</a>. + + +<h2><a name="hill_climb">Hill-climbing</h2> + +Hill-climbing starts at a specific point in space, +considers all nearest neighbors, and moves to the neighbor +that has the highest score; if no neighbors have higher +score than the current point (i.e., we have reached a local maximum), +the algorithm stops. One can then restart in another part of the space. +<p> +A common definition of "neighbor" is all graphs that can be +generated from the current graph by adding, deleting or reversing a +single arc, subject to the acyclicity constraint. +Other neighborhoods are possible: see +<a href="http://research.microsoft.com/~dmax/publications/jmlr02.pdf"> +Optimal Structure Identification with Greedy Search</a>, Max +Chickering, JMLR 2002. + +<!-- +Note: This algorithm is currently (Feb '02) being implemented by Qian +Diao. +--> + + +<h2><a name="mcmc">MCMC</h2> + +We can use a Markov Chain Monte Carlo (MCMC) algorithm called +Metropolis-Hastings (MH) to search the space of all +DAGs. +The standard proposal distribution is to consider moving to all +nearest neighbors in the sense defined <a href="#hill_climb">above</a>. +<p> +The function can be called +as in the following example. +<pre> +[sampled_graphs, accept_ratio] = learn_struct_mcmc(data, ns, 'nsamples', 100, 'burnin', 10); +</pre> +We can convert our set of sampled graphs to a histogram +(empirical posterior over all the DAGs) thus +<pre> +all_dags = mk_all_dags(N); +mcmc_post = mcmc_sample_to_hist(sampled_graphs, all_dags); +</pre> +To see how well this performs, let us compute the exact posterior exhaustively. +<p> +<pre> +score = score_dags(data, ns, all_dags); +post = normalise(exp(score)); % assuming uniform structural prior +</pre> +We plot the results below. +(The data set was 100 samples drawn from a random 4 node bnet; see the +file BNT/examples/static/mcmc1.) +<pre> +subplot(2,1,1) +bar(post) +subplot(2,1,2) +bar(mcmc_post) +</pre> +<img src="Figures/mcmc_post.jpg" width="800" height="500"> +<p> +We can also plot the acceptance ratio versus number of MCMC steps, +as a crude convergence diagnostic. +<pre> +clf +plot(accept_ratio) +</pre> +<img src="Figures/mcmc_accept.jpg" width="800" height="300"> +<p> +Even though the number of samples needed by MCMC is theoretically +polynomial (not exponential) in the dimensionality of the search space, in practice it has been +found that MCMC does not converge in reasonable time for graphs with +more than about 10 nodes. + + + + +<h2><a name="active">Active structure learning</h2> + +As was mentioned <a href="#markov_equiv">above</a>, +one can only learn a DAG up to Markov equivalence, even given infinite data. +If one is interested in learning the structure of a causal network, +one needs interventional data. +(By "intervention" we mean forcing a node to take on a specific value, +thereby effectively severing its incoming arcs.) +<p> +Most of the scoring functions accept an optional argument +that specifies whether a node was observed to have a certain value, or +was forced to have that value: we set clamped(i,m)=1 if node i was +forced in training case m. e.g., see the file +BNT/examples/static/cooper_yoo. +<p> +An interesting question is to decide which interventions to perform +(c.f., design of experiments). For details, see the following tech +report +<ul> +<li> <a href = "../../Papers/alearn.ps.gz"> +Active learning of causal Bayes net structure</a>, Kevin Murphy, March +2001. +</ul> + + +<h2><a name="struct_em">Structural EM</h2> + +Computing the Bayesian score when there is partial observability is +computationally challenging, because the parameter posterior becomes +multimodal (the hidden nodes induce a mixture distribution). +One therefore needs to use approximations such as BIC. +Unfortunately, search algorithms are still expensive, because we need +to run EM at each step to compute the MLE, which is needed to compute +the score of each model. An alternative approach is +to do the local search steps inside of the M step of EM, which is more +efficient since the data has been "filled in" - this is +called the structural EM algorithm (Friedman 1997), and provably +converges to a local maximum of the BIC score. +<p> +Wei Hu has implemented SEM for discrete nodes. +You can download his package from +<a href="../SEM.zip">here</a>. +Please address all questions about this code to +wei.hu@intel.com. +See also <a href="#phl">Phl's implementation of SEM</a>. + +<!-- +<h2><a name="reveal">REVEAL algorithm</h2> + +A simple way to learn the structure of a fully observed, discrete, +factored DBN from a time series is described <a +href="usage_dbn.html#struct_learn">here</a>. +--> + + +<h2><a name="graphdraw">Visualizing the graph</h2> + +Click <a href="graphviz.html">here</a> for more information +on graph visualization. + +<h2><a name = "constraint">Constraint-based methods</h2> + +The IC algorithm (Pearl and Verma, 1991), +and the faster, but otherwise equivalent, PC algorithm (Spirtes, Glymour, and Scheines 1993), +computes many conditional independence tests, +and combines these constraints into a +PDAG to represent the whole +<a href="#markov_equiv">Markov equivalence class</a>. +<p> +IC*/FCI extend IC/PC to handle latent variables: see <a href="#ic_star">below</a>. +(IC stands for inductive causation; PC stands for Peter and Clark, +the first names of Spirtes and Glymour; FCI stands for fast causal +inference. +What we, following Pearl (2000), call IC* was called +IC in the original Pearl and Verma paper.) +For details, see +<ul> +<li> +<a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Causation, +Prediction, and Search</a>, Spirtes, Glymour and +Scheines (SGS), 2001 (2nd edition), MIT Press. +<li> +<a href="http://bayes.cs.ucla.edu/BOOK-2K/index.html">Causality: Models, Reasoning and Inference</a>, J. Pearl, +2000, Cambridge University Press. +</ul> + +<p> + +The PC algorithm takes as arguments a function f, the number of nodes N, +the maximum fan in K, and additional arguments A which are passed to f. +The function f(X,Y,S,A) returns 1 if X is conditionally independent of Y given S, and 0 +otherwise. +For example, suppose we cheat by +passing in a CI "oracle" which has access to the true DAG; the oracle +tests for d-separation in this DAG, i.e., +f(X,Y,S) calls dsep(X,Y,S,dag). We can to this as follows. +<pre> +pdag = learn_struct_pdag_pc('dsep', N, max_fan_in, dag); +</pre> +pdag(i,j) = -1 if there is definitely an i->j arc, +and pdag(i,j) = 1 if there is either an i->j or and i<-j arc. +<p> +Applied to the sprinkler network, this returns +<pre> +pdag = + 0 1 1 0 + 1 0 0 -1 + 1 0 0 -1 + 0 0 0 0 +</pre> +So as expected, we see that the V-structure at the W node is uniquely identified, +but the other arcs have ambiguous orientation. +<p> +We now give an example from p141 (1st edn) / p103 (2nd end) of the SGS +book. +This example concerns the female orgasm. +We are given a correlation matrix C between 7 measured factors (such +as subjective experiences of coital and masturbatory experiences), +derived from 281 samples, and want to learn a causal model of the +data. We will not discuss the merits of this type of work here, but +merely show how to reproduce the results in the SGS book. +Their program, +<a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Tetrad</a>, +makes use of the Fisher Z-test for conditional +independence, so we do the same: +<pre> +max_fan_in = 4; +nsamples = 281; +alpha = 0.05; +pdag = learn_struct_pdag_pc('cond_indep_fisher_z', n, max_fan_in, C, nsamples, alpha); +</pre> +In this case, the CI test is +<pre> +f(X,Y,S) = cond_indep_fisher_z(X,Y,S, C,nsamples,alpha) +</pre> +The results match those of Fig 12a of SGS apart from two edge +differences; presumably this is due to rounding error (although it +could be a bug, either in BNT or in Tetrad). +This example can be found in the file BNT/examples/static/pc2.m. + +<p> + +The IC* algorithm (Pearl and Verma, 1991), +and the faster FCI algorithm (Spirtes, Glymour, and Scheines 1993), +are like the IC/PC algorithm, except that they can detect the presence +of latent variables. +See the file <tt>learn_struct_pdag_ic_star</tt> written by Tamar +Kushnir. The output is a matrix P, defined as follows +(see Pearl (2000), p52 for details): +<pre> +% P(i,j) = -1 if there is either a latent variable L such that i <-L->j OR there is a directed edge from i->j. +% P(i,j) = -2 if there is a marked directed i-*>j edge. +% P(i,j) = P(j,i) = 1 if there is and undirected edge i--j +% P(i,j) = P(j,i) = 2 if there is a latent variable L such that i<-L->j. +</pre> + + +<h2><a name="phl">Philippe Leray's structure learning package</h2> + +Philippe Leray has written a +<a href="http://bnt.insa-rouen.fr/ajouts.html"> +structure learning package</a> that uses BNT. + +It currently (Juen 2003) has the following features: +<ul> +<li>PC with Chi2 statistical test +<li> MWST : Maximum weighted Spanning Tree +<li> Hill Climbing +<li> Greedy Search +<li> Structural EM +<li> hist_ic : optimal Histogram based on IC information criterion +<li> cpdag_to_dag +<li> dag_to_cpdag +<li> ... +</ul> + + +</a> + + +<!-- +<h2><a name="read_learning">Further reading on learning</h2> + +I recommend the following tutorials for more details on learning. +<ul> +<li> <a +href="http://www.cs.berkeley.edu/~murphyk/Papers/intel.ps.gz">My short +tutorial</a> on graphical models, which contains an overview of learning. + +<li> +<A HREF="ftp://ftp.research.microsoft.com/pub/tr/TR-95-06.PS"> +A tutorial on learning with Bayesian networks</a>, D. Heckerman, +Microsoft Research Tech Report, 1995. + +<li> <A HREF="http://www-cad.eecs.berkeley.edu/~wray/Mirror/lwgmja"> +Operations for Learning with Graphical Models</a>, +W. L. Buntine, JAIR'94, 159--225. +</ul> +<p> +--> + + + + + +<h1><a name="engines">Inference engines</h1> + +Up until now, we have used the junction tree algorithm for inference. +However, sometimes this is too slow, or not even applicable. +In general, there are many inference algorithms each of which make +different tradeoffs between speed, accuracy, complexity and +generality. Furthermore, there might be many implementations of the +same algorithm; for instance, a general purpose, readable version, +and a highly-optimized, specialized one. +To cope with this variety, we treat each inference algorithm as an +object, which we call an inference engine. + +<p> +An inference engine is an object that contains a bnet and supports the +'enter_evidence' and 'marginal_nodes' methods. The engine constructor +takes the bnet as argument and may do some model-specific processing. +When 'enter_evidence' is called, the engine may do some +evidence-specific processing. Finally, when 'marginal_nodes' is +called, the engine may do some query-specific processing. + +<p> +The amount of work done when each stage is specified -- structure, +parameters, evidence, and query -- depends on the engine. The cost of +work done early in this sequence can be amortized. On the other hand, +one can make better optimizations if one waits until later in the +sequence. +For example, the parameters might imply +conditional indpendencies that are not evident in the graph structure, +but can nevertheless be exploited; the evidence indicates which nodes +are observed and hence can effectively be disconnected from the +graph; and the query might indicate that large parts of the network +are d-separated from the query nodes. (Since it is not the actual +<em>values</em> of the evidence that matters, just which nodes are observed, +many engines allow you to specify which nodes will be observed when they are constructed, +i.e., before calling 'enter_evidence'. Some engines can still cope if +the actual pattern of evidence is different, e.g., if there is missing +data.) +<p> + +Although being maximally lazy (i.e., only doing work when a query is +issued) may seem desirable, +this is not always the most efficient. +For example, +when learning using EM, we need to call marginal_nodes N times, where N is the +number of nodes. <a href="varelim">Variable elimination</a> would end +up repeating a lot of work +each time marginal_nodes is called, making it inefficient for +learning. The junction tree algorithm, by contrast, uses dynamic +programming to avoid this redundant computation --- it calculates all +marginals in two passes during 'enter_evidence', so calling +'marginal_nodes' takes constant time. +<p> +We will discuss some of the inference algorithms implemented in BNT +below, and finish with a <a href="#engine_summary">summary</a> of all +of them. + + + + + + + +<h2><a name="varelim">Variable elimination</h2> + +The variable elimination algorithm, also known as bucket elimination +or peeling, is one of the simplest inference algorithms. +The basic idea is to "push sums inside of products"; this is explained +in more detail +<a +href="http://HTTP.CS.Berkeley.EDU/~murphyk/Bayes/bayes.html#infer">here</a>. +<p> +The principle of distributing sums over products can be generalized +greatly to apply to any commutative semiring. +This forms the basis of many common algorithms, such as Viterbi +decoding and the Fast Fourier Transform. For details, see + +<ul> +<li> R. McEliece and S. M. Aji, 2000. +<!--<a href="http://www.systems.caltech.edu/EE/Faculty/rjm/papers/GDL.ps">--> +<a href="GDL.pdf"> +The Generalized Distributive Law</a>, +IEEE Trans. Inform. Theory, vol. 46, no. 2 (March 2000), +pp. 325--343. + + +<li> +F. R. Kschischang, B. J. Frey and H.-A. Loeliger, 2001. +<a href="http://www.cs.toronto.edu/~frey/papers/fgspa.abs.html"> +Factor graphs and the sum-product algorithm</a> +IEEE Transactions on Information Theory, February, 2001. + +</ul> + +<p> +Choosing an order in which to sum out the variables so as to minimize +computational cost is known to be NP-hard. +The implementation of this algorithm in +<tt>var_elim_inf_engine</tt> makes no attempt to optimize this +ordering (in contrast, say, to <tt>jtree_inf_engine</tt>, which uses a +greedy search procedure to find a good ordering). +<p> +Note: unlike most algorithms, var_elim does all its computational work +inside of <tt>marginal_nodes</tt>, not inside of +<tt>enter_evidence</tt>. + + + + +<h2><a name="global">Global inference methods</h2> + +The simplest inference algorithm of all is to explicitely construct +the joint distribution over all the nodes, and then to marginalize it. +This is implemented in <tt>global_joint_inf_engine</tt>. +Since the size of the joint is exponential in the +number of discrete (hidden) nodes, this is not a very practical algorithm. +It is included merely for pedagogical and debugging purposes. +<p> +Three specialized versions of this algorithm have also been implemented, +corresponding to the cases where all the nodes are discrete (D), all +are Gaussian (G), and some are discrete and some Gaussian (CG). +They are called <tt>enumerative_inf_engine</tt>, +<tt>gaussian_inf_engine</tt>, +and <tt>cond_gauss_inf_engine</tt> respectively. +<p> +Note: unlike most algorithms, these global inference algorithms do all their computational work +inside of <tt>marginal_nodes</tt>, not inside of +<tt>enter_evidence</tt>. + + +<h2><a name="quickscore">Quickscore</h2> + +The junction tree algorithm is quite slow on the <a href="#qmr">QMR</a> network, +since the cliques are so big. +One simple trick we can use is to notice that hidden leaves do not +affect the posteriors on the roots, and hence do not need to be +included in the network. +A second trick is to notice that the negative findings can be +"absorbed" into the prior: +see the file +BNT/examples/static/mk_minimal_qmr_bnet for details. +<p> + +A much more significant speedup is obtained by exploiting special +properties of the noisy-or node, as done by the quickscore +algorithm. For details, see +<ul> +<li> Heckerman, "A tractable inference algorithm for diagnosing multiple diseases", UAI 89. +<li> Rish and Dechter, "On the impact of causal independence", UCI +tech report, 1998. +</ul> + +This has been implemented in BNT as a special-purpose inference +engine, which can be created and used as follows: +<pre> +engine = quickscore_inf_engine(inhibit, leak, prior); +engine = enter_evidence(engine, pos, neg); +m = marginal_nodes(engine, i); +</pre> + + +<h2><a name="belprop">Belief propagation</h2> + +Even using quickscore, exact inference takes time that is exponential +in the number of positive findings. +Hence for large networks we need to resort to approximate inference techniques. +See for example +<ul> +<li> T. Jaakkola and M. Jordan, "Variational probabilistic inference and the +QMR-DT network", JAIR 10, 1999. + +<li> K. Murphy, Y. Weiss and M. Jordan, "Loopy belief propagation for approximate inference: an empirical study", + UAI 99. +</ul> +The latter approximation +entails applying Pearl's belief propagation algorithm to a model even +if it has loops (hence the name loopy belief propagation). +Pearl's algorithm, implemented as <tt>pearl_inf_engine</tt>, gives +exact results when applied to singly-connected graphs +(a.k.a. polytrees, since +the underlying undirected topology is a tree, but a node may have +multiple parents). +To apply this algorithm to a graph with loops, +use <tt>pearl_inf_engine</tt>. +This can use a centralized or distributed message passing protocol. +You can use it as in the following example. +<pre> +engine = pearl_inf_engine(bnet, 'max_iter', 30); +engine = enter_evidence(engine, evidence); +m = marginal_nodes(engine, i); +</pre> +We found that this algorithm often converges, and when it does, often +is very accurate, but it depends on the precise setting of the +parameter values of the network. +(See the file BNT/examples/static/qmr1 to repeat the experiment for yourself.) +Understanding when and why belief propagation converges/ works +is a topic of ongoing research. +<p> +<tt>pearl_inf_engine</tt> can exploit special structure in noisy-or +and gmux nodes to compute messages efficiently. +<p> +<tt>belprop_inf_engine</tt> is like pearl, but uses potentials to +represent messages. Hence this is slower. +<p> +<tt>belprop_fg_inf_engine</tt> is like belprop, +but is designed for factor graphs. + + + +<h2><a name="sampling">Sampling</h2> + +BNT now (Mar '02) has two sampling (Monte Carlo) inference algorithms: +<ul> +<li> <tt>likelihood_weighting_inf_engine</tt> which does importance +sampling and can handle any node type. +<li> <tt>gibbs_sampling_inf_engine</tt>, written by Bhaskara Marthi. +Currently this can only handle tabular CPDs. +For a much faster and more powerful Gibbs sampling program, see +<a href="http://www.mrc-bsu.cam.ac.uk/bugs">BUGS</a>. +</ul> +Note: To generate samples from a network (which is not the same as inference!), +use <tt>sample_bnet</tt>. + + + +<h2><a name="engine_summary">Summary of inference engines</h2> + + +The inference engines differ in many ways. Here are +some of the major "axes": +<ul> +<li> Works for all topologies or makes restrictions? +<li> Works for all node types or makes restrictions? +<li> Exact or approximate inference? +</ul> + +<p> +In terms of topology, most engines handle any kind of DAG. +<tt>belprop_fg</tt> does approximate inference on factor graphs (FG), which +can be used to represent directed, undirected, and mixed (chain) +graphs. +(In the future, we plan to support exact inference on chain graphs.) +<tt>quickscore</tt> only works on QMR-like models. +<p> +In terms of node types: algorithms that use potentials can handle +discrete (D), Gaussian (G) or conditional Gaussian (CG) models. +Sampling algorithms can essentially handle any kind of node (distribution). +Other algorithms make more restrictive assumptions in exchange for +speed. +<p> +Finally, most algorithms are designed to give the exact answer. +The belief propagation algorithms are exact if applied to trees, and +in some other cases. +Sampling is considered approximate, even though, in the limit of an +infinite number of samples, it gives the exact answer. + +<p> + +Here is a summary of the properties +of all the engines in BNT which work on static networks. +<p> +<table> +<table border units = pixels><tr> +<td align=left width=0>Name +<td align=left width=0>Exact? +<td align=left width=0>Node type? +<td align=left width=0>topology +<tr> +<tr> +<td align=left> belprop +<td align=left> approx +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> belprop_fg +<td align=left> approx +<td align=left> D +<td align=left> factor graph +<tr> +<td align=left> cond_gauss +<td align=left> exact +<td align=left> CG +<td align=left> DAG +<tr> +<td align=left> enumerative +<td align=left> exact +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> gaussian +<td align=left> exact +<td align=left> G +<td align=left> DAG +<tr> +<td align=left> gibbs +<td align=left> approx +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> global_joint +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +<tr> +<td align=left> jtree +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +b<tr> +<td align=left> likelihood_weighting +<td align=left> approx +<td align=left> any +<td align=left> DAG +<tr> +<td align=left> pearl +<td align=left> approx +<td align=left> D,G +<td align=left> DAG +<tr> +<td align=left> pearl +<td align=left> exact +<td align=left> D,G +<td align=left> polytree +<tr> +<td align=left> quickscore +<td align=left> exact +<td align=left> noisy-or +<td align=left> QMR +<tr> +<td align=left> stab_cond_gauss +<td align=left> exact +<td align=left> CG +<td align=left> DAG +<tr> +<td align=left> var_elim +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +</table> + + + +<h1><a name="influence">Influence diagrams/ decision making</h1> + +BNT implements an exact algorithm for solving LIMIDs (limited memory +influence diagrams), described in +<ul> +<li> S. L. Lauritzen and D. Nilsson. +<a href="http://www.math.auc.dk/~steffen/papers/limids.pdf"> +Representing and solving decision problems with limited +information</a> +Management Science, 47, 1238 - 1251. September 2001. +</ul> +LIMIDs explicitely show all information arcs, rather than implicitely +assuming no forgetting. This allows them to model forgetful +controllers. +<p> +See the examples in <tt>BNT/examples/limids</tt> for details. + + + + +<h1>DBNs, HMMs, Kalman filters and all that</h1> + +Click <a href="usage_dbn.html">here</a> for documentation about how to +use BNT for dynamical systems and sequence data. + + +</BODY> diff --git a/sourcecodes/bnt-master/docs/usage_02nov13.html b/sourcecodes/bnt-master/docs/usage_02nov13.html new file mode 100644 index 00000000..a1e24294 --- /dev/null +++ b/sourcecodes/bnt-master/docs/usage_02nov13.html @@ -0,0 +1,3215 @@ +<HEAD> +<TITLE>How to use the Bayes Net Toolbox</TITLE> +</HEAD> + +<BODY BGCOLOR="#FFFFFF"> +<!-- white background is better for the pictures and equations --> + +<h1>How to use the Bayes Net Toolbox</h1> + +This documentation was last updated on 13 November 2002. +<br> +Click <a href="changelog.html">here</a> for a list of changes made to +BNT. +<br> +Click +<a href="http://bnt.insa-rouen.fr/">here</a> +for a French version of this documentation (which might not +be up-to-date). + + +<p> + +<ul> +<li> <a href="#install">Installation</a> +<ul> +<li> <a href="#installM">Installing the Matlab code</a> +<li> <a href="#installC">Installing the C code</a> +<li> <a href="../matlab_tips.html">Useful Matlab tips</a>. +</ul> + +<li> <a href="#basics">Creating your first Bayes net</a> + <ul> + <li> <a href="#basics">Creating a model by hand</a> + <li> <a href="#file">Loading a model from a file</a> + <li> <a href="http://bnt.insa-rouen.fr/ajouts.html">Creating a model using a GUI</a> + </ul> + +<li> <a href="#inference">Inference</a> + <ul> + <li> <a href="#marginal">Computing marginal distributions</a> + <li> <a href="#joint">Computing joint distributions</a> + <li> <a href="#soft">Soft/virtual evidence</a> + <li> <a href="#mpe">Most probable explanation</a> + </ul> + +<li> <a href="#cpd">Conditional Probability Distributions</a> + <ul> + <li> <a href="#tabular">Tabular (multinomial) nodes</a> + <li> <a href="#noisyor">Noisy-or nodes</a> + <li> <a href="#deterministic">Other (noisy) deterministic nodes</a> + <li> <a href="#softmax">Softmax (multinomial logit) nodes</a> + <li> <a href="#mlp">Neural network nodes</a> + <li> <a href="#root">Root nodes</a> + <li> <a href="#gaussian">Gaussian nodes</a> + <li> <a href="#glm">Generalized linear model nodes</a> + <li> <a href="#dtree">Classification/regression tree nodes</a> + <li> <a href="#nongauss">Other continuous distributions</a> + <li> <a href="#cpd_summary">Summary of CPD types</a> + </ul> + +<li> <a href="#examples">Example models</a> + <ul> + <li> <a + href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt"> +Gaussian mixture models</a> + <li> <a href="#pca">PCA, ICA, and all that</a> + <li> <a href="#mixep">Mixtures of experts</a> + <li> <a href="#hme">Hierarchical mixtures of experts</a> + <li> <a href="#qmr">QMR</a> + <li> <a href="#cg_model">Conditional Gaussian models</a> + <li> <a href="#hybrid">Other hybrid models</a> + </ul> + +<li> <a href="#param_learning">Parameter learning</a> + <ul> + <li> <a href="#load_data">Loading data from a file</a> + <li> <a href="#mle_complete">Maximum likelihood parameter estimation from complete data</a> + <li> <a href="#prior">Parameter priors</a> + <li> <a href="#bayes_learn">(Sequential) Bayesian parameter updating from complete data</a> + <li> <a href="#em">Maximum likelihood parameter estimation with missing values (EM)</a> + <li> <a href="#tying">Parameter tying</a> + </ul> + +<li> <a href="#structure_learning">Structure learning</a> + <ul> + <li> <a href="#enumerate">Exhaustive search</a> + <li> <a href="#K2">K2</a> + <li> <a href="#hill_climb">Hill-climbing</a> + <li> <a href="#mcmc">MCMC</a> + <li> <a href="#active">Active learning</a> + <li> <a href="#struct_em">Structural EM</a> + <li> <a href="#graphdraw">Visualizing the learned graph structure</a> + <li> <a href="#constraint">Constraint-based methods</a> + </ul> + + +<li> <a href="#engines">Inference engines</a> + <ul> + <li> <a href="#jtree">Junction tree</a> + <li> <a href="#varelim">Variable elimination</a> + <li> <a href="#global">Global inference methods</a> + <li> <a href="#quickscore">Quickscore</a> + <li> <a href="#belprop">Belief propagation</a> + <li> <a href="#sampling">Sampling (Monte Carlo)</a> + <li> <a href="#engine_summary">Summary of inference engines</a> + </ul> + + +<li> <a href="#influence">Influence diagrams/ decision making</a> + + +<li> <a href="usage_dbn.html">DBNs, HMMs, Kalman filters and all that</a> +</ul> + +</ul> + + + + +<h1><a name="install">Installation</h1> + +<h2><a name="installM">Installing the Matlab code</h2> + +<ul> +<li> <a href="bnt_download.html">Download</a> the BNT.zip file. + +<p> +<li> Unpack the file. In Unix, type +<!--"tar xvf BNT.tar".--> +"unzip FullBNT.zip". +In Windows, use +a program like <a href="http://www.winzip.com">Winzip</a>. This will +create a directory called FullBNT, which contains BNT and other libraries. + +<p> +<li> Read the file <tt>BNT/README</tt> to make sure the date +matches the one on the top of <a href=bnt.html>the BNT home page</a>. +If not, you may need to press 'refresh' on your browser, and download +again, to get the most recent version. + +<p> +<li> <b>Edit the file "BNT/add_BNT_to_path.m"</b> so it contains the correct +pathname. +For example, in Windows, +I download FullBNT.zip into C:\kpmurphy\matlab, and +then comment out the second line (with the % character), and uncomment +the third line, which reads +<pre> +BNT_HOME = 'C:\kpmurphy\matlab\FullBNT'; +</pre> + +<p> +<li> Start up Matlab. + +<p> +<li> Type "ver" at the Matlab prompt (">>"). +<b>You need Matlab version 5.2 or newer to run BNT</b>. +(Versions 5.0 and 5.1 have a memory leak which seems to sometimes +crash BNT.) + +<p> +<li> Move to the BNT directory. +For example, in Windows, I type +<pre> +>> cd C:\kpmurphy\matlab\FullBNT\BNT +</pre> + +<p> +<li> Type "add_BNT_to_path". +This executes the command +<tt>addpath(genpath(BNT_HOME))</tt>, +which adds all directories below FullBNT to the matlab path. + +<p> +<li> Type "test_BNT". + +If all goes well, this will produce a bunch of numbers and maybe some +warning messages (which you can ignore), but no error messages. +(The warnings should only be of the form +"Warning: Maximum number of iterations has been exceeded", and are +produced by Netlab.) + +<p> +<li> <a href="http://groups.yahoo.com/group/BayesNetToolbox/join"> +Join the BNT email list</a> + +</ul> + + +If you are new to Matlab, you might like to check out +<a href="matlab_tips.html">some useful Matlab tips</a>. +For instance, this explains how to create a startup file, which can be +used to set your path variable automatically, so you can avoid having +to type the above commands every time. + + + + +<h2><a name="installC">Installing the C code</h2> + +Some BNT functions also have C implementations. +<b>It is not necessary to install the C code</b>, but it can result in a speedup +of a factor of 5-10. +To install all the C code, +edit installC_BNT.m so it contains the right path, +then type <tt>installC_BNT</tt>. +To uninstall all the C code, +edit uninstallC_BNT.m so it contains the right path, +then type <tt>uninstallC_BNT</tt>. +For an up-to-date list of the files which have C implementations, see +BNT/installC_BNT.m. + +<p> +mex is a script that lets you call C code from Matlab - it does not compile matlab to +C (see mcc below). +If your C/C++ compiler is set up correctly, mex should work out of +the box. +If not, you might need to type +<p> +<tt> mex -setup</tt> +<p> +before calling installC. +<p> +To make mex call gcc on Windows, +you must install <a +href="http://www.mrc-cbu.cam.ac.uk/Imaging/gnumex20.html">gnumex</a>. +You can use the <a href="http://www.mingw.org/">minimalist GNU for +Windows</a> version of gcc, or +the <a href="http://sources.redhat.com/cygwin/">cygwin</a> version. +<p> +In general, typing +'mex foo.c' from inside Matlab creates a file called +'foo.mexglx' or 'foo.dll' (the exact file +extension is system dependent - on Linux it is 'mexglx', on Windows it is '.dll'). +The resulting file will hide the original 'foo.m' (if it existed), i.e., +typing 'foo' at the prompt will call the compiled C version. +To reveal the original matlab version, just delete foo.mexglx (this is +what uninstallC does). +<p> +Sometimes it takes time for Matlab to realize that the file has +changed from matlab to C or vice versa; try typing 'clear all' or +restarting Matlab to refresh it. +To find out which version of a file you are running, type +'which foo'. +<p> +<a href="http://www.mathworks.com/products/compiler">mcc</a>, the +Matlab to C compiler, is a separate product, +and is quite different from mex. It does not yet support +objects/classes, which is why we can't compile all of BNT to C automatically. +Also, hand-written C code is usually much +better than the C code generated by mcc. + + +<p> +Acknowledgements: +Although I wrote some of the C code, most of +the C code (e.g., for jtree and dpot) was written by Wei Hu; +the triangulation C code was written by Ilya Shpitser. + + +<h1><a name="basics">Creating your first Bayes net</h1> + +To define a Bayes net, you must specify the graph structure and then +the parameters. We look at each in turn, using a simple example +(adapted from Russell and +Norvig, "Artificial Intelligence: a Modern Approach", Prentice Hall, +1995, p454). + + +<h2>Graph structure</h2> + + +Consider the following network. + +<p> +<center> +<IMG SRC="Figures/sprinkler.gif"> +</center> +<p> + +<P> +To specify this directed acyclic graph (dag), we create an adjacency matrix: +<PRE> +N = 4; +dag = zeros(N,N); +C = 1; S = 2; R = 3; W = 4; +dag(C,[R S]) = 1; +dag(R,W) = 1; +dag(S,W)=1; +</PRE> +<P> +We have numbered the nodes as follows: +Cloudy = 1, Sprinkler = 2, Rain = 3, WetGrass = 4. +<b>The nodes must always be numbered in topological order, i.e., +ancestors before descendants.</b> +For a more complicated graph, this is a little inconvenient: we will +see how to get around this <a href="usage_dbn.html#bat">below</a>. +<p> +In Matlab 6, you can use logical arrays instead of double arrays, +which are 4 times smaller: +<pre> +dag = false(N,N); +dag(C,[R S]) = true; +... +</pre> +<p> +A preliminary attempt to make a <b>GUI</b> +has been writte by Philippe LeRay and can be downloaded +from <a href="http://bnt.insa-rouen.fr/ajouts.html">here</a>. +<p> +You can visualize the resulting graph structure using +the methods discussed <a href="#graphdraw">below</a>. + +<h2>Creating the Bayes net shell</h2> + +In addition to specifying the graph structure, +we must specify the size and type of each node. +If a node is discrete, its size is the +number of possible values +each node can take on; if a node is continuous, +it can be a vector, and its size is the length of this vector. +In this case, we will assume all nodes are discrete and binary. +<PRE> +discrete_nodes = 1:N; +node_sizes = 2*ones(1,N); +</pre> +If the nodes were not binary, you could type e.g., +<pre> +node_sizes = [4 2 3 5]; +</pre> +meaning that Cloudy has 4 possible values, +Sprinkler has 2 possible values, etc. +Note that these are cardinal values, not ordinal, i.e., +they are not ordered in any way, like 'low', 'medium', 'high'. +<p> +We are now ready to make the Bayes net: +<pre> +bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes); +</PRE> +By default, all nodes are assumed to be discrete, so we can also just +write +<pre> +bnet = mk_bnet(dag, node_sizes); +</PRE> +You may also specify which nodes will be observed. +If you don't know, or if this not fixed in advance, +just use the empty list (the default). +<pre> +onodes = []; +bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes, 'observed', onodes); +</PRE> +Note that optional arguments are specified using a name/value syntax. +This is common for many BNT functions. +In general, to find out more about a function (e.g., which optional +arguments it takes), please see its +documentation string by typing +<pre> +help mk_bnet +</pre> +See also other <a href="matlab_tips.html">useful Matlab tips</a>. +<p> +It is possible to associate names with nodes, as follows: +<pre> +bnet = mk_bnet(dag, node_sizes, 'names', {'cloudy','S','R','W'}, 'discrete', 1:4); +</pre> +You can then refer to a node by its name: +<pre> +C = bnet.names{'cloudy'}; % bnet.names is an associative array +bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]); +</pre> + + +<h2><a name="cpt">Parameters</h2> + +A model consists of the graph structure and the parameters. +The parameters are represented by CPD objects (CPD = Conditional +Probability Distribution), which define the probability distribution +of a node given its parents. +(We will use the terms "node" and "random variable" interchangeably.) +The simplest kind of CPD is a table (multi-dimensional array), which +is suitable when all the nodes are discrete-valued. Note that the discrete +values are not assumed to be ordered in any way; that is, they +represent categorical quantities, like male and female, rather than +ordinal quantities, like low, medium and high. +(We will discuss CPDs in more detail <a href="#cpd">below</a>.) +<p> +Tabular CPDs, also called CPTs (conditional probability tables), +are stored as multidimensional arrays, where the dimensions +are arranged in the same order as the nodes, e.g., the CPT for node 4 +(WetGrass) is indexed by Sprinkler (2), Rain (3) and then WetGrass (4) itself. +Hence the child is always the last dimension. +If a node has no parents, its CPT is a column vector representing its +prior. +Note that in Matlab (unlike C), arrays are indexed +from 1, and are layed out in memory such that the first index toggles +fastest, e.g., the CPT for node 4 (WetGrass) is as follows +<P> +<P><IMG ALIGN=BOTTOM SRC="Figures/CPTgrass.gif"><P> +<P> +where we have used the convention that false==1, true==2. +We can create this CPT in Matlab as follows +<PRE> +CPT = zeros(2,2,2); +CPT(1,1,1) = 1.0; +CPT(2,1,1) = 0.1; +... +</PRE> +Here is an easier way: +<PRE> +CPT = reshape([1 0.1 0.1 0.01 0 0.9 0.9 0.99], [2 2 2]); +</PRE> +In fact, we don't need to reshape the array, since the CPD constructor +will do that for us. So we can just write +<pre> +bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [1 0.1 0.1 0.01 0 0.9 0.9 0.99]); +</pre> +The other nodes are created similarly (using the old syntax for +optional parameters) +<PRE> +bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]); +bnet.CPD{R} = tabular_CPD(bnet, R, [0.8 0.2 0.2 0.8]); +bnet.CPD{S} = tabular_CPD(bnet, S, [0.5 0.9 0.5 0.1]); +bnet.CPD{W} = tabular_CPD(bnet, W, [1 0.1 0.1 0.01 0 0.9 0.9 0.99]); +</PRE> + + +<h2><a name="rnd_cpt">Random Parameters</h2> + +If we do not specify the CPT, random parameters will be +created, i.e., each "row" of the CPT will be drawn from the uniform distribution. +To ensure repeatable results, use +<pre> +rand('state', seed); +randn('state', seed); +</pre> +To control the degree of randomness (entropy), +you can sample each row of the CPT from a Dirichlet(p,p,...) distribution. +If p << 1, this encourages "deterministic" CPTs (one entry near 1, the rest near 0). +If p = 1, each entry is drawn from U[0,1]. +If p >> 1, the entries will all be near 1/k, where k is the arity of +this node, i.e., each row will be nearly uniform. +You can do this as follows, assuming this node +is number i, and ns is the node_sizes. +<pre> +k = ns(i); +ps = parents(dag, i); +psz = prod(ns(ps)); +CPT = sample_dirichlet(p*ones(1,k), psz); +bnet.CPD{i} = tabular_CPD(bnet, i, 'CPT', CPT); +</pre> + + +<h2><a name="file">Loading a network from a file</h2> + +If you already have a Bayes net represented in the XML-based +<a href="http://www.cs.cmu.edu/afs/cs/user/fgcozman/www/Research/InterchangeFormat/"> +Bayes Net Interchange Format (BNIF)</a> (e.g., downloaded from the +<a +href="http://www.cs.huji.ac.il/labs/compbio/Repository"> +Bayes Net repository</a>), +you can convert it to BNT format using +the +<a href="http://www.digitas.harvard.edu/~ken/bif2bnt/">BIF-BNT Java +program</a> written by Ken Shan. +(This is not necessarily up-to-date.) + + +<h2>Creating a model using a GUI</h2> + +Click <a href="http://bnt.insa-rouen.fr/ajouts.html">here</a>. + + + +<h1><a name="inference">Inference</h1> + +Having created the BN, we can now use it for inference. +There are many different algorithms for doing inference in Bayes nets, +that make different tradeoffs between speed, +complexity, generality, and accuracy. +BNT therefore offers a variety of different inference +"engines". We will discuss these +in more detail <a href="#engines">below</a>. +For now, we will use the junction tree +engine, which is the mother of all exact inference algorithms. +This can be created as follows. +<pre> +engine = jtree_inf_engine(bnet); +</pre> +The other engines have similar constructors, but might take +additional, algorithm-specific parameters. +All engines are used in the same way, once they have been created. +We illustrate this in the following sections. + + +<h2><a name="marginal">Computing marginal distributions</h2> + +Suppose we want to compute the probability that the sprinker was on +given that the grass is wet. +The evidence consists of the fact that W=2. All the other nodes +are hidden (unobserved). We can specify this as follows. +<pre> +evidence = cell(1,N); +evidence{W} = 2; +</pre> +We use a 1D cell array instead of a vector to +cope with the fact that nodes can be vectors of different lengths. +In addition, the value [] can be used +to denote 'no evidence', instead of having to specify the observation +pattern as a separate argument. +(Click <a href="cellarray.html">here</a> for a quick tutorial on cell +arrays in matlab.) +<p> +We are now ready to add the evidence to the engine. +<pre> +[engine, loglik] = enter_evidence(engine, evidence); +</pre> +The behavior of this function is algorithm-specific, and is discussed +in more detail <a href="#engines">below</a>. +In the case of the jtree engine, +enter_evidence implements a two-pass message-passing scheme. +The first return argument contains the modified engine, which +incorporates the evidence. The second return argument contains the +log-likelihood of the evidence. (Not all engines are capable of +computing the log-likelihood.) +<p> +Finally, we can compute p=P(S=2|W=2) as follows. +<PRE> +marg = marginal_nodes(engine, S); +marg.T +ans = + 0.57024 + 0.42976 +p = marg.T(2); +</PRE> +We see that p = 0.4298. +<p> +Now let us add the evidence that it was raining, and see what +difference it makes. +<PRE> +evidence{R} = 2; +[engine, loglik] = enter_evidence(engine, evidence); +marg = marginal_nodes(engine, S); +p = marg.T(2); +</PRE> +We find that p = P(S=2|W=2,R=2) = 0.1945, +which is lower than +before, because the rain can ``explain away'' the +fact that the grass is wet. +<p> +You can plot a marginal distribution over a discrete variable +as a barchart using the built 'bar' function: +<pre> +bar(marg.T) +</pre> +This is what it looks like + +<p> +<center> +<IMG SRC="Figures/sprinkler_bar.gif"> +</center> +<p> + +<h2><a name="observed">Observed nodes</h2> + +What happens if we ask for the marginal on an observed node, e.g. P(W|W=2)? +An observed discrete node effectively only has 1 value (the observed + one) --- all other values would result in 0 probability. +For efficiency, BNT treats observed (discrete) nodes as if they were + set to 1, as we see below: +<pre> +evidence = cell(1,N); +evidence{W} = 2; +engine = enter_evidence(engine, evidence); +m = marginal_nodes(engine, W); +m.T +ans = + 1 +</pre> +This can get a little confusing, since we assigned W=2. +So we can ask BNT to add the evidence back in by passing in an optional argument: +<pre> +m = marginal_nodes(engine, W, 1); +m.T +ans = + 0 + 1 +</pre> +This shows that P(W=1|W=2) = 0 and P(W=2|W=2) = 1. + + + +<h2><a name="joint">Computing joint distributions</h2> + +We can compute the joint probability on a set of nodes as in the +following example. +<pre> +evidence = cell(1,N); +[engine, ll] = enter_evidence(engine, evidence); +m = marginal_nodes(engine, [S R W]); +</pre> +m is a structure. The 'T' field is a multi-dimensional array (in +this case, 3-dimensional) that contains the joint probability +distribution on the specified nodes. +<pre> +>> m.T +ans(:,:,1) = + 0.2900 0.0410 + 0.0210 0.0009 +ans(:,:,2) = + 0 0.3690 + 0.1890 0.0891 +</pre> +We see that P(S=1,R=1,W=2) = 0, since it is impossible for the grass +to be wet if both the rain and sprinkler are off. +<p> +Let us now add some evidence to R. +<pre> +evidence{R} = 2; +[engine, ll] = enter_evidence(engine, evidence); +m = marginal_nodes(engine, [S R W]) +m = + domain: [2 3 4] + T: [2x1x2 double] +>> m.T +m.T +ans(:,:,1) = + 0.0820 + 0.0018 +ans(:,:,2) = + 0.7380 + 0.1782 +</pre> +The joint T(i,j,k) = P(S=i,R=j,W=k|evidence) +should have T(i,1,k) = 0 for all i,k, since R=1 is incompatible +with the evidence that R=2. +Instead of creating large tables with many 0s, BNT sets the effective +size of observed (discrete) nodes to 1, as explained above. +This is why m.T has size 2x1x2. +To get a 2x2x2 table, type +<pre> +m = marginal_nodes(engine, [S R W], 1) +m = + domain: [2 3 4] + T: [2x2x2 double] +>> m.T +m.T +ans(:,:,1) = + 0 0.082 + 0 0.0018 +ans(:,:,2) = + 0 0.738 + 0 0.1782 +</pre> + +<p> +Note: It is not always possible to compute the joint on arbitrary +sets of nodes: it depends on which inference engine you use, as discussed +in more detail <a href="#engines">below</a>. + + +<h2><a name="soft">Soft/virtual evidence</h2> + +Sometimes a node is not observed, but we have some distribution over +its possible values; this is often called "soft" or "virtual" +evidence. +One can use this as follows +<pre> +[engine, loglik] = enter_evidence(engine, evidence, 'soft', soft_evidence); +</pre> +where soft_evidence{i} is either [] (if node i has no soft evidence) +or is a vector representing the probability distribution over i's +possible values. +For example, if we don't know i's exact value, but we know its +likelihood ratio is 60/40, we can write evidence{i} = [] and +soft_evidence{i} = [0.6 0.4]. +<p> +Currently only jtree_inf_engine supports this option. +It assumes that all hidden nodes, and all nodes for +which we have soft evidence, are discrete. +For a longer example, see BNT/examples/static/softev1.m. + + +<h2><a name="mpe">Most probable explanation</h2> + +To compute the most probable explanation (MPE) of the evidence (i.e., +the most probable assignment, or a mode of the joint), use +<pre> +[mpe, ll] = calc_mpe(engine, evidence); +</pre> +mpe{i} is the most likely value of node i. +This calls enter_evidence with the 'maximize' flag set to 1, which +causes the engine to do max-product instead of sum-product. +The resulting max-marginals are then thresholded. +If there is more than one maximum probability assignment, we must take + care to break ties in a consistent manner (thresholding the + max-marginals may give the wrong result). To force this behavior, + type +<pre> +[mpe, ll] = calc_mpe(engine, evidence, 1); +</pre> +Note that computing the MPE is someties called abductive reasoning. + +<p> +You can also use <tt>calc_mpe_bucket</tt> written by Ron Zohar, +that does a forwards max-product pass, and then a backwards traceback +pass, which is how Viterbi is traditionally implemented. + + + +<h1><a name="cpd">Conditional Probability Distributions</h1> + +A Conditional Probability Distributions (CPD) +defines P(X(i) | X(Pa(i))), where X(i) is the i'th node, and X(Pa(i)) +are the parents of node i. There are many ways to represent this +distribution, which depend in part on whether X(i) and X(Pa(i)) are +discrete, continuous, or a combination. +We will discuss various representations below. + + +<h2><a name="tabular">Tabular nodes</h2> + +If the CPD is represented as a table (i.e., if it is a multinomial +distribution), it has a number of parameters that is exponential in +the number of parents. See the example <a href="#cpt">above</a>. + + +<h2><a name="noisyor">Noisy-or nodes</h2> + +A noisy-OR node is like a regular logical OR gate except that +sometimes the effects of parents that are on get inhibited. +Let the prob. that parent i gets inhibited be q(i). +Then a node, C, with 2 parents, A and B, has the following CPD, where +we use F and T to represent off and on (1 and 2 in BNT). +<pre> +A B P(C=off) P(C=on) +--------------------------- +F F 1.0 0.0 +T F q(A) 1-q(A) +F T q(B) 1-q(B) +T T q(A)q(B) q-q(A)q(B) +</pre> +Thus we see that the causes get inhibited independently. +It is common to associate a "leak" node with a noisy-or CPD, which is +like a parent that is always on. This can account for all other unmodelled +causes which might turn the node on. +<p> +The noisy-or distribution is similar to the logistic distribution. +To see this, let the nodes, S(i), have values in {0,1}, and let q(i,j) +be the prob. that j inhibits i. Then +<pre> +Pr(S(i)=1 | parents(S(i))) = 1 - prod_{j} q(i,j)^S(j) +</pre> +Now define w(i,j) = -ln q(i,j) and rho(x) = 1-exp(-x). Then +<pre> +Pr(S(i)=1 | parents(S(i))) = rho(sum_j w(i,j) S(j)) +</pre> +For a sigmoid node, we have +<pre> +Pr(S(i)=1 | parents(S(i))) = sigma(-sum_j w(i,j) S(j)) +</pre> +where sigma(x) = 1/(1+exp(-x)). Hence they differ in the choice of +the activation function (although both are monotonically increasing). +In addition, in the case of a noisy-or, the weights are constrained to be +positive, since they derive from probabilities q(i,j). +In both cases, the number of parameters is <em>linear</em> in the +number of parents, unlike the case of a multinomial distribution, +where the number of parameters is exponential in the number of parents. +We will see an example of noisy-OR nodes <a href="#qmr">below</a>. + + +<h2><a name="deterministic">Other (noisy) deterministic nodes</h2> + +Deterministic CPDs for discrete random variables can be created using +the deterministic_CPD class. It is also possible to 'flip' the output +of the function with some probability, to simulate noise. +The boolean_CPD class is just a special case of a +deterministic CPD, where the parents and child are all binary. +<p> +Both of these classes are just "syntactic sugar" for the tabular_CPD +class. + + + +<h2><a name="softmax">Softmax nodes</h2> + +If we have a discrete node with a continuous parent, +we can define its CPD using a softmax function +(also known as the multinomial logit function). +This acts like a soft thresholding operator, and is defined as follows: +<pre> + exp(w(:,i)'*x + b(i)) +Pr(Q=i | X=x) = ----------------------------- + sum_j exp(w(:,j)'*x + b(j)) + +</pre> +The parameters of a softmax node, w(:,i) and b(i), i=1..|Q|, have the +following interpretation: w(:,i)-w(:,j) is the normal vector to the +decision boundary between classes i and j, +and b(i)-b(j) is its offset (bias). For example, suppose +X is a 2-vector, and Q is binary. Then +<pre> +w = [1 -1; + 0 0]; + +b = [0 0]; +</pre> +means class 1 are points in the 2D plane with positive x coordinate, +and class 2 are points in the 2D plane with negative x coordinate. +If w has large magnitude, the decision boundary is sharp, otherwise it +is soft. +In the special case that Q is binary (0/1), the softmax function reduces to the logistic +(sigmoid) function. +<p> +Fitting a softmax function can be done using the iteratively reweighted +least squares (IRLS) algorithm. +We use the implementation from +<a href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>. +Note that since +the softmax distribution is not in the exponential family, it does not +have finite sufficient statistics, and hence we must store all the +training data in uncompressed form. +If this takes too much space, one should use online (stochastic) gradient +descent (not implemented in BNT). +<p> +If a softmax node also has discrete parents, +we use a different set of w/b parameters for each combination of +parent values, as in the <a href="#gaussian">conditional linear +Gaussian CPD</a>. +This feature was implemented by Pierpaolo Brutti. +He is currently extending it so that discrete parents can be treated +as if they were continuous, by adding indicator variables to the X +vector. +<p> +We will see an example of softmax nodes <a href="#mixexp">below</a>. + + +<h2><a name="mlp">Neural network nodes</h2> + +Pierpaolo Brutti has implemented the mlp_CPD class, which uses a multi layer perceptron +to implement a mapping from continuous parents to discrete children, +similar to the softmax function. +(If there are also discrete parents, it creates a mixture of MLPs.) +It uses code from <a +href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>. +This is work in progress. + +<h2><a name="root">Root nodes</h2> + +A root node has no parents and no parameters; it can be used to model +an observed, exogeneous input variable, i.e., one which is "outside" +the model. +This is useful for conditional density models. +We will see an example of root nodes <a href="#mixexp">below</a>. + + +<h2><a name="gaussian">Gaussian nodes</h2> + +We now consider a distribution suitable for the continuous-valued nodes. +Suppose the node is called Y, its continuous parents (if any) are +called X, and its discrete parents (if any) are called Q. +The distribution on Y is defined as follows: +<pre> +- no parents: Y ~ N(mu, Sigma) +- cts parents : Y|X=x ~ N(mu + W x, Sigma) +- discrete parents: Y|Q=i ~ N(mu(:,i), Sigma(:,:,i)) +- cts and discrete parents: Y|X=x,Q=i ~ N(mu(:,i) + W(:,:,i) * x, Sigma(:,:,i)) +</pre> +where N(mu, Sigma) denotes a Normal distribution with mean mu and +covariance Sigma. Let |X|, |Y| and |Q| denote the sizes of X, Y and Q +respectively. +If there are no discrete parents, |Q|=1; if there is +more than one, then |Q| = a vector of the sizes of each discrete parent. +If there are no continuous parents, |X|=0; if there is more than one, +then |X| = the sum of their sizes. +Then mu is a |Y|*|Q| vector, Sigma is a |Y|*|Y|*|Q| positive +semi-definite matrix, and W is a |Y|*|X|*|Q| regression (weight) +matrix. +<p> +We can create a Gaussian node with random parameters as follows. +<pre> +bnet.CPD{i} = gaussian_CPD(bnet, i); +</pre> +We can specify the value of one or more of the parameters as in the +following example, in which |Y|=2, and |Q|=1. +<pre> +bnet.CPD{i} = gaussian_CPD(bnet, i, 'mean', [0; 0], 'weights', randn(Y,X), 'cov', eye(Y)); +</pre> +<p> +We will see an example of conditional linear Gaussian nodes <a +href="#cg_model">below</a>. +<p> +<b>When learning Gaussians from data</b>, it is helpful to ensure the +data has a small magnitde +(see e.g., KPMstats/standardize) to prevent numerical problems. +Unless you have a lot of data, it is also a very good idea to use +diagonal instead of full covariance matrices. +(BNT does not currently support spherical covariances, although it +would be easy to add, since KPMstats/clg_Mstep supports this option; +you would just need to modify gaussian_CPD/update_ess to accumulate +weighted inner products.) + + + +<h2><a name="nongauss">Other continuous distributions</h2> + +Currently BNT does not support any CPDs for continuous nodes other +than the Gaussian. +However, you can use a mixture of Gaussians to +approximate other continuous distributions. We will see some an example +of this with the IFA model <a href="#pca">below</a>. + + +<h2><a name="glm">Generalized linear model nodes</h2> + +In the future, we may incorporate some of the functionality of +<a href = +"http://www.sci.usq.edu.au/staff/dunn/glmlab/glmlab.html">glmlab</a> +into BNT. + + +<h2><a name="dtree">Classification/regression tree nodes</h2> + +We plan to add classification and regression trees to define CPDs for +discrete and continuous nodes, respectively. +Trees have many advantages: they are easy to interpret, they can do +feature selection, they can +handle discrete and continuous inputs, they do not make strong +assumptions about the form of the distribution, the number of +parameters can grow in a data-dependent way (i.e., they are +semi-parametric), they can handle missing data, etc. +However, they are not yet implemented. +<!-- +Yimin Zhang is currently (Feb '02) implementing this. +--> + + +<h2><a name="cpd_summary">Summary of CPD types</h2> + +We list all the different types of CPDs supported by BNT. +For each CPD, we specify if the child and parents can be discrete (D) or +continuous (C) (Binary (B) nodes are a special case). +We also specify which methods each class supports. +If a method is inherited, the name of the parent class is mentioned. +If a parent class calls a child method, this is mentioned. +<p> +The <tt>CPD_to_CPT</tt> method converts a CPD to a table; this +requires that the child and all parents are discrete. +The CPT might be exponentially big... +<tt>convert_to_table</tt> evaluates a CPD with evidence, and +represents the the resulting potential as an array. +This requires that the child is discrete, and any continuous parents +are observed. +<tt>convert_to_pot</tt> evaluates a CPD with evidence, and +represents the resulting potential as a dpot, gpot, cgpot or upot, as +requested. (d=discrete, g=Gaussian, cg = conditional Gaussian, u = +utility). + +<p> +When we sample a node, all the parents are observed. +When we compute the (log) probability of a node, all the parents and +the child are observed. +<p> +We also specify if the parameters are learnable. +For learning with EM, we require +the methods <tt>reset_ess</tt>, <tt>update_ess</tt> and +<tt>maximize_params</tt>. +For learning from fully observed data, we require +the method <tt>learn_params</tt>. +By default, all classes inherit this from generic_CPD, which simply +calls <tt>update_ess</tt> N times, once for each data case, followed +by <tt>maximize_params</tt>, i.e., it is like EM, without the E step. +Some classes implement a batch formula, which is quicker. +<p> +Bayesian learning means computing a posterior over the parameters +given fully observed data. +<p> +Pearl means we implement the methods <tt>compute_pi</tt> and +<tt>compute_lambda_msg</tt>, used by +<tt>pearl_inf_engine</tt>, which runs on directed graphs. +<tt>belprop_inf_engine</tt> only needs <tt>convert_to_pot</tt>.H +The pearl methods can exploit special properties of the CPDs for +computing the messages efficiently, whereas belprop does not. +<p> +The only method implemented by generic_CPD is <tt>adjustable_CPD</tt>, +which is not shown, since it is not very interesting. + + +<p> + + +<table> +<table border units = pixels><tr> +<td align=center>Name +<td align=center>Child +<td align=center>Parents +<td align=center>Comments +<td align=center>CPD_to_CPT +<td align=center>conv_to_table +<td align=center>conv_to_pot +<td align=center>sample +<td align=center>prob +<td align=center>learn +<td align=center>Bayes +<td align=center>Pearl + + +<tr> +<!-- Name--><td> +<!-- Child--><td> +<!-- Parents--><td> +<!-- Comments--><td> +<!-- CPD_to_CPT--><td> +<!-- conv_to_table--><td> +<!-- conv_to_pot--><td> +<!-- sample--><td> +<!-- prob--><td> +<!-- learn--><td> +<!-- Bayes--><td> +<!-- Pearl--><td> + +<tr> +<!-- Name--><td>boolean +<!-- Child--><td>B +<!-- Parents--><td>B +<!-- Comments--><td>Syntactic sugar for tabular +<!-- CPD_to_CPT--><td>- +<!-- conv_to_table--><td>- +<!-- conv_to_pot--><td>- +<!-- sample--><td>- +<!-- prob--><td>- +<!-- learn--><td>- +<!-- Bayes--><td>- +<!-- Pearl--><td>- + +<tr> +<!-- Name--><td>deterministic +<!-- Child--><td>D +<!-- Parents--><td>D +<!-- Comments--><td>Syntactic sugar for tabular +<!-- CPD_to_CPT--><td>- +<!-- conv_to_table--><td>- +<!-- conv_to_pot--><td>- +<!-- sample--><td>- +<!-- prob--><td>- +<!-- learn--><td>- +<!-- Bayes--><td>- +<!-- Pearl--><td>- + +<tr> +<!-- Name--><td>Discrete +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>Virtual class +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Calls CPD_to_CPT +<!-- conv_to_pot--><td>Calls conv_to_table +<!-- sample--><td>Calls conv_to_table +<!-- prob--><td>Calls conv_to_table +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + +<tr> +<!-- Name--><td>Gaussian +<!-- Child--><td>C +<!-- Parents--><td>C/D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>Y +<!-- prob--><td>Y +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + +<tr> +<!-- Name--><td>gmux +<!-- Child--><td>C +<!-- Parents--><td>C/D +<!-- Comments--><td>multiplexer +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>N +<!-- prob--><td>N +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>Y + + +<tr> +<!-- Name--><td>MLP +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>multi layer perceptron +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Y +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>noisy-or +<!-- Child--><td>B +<!-- Parents--><td>B +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>Y +<!-- conv_to_table--><td>Inherits from discrete +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>Y + + +<tr> +<!-- Name--><td>root +<!-- Child--><td>C/D +<!-- Parents--><td>none +<!-- Comments--><td>no params +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>Y +<!-- prob--><td>Y +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>softmax +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Y +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>generic +<!-- Child--><td>C/D +<!-- Parents--><td>C/D +<!-- Comments--><td>Virtual class +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>N +<!-- sample--><td>N +<!-- prob--><td>N +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>Tabular +<!-- Child--><td>D +<!-- Parents--><td>D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>Y +<!-- conv_to_table--><td>Inherits from discrete +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>Y +<!-- Pearl--><td>Y + +</table> + + + +<h1><a name="examples">Example models</h1> + + +<h2>Gaussian mixture models</h2> + +Richard W. DeVaul has made a detailed tutorial on how to fit mixtures +of Gaussians using BNT. Available +<a href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt">here</a>. + + +<h2><a name="pca">PCA, ICA, and all that </h2> + +In Figure (a) below, we show how Factor Analysis can be thought of as a +graphical model. Here, X has an N(0,I) prior, and +Y|X=x ~ N(mu + Wx, Psi), +where Psi is diagonal and W is called the "factor loading matrix". +Since the noise on both X and Y is diagonal, the components of these +vectors are uncorrelated, and hence can be represented as individual +scalar nodes, as we show in (b). +(This is useful if parts of the observations on the Y vector are occasionally missing.) +We usually take k=|X| << |Y|=D, so the model tries to explain +many observations using a low-dimensional subspace. + + +<center> +<table> +<tr> +<td><img src="Figures/fa.gif"> +<td><img src="Figures/fa_scalar.gif"> +<td><img src="Figures/mfa.gif"> +<td><img src="Figures/ifa.gif"> +<tr> +<td align=center> (a) +<td align=center> (b) +<td align=center> (c) +<td align=center> (d) +</table> +</center> + +<p> +We can create this model in BNT as follows. +<pre> +ns = [k D]; +dag = zeros(2,2); +dag(1,2) = 1; +bnet = mk_bnet(dag, ns, 'discrete', []); +bnet.CPD{1} = gaussian_CPD(bnet, 1, 'mean', zeros(k,1), 'cov', eye(k), ... + 'cov_type', 'diag', 'clamp_mean', 1, 'clamp_cov', 1); +bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(D,1), 'cov', diag(Psi0), 'weights', W0, ... + 'cov_type', 'diag', 'clamp_mean', 1); +</pre> + +The root node is clamped to the N(0,I) distribution, so that we will +not update these parameters during learning. +The mean of the leaf node is clamped to 0, +since we assume the data has been centered (had its mean subtracted +off); this is just for simplicity. +Finally, the covariance of the leaf node is constrained to be +diagonal. W0 and Psi0 are the initial parameter guesses. + +<p> +We can fit this model (i.e., estimate its parameters in a maximum +likelihood (ML) sense) using EM, as we +explain <a href="#em">below</a>. +Not surprisingly, the ML estimates for mu and Psi turn out to be +identical to the +sample mean and variance, which can be computed directly as +<pre> +mu_ML = mean(data); +Psi_ML = diag(cov(data)); +</pre> +Note that W can only be identified up to a rotation matrix, because of +the spherical symmetry of the source. + +<p> +If we restrict Psi to be spherical, i.e., Psi = sigma*I, +there is a closed-form solution for W as well, +i.e., we do not need to use EM. +In particular, W contains the first |X| eigenvectors of the sample covariance +matrix, with scalings determined by the eigenvalues and sigma. +Classical PCA can be obtained by taking the sigma->0 limit. +For details, see + +<ul> +<li> <a href="ftp://hope.caltech.edu/pub/roweis/Empca/empca.ps"> +"EM algorithms for PCA and SPCA"</a>, Sam Roweis, NIPS 97. +(<a href="ftp://hope.caltech.edu/pub/roweis/Code/empca.tar.gz"> +Matlab software</a>) + +<p> +<li> +<a +href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003> +"Mixtures of probabilistic principal component analyzers"</a>, +Tipping and Bishop, Neural Computation 11(2):443--482, 1999. +</ul> + +<p> +By adding a hidden discrete variable, we can create mixtures of FA +models, as shown in (c). +Now we can explain the data using a set of subspaces. +We can create this model in BNT as follows. +<pre> +ns = [M k D]; +dag = zeros(3); +dag(1,3) = 1; +dag(2,3) = 1; +bnet = mk_bnet(dag, ns, 'discrete', 1); +bnet.CPD{1} = tabular_CPD(bnet, 1, Pi0); +bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(k, 1), 'cov', eye(k), 'cov_type', 'diag', ... + 'clamp_mean', 1, 'clamp_cov', 1); +bnet.CPD{3} = gaussian_CPD(bnet, 3, 'mean', Mu0', 'cov', repmat(diag(Psi0), [1 1 M]), ... + 'weights', W0, 'cov_type', 'diag', 'tied_cov', 1); +</pre> +Notice how the covariance matrix for Y is the same for all values of +Q; that is, the noise level in each sub-space is assumed the same. +However, we allow the offset, mu, to vary. +For details, see +<ul> + +<LI> +<a HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/tr-96-1.ps.gz"> The EM +Algorithm for Mixtures of Factor Analyzers </A>, +Ghahramani, Z. and Hinton, G.E. (1996), +University of Toronto +Technical Report CRG-TR-96-1. +(<A HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/mfa.tar.gz">Matlab software</A>) + +<p> +<li> +<a +href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003> +"Mixtures of probabilistic principal component analyzers"</a>, +Tipping and Bishop, Neural Computation 11(2):443--482, 1999. +</ul> + +<p> +I have included Zoubin's specialized MFA code (with his permission) +with the toolbox, so you can check that BNT gives the same results: +see 'BNT/examples/static/mfa1.m'. + +<p> +Independent Factor Analysis (IFA) generalizes FA by allowing a +non-Gaussian prior on each component of X. +(Note that we can approximate a non-Gaussian prior using a mixture of +Gaussians.) +This means that the likelihood function is no longer rotationally +invariant, so we can uniquely identify W and the hidden +sources X. +IFA also allows a non-diagonal Psi (i.e. correlations between the components of Y). +We recover classical Independent Components Analysis (ICA) +in the Psi -> 0 limit, and by assuming that |X|=|Y|, so that the +weight matrix W is square and invertible. +For details, see +<ul> +<li> +<a href="http://www.gatsby.ucl.ac.uk/~hagai/ifa.ps">Independent Factor +Analysis</a>, H. Attias, Neural Computation 11: 803--851, 1998. +</ul> + + + +<h2><a name="mixexp">Mixtures of experts</h2> + +As an example of the use of the softmax function, +we introduce the Mixture of Experts model. +<!-- +We also show +the Hierarchical Mixture of Experts model, where the hierarchy has two +levels. +(This is essentially a probabilistic decision tree of height two.) +--> +As before, +circles denote continuous-valued nodes, +squares denote discrete nodes, clear +means hidden, and shaded means observed. +<p> +<center> +<table> +<tr> +<td><img src="Figures/mixexp.gif"> +<!-- +<td><img src="Figures/hme.gif"> +--> +</table> +</center> +<p> +X is the observed +input, Y is the output, and +the Q nodes are hidden "gating" nodes, which select the appropriate +set of parameters for Y. During training, Y is assumed observed, +but for testing, the goal is to predict Y given X. +Note that this is a <em>conditional</em> density model, so we don't +associate any parameters with X. +Hence X's CPD will be a root CPD, which is a way of modelling +exogenous nodes. +If the output is a continuous-valued quantity, +we assume the "experts" are linear-regression units, +and set Y's CPD to linear-Gaussian. +If the output is discrete, we set Y's CPD to a softmax function. +The Q CPDs will always be softmax functions. + +<p> +As a concrete example, consider the mixture of experts model where X and Y are +scalars, and Q is binary. +This is just piecewise linear regression, where +we have two line segments, i.e., +<P> +<IMG ALIGN=BOTTOM SRC="Eqns/lin_reg_eqn.gif"> +<P> +We can create this model with random parameters as follows. +(This code is bundled in BNT/examples/static/mixexp2.m.) +<PRE> +X = 1; +Q = 2; +Y = 3; +dag = zeros(3,3); +dag(X,[Q Y]) = 1 +dag(Q,Y) = 1; +ns = [1 2 1]; % make X and Y scalars, and have 2 experts +onodes = [1 3]; +bnet = mk_bnet(dag, ns, 'discrete', 2, 'observed', onodes); + +rand('state', 0); +randn('state', 0); +bnet.CPD{1} = root_CPD(bnet, 1); +bnet.CPD{2} = softmax_CPD(bnet, 2); +bnet.CPD{3} = gaussian_CPD(bnet, 3); +</PRE> +Now let us fit this model using <a href="#em">EM</a>. +First we <a href="#load_data">load the data</a> (1000 training cases) and plot them. +<P> +<PRE> +data = load('/examples/static/Misc/mixexp_data.txt', '-ascii'); +plot(data(:,1), data(:,2), '.'); +</PRE> +<p> +<center> +<IMG SRC="Figures/mixexp_data.gif"> +</center> +<p> +This is what the model looks like before training. +(Thanks to Thomas Hofman for writing this plotting routine.) +<p> +<center> +<IMG SRC="Figures/mixexp_before.gif"> +</center> +<p> +Now let's train the model, and plot the final performance. +(We will discuss how to train models in more detail <a href="#param_learning">below</a>.) +<P> +<PRE> +ncases = size(data, 1); % each row of data is a training case +cases = cell(3, ncases); +cases([1 3], :) = num2cell(data'); % each column of cases is a training case +engine = jtree_inf_engine(bnet); +max_iter = 20; +[bnet2, LLtrace] = learn_params_em(engine, cases, max_iter); +</PRE> +(We specify which nodes will be observed when we create the engine. +Hence BNT knows that the hidden nodes are all discrete. +For complex models, this can lead to a significant speedup.) +Below we show what the model looks like after 16 iterations of EM +(with 100 IRLS iterations per M step), when it converged +using the default convergence tolerance (that the +fractional change in the log-likelihood be less than 1e-3). +Before learning, the log-likelihood was +-322.927442; afterwards, it was -13.728778. +<p> +<center> +<IMG SRC="Figures/mixexp_after.gif"> +</center> +(See BNT/examples/static/mixexp2.m for details of the code.) + + + +<h2><a name="hme">Hierarchical mixtures of experts</h2> + +A hierarchical mixture of experts (HME) extends the mixture of experts +model by having more than one hidden node. A two-level example is shown below, along +with its more traditional representation as a neural network. +This is like a (balanced) probabilistic decision tree of height 2. +<p> +<center> +<IMG SRC="Figures/HMEforMatlab.jpg"> +</center> +<p> +<a href="mailto:pbrutti@stat.cmu.edu">Pierpaolo Brutti</a> +has written an extensive set of routines for HMEs, +which are bundled with BNT: see the examples/static/HME directory. +These routines allow you to choose the number of hidden (gating) +layers, and the form of the experts (softmax or MLP). +See the file hmemenu, which provides a demo. +For example, the figure below shows the decision boundaries learned +for a ternary classification problem, using a 2 level HME with softmax +gates and softmax experts; the training set is on the left, the +testing set on the right. +<p> +<center> +<!--<IMG SRC="Figures/hme_dec_boundary.gif">--> +<IMG SRC="Figures/hme_dec_boundary.png"> +</center> +<p> + + +<p> +For more details, see the following: +<ul> + +<li> <a href="http://www.cs.berkeley.edu/~jordan/papers/hierarchies.ps.Z"> +Hierarchical mixtures of experts and the EM algorithm</a> +M. I. Jordan and R. A. Jacobs. Neural Computation, 6, 181-214, 1994. + +<li> <a href = +"http://www.cs.berkeley.edu/~dmartin/software">David Martin's +matlab code for HME</a> + +<li> <a +href="http://www.cs.berkeley.edu/~jordan/papers/uai.ps.Z">Why the +logistic function? A tutorial discussion on +probabilities and neural networks.</a> M. I. Jordan. MIT Computational +Cognitive Science Report 9503, August 1995. + +<li> "Generalized Linear Models", McCullagh and Nelder, Chapman and +Halll, 1983. + +<li> +"Improved learning algorithms for mixtures of experts in multiclass +classification". +K. Chen, L. Xu, H. Chi. +Neural Networks (1999) 12: 1229-1252. + +<li> <a href="http://www.oigeeza.com/steve/"> +Classification Using Hierarchical Mixtures of Experts</a> +S.R. Waterhouse and A.J. Robinson. +In Proc. IEEE Workshop on Neural Network for Signal Processing IV (1994), pp. 177-186 + +<li> <a href="http://www.idiap.ch/~perry/"> +Localized mixtures of experts</a>, +P. Moerland, 1998. + +<li> "Nonlinear gated experts for time series", +A.S. Weigend and M. Mangeas, 1995. + +</ul> + + +<h2><a name="qmr">QMR</h2> + +Bayes nets originally arose out of an attempt to add probabilities to +expert systems, and this is still the most common use for BNs. +A famous example is +QMR-DT, a decision-theoretic reformulation of the Quick Medical +Reference (QMR) model. +<p> +<center> +<IMG ALIGN=BOTTOM SRC="Figures/qmr.gif"> +</center> +Here, the top layer represents hidden disease nodes, and the bottom +layer represents observed symptom nodes. +The goal is to infer the posterior probability of each disease given +all the symptoms (which can be present, absent or unknown). +Each node in the top layer has a Bernoulli prior (with a low prior +probability that the disease is present). +Since each node in the bottom layer has a high fan-in, we use a +noisy-OR parameterization; each disease has an independent chance of +causing each symptom. +The real QMR-DT model is copyright, but +we can create a random QMR-like model as follows. +<pre> +function bnet = mk_qmr_bnet(G, inhibit, leak, prior) +% MK_QMR_BNET Make a QMR model +% bnet = mk_qmr_bnet(G, inhibit, leak, prior) +% +% G(i,j) = 1 iff there is an arc from disease i to finding j +% inhibit(i,j) = inhibition probability on i->j arc +% leak(j) = inhibition prob. on leak->j arc +% prior(i) = prob. disease i is on + +[Ndiseases Nfindings] = size(inhibit); +N = Ndiseases + Nfindings; +finding_node = Ndiseases+1:N; +ns = 2*ones(1,N); +dag = zeros(N,N); +dag(1:Ndiseases, finding_node) = G; +bnet = mk_bnet(dag, ns, 'observed', finding_node); + +for d=1:Ndiseases + CPT = [1-prior(d) prior(d)]; + bnet.CPD{d} = tabular_CPD(bnet, d, CPT'); +end + +for i=1:Nfindings + fnode = finding_node(i); + ps = parents(G, i); + bnet.CPD{fnode} = noisyor_CPD(bnet, fnode, leak(i), inhibit(ps, i)); +end +</pre> +In the file BNT/examples/static/qmr1, we create a random bipartite +graph G, with 5 diseases and 10 findings, and random parameters. +(In general, to create a random dag, use 'mk_random_dag'.) +We can visualize the resulting graph structure using +the methods discussed <a href="#graphdraw">below</a>, with the +following results: +<p> +<img src="Figures/qmr.rnd.jpg"> + +<p> +Now let us put some random evidence on all the leaves except the very +first and very last, and compute the disease posteriors. +<pre> +pos = 2:floor(Nfindings/2); +neg = (pos(end)+1):(Nfindings-1); +onodes = myunion(pos, neg); +evidence = cell(1, N); +evidence(findings(pos)) = num2cell(repmat(2, 1, length(pos))); +evidence(findings(neg)) = num2cell(repmat(1, 1, length(neg))); + +engine = jtree_inf_engine(bnet); +[engine, ll] = enter_evidence(engine, evidence); +post = zeros(1, Ndiseases); +for i=diseases(:)' + m = marginal_nodes(engine, i); + post(i) = m.T(2); +end +</pre> +Junction tree can be quite slow on large QMR models. +Fortunately, it is possible to exploit properties of the noisy-OR +function to speed up exact inference using an algorithm called +<a href="#quickscore">quickscore</a>, discussed below. + + + + + +<h2><a name="cg_model">Conditional Gaussian models</h2> + +A conditional Gaussian model is one in which, conditioned on all the discrete +nodes, the distribution over the remaining (continuous) nodes is +multivariate Gaussian. This means we can have arcs from discrete (D) +to continuous (C) nodes, but not vice versa. +(We <em>are</em> allowed C->D arcs if the continuous nodes are observed, +as in the <a href="#mixexp">mixture of experts</a> model, +since this distribution can be represented with a discrete potential.) +<p> +We now give an example of a CG model, from +the paper "Propagation of Probabilities, Means amd +Variances in Mixed Graphical Association Models", Steffen Lauritzen, +JASA 87(420):1098--1108, 1992 (reprinted in the book "Probabilistic Networks and Expert +Systems", R. G. Cowell, A. P. Dawid, S. L. Lauritzen and +D. J. Spiegelhalter, Springer, 1999.) + +<h3>Specifying the graph</h3> + +Consider the model of waste emissions from an incinerator plant shown below. +We follow the standard convention that shaded nodes are observed, +clear nodes are hidden. +We also use the non-standard convention that +square nodes are discrete (tabular) and round nodes are +Gaussian. + +<p> +<center> +<IMG SRC="Figures/cg1.gif"> +</center> +<p> + +We can create this model as follows. +<pre> +F = 1; W = 2; E = 3; B = 4; C = 5; D = 6; Min = 7; Mout = 8; L = 9; +n = 9; + +dag = zeros(n); +dag(F,E)=1; +dag(W,[E Min D]) = 1; +dag(E,D)=1; +dag(B,[C D])=1; +dag(D,[L Mout])=1; +dag(Min,Mout)=1; + +% node sizes - all cts nodes are scalar, all discrete nodes are binary +ns = ones(1, n); +dnodes = [F W B]; +cnodes = mysetdiff(1:n, dnodes); +ns(dnodes) = 2; + +bnet = mk_bnet(dag, ns, 'discrete', dnodes); +</pre> +'dnodes' is a list of the discrete nodes; 'cnodes' is the continuous +nodes. 'mysetdiff' is a faster version of the built-in 'setdiff'. +<p> + + +<h3>Specifying the parameters</h3> + +The parameters of the discrete nodes can be specified as follows. +<pre> +bnet.CPD{B} = tabular_CPD(bnet, B, 'CPT', [0.85 0.15]); % 1=stable, 2=unstable +bnet.CPD{F} = tabular_CPD(bnet, F, 'CPT', [0.95 0.05]); % 1=intact, 2=defect +bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [2/7 5/7]); % 1=industrial, 2=household +</pre> + +<p> +The parameters of the continuous nodes can be specified as follows. +<pre> +bnet.CPD{E} = gaussian_CPD(bnet, E, 'mean', [-3.9 -0.4 -3.2 -0.5], ... + 'cov', [0.00002 0.0001 0.00002 0.0001]); +bnet.CPD{D} = gaussian_CPD(bnet, D, 'mean', [6.5 6.0 7.5 7.0], ... + 'cov', [0.03 0.04 0.1 0.1], 'weights', [1 1 1 1]); +bnet.CPD{C} = gaussian_CPD(bnet, C, 'mean', [-2 -1], 'cov', [0.1 0.3]); +bnet.CPD{L} = gaussian_CPD(bnet, L, 'mean', 3, 'cov', 0.25, 'weights', -0.5); +bnet.CPD{Min} = gaussian_CPD(bnet, Min, 'mean', [0.5 -0.5], 'cov', [0.01 0.005]); +bnet.CPD{Mout} = gaussian_CPD(bnet, Mout, 'mean', 0, 'cov', 0.002, 'weights', [1 1]); +</pre> + + +<h3><a name="cg_infer">Inference</h3> + +<!--Let us perform inference in the <a href="#cg_model">waste incinerator example</a>.--> +First we compute the unconditional marginals. +<pre> +engine = jtree_inf_engine(bnet); +evidence = cell(1,n); +[engine, ll] = enter_evidence(engine, evidence); +marg = marginal_nodes(engine, E); +</pre> +<!--(Of course, we could use <tt>cond_gauss_inf_engine</tt> instead of jtree.)--> +'marg' is a structure that contains the fields 'mu' and 'Sigma', which +contain the mean and (co)variance of the marginal on E. +In this case, they are both scalars. +Let us check they match the published figures (to 2 decimal places). +<!--(We can't expect +more precision than this in general because I have implemented the algorithm of +Lauritzen (1992), which can be numerically unstable.)--> +<pre> +tol = 1e-2; +assert(approxeq(marg.mu, -3.25, tol)); +assert(approxeq(sqrt(marg.Sigma), 0.709, tol)); +</pre> +We can compute the other posteriors similarly. +Now let us add some evidence. +<pre> +evidence = cell(1,n); +evidence{W} = 1; % industrial +evidence{L} = 1.1; +evidence{C} = -0.9; +[engine, ll] = enter_evidence(engine, evidence); +</pre> +Now we find +<pre> +marg = marginal_nodes(engine, E); +assert(approxeq(marg.mu, -3.8983, tol)); +assert(approxeq(sqrt(marg.Sigma), 0.0763, tol)); +</pre> + + +We can also compute the joint probability on a set of nodes. +For example, P(D, Mout | evidence) is a 2D Gaussian: +<pre> +marg = marginal_nodes(engine, [D Mout]) +marg = + domain: [6 8] + mu: [2x1 double] + Sigma: [2x2 double] + T: 1.0000 +</pre> +The mean is +<pre> +marg.mu +ans = + 3.6077 + 4.1077 +</pre> +and the covariance matrix is +<pre> +marg.Sigma +ans = + 0.1062 0.1062 + 0.1062 0.1182 +</pre> +It is easy to visualize this posterior using standard Matlab plotting +functions, e.g., +<pre> +gaussplot2d(marg.mu, marg.Sigma); +</pre> +produces the following picture. + +<p> +<center> +<IMG SRC="Figures/gaussplot.png"> +</center> +<p> + + +The T field indicates that the mixing weight of this Gaussian +component is 1.0. +If the joint contains discrete and continuous variables, the result +will be a mixture of Gaussians, e.g., +<pre> +marg = marginal_nodes(engine, [F E]) + domain: [1 3] + mu: [-3.9000 -0.4003] + Sigma: [1x1x2 double] + T: [0.9995 4.7373e-04] +</pre> +The interpretation is +Sigma(i,j,k) = Cov[ E(i) E(j) | F=k ]. +In this case, E is a scalar, so i=j=1; k specifies the mixture component. +<p> +We saw in the sprinkler network that BNT sets the effective size of +observed discrete nodes to 1, since they only have one legal value. +For continuous nodes, BNT sets their length to 0, +since they have been reduced to a point. +For example, +<pre> +marg = marginal_nodes(engine, [B C]) + domain: [4 5] + mu: [] + Sigma: [] + T: [0.0123 0.9877] +</pre> +It is simple to post-process the output of marginal_nodes. +For example, the file BNT/examples/static/cg1 sets the mu term of +observed nodes to their observed value, and the Sigma term to 0 (since +observed nodes have no variance). + +<p> +Note that the implemented version of the junction tree is numerically +unstable when using CG potentials +(which is why, in the example above, we only required our answers to agree with +the published ones to 2dp.) +This is why you might want to use <tt>stab_cond_gauss_inf_engine</tt>, +implemented by Shan Huang. This is described in + +<ul> +<li> "Stable Local Computation with Conditional Gaussian Distributions", +S. Lauritzen and F. Jensen, Tech Report R-99-2014, +Dept. Math. Sciences, Allborg Univ., 1999. +</ul> + +However, even the numerically stable version +can be computationally intractable if there are many hidden discrete +nodes, because the number of mixture components grows exponentially e.g., in a +<a href="usage_dbn.html#lds">switching linear dynamical system</a>. +In general, one must resort to approximate inference techniques: see +the discussion on <a href="#engines">inference engines</a> below. + + +<h2><a name="hybrid">Other hybrid models</h2> + +When we have C->D arcs, where C is hidden, we need to use +approximate inference. +One approach (not implemented in BNT) is described in +<ul> +<li> <a +href="http://www.cs.berkeley.edu/~murphyk/Papers/hybrid_uai99.ps.gz">A +Variational Approximation for Bayesian Networks with +Discrete and Continuous Latent Variables</a>, +K. Murphy, UAI 99. +</ul> +Of course, one can always use <a href="#sampling">sampling</a> methods +for approximate inference in such models. + + + +<h1><a name="param_learning">Parameter Learning</h1> + +The parameter estimation routines in BNT can be classified into 4 +types, depending on whether the goal is to compute +a full (Bayesian) posterior over the parameters or just a point +estimate (e.g., Maximum Likelihood or Maximum A Posteriori), +and whether all the variables are fully observed or there is missing +data/ hidden variables (partial observability). +<p> + +<TABLE BORDER> +<tr> + <TH></TH> + <th>Full obs</th> + <th>Partial obs</th> +</tr> +<tr> + <th>Point</th> + <td><tt>learn_params</tt></td> + <td><tt>learn_params_em</tt></td> +</tr> +<tr> + <th>Bayes</th> + <td><tt>bayes_update_params</tt></td> + <td>not yet supported</td> +</tr> +</table> + + +<h2><a name="load_data">Loading data from a file</h2> + +To load numeric data from an ASCII text file called 'dat.txt', where each row is a +case and columns are separated by white-space, such as +<pre> +011979 1626.5 0.0 +021979 1367.0 0.0 +... +</pre> +you can use +<pre> +data = load('dat.txt'); +</pre> +or +<pre> +load dat.txt -ascii +</pre> +In the latter case, the data is stored in a variable called 'dat' (the +filename minus the extension). +Alternatively, suppose the data is stored in a .csv file (has commas +separating the columns, and contains a header line), such as +<pre> +header info goes here +ORD,011979,1626.5,0.0 +DSM,021979,1367.0,0.0 +... +</pre> +You can load this using +<pre> +[a,b,c,d] = textread('dat.txt', '%s %d %f %f', 'delimiter', ',', 'headerlines', 1); +</pre> +If your file is not in either of these formats, you can either use Perl to convert +it to this format, or use the Matlab scanf command. +Type +<tt> +help iofun +</tt> +for more information on Matlab's file functions. +<!-- +<p> +To load data directly from Excel, +you should buy the +<a href="http://www.mathworks.com/products/excellink/">Excel Link</a>. +To load data directly from a relational database, +you should buy the +<a href="http://www.mathworks.com/products/database">Database +toolbox</a>. +--> +<p> +BNT learning routines require data to be stored in a cell array. +data{i,m} is the value of node i in case (example) m, i.e., each +<em>column</em> is a case. +If node i is not observed in case m (missing value), set +data{i,m} = []. +(Not all the learning routines can cope with such missing values, however.) +In the special case that all the nodes are observed and are +scalar-valued (as opposed to vector-valued), the data can be +stored in a matrix (as opposed to a cell-array). +<p> +Suppose, as in the <a href="#mixexp">mixture of experts example</a>, +that we have 3 nodes in the graph: X(1) is the observed input, X(3) is +the observed output, and X(2) is a hidden (gating) node. We can +create the dataset as follows. +<pre> +data = load('dat.txt'); +ncases = size(data, 1); +cases = cell(3, ncases); +cases([1 3], :) = num2cell(data'); +</pre> +Notice how we transposed the data, to convert rows into columns. +Also, cases{2,m} = [] for all m, since X(2) is always hidden. + + +<h2><a name="mle_complete">Maximum likelihood parameter estimation from complete data</h2> + +As an example, let's generate some data from the sprinkler network, randomize the parameters, +and then try to recover the original model. +First we create some training data using forwards sampling. +<pre> +samples = cell(N, nsamples); +for i=1:nsamples + samples(:,i) = sample_bnet(bnet); +end +</pre> +samples{j,i} contains the value of the j'th node in case i. +sample_bnet returns a cell array because, in general, each node might +be a vector of different length. +In this case, all nodes are discrete (and hence scalars), so we +could have used a regular array instead (which can be quicker): +<pre> +data = cell2num(samples); +</pre +So now data(j,i) = samples{j,i}. +<p> +Now we create a network with random parameters. +(The initial values of bnet2 don't matter in this case, since we can find the +globally optimal MLE independent of where we start.) +<pre> +% Make a tabula rasa +bnet2 = mk_bnet(dag, node_sizes); +seed = 0; +rand('state', seed); +bnet2.CPD{C} = tabular_CPD(bnet2, C); +bnet2.CPD{R} = tabular_CPD(bnet2, R); +bnet2.CPD{S} = tabular_CPD(bnet2, S); +bnet2.CPD{W} = tabular_CPD(bnet2, W); +</pre> +Finally, we find the maximum likelihood estimates of the parameters. +<pre> +bnet3 = learn_params(bnet2, samples); +</pre> +To view the learned parameters, we use a little Matlab hackery. +<pre> +CPT3 = cell(1,N); +for i=1:N + s=struct(bnet3.CPD{i}); % violate object privacy + CPT3{i}=s.CPT; +end +</pre> +Here are the parameters learned for node 4. +<pre> +dispcpt(CPT3{4}) +1 1 : 1.0000 0.0000 +2 1 : 0.2000 0.8000 +1 2 : 0.2273 0.7727 +2 2 : 0.0000 1.0000 +</pre> +So we see that the learned parameters are fairly close to the "true" +ones, which we display below. +<pre> +dispcpt(CPT{4}) +1 1 : 1.0000 0.0000 +2 1 : 0.1000 0.9000 +1 2 : 0.1000 0.9000 +2 2 : 0.0100 0.9900 +</pre> +We can get better results by using a larger training set, or using +informative priors (see <a href="#prior">below</a>). + + + +<h2><a name="prior">Parameter priors</h2> + +Currently, only tabular CPDs can have priors on their parameters. +The conjugate prior for a multinomial is the Dirichlet. +(For binary random variables, the multinomial is the same as the +Bernoulli, and the Dirichlet is the same as the Beta.) +<p> +The Dirichlet has a simple interpretation in terms of pseudo counts. +If we let N_ijk = the num. times X_i=k and Pa_i=j occurs in the +training set, where Pa_i are the parents of X_i, +then the maximum likelihood (ML) estimate is +T_ijk = N_ijk / N_ij (where N_ij = sum_k' N_ijk'), which will be 0 if N_ijk=0. +To prevent us from declaring that (X_i=k, Pa_i=j) is impossible just because this +event was not seen in the training set, +we can pretend we saw value k of X_i, for each value j of Pa_i some number (alpha_ijk) +of times in the past. +The MAP (maximum a posterior) estimate is then +<pre> +T_ijk = (N_ijk + alpha_ijk) / (N_ij + alpha_ij) +</pre> +and is never 0 if all alpha_ijk > 0. +For example, consider the network A->B, where A is binary and B has 3 +values. +A uniform prior for B has the form +<pre> + B=1 B=2 B=3 +A=1 1 1 1 +A=2 1 1 1 +</pre> +which can be created using +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'unif'); +</pre> +This prior does not satisfy the likelihood equivalence principle, +which says that <a href="#markov_equiv">Markov equivalent</a> models +should have the same marginal likelihood. +A prior that does satisfy this principle is shown below. +Heckerman (1995) calls this the +BDeu prior (likelihood equivalent uniform Bayesian Dirichlet). +<pre> + B=1 B=2 B=3 +A=1 1/6 1/6 1/6 +A=2 1/6 1/6 1/6 +</pre> +where we put N/(q*r) in each bin; N is the equivalent sample size, +r=|A|, q = |B|. +This can be created as follows +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'BDeu'); +</pre> +Here, 1 is the equivalent sample size, and is the strength of the +prior. +You can change this using +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', ... + 'BDeu', 'dirichlet_weight', 10); +</pre> +<!--where counts is an array of pseudo-counts of the same size as the +CPT.--> +<!-- +<p> +When you specify a prior, you should set row i of the CPT to the +normalized version of row i of the pseudo-count matrix, i.e., to the +expected values of the parameters. This will ensure that computing the +marginal likelihood sequentially (see <a +href="#bayes_learn">below</a>) and in batch form gives the same +results. +To do this, proceed as follows. +<pre> +tabular_CPD(bnet, i, 'prior', counts, 'CPT', mk_stochastic(counts)); +</pre> +For a non-informative prior, you can just write +<pre> +tabular_CPD(bnet, i, 'prior', 'unif', 'CPT', 'unif'); +</pre> +--> + + +<h2><a name="bayes_learn">(Sequential) Bayesian parameter updating from complete data</h2> + +If we use conjugate priors and have fully observed data, we can +compute the posterior over the parameters in batch form as follows. +<pre> +cases = sample_bnet(bnet, nsamples); +bnet = bayes_update_params(bnet, cases); +LL = log_marg_lik_complete(bnet, cases); +</pre> +bnet.CPD{i}.prior contains the new Dirichlet pseudocounts, +and bnet.CPD{i}.CPT is set to the mean of the posterior (the +normalized counts). +(Hence if the initial pseudo counts are 0, +<tt>bayes_update_params</tt> and <tt>learn_params</tt> will give the +same result.) + + + + +<p> +We can compute the same result sequentially (on-line) as follows. +<pre> +LL = 0; +for m=1:nsamples + LL = LL + log_marg_lik_complete(bnet, cases(:,m)); + bnet = bayes_update_params(bnet, cases(:,m)); +end +</pre> + +The file <tt>BNT/examples/static/StructLearn/model_select1</tt> has an example of +sequential model selection which uses the same idea. +We generate data from the model A->B +and compute the posterior prob of all 3 dags on 2 nodes: + (1) A B, (2) A <- B , (3) A -> B +Models 2 and 3 are <a href="#markov_equiv">Markov equivalent</a>, and therefore indistinguishable from +observational data alone, so we expect their posteriors to be the same +(assuming a prior which satisfies likelihood equivalence). +If we use random parameters, the "true" model only gets a higher posterior after 2000 trials! +However, if we make B a noisy NOT gate, the true model "wins" after 12 +trials, as shown below (red = model 1, blue/green (superimposed) +represents models 2/3). +<p> +<img src="Figures/model_select.png"> +<p> +The use of marginal likelihood for model selection is discussed in +greater detail in the +section on <a href="structure_learning">structure learning</a>. + + + + +<h2><a name="em">Maximum likelihood parameter estimation with missing values</h2> + +Now we consider learning when some values are not observed. +Let us randomly hide half the values generated from the water +sprinkler example. +<pre> +samples2 = samples; +hide = rand(N, nsamples) > 0.5; +[I,J]=find(hide); +for k=1:length(I) + samples2{I(k), J(k)} = []; +end +</pre> +samples2{i,l} is the value of node i in training case l, or [] if unobserved. +<p> +Now we will compute the MLEs using the EM algorithm. +We need to use an inference algorithm to compute the expected +sufficient statistics in the E step; the M (maximization) step is as +above. +<pre> +engine2 = jtree_inf_engine(bnet2); +max_iter = 10; +[bnet4, LLtrace] = learn_params_em(engine2, samples2, max_iter); +</pre> +LLtrace(i) is the log-likelihood at iteration i. We can plot this as +follows: +<pre> +plot(LLtrace, 'x-') +</pre> +Let's display the results after 10 iterations of EM. +<pre> +celldisp(CPT4) +CPT4{1} = + 0.6616 + 0.3384 +CPT4{2} = + 0.6510 0.3490 + 0.8751 0.1249 +CPT4{3} = + 0.8366 0.1634 + 0.0197 0.9803 +CPT4{4} = +(:,:,1) = + 0.8276 0.0546 + 0.5452 0.1658 +(:,:,2) = + 0.1724 0.9454 + 0.4548 0.8342 +</pre> +We can get improved performance by using one or more of the following +methods: +<ul> +<li> Increasing the size of the training set. +<li> Decreasing the amount of hidden data. +<li> Running EM for longer. +<li> Using informative priors. +<li> Initialising EM from multiple starting points. +</ul> + +Click <a href="#gaussian">here</a> for a discussion of learning +Gaussians, which can cause numerical problems. + +<h2><a name="tying">Parameter tying</h2> + +In networks with repeated structure (e.g., chains and grids), it is +common to assume that the parameters are the same at every node. This +is called parameter tying, and reduces the amount of data needed for +learning. +<p> +When we have tied parameters, there is no longer a one-to-one +correspondence between nodes and CPDs. +Rather, each CPD species the parameters for a whole equivalence class +of nodes. +It is easiest to see this by example. +Consider the following <a href="usage_dbn.html#hmm">hidden Markov +model (HMM)</a> +<p> +<img src="Figures/hmm3.gif"> +<p> +<!-- +We can create this graph structure, assuming we have T time-slices, +as follows. +(We number the nodes as shown in the figure, but we could equally well +number the hidden nodes 1:T, and the observed nodes T+1:2T.) +<pre> +N = 2*T; +dag = zeros(N); +hnodes = 1:2:2*T; +for i=1:T-1 + dag(hnodes(i), hnodes(i+1))=1; +end +onodes = 2:2:2*T; +for i=1:T + dag(hnodes(i), onodes(i)) = 1; +end +</pre> +<p> +The hidden nodes are always discrete, and have Q possible values each, +but the observed nodes can be discrete or continuous, and have O possible values/length. +<pre> +if cts_obs + dnodes = hnodes; +else + dnodes = 1:N; +end +ns = ones(1,N); +ns(hnodes) = Q; +ns(onodes) = O; +</pre> +--> +When HMMs are used for semi-infinite processes like speech recognition, +we assume the transition matrix +P(H(t+1)|H(t)) is the same for all t; this is called a time-invariant +or homogenous Markov chain. +Hence hidden nodes 2, 3, ..., T +are all in the same equivalence class, say class Hclass. +Similarly, the observation matrix P(O(t)|H(t)) is assumed to be the +same for all t, so the observed nodes are all in the same equivalence +class, say class Oclass. +Finally, the prior term P(H(1)) is in a class all by itself, say class +H1class. +This is illustrated below, where we explicitly represent the +parameters as random variables (dotted nodes). +<p> +<img src="Figures/hmm4_params.gif"> +<p> +In BNT, we cannot represent parameters as random variables (nodes). +Instead, we "hide" the +parameters inside one CPD for each equivalence class, +and then specify that the other CPDs should share these parameters, as +follows. +<pre> +hnodes = 1:2:2*T; +onodes = 2:2:2*T; +H1class = 1; Hclass = 2; Oclass = 3; +eclass = ones(1,N); +eclass(hnodes(2:end)) = Hclass; +eclass(hnodes(1)) = H1class; +eclass(onodes) = Oclass; +% create dag and ns in the usual way +bnet = mk_bnet(dag, ns, 'discrete', dnodes, 'equiv_class', eclass); +</pre> +Finally, we define the parameters for each equivalence class: +<pre> +bnet.CPD{H1class} = tabular_CPD(bnet, hnodes(1)); % prior +bnet.CPD{Hclass} = tabular_CPD(bnet, hnodes(2)); % transition matrix +if cts_obs + bnet.CPD{Oclass} = gaussian_CPD(bnet, onodes(1)); +else + bnet.CPD{Oclass} = tabular_CPD(bnet, onodes(1)); +end +</pre> +In general, if bnet.CPD{e} = xxx_CPD(bnet, j), then j should be a +member of e's equivalence class; that is, it is not always the case +that e == j. You can use bnet.rep_of_eclass(e) to return the +representative of equivalence class e. +BNT will look up the parents of j to determine the size +of the CPT to use. It assumes that this is the same for all members of +the equivalence class. +Click <a href="http://www.cs.berkeley.edu/~murphyk/Bayes/param_tieing.html">here</a> for +a more complex example of parameter tying. +<p> +Note: +Normally one would define an HMM as a +<a href = "usage_dbn.html">Dynamic Bayes Net</a> +(see the function BNT/examples/dynamic/mk_chmm.m). +However, one can define an HMM as a static BN using the function +BNT/examples/static/Models/mk_hmm_bnet.m. + + + +<h1><a name="structure_learning">Structure learning</h1> + +Update (9/29/03): +Phillipe LeRay is developing some additional structure learning code +on top of BNT. Click <a +href="http://bnt.insa-rouen.fr/ajouts.html">here</a> +for details. + +<p> + +There are two very different approaches to structure learning: +constraint-based and search-and-score. +In the <a href="#constraint">constraint-based approach</a>, +we start with a fully connected graph, and remove edges if certain +conditional independencies are measured in the data. +This has the disadvantage that repeated independence tests lose +statistical power. +<p> +In the more popular search-and-score approach, +we perform a search through the space of possible DAGs, and either +return the best one found (a point estimate), or return a sample of the +models found (an approximation to the Bayesian posterior). +<p> +Unfortunately, the number of DAGs as a function of the number of +nodes, G(n), is super-exponential in n. +A closed form formula for G(n) is not known, but the first few values +are shown below (from Cooper, 1999). + +<table> +<tr> <th>n</th> <th align=left>G(n)</th> </tr> +<tr> <td>1</td> <td>1</td> </tr> +<tr> <td>2</td> <td>3</td> </tr> +<tr> <td>3</td> <td>25</td> </tr> +<tr> <td>4</td> <td>543</td> </tr> +<tr> <td>5</td> <td>29,281</td> </tr> +<tr> <td>6</td> <td>3,781,503</td> </tr> +<tr> <td>7</td> <td>1.1 x 10^9</td> </tr> +<tr> <td>8</td> <td>7.8 x 10^11</td> </tr> +<tr> <td>9</td> <td>1.2 x 10^15</td> </tr> +<tr> <td>10</td> <td>4.2 x 10^18</td> </tr> +</table> + +Since the number of DAGs is super-exponential in the number of nodes, +we cannot exhaustively search the space, so we either use a local +search algorithm (e.g., greedy hill climbining, perhaps with multiple +restarts) or a global search algorithm (e.g., Markov Chain Monte +Carlo). +<p> +If we know a total ordering on the nodes, +finding the best structure amounts to picking the best set of parents +for each node independently. +This is what the K2 algorithm does. +If the ordering is unknown, we can search over orderings, +which is more efficient than searching over DAGs (Koller and Friedman, 2000). +<p> +In addition to the search procedure, we must specify the scoring +function. There are two popular choices. The Bayesian score integrates +out the parameters, i.e., it is the marginal likelihood of the model. +The BIC (Bayesian Information Criterion) is defined as +log P(D|theta_hat) - 0.5*d*log(N), where D is the data, theta_hat is +the ML estimate of the parameters, d is the number of parameters, and +N is the number of data cases. +The BIC method has the advantage of not requiring a prior. +<p> +BIC can be derived as a large sample +approximation to the marginal likelihood. +(It is also equal to the Minimum Description Length of a model.) +However, in practice, the sample size does not need to be very large +for the approximation to be good. +For example, in the figure below, we plot the ratio between the log marginal likelihood +and the BIC score against data-set size; we see that the ratio rapidly +approaches 1, especially for non-informative priors. +(This plot was generated by the file BNT/examples/static/bic1.m. It +uses the water sprinkler BN with BDeu Dirichlet priors with different +equivalent sample sizes.) + +<p> +<center> +<IMG SRC="Figures/bic.png"> +</center> +<p> + +<p> +As with parameter learning, handling missing data/ hidden variables is +much harder than the fully observed case. +The structure learning routines in BNT can therefore be classified into 4 +types, analogously to the parameter learning case. +<p> + +<TABLE BORDER> +<tr> + <TH></TH> + <th>Full obs</th> + <th>Partial obs</th> +</tr> +<tr> + <th>Point</th> + <td><tt>learn_struct_K2</tt> <br> +<!-- <tt>learn_struct_hill_climb</tt></td> --> + <td><tt>not yet supported</tt></td> +</tr> +<tr> + <th>Bayes</th> + <td><tt>learn_struct_mcmc</tt></td> + <td>not yet supported</td> +</tr> +</table> + + +<h2><a name="markov_equiv">Markov equivalence</h2> + +If two DAGs encode the same conditional independencies, they are +called Markov equivalent. The set of all DAGs can be paritioned into +Markov equivalence classes. Graphs within the same class can +have +the direction of some of their arcs reversed without changing any of +the CI relationships. +Each class can be represented by a PDAG +(partially directed acyclic graph) called an essential graph or +pattern. This specifies which edges must be oriented in a certain +direction, and which may be reversed. + +<p> +When learning graph structure from observational data, +the best one can hope to do is to identify the model up to Markov +equivalence. To distinguish amongst graphs within the same equivalence +class, one needs interventional data: see the discussion on <a +href="#active">active learning</a> below. + + + +<h2><a name="enumerate">Exhaustive search</h2> + +The brute-force approach to structure learning is to enumerate all +possible DAGs, and score each one. This provides a "gold standard" +with which to compare other algorithms. We can do this as follows. +<pre> +dags = mk_all_dags(N); +score = score_dags(data, ns, dags); +</pre> +where data(i,m) is the value of node i in case m, +and ns(i) is the size of node i. +If the DAGs have a lot of families in common, we can cache the sufficient statistics, +making this potentially more efficient than scoring the DAGs one at a time. +(Caching is not currently implemented, however.) +<p> +By default, we use the Bayesian scoring metric, and assume CPDs are +represented by tables with BDeu(1) priors. +We can override these defaults as follows. +If we want to use uniform priors, we can say +<pre> +params = cell(1,N); +for i=1:N + params{i} = {'prior', 'unif'}; +end +score = score_dags(data, ns, dags, 'params', params); +</pre> +params{i} is a cell-array, containing optional arguments that are +passed to the constructor for CPD i. +<p> +Now suppose we want to use different node types, e.g., +Suppose nodes 1 and 2 are Gaussian, and nodes 3 and 4 softmax (both +these CPDs can support discrete and continuous parents, which is +necessary since all other nodes will be considered as parents). +The Bayesian scoring metric currently only works for tabular CPDs, so +we will use BIC: +<pre> +score = score_dags(data, ns, dags, 'discrete', [3 4], 'params', [], + 'type', {'gaussian', 'gaussian', 'softmax', softmax'}, 'scoring_fn', 'bic') +</pre> +In practice, one can't enumerate all possible DAGs for N > 5, +but one can evaluate any reasonably-sized set of hypotheses in this +way (e.g., nearest neighbors of your current best guess). +Think of this as "computer assisted model refinement" as opposed to de +novo learning. + + +<h2><a name="K2">K2</h2> + +The K2 algorithm (Cooper and Herskovits, 1992) is a greedy search algorithm that works as follows. +Initially each node has no parents. It then adds incrementally that parent whose addition most +increases the score of the resulting structure. When the addition of no single +parent can increase the score, it stops adding parents to the node. +Since we are using a fixed ordering, we do not need to check for +cycles, and can choose the parents for each node independently. +<p> +The original paper used the Bayesian scoring +metric with tabular CPDs and Dirichlet priors. +BNT generalizes this to allow any kind of CPD, and either the Bayesian +scoring metric or BIC, as in the example <a href="#enumerate">above</a>. +In addition, you can specify +an optional upper bound on the number of parents for each node. +The file BNT/examples/static/k2demo1.m gives an example of how to use K2. +We use the water sprinkler network and sample 100 cases from it as before. +Then we see how much data it takes to recover the generating structure: +<pre> +order = [C S R W]; +max_fan_in = 2; +sz = 5:5:100; +for i=1:length(sz) + dag2 = learn_struct_K2(data(:,1:sz(i)), node_sizes, order, 'max_fan_in', max_fan_in); + correct(i) = isequal(dag, dag2); +end +</pre> +Here are the results. +<pre> +correct = + Columns 1 through 12 + 0 0 0 0 0 0 0 1 0 1 1 1 + Columns 13 through 20 + 1 1 1 1 1 1 1 1 +</pre> +So we see it takes about sz(10)=50 cases. (BIC behaves similarly, +showing that the prior doesn't matter too much.) +In general, we cannot hope to recover the "true" generating structure, +only one that is in its <a href="#markov_equiv">Markov equivalence +class</a>. + + +<h2><a name="hill_climb">Hill-climbing</h2> + +Hill-climbing starts at a specific point in space, +considers all nearest neighbors, and moves to the neighbor +that has the highest score; if no neighbors have higher +score than the current point (i.e., we have reached a local maximum), +the algorithm stops. One can then restart in another part of the space. +<p> +A common definition of "neighbor" is all graphs that can be +generated from the current graph by adding, deleting or reversing a +single arc, subject to the acyclicity constraint. +Other neighborhoods are possible: see +<a href="http://research.microsoft.com/~dmax/publications/jmlr02.pdf"> +Optimal Structure Identification with Greedy Search</a>, Max +Chickering, JMLR 2002. + +<!-- +Note: This algorithm is currently (Feb '02) being implemented by Qian +Diao. +--> + + +<h2><a name="mcmc">MCMC</h2> + +We can use a Markov Chain Monte Carlo (MCMC) algorithm called +Metropolis-Hastings (MH) to search the space of all +DAGs. +The standard proposal distribution is to consider moving to all +nearest neighbors in the sense defined <a href="#hill_climb">above</a>. +<p> +The function can be called +as in the following example. +<pre> +[sampled_graphs, accept_ratio] = learn_struct_mcmc(data, ns, 'nsamples', 100, 'burnin', 10); +</pre> +We can convert our set of sampled graphs to a histogram +(empirical posterior over all the DAGs) thus +<pre> +all_dags = mk_all_dags(N); +mcmc_post = mcmc_sample_to_hist(sampled_graphs, all_dags); +</pre> +To see how well this performs, let us compute the exact posterior exhaustively. +<p> +<pre> +score = score_dags(data, ns, all_dags); +post = normalise(exp(score)); % assuming uniform structural prior +</pre> +We plot the results below. +(The data set was 100 samples drawn from a random 4 node bnet; see the +file BNT/examples/static/mcmc1.) +<pre> +subplot(2,1,1) +bar(post) +subplot(2,1,2) +bar(mcmc_post) +</pre> +<img src="Figures/mcmc_post.jpg" width="800" height="500"> +<p> +We can also plot the acceptance ratio versus number of MCMC steps, +as a crude convergence diagnostic. +<pre> +clf +plot(accept_ratio) +</pre> +<img src="Figures/mcmc_accept.jpg" width="800" height="300"> +<p> +Even though the number of samples needed by MCMC is theoretically +polynomial (not exponential) in the dimensionality of the search space, in practice it has been +found that MCMC does not converge in reasonable time for graphs with +more than about 10 nodes. + + + + +<h2><a name="active">Active structure learning</h2> + +As was mentioned <a href="#markov_equiv">above</a>, +one can only learn a DAG up to Markov equivalence, even given infinite data. +If one is interested in learning the structure of a causal network, +one needs interventional data. +(By "intervention" we mean forcing a node to take on a specific value, +thereby effectively severing its incoming arcs.) +<p> +Most of the scoring functions accept an optional argument +that specifies whether a node was observed to have a certain value, or +was forced to have that value: we set clamped(i,m)=1 if node i was +forced in training case m. e.g., see the file +BNT/examples/static/cooper_yoo. +<p> +An interesting question is to decide which interventions to perform +(c.f., design of experiments). For details, see the following tech +report +<ul> +<li> <a href = "../../Papers/alearn.ps.gz"> +Active learning of causal Bayes net structure</a>, Kevin Murphy, March +2001. +</ul> + + +<h2><a name="struct_em">Structural EM</h2> + +Computing the Bayesian score when there is partial observability is +computationally challenging, because the parameter posterior becomes +multimodal (the hidden nodes induce a mixture distribution). +One therefore needs to use approximations such as BIC. +Unfortunately, search algorithms are still expensive, because we need +to run EM at each step to compute the MLE, which is needed to compute +the score of each model. An alternative approach is +to do the local search steps inside of the M step of EM, which is more +efficient since the data has been "filled in" - this is +called the structural EM algorithm (Friedman 1997), and provably +converges to a local maximum of the BIC score. +<p> +Wei Hu has implemented SEM for discrete nodes. +You can download his package from +<a href="../SEM.zip">here</a>. +Please address all questions about this code to +wei.hu@intel.com. +See also <a href="#phl">Phl's implementation of SEM</a>. + +<!-- +<h2><a name="reveal">REVEAL algorithm</h2> + +A simple way to learn the structure of a fully observed, discrete, +factored DBN from a time series is described <a +href="usage_dbn.html#struct_learn">here</a>. +--> + + +<h2><a name="graphdraw">Visualizing the graph</h2> + +You can visualize an arbitrary graph (such as one learned using the +structure learning routines) with Matlab code contributed by +<a href="http://www.mbfys.kun.nl/~cemgil/matlab/layout.html">Ali +Taylan Cemgil</a> +from the University of Nijmegen. +For static BNs, call it as follows: +<pre> +draw_graph(bnet.dag); +</pre> +For example, this is the output produced on a +<a href="#qmr">random QMR-like model</a>: +<p> +<img src="Figures/qmr.rnd.jpg"> +<p> +If you install the excellent <a +href="http://www.research.att.com/sw/tools/graphviz">graphhviz</a>, an +open-source graph visualization package from AT&T, +you can create a much better visualization as follows +<pre> +graph_to_dot(bnet.dag) +</pre> +This works by converting the adjacency matrix to a file suitable +for input to graphviz (using the dot format), +then converting the output of graphviz to postscript, and displaying the results using +ghostview. +You can do each of these steps separately for more control, as shown +below. +<pre> +graph_to_dot(bnet.dag, 'filename', 'foo.dot'); +dot -Tps foo.dot -o foo.ps +ghostview foo.ps & +</pre> + +<h2><a name = "constraint">Constraint-based methods</h2> + +The IC algorithm (Pearl and Verma, 1991), +and the faster, but otherwise equivalent, PC algorithm (Spirtes, Glymour, and Scheines 1993), +computes many conditional independence tests, +and combines these constraints into a +PDAG to represent the whole +<a href="#markov_equiv">Markov equivalence class</a>. +<p> +IC*/FCI extend IC/PC to handle latent variables: see <a href="#ic_star">below</a>. +(IC stands for inductive causation; PC stands for Peter and Clark, +the first names of Spirtes and Glymour; FCI stands for fast causal +inference. +What we, following Pearl (2000), call IC* was called +IC in the original Pearl and Verma paper.) +For details, see +<ul> +<li> +<a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Causation, +Prediction, and Search</a>, Spirtes, Glymour and +Scheines (SGS), 2001 (2nd edition), MIT Press. +<li> +<a href="http://bayes.cs.ucla.edu/BOOK-2K/index.html">Causality: Models, Reasoning and Inference</a>, J. Pearl, +2000, Cambridge University Press. +</ul> + +<p> + +The PC algorithm takes as arguments a function f, the number of nodes N, +the maximum fan in K, and additional arguments A which are passed to f. +The function f(X,Y,S,A) returns 1 if X is conditionally independent of Y given S, and 0 +otherwise. +For example, suppose we cheat by +passing in a CI "oracle" which has access to the true DAG; the oracle +tests for d-separation in this DAG, i.e., +f(X,Y,S) calls dsep(X,Y,S,dag). We can to this as follows. +<pre> +pdag = learn_struct_pdag_pc('dsep', N, max_fan_in, dag); +</pre> +pdag(i,j) = -1 if there is definitely an i->j arc, +and pdag(i,j) = 1 if there is either an i->j or and i<-j arc. +<p> +Applied to the sprinkler network, this returns +<pre> +pdag = + 0 1 1 0 + 1 0 0 -1 + 1 0 0 -1 + 0 0 0 0 +</pre> +So as expected, we see that the V-structure at the W node is uniquely identified, +but the other arcs have ambiguous orientation. +<p> +We now give an example from p141 (1st edn) / p103 (2nd end) of the SGS +book. +This example concerns the female orgasm. +We are given a correlation matrix C between 7 measured factors (such +as subjective experiences of coital and masturbatory experiences), +derived from 281 samples, and want to learn a causal model of the +data. We will not discuss the merits of this type of work here, but +merely show how to reproduce the results in the SGS book. +Their program, +<a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Tetrad</a>, +makes use of the Fisher Z-test for conditional +independence, so we do the same: +<pre> +max_fan_in = 4; +nsamples = 281; +alpha = 0.05; +pdag = learn_struct_pdag_pc('cond_indep_fisher_z', n, max_fan_in, C, nsamples, alpha); +</pre> +In this case, the CI test is +<pre> +f(X,Y,S) = cond_indep_fisher_z(X,Y,S, C,nsamples,alpha) +</pre> +The results match those of Fig 12a of SGS apart from two edge +differences; presumably this is due to rounding error (although it +could be a bug, either in BNT or in Tetrad). +This example can be found in the file BNT/examples/static/pc2.m. + +<p> + +The IC* algorithm (Pearl and Verma, 1991), +and the faster FCI algorithm (Spirtes, Glymour, and Scheines 1993), +are like the IC/PC algorithm, except that they can detect the presence +of latent variables. +See the file <tt>learn_struct_pdag_ic_star</tt> written by Tamar +Kushnir. The output is a matrix P, defined as follows +(see Pearl (2000), p52 for details): +<pre> +% P(i,j) = -1 if there is either a latent variable L such that i <-L->j OR there is a directed edge from i->j. +% P(i,j) = -2 if there is a marked directed i-*>j edge. +% P(i,j) = P(j,i) = 1 if there is and undirected edge i--j +% P(i,j) = P(j,i) = 2 if there is a latent variable L such that i<-L->j. +</pre> + + +<h2><a name="phl">Philippe Leray's structure learning package</h2> + +Philippe Leray has written a +<a href="http://bnt.insa-rouen.fr/ajouts.html"> +structure learning package</a> that uses BNT. + +It currently (Juen 2003) has the following features: +<ul> +<li>PC with Chi2 statistical test +<li> MWST : Maximum weighted Spanning Tree +<li> Hill Climbing +<li> Greedy Search +<li> Structural EM +<li> hist_ic : optimal Histogram based on IC information criterion +<li> cpdag_to_dag +<li> dag_to_cpdag +<li> ... +</ul> + + +</a> + + +<!-- +<h2><a name="read_learning">Further reading on learning</h2> + +I recommend the following tutorials for more details on learning. +<ul> +<li> <a +href="http://www.cs.berkeley.edu/~murphyk/Papers/intel.ps.gz">My short +tutorial</a> on graphical models, which contains an overview of learning. + +<li> +<A HREF="ftp://ftp.research.microsoft.com/pub/tr/TR-95-06.PS"> +A tutorial on learning with Bayesian networks</a>, D. Heckerman, +Microsoft Research Tech Report, 1995. + +<li> <A HREF="http://www-cad.eecs.berkeley.edu/~wray/Mirror/lwgmja"> +Operations for Learning with Graphical Models</a>, +W. L. Buntine, JAIR'94, 159--225. +</ul> +<p> +--> + + + + + +<h1><a name="engines">Inference engines</h1> + +Up until now, we have used the junction tree algorithm for inference. +However, sometimes this is too slow, or not even applicable. +In general, there are many inference algorithms each of which make +different tradeoffs between speed, accuracy, complexity and +generality. Furthermore, there might be many implementations of the +same algorithm; for instance, a general purpose, readable version, +and a highly-optimized, specialized one. +To cope with this variety, we treat each inference algorithm as an +object, which we call an inference engine. + +<p> +An inference engine is an object that contains a bnet and supports the +'enter_evidence' and 'marginal_nodes' methods. The engine constructor +takes the bnet as argument and may do some model-specific processing. +When 'enter_evidence' is called, the engine may do some +evidence-specific processing. Finally, when 'marginal_nodes' is +called, the engine may do some query-specific processing. + +<p> +The amount of work done when each stage is specified -- structure, +parameters, evidence, and query -- depends on the engine. The cost of +work done early in this sequence can be amortized. On the other hand, +one can make better optimizations if one waits until later in the +sequence. +For example, the parameters might imply +conditional indpendencies that are not evident in the graph structure, +but can nevertheless be exploited; the evidence indicates which nodes +are observed and hence can effectively be disconnected from the +graph; and the query might indicate that large parts of the network +are d-separated from the query nodes. (Since it is not the actual +<em>values</em> of the evidence that matters, just which nodes are observed, +many engines allow you to specify which nodes will be observed when they are constructed, +i.e., before calling 'enter_evidence'. Some engines can still cope if +the actual pattern of evidence is different, e.g., if there is missing +data.) +<p> + +Although being maximally lazy (i.e., only doing work when a query is +issued) may seem desirable, +this is not always the most efficient. +For example, +when learning using EM, we need to call marginal_nodes N times, where N is the +number of nodes. <a href="varelim">Variable elimination</a> would end +up repeating a lot of work +each time marginal_nodes is called, making it inefficient for +learning. The junction tree algorithm, by contrast, uses dynamic +programming to avoid this redundant computation --- it calculates all +marginals in two passes during 'enter_evidence', so calling +'marginal_nodes' takes constant time. +<p> +We will discuss some of the inference algorithms implemented in BNT +below, and finish with a <a href="#engine_summary">summary</a> of all +of them. + + + + + + + +<h2><a name="varelim">Variable elimination</h2> + +The variable elimination algorithm, also known as bucket elimination +or peeling, is one of the simplest inference algorithms. +The basic idea is to "push sums inside of products"; this is explained +in more detail +<a +href="http://HTTP.CS.Berkeley.EDU/~murphyk/Bayes/bayes.html#infer">here</a>. +<p> +The principle of distributing sums over products can be generalized +greatly to apply to any commutative semiring. +This forms the basis of many common algorithms, such as Viterbi +decoding and the Fast Fourier Transform. For details, see + +<ul> +<li> R. McEliece and S. M. Aji, 2000. +<!--<a href="http://www.systems.caltech.edu/EE/Faculty/rjm/papers/GDL.ps">--> +<a href="GDL.pdf"> +The Generalized Distributive Law</a>, +IEEE Trans. Inform. Theory, vol. 46, no. 2 (March 2000), +pp. 325--343. + + +<li> +F. R. Kschischang, B. J. Frey and H.-A. Loeliger, 2001. +<a href="http://www.cs.toronto.edu/~frey/papers/fgspa.abs.html"> +Factor graphs and the sum-product algorithm</a> +IEEE Transactions on Information Theory, February, 2001. + +</ul> + +<p> +Choosing an order in which to sum out the variables so as to minimize +computational cost is known to be NP-hard. +The implementation of this algorithm in +<tt>var_elim_inf_engine</tt> makes no attempt to optimize this +ordering (in contrast, say, to <tt>jtree_inf_engine</tt>, which uses a +greedy search procedure to find a good ordering). +<p> +Note: unlike most algorithms, var_elim does all its computational work +inside of <tt>marginal_nodes</tt>, not inside of +<tt>enter_evidence</tt>. + + + + +<h2><a name="global">Global inference methods</h2> + +The simplest inference algorithm of all is to explicitely construct +the joint distribution over all the nodes, and then to marginalize it. +This is implemented in <tt>global_joint_inf_engine</tt>. +Since the size of the joint is exponential in the +number of discrete (hidden) nodes, this is not a very practical algorithm. +It is included merely for pedagogical and debugging purposes. +<p> +Three specialized versions of this algorithm have also been implemented, +corresponding to the cases where all the nodes are discrete (D), all +are Gaussian (G), and some are discrete and some Gaussian (CG). +They are called <tt>enumerative_inf_engine</tt>, +<tt>gaussian_inf_engine</tt>, +and <tt>cond_gauss_inf_engine</tt> respectively. +<p> +Note: unlike most algorithms, these global inference algorithms do all their computational work +inside of <tt>marginal_nodes</tt>, not inside of +<tt>enter_evidence</tt>. + + +<h2><a name="quickscore">Quickscore</h2> + +The junction tree algorithm is quite slow on the <a href="#qmr">QMR</a> network, +since the cliques are so big. +One simple trick we can use is to notice that hidden leaves do not +affect the posteriors on the roots, and hence do not need to be +included in the network. +A second trick is to notice that the negative findings can be +"absorbed" into the prior: +see the file +BNT/examples/static/mk_minimal_qmr_bnet for details. +<p> + +A much more significant speedup is obtained by exploiting special +properties of the noisy-or node, as done by the quickscore +algorithm. For details, see +<ul> +<li> Heckerman, "A tractable inference algorithm for diagnosing multiple diseases", UAI 89. +<li> Rish and Dechter, "On the impact of causal independence", UCI +tech report, 1998. +</ul> + +This has been implemented in BNT as a special-purpose inference +engine, which can be created and used as follows: +<pre> +engine = quickscore_inf_engine(inhibit, leak, prior); +engine = enter_evidence(engine, pos, neg); +m = marginal_nodes(engine, i); +</pre> + + +<h2><a name="belprop">Belief propagation</h2> + +Even using quickscore, exact inference takes time that is exponential +in the number of positive findings. +Hence for large networks we need to resort to approximate inference techniques. +See for example +<ul> +<li> T. Jaakkola and M. Jordan, "Variational probabilistic inference and the +QMR-DT network", JAIR 10, 1999. + +<li> K. Murphy, Y. Weiss and M. Jordan, "Loopy belief propagation for approximate inference: an empirical study", + UAI 99. +</ul> +The latter approximation +entails applying Pearl's belief propagation algorithm to a model even +if it has loops (hence the name loopy belief propagation). +Pearl's algorithm, implemented as <tt>pearl_inf_engine</tt>, gives +exact results when applied to singly-connected graphs +(a.k.a. polytrees, since +the underlying undirected topology is a tree, but a node may have +multiple parents). +To apply this algorithm to a graph with loops, +use <tt>pearl_inf_engine</tt>. +This can use a centralized or distributed message passing protocol. +You can use it as in the following example. +<pre> +engine = pearl_inf_engine(bnet, 'max_iter', 30); +engine = enter_evidence(engine, evidence); +m = marginal_nodes(engine, i); +</pre> +We found that this algorithm often converges, and when it does, often +is very accurate, but it depends on the precise setting of the +parameter values of the network. +(See the file BNT/examples/static/qmr1 to repeat the experiment for yourself.) +Understanding when and why belief propagation converges/ works +is a topic of ongoing research. +<p> +<tt>pearl_inf_engine</tt> can exploit special structure in noisy-or +and gmux nodes to compute messages efficiently. +<p> +<tt>belprop_inf_engine</tt> is like pearl, but uses potentials to +represent messages. Hence this is slower. +<p> +<tt>belprop_fg_inf_engine</tt> is like belprop, +but is designed for factor graphs. + + + +<h2><a name="sampling">Sampling</h2> + +BNT now (Mar '02) has two sampling (Monte Carlo) inference algorithms: +<ul> +<li> <tt>likelihood_weighting_inf_engine</tt> which does importance +sampling and can handle any node type. +<li> <tt>gibbs_sampling_inf_engine</tt>, written by Bhaskara Marthi. +Currently this can only handle tabular CPDs. +For a much faster and more powerful Gibbs sampling program, see +<a href="http://www.mrc-bsu.cam.ac.uk/bugs">BUGS</a>. +</ul> +Note: To generate samples from a network (which is not the same as inference!), +use <tt>sample_bnet</tt>. + + + +<h2><a name="engine_summary">Summary of inference engines</h2> + + +The inference engines differ in many ways. Here are +some of the major "axes": +<ul> +<li> Works for all topologies or makes restrictions? +<li> Works for all node types or makes restrictions? +<li> Exact or approximate inference? +</ul> + +<p> +In terms of topology, most engines handle any kind of DAG. +<tt>belprop_fg</tt> does approximate inference on factor graphs (FG), which +can be used to represent directed, undirected, and mixed (chain) +graphs. +(In the future, we plan to support exact inference on chain graphs.) +<tt>quickscore</tt> only works on QMR-like models. +<p> +In terms of node types: algorithms that use potentials can handle +discrete (D), Gaussian (G) or conditional Gaussian (CG) models. +Sampling algorithms can essentially handle any kind of node (distribution). +Other algorithms make more restrictive assumptions in exchange for +speed. +<p> +Finally, most algorithms are designed to give the exact answer. +The belief propagation algorithms are exact if applied to trees, and +in some other cases. +Sampling is considered approximate, even though, in the limit of an +infinite number of samples, it gives the exact answer. + +<p> + +Here is a summary of the properties +of all the engines in BNT which work on static networks. +<p> +<table> +<table border units = pixels><tr> +<td align=left width=0>Name +<td align=left width=0>Exact? +<td align=left width=0>Node type? +<td align=left width=0>topology +<tr> +<tr> +<td align=left> belprop +<td align=left> approx +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> belprop_fg +<td align=left> approx +<td align=left> D +<td align=left> factor graph +<tr> +<td align=left> cond_gauss +<td align=left> exact +<td align=left> CG +<td align=left> DAG +<tr> +<td align=left> enumerative +<td align=left> exact +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> gaussian +<td align=left> exact +<td align=left> G +<td align=left> DAG +<tr> +<td align=left> gibbs +<td align=left> approx +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> global_joint +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +<tr> +<td align=left> jtree +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +b<tr> +<td align=left> likelihood_weighting +<td align=left> approx +<td align=left> any +<td align=left> DAG +<tr> +<td align=left> pearl +<td align=left> approx +<td align=left> D,G +<td align=left> DAG +<tr> +<td align=left> pearl +<td align=left> exact +<td align=left> D,G +<td align=left> polytree +<tr> +<td align=left> quickscore +<td align=left> exact +<td align=left> noisy-or +<td align=left> QMR +<tr> +<td align=left> stab_cond_gauss +<td align=left> exact +<td align=left> CG +<td align=left> DAG +<tr> +<td align=left> var_elim +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +</table> + + + +<h1><a name="influence">Influence diagrams/ decision making</h1> + +BNT implements an exact algorithm for solving LIMIDs (limited memory +influence diagrams), described in +<ul> +<li> S. L. Lauritzen and D. Nilsson. +<a href="http://www.math.auc.dk/~steffen/papers/limids.pdf"> +Representing and solving decision problems with limited +information</a> +Management Science, 47, 1238 - 1251. September 2001. +</ul> +LIMIDs explicitely show all information arcs, rather than implicitely +assuming no forgetting. This allows them to model forgetful +controllers. +<p> +See the examples in <tt>BNT/examples/limids</tt> for details. + + + + +<h1>DBNs, HMMs, Kalman filters and all that</h1> + +Click <a href="usage_dbn.html">here</a> for documentation about how to +use BNT for dynamical systems and sequence data. + + +</BODY> diff --git a/sourcecodes/bnt-master/docs/usage_cropped.html b/sourcecodes/bnt-master/docs/usage_cropped.html new file mode 100644 index 00000000..11bb4e12 --- /dev/null +++ b/sourcecodes/bnt-master/docs/usage_cropped.html @@ -0,0 +1,3232 @@ +<HEAD> +<TITLE>How to use the Bayes Net Toolbox</TITLE> +</HEAD> + +<BODY BGCOLOR="#FFFFFF"> +<!-- white background is better for the pictures and equations --> + +<h1>How to use the Bayes Net Toolbox</h1> + +This documentation was last updated on 13 November 2002. +<br> +Click <a href="changelog.html">here</a> for a list of changes made to +BNT. +<br> +Click +<a href="http://bnt.insa-rouen.fr/">here</a> +for a French version of this documentation (which might not +be up-to-date). + + +<p> + +<ul> +<li> <a href="#install">Installation</a> +<ul> +<li> <a href="#installM">Installing the Matlab code</a> +<li> <a href="#installC">Installing the C code</a> +<li> <a href="../matlab_tips.html">Useful Matlab tips</a>. +</ul> + +<li> <a href="#basics">Creating your first Bayes net</a> + <ul> + <li> <a href="#basics">Creating a model by hand</a> + <li> <a href="#file">Loading a model from a file</a> + <li> <a href="http://bnt.insa-rouen.fr/ajouts.html">Creating a model using a GUI</a> + </ul> + +<li> <a href="#inference">Inference</a> + <ul> + <li> <a href="#marginal">Computing marginal distributions</a> + <li> <a href="#joint">Computing joint distributions</a> + <li> <a href="#soft">Soft/virtual evidence</a> + <li> <a href="#mpe">Most probable explanation</a> + </ul> + +<li> <a href="#cpd">Conditional Probability Distributions</a> + <ul> + <li> <a href="#tabular">Tabular (multinomial) nodes</a> + <li> <a href="#noisyor">Noisy-or nodes</a> + <li> <a href="#deterministic">Other (noisy) deterministic nodes</a> + <li> <a href="#softmax">Softmax (multinomial logit) nodes</a> + <li> <a href="#mlp">Neural network nodes</a> + <li> <a href="#root">Root nodes</a> + <li> <a href="#gaussian">Gaussian nodes</a> + <li> <a href="#glm">Generalized linear model nodes</a> + <li> <a href="#dtree">Classification/regression tree nodes</a> + <li> <a href="#nongauss">Other continuous distributions</a> + <li> <a href="#cpd_summary">Summary of CPD types</a> + </ul> + +<li> <a href="#examples">Example models</a> + <ul> + <li> <a + href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt"> +Gaussian mixture models</a> + <li> <a href="#pca">PCA, ICA, and all that</a> + <li> <a href="#mixep">Mixtures of experts</a> + <li> <a href="#hme">Hierarchical mixtures of experts</a> + <li> <a href="#qmr">QMR</a> + <li> <a href="#cg_model">Conditional Gaussian models</a> + <li> <a href="#hybrid">Other hybrid models</a> + </ul> + +<li> <a href="#param_learning">Parameter learning</a> + <ul> + <li> <a href="#load_data">Loading data from a file</a> + <li> <a href="#mle_complete">Maximum likelihood parameter estimation from complete data</a> + <li> <a href="#prior">Parameter priors</a> + <li> <a href="#bayes_learn">(Sequential) Bayesian parameter updating from complete data</a> + <li> <a href="#em">Maximum likelihood parameter estimation with missing values (EM)</a> + <li> <a href="#tying">Parameter tying</a> + </ul> + +<li> <a href="#structure_learning">Structure learning</a> + <ul> + <li> <a href="#enumerate">Exhaustive search</a> + <li> <a href="#K2">K2</a> + <li> <a href="#hill_climb">Hill-climbing</a> + <li> <a href="#mcmc">MCMC</a> + <li> <a href="#active">Active learning</a> + <li> <a href="#struct_em">Structural EM</a> + <li> <a href="#graphdraw">Visualizing the learned graph structure</a> + <li> <a href="#constraint">Constraint-based methods</a> + </ul> + + +<li> <a href="#engines">Inference engines</a> + <ul> + <li> <a href="#jtree">Junction tree</a> + <li> <a href="#varelim">Variable elimination</a> + <li> <a href="#global">Global inference methods</a> + <li> <a href="#quickscore">Quickscore</a> + <li> <a href="#belprop">Belief propagation</a> + <li> <a href="#sampling">Sampling (Monte Carlo)</a> + <li> <a href="#engine_summary">Summary of inference engines</a> + </ul> + + +<li> <a href="#influence">Influence diagrams/ decision making</a> + + +<li> <a href="usage_dbn.html">DBNs, HMMs, Kalman filters and all that</a> +</ul> + +</ul> + + + + +<h1><a name="install">Installation</h1> + +<h2><a name="installM">Installing the Matlab code</h2> + +<ul> +<li> <a href="bnt_download.html">Download</a> the BNT.zip file. + +<p> +<li> Unpack the file. In Unix, type +<!--"tar xvf BNT.tar".--> +"unzip FullBNT.zip". +In Windows, use +a program like <a href="http://www.winzip.com">Winzip</a>. This will +create a directory called FullBNT, which contains BNT and other libraries. + +<p> +<li> Read the file <tt>BNT/README</tt> to make sure the date +matches the one on the top of <a href=bnt.html>the BNT home page</a>. +If not, you may need to press 'refresh' on your browser, and download +again, to get the most recent version. + +<p> +<li> <b>Edit the file "BNT/add_BNT_to_path.m"</b> so it contains the correct +pathname. +For example, in Windows, +I download FullBNT.zip into C:\kpmurphy\matlab, and +then comment out the second line (with the % character), and uncomment +the third line, which reads +<pre> +BNT_HOME = 'C:\kpmurphy\matlab\FullBNT'; +</pre> + +<p> +<li> Start up Matlab. + +<p> +<li> Type "ver" at the Matlab prompt (">>"). +<b>You need Matlab version 5.2 or newer to run BNT</b>. +(Versions 5.0 and 5.1 have a memory leak which seems to sometimes +crash BNT.) + +<p> +<li> Move to the BNT directory. +For example, in Windows, I type +<pre> +>> cd C:\kpmurphy\matlab\FullBNT\BNT +</pre> + +<p> +<li> Type "add_BNT_to_path". +This executes the command +<tt>addpath(genpath(BNT_HOME))</tt>, +which adds all directories below FullBNT to the matlab path. + +<p> +<li> Type "test_BNT". +<b>If all goes well, this will just produce a bunch of numbers and pictures.</b> +It may produce some +warning messages (which you can ignore), but should not produce any error messages. +(The warnings should only be of the form +"Warning: Maximum number of iterations has been exceeded", and are +produced by Netlab.) + +<p> +<li> <a href="http://groups.yahoo.com/group/BayesNetToolbox/join"> +Join the BNT email list</a> + +</ul> + + +If you are new to Matlab, you might like to check out +<a href="matlab_tips.html">some useful Matlab tips</a>. +For instance, this explains how to create a startup file, which can be +used to set your path variable automatically, so you can avoid having +to type the above commands every time. + + + + +<h2><a name="installC">Installing the C code</h2> + +Some BNT functions also have C implementations. +<b>It is not necessary to install the C code</b>, but it can result in a speedup +of a factor of 2-5 of certain simple functions. +To install all the C code, +edit installC_BNT.m so it contains the right path, +then type <tt>installC_BNT</tt>. +To uninstall all the C code, +edit uninstallC_BNT.m so it contains the right path, +then type <tt>uninstallC_BNT</tt>. +For an up-to-date list of the files which have C implementations, see +BNT/installC_BNT.m. + +<p> +mex is a script that lets you call C code from Matlab - it does not compile matlab to +C (see mcc below). +If your C/C++ compiler is set up correctly, mex should work out of +the box. (Matlab 6 now ships with its own minimal C compiler.) +If not, you might need to type +<p> +<tt> mex -setup</tt> +<p> +before calling installC. +<p> +To make mex call gcc on Windows, +you must install <a +href="http://www.mrc-cbu.cam.ac.uk/Imaging/gnumex20.html">gnumex</a>. +You can use the <a href="http://www.mingw.org/">minimalist GNU for +Windows</a> version of gcc, or +the <a href="http://sources.redhat.com/cygwin/">cygwin</a> version. +<p> +In general, typing +'mex foo.c' from inside Matlab creates a file called +'foo.mexglx' or 'foo.dll' (the exact file +extension is system dependent - on Linux it is 'mexglx', on Windows it is '.dll'). +The resulting file will hide the original 'foo.m' (if it existed), i.e., +typing 'foo' at the prompt will call the compiled C version. +To reveal the original matlab version, just delete foo.mexglx (this is +what uninstallC does). +<p> +Sometimes it takes time for Matlab to realize that the file has +changed from matlab to C or vice versa; try typing 'clear all' or +restarting Matlab to refresh it. +To find out which version of a file you are running, type +'which foo'. +<p> +<a href="http://www.mathworks.com/products/compiler">mcc</a>, the +Matlab to C compiler, is a separate product, +and is quite different from mex. It does not yet support +objects/classes, which is why we can't compile all of BNT to C automatically. +Also, hand-written C code is usually much +better than the C code generated by mcc. + + +<p> +Acknowledgements: +Although I wrote some of the C code, most of +the C code (e.g., for jtree and dpot) was written by Wei Hu; +the triangulation C code was written by Ilya Shpitser. + + +<h1><a name="basics">Creating your first Bayes net</h1> + +To define a Bayes net, you must specify the graph structure and then +the parameters. We look at each in turn, using a simple example +(adapted from Russell and +Norvig, "Artificial Intelligence: a Modern Approach", Prentice Hall, +1995, p454). + + +<h2>Graph structure</h2> + + +Consider the following network. + +<p> +<center> +<IMG SRC="Figures/sprinkler.gif"> +</center> +<p> + +<P> +To specify this directed acyclic graph (dag), we create an adjacency matrix: +<PRE> +N = 4; +dag = zeros(N,N); +C = 1; S = 2; R = 3; W = 4; +dag(C,[R S]) = 1; +dag(R,W) = 1; +dag(S,W)=1; +</PRE> +<P> +We have numbered the nodes as follows: +Cloudy = 1, Sprinkler = 2, Rain = 3, WetGrass = 4. +<b>The nodes must always be numbered in topological order, i.e., +ancestors before descendants.</b> +For a more complicated graph, this is a little inconvenient: we will +see how to get around this <a href="usage_dbn.html#bat">below</a>. +<p> +<!-- +In Matlab 6, you can use logical arrays instead of double arrays, +which are 4 times smaller: +<pre> +dag = false(N,N); +dag(C,[R S]) = true; +... +</pre> +-- +<p> +A preliminary attempt to make a <b>GUI</b> +has been writte by Philippe LeRay and can be downloaded +from <a href="http://bnt.insa-rouen.fr/ajouts.html">here</a>. +<p> +You can visualize the resulting graph structure using +the methods discussed <a href="#graphdraw">below</a>. + +<h2>Creating the Bayes net shell</h2> + +In addition to specifying the graph structure, +we must specify the size and type of each node. +If a node is discrete, its size is the +number of possible values +each node can take on; if a node is continuous, +it can be a vector, and its size is the length of this vector. +In this case, we will assume all nodes are discrete and binary. +<PRE> +discrete_nodes = 1:N; +node_sizes = 2*ones(1,N); +</pre> +If the nodes were not binary, you could type e.g., +<pre> +node_sizes = [4 2 3 5]; +</pre> +meaning that Cloudy has 4 possible values, +Sprinkler has 2 possible values, etc. +Note that these are cardinal values, not ordinal, i.e., +they are not ordered in any way, like 'low', 'medium', 'high'. +<p> +We are now ready to make the Bayes net: +<pre> +bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes); +</PRE> +By default, all nodes are assumed to be discrete, so we can also just +write +<pre> +bnet = mk_bnet(dag, node_sizes); +</PRE> +You may also specify which nodes will be observed. +If you don't know, or if this not fixed in advance, +just use the empty list (the default). +<pre> +onodes = []; +bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes, 'observed', onodes); +</PRE> +Note that optional arguments are specified using a name/value syntax. +This is common for many BNT functions. +In general, to find out more about a function (e.g., which optional +arguments it takes), please see its +documentation string by typing +<pre> +help mk_bnet +</pre> +See also other <a href="matlab_tips.html">useful Matlab tips</a>. +<p> +It is possible to associate names with nodes, as follows: +<pre> +bnet = mk_bnet(dag, node_sizes, 'names', {'cloudy','S','R','W'}, 'discrete', 1:4); +</pre> +You can then refer to a node by its name: +<pre> +C = bnet.names{'cloudy'}; % bnet.names is an associative array +bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]); +</pre> + + +<h2><a name="cpt">Parameters</h2> + +A model consists of the graph structure and the parameters. +The parameters are represented by CPD objects (CPD = Conditional +Probability Distribution), which define the probability distribution +of a node given its parents. +(We will use the terms "node" and "random variable" interchangeably.) +The simplest kind of CPD is a table (multi-dimensional array), which +is suitable when all the nodes are discrete-valued. Note that the discrete +values are not assumed to be ordered in any way; that is, they +represent categorical quantities, like male and female, rather than +ordinal quantities, like low, medium and high. +(We will discuss CPDs in more detail <a href="#cpd">below</a>.) +<p> +Tabular CPDs, also called CPTs (conditional probability tables), +are stored as multidimensional arrays, where the dimensions +are arranged in the same order as the nodes, e.g., the CPT for node 4 +(WetGrass) is indexed by Sprinkler (2), Rain (3) and then WetGrass (4) itself. +Hence the child is always the last dimension. +If a node has no parents, its CPT is a column vector representing its +prior. +Note that in Matlab (unlike C), arrays are indexed +from 1, and are layed out in memory such that the first index toggles +fastest, e.g., the CPT for node 4 (WetGrass) is as follows +<P> +<P><IMG ALIGN=BOTTOM SRC="Figures/CPTgrass.gif"><P> +<P> +where we have used the convention that false==1, true==2. +We can create this CPT in Matlab as follows +<PRE> +CPT = zeros(2,2,2); +CPT(1,1,1) = 1.0; +CPT(2,1,1) = 0.1; +... +</PRE> +Here is an easier way: +<PRE> +CPT = reshape([1 0.1 0.1 0.01 0 0.9 0.9 0.99], [2 2 2]); +</PRE> +In fact, we don't need to reshape the array, since the CPD constructor +will do that for us. So we can just write +<pre> +bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [1 0.1 0.1 0.01 0 0.9 0.9 0.99]); +</pre> +The other nodes are created similarly (using the old syntax for +optional parameters) +<PRE> +bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]); +bnet.CPD{R} = tabular_CPD(bnet, R, [0.8 0.2 0.2 0.8]); +bnet.CPD{S} = tabular_CPD(bnet, S, [0.5 0.9 0.5 0.1]); +bnet.CPD{W} = tabular_CPD(bnet, W, [1 0.1 0.1 0.01 0 0.9 0.9 0.99]); +</PRE> + + +<h2><a name="rnd_cpt">Random Parameters</h2> + +If we do not specify the CPT, random parameters will be +created, i.e., each "row" of the CPT will be drawn from the uniform distribution. +To ensure repeatable results, use +<pre> +rand('state', seed); +randn('state', seed); +</pre> +To control the degree of randomness (entropy), +you can sample each row of the CPT from a Dirichlet(p,p,...) distribution. +If p << 1, this encourages "deterministic" CPTs (one entry near 1, the rest near 0). +If p = 1, each entry is drawn from U[0,1]. +If p >> 1, the entries will all be near 1/k, where k is the arity of +this node, i.e., each row will be nearly uniform. +You can do this as follows, assuming this node +is number i, and ns is the node_sizes. +<pre> +k = ns(i); +ps = parents(dag, i); +psz = prod(ns(ps)); +CPT = sample_dirichlet(p*ones(1,k), psz); +bnet.CPD{i} = tabular_CPD(bnet, i, 'CPT', CPT); +</pre> + + +<h2><a name="file">Loading a network from a file</h2> + +If you already have a Bayes net represented in the XML-based +<a href="http://www.cs.cmu.edu/afs/cs/user/fgcozman/www/Research/InterchangeFormat/"> +Bayes Net Interchange Format (BNIF)</a> (e.g., downloaded from the +<a +href="http://www.cs.huji.ac.il/labs/compbio/Repository"> +Bayes Net repository</a>), +you can convert it to BNT format using +the +<a href="http://www.digitas.harvard.edu/~ken/bif2bnt/">BIF-BNT Java +program</a> written by Ken Shan. +(This is not necessarily up-to-date.) + + +<h2>Creating a model using a GUI</h2> + +Click <a href="http://bnt.insa-rouen.fr/ajouts.html">here</a>. + + + +<h1><a name="inference">Inference</h1> + +Having created the BN, we can now use it for inference. +There are many different algorithms for doing inference in Bayes nets, +that make different tradeoffs between speed, +complexity, generality, and accuracy. +BNT therefore offers a variety of different inference +"engines". We will discuss these +in more detail <a href="#engines">below</a>. +For now, we will use the junction tree +engine, which is the mother of all exact inference algorithms. +This can be created as follows. +<pre> +engine = jtree_inf_engine(bnet); +</pre> +The other engines have similar constructors, but might take +additional, algorithm-specific parameters. +All engines are used in the same way, once they have been created. +We illustrate this in the following sections. + + +<h2><a name="marginal">Computing marginal distributions</h2> + +Suppose we want to compute the probability that the sprinker was on +given that the grass is wet. +The evidence consists of the fact that W=2. All the other nodes +are hidden (unobserved). We can specify this as follows. +<pre> +evidence = cell(1,N); +evidence{W} = 2; +</pre> +We use a 1D cell array instead of a vector to +cope with the fact that nodes can be vectors of different lengths. +In addition, the value [] can be used +to denote 'no evidence', instead of having to specify the observation +pattern as a separate argument. +(Click <a href="cellarray.html">here</a> for a quick tutorial on cell +arrays in matlab.) +<p> +We are now ready to add the evidence to the engine. +<pre> +[engine, loglik] = enter_evidence(engine, evidence); +</pre> +The behavior of this function is algorithm-specific, and is discussed +in more detail <a href="#engines">below</a>. +In the case of the jtree engine, +enter_evidence implements a two-pass message-passing scheme. +The first return argument contains the modified engine, which +incorporates the evidence. The second return argument contains the +log-likelihood of the evidence. (Not all engines are capable of +computing the log-likelihood.) +<p> +Finally, we can compute p=P(S=2|W=2) as follows. +<PRE> +marg = marginal_nodes(engine, S); +marg.T +ans = + 0.57024 + 0.42976 +p = marg.T(2); +</PRE> +We see that p = 0.4298. +<p> +Now let us add the evidence that it was raining, and see what +difference it makes. +<PRE> +evidence{R} = 2; +[engine, loglik] = enter_evidence(engine, evidence); +marg = marginal_nodes(engine, S); +p = marg.T(2); +</PRE> +We find that p = P(S=2|W=2,R=2) = 0.1945, +which is lower than +before, because the rain can ``explain away'' the +fact that the grass is wet. +<p> +You can plot a marginal distribution over a discrete variable +as a barchart using the built 'bar' function: +<pre> +bar(marg.T) +</pre> +This is what it looks like + +<p> +<center> +<IMG SRC="Figures/sprinkler_bar.gif"> +</center> +<p> + +<h2><a name="observed">Observed nodes</h2> + +What happens if we ask for the marginal on an observed node, e.g. P(W|W=2)? +An observed discrete node effectively only has 1 value (the observed + one) --- all other values would result in 0 probability. +For efficiency, BNT treats observed (discrete) nodes as if they were + set to 1, as we see below: +<pre> +evidence = cell(1,N); +evidence{W} = 2; +engine = enter_evidence(engine, evidence); +m = marginal_nodes(engine, W); +m.T +ans = + 1 +</pre> +This can get a little confusing, since we assigned W=2. +So we can ask BNT to add the evidence back in by passing in an optional argument: +<pre> +m = marginal_nodes(engine, W, 1); +m.T +ans = + 0 + 1 +</pre> +This shows that P(W=1|W=2) = 0 and P(W=2|W=2) = 1. + + + +<h2><a name="joint">Computing joint distributions</h2> + +We can compute the joint probability on a set of nodes as in the +following example. +<pre> +evidence = cell(1,N); +[engine, ll] = enter_evidence(engine, evidence); +m = marginal_nodes(engine, [S R W]); +</pre> +m is a structure. The 'T' field is a multi-dimensional array (in +this case, 3-dimensional) that contains the joint probability +distribution on the specified nodes. +<pre> +>> m.T +ans(:,:,1) = + 0.2900 0.0410 + 0.0210 0.0009 +ans(:,:,2) = + 0 0.3690 + 0.1890 0.0891 +</pre> +We see that P(S=1,R=1,W=2) = 0, since it is impossible for the grass +to be wet if both the rain and sprinkler are off. +<p> +Let us now add some evidence to R. +<pre> +evidence{R} = 2; +[engine, ll] = enter_evidence(engine, evidence); +m = marginal_nodes(engine, [S R W]) +m = + domain: [2 3 4] + T: [2x1x2 double] +>> m.T +m.T +ans(:,:,1) = + 0.0820 + 0.0018 +ans(:,:,2) = + 0.7380 + 0.1782 +</pre> +The joint T(i,j,k) = P(S=i,R=j,W=k|evidence) +should have T(i,1,k) = 0 for all i,k, since R=1 is incompatible +with the evidence that R=2. +Instead of creating large tables with many 0s, BNT sets the effective +size of observed (discrete) nodes to 1, as explained above. +This is why m.T has size 2x1x2. +To get a 2x2x2 table, type +<pre> +m = marginal_nodes(engine, [S R W], 1) +m = + domain: [2 3 4] + T: [2x2x2 double] +>> m.T +m.T +ans(:,:,1) = + 0 0.082 + 0 0.0018 +ans(:,:,2) = + 0 0.738 + 0 0.1782 +</pre> + +<p> +Note: It is not always possible to compute the joint on arbitrary +sets of nodes: it depends on which inference engine you use, as discussed +in more detail <a href="#engines">below</a>. + + +<h2><a name="soft">Soft/virtual evidence</h2> + +Sometimes a node is not observed, but we have some distribution over +its possible values; this is often called "soft" or "virtual" +evidence. +One can use this as follows +<pre> +[engine, loglik] = enter_evidence(engine, evidence, 'soft', soft_evidence); +</pre> +where soft_evidence{i} is either [] (if node i has no soft evidence) +or is a vector representing the probability distribution over i's +possible values. +For example, if we don't know i's exact value, but we know its +likelihood ratio is 60/40, we can write evidence{i} = [] and +soft_evidence{i} = [0.6 0.4]. +<p> +Currently only jtree_inf_engine supports this option. +It assumes that all hidden nodes, and all nodes for +which we have soft evidence, are discrete. +For a longer example, see BNT/examples/static/softev1.m. + + +<h2><a name="mpe">Most probable explanation</h2> + +To compute the most probable explanation (MPE) of the evidence (i.e., +the most probable assignment, or a mode of the joint), use +<pre> +[mpe, ll] = calc_mpe(engine, evidence); +</pre> +mpe{i} is the most likely value of node i. +This calls enter_evidence with the 'maximize' flag set to 1, which +causes the engine to do max-product instead of sum-product. +The resulting max-marginals are then thresholded. +If there is more than one maximum probability assignment, we must take + care to break ties in a consistent manner (thresholding the + max-marginals may give the wrong result). To force this behavior, + type +<pre> +[mpe, ll] = calc_mpe(engine, evidence, 1); +</pre> +Note that computing the MPE is someties called abductive reasoning. + +<p> +You can also use <tt>calc_mpe_bucket</tt> written by Ron Zohar, +that does a forwards max-product pass, and then a backwards traceback +pass, which is how Viterbi is traditionally implemented. + + + +<h1><a name="cpd">Conditional Probability Distributions</h1> + +A Conditional Probability Distributions (CPD) +defines P(X(i) | X(Pa(i))), where X(i) is the i'th node, and X(Pa(i)) +are the parents of node i. There are many ways to represent this +distribution, which depend in part on whether X(i) and X(Pa(i)) are +discrete, continuous, or a combination. +We will discuss various representations below. + + +<h2><a name="tabular">Tabular nodes</h2> + +If the CPD is represented as a table (i.e., if it is a multinomial +distribution), it has a number of parameters that is exponential in +the number of parents. See the example <a href="#cpt">above</a>. + + +<h2><a name="noisyor">Noisy-or nodes</h2> + +A noisy-OR node is like a regular logical OR gate except that +sometimes the effects of parents that are on get inhibited. +Let the prob. that parent i gets inhibited be q(i). +Then a node, C, with 2 parents, A and B, has the following CPD, where +we use F and T to represent off and on (1 and 2 in BNT). +<pre> +A B P(C=off) P(C=on) +--------------------------- +F F 1.0 0.0 +T F q(A) 1-q(A) +F T q(B) 1-q(B) +T T q(A)q(B) q-q(A)q(B) +</pre> +Thus we see that the causes get inhibited independently. +It is common to associate a "leak" node with a noisy-or CPD, which is +like a parent that is always on. This can account for all other unmodelled +causes which might turn the node on. +<p> +The noisy-or distribution is similar to the logistic distribution. +To see this, let the nodes, S(i), have values in {0,1}, and let q(i,j) +be the prob. that j inhibits i. Then +<pre> +Pr(S(i)=1 | parents(S(i))) = 1 - prod_{j} q(i,j)^S(j) +</pre> +Now define w(i,j) = -ln q(i,j) and rho(x) = 1-exp(-x). Then +<pre> +Pr(S(i)=1 | parents(S(i))) = rho(sum_j w(i,j) S(j)) +</pre> +For a sigmoid node, we have +<pre> +Pr(S(i)=1 | parents(S(i))) = sigma(-sum_j w(i,j) S(j)) +</pre> +where sigma(x) = 1/(1+exp(-x)). Hence they differ in the choice of +the activation function (although both are monotonically increasing). +In addition, in the case of a noisy-or, the weights are constrained to be +positive, since they derive from probabilities q(i,j). +In both cases, the number of parameters is <em>linear</em> in the +number of parents, unlike the case of a multinomial distribution, +where the number of parameters is exponential in the number of parents. +We will see an example of noisy-OR nodes <a href="#qmr">below</a>. + + +<h2><a name="deterministic">Other (noisy) deterministic nodes</h2> + +Deterministic CPDs for discrete random variables can be created using +the deterministic_CPD class. It is also possible to 'flip' the output +of the function with some probability, to simulate noise. +The boolean_CPD class is just a special case of a +deterministic CPD, where the parents and child are all binary. +<p> +Both of these classes are just "syntactic sugar" for the tabular_CPD +class. + + + +<h2><a name="softmax">Softmax nodes</h2> + +If we have a discrete node with a continuous parent, +we can define its CPD using a softmax function +(also known as the multinomial logit function). +This acts like a soft thresholding operator, and is defined as follows: +<pre> + exp(w(:,i)'*x + b(i)) +Pr(Q=i | X=x) = ----------------------------- + sum_j exp(w(:,j)'*x + b(j)) + +</pre> +The parameters of a softmax node, w(:,i) and b(i), i=1..|Q|, have the +following interpretation: w(:,i)-w(:,j) is the normal vector to the +decision boundary between classes i and j, +and b(i)-b(j) is its offset (bias). For example, suppose +X is a 2-vector, and Q is binary. Then +<pre> +w = [1 -1; + 0 0]; + +b = [0 0]; +</pre> +means class 1 are points in the 2D plane with positive x coordinate, +and class 2 are points in the 2D plane with negative x coordinate. +If w has large magnitude, the decision boundary is sharp, otherwise it +is soft. +In the special case that Q is binary (0/1), the softmax function reduces to the logistic +(sigmoid) function. +<p> +Fitting a softmax function can be done using the iteratively reweighted +least squares (IRLS) algorithm. +We use the implementation from +<a href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>. +Note that since +the softmax distribution is not in the exponential family, it does not +have finite sufficient statistics, and hence we must store all the +training data in uncompressed form. +If this takes too much space, one should use online (stochastic) gradient +descent (not implemented in BNT). +<p> +If a softmax node also has discrete parents, +we use a different set of w/b parameters for each combination of +parent values, as in the <a href="#gaussian">conditional linear +Gaussian CPD</a>. +This feature was implemented by Pierpaolo Brutti. +He is currently extending it so that discrete parents can be treated +as if they were continuous, by adding indicator variables to the X +vector. +<p> +We will see an example of softmax nodes <a href="#mixexp">below</a>. + + +<h2><a name="mlp">Neural network nodes</h2> + +Pierpaolo Brutti has implemented the mlp_CPD class, which uses a multi layer perceptron +to implement a mapping from continuous parents to discrete children, +similar to the softmax function. +(If there are also discrete parents, it creates a mixture of MLPs.) +It uses code from <a +href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>. +This is work in progress. + +<h2><a name="root">Root nodes</h2> + +A root node has no parents and no parameters; it can be used to model +an observed, exogeneous input variable, i.e., one which is "outside" +the model. +This is useful for conditional density models. +We will see an example of root nodes <a href="#mixexp">below</a>. + + +<h2><a name="gaussian">Gaussian nodes</h2> + +We now consider a distribution suitable for the continuous-valued nodes. +Suppose the node is called Y, its continuous parents (if any) are +called X, and its discrete parents (if any) are called Q. +The distribution on Y is defined as follows: +<pre> +- no parents: Y ~ N(mu, Sigma) +- cts parents : Y|X=x ~ N(mu + W x, Sigma) +- discrete parents: Y|Q=i ~ N(mu(:,i), Sigma(:,:,i)) +- cts and discrete parents: Y|X=x,Q=i ~ N(mu(:,i) + W(:,:,i) * x, Sigma(:,:,i)) +</pre> +where N(mu, Sigma) denotes a Normal distribution with mean mu and +covariance Sigma. Let |X|, |Y| and |Q| denote the sizes of X, Y and Q +respectively. +If there are no discrete parents, |Q|=1; if there is +more than one, then |Q| = a vector of the sizes of each discrete parent. +If there are no continuous parents, |X|=0; if there is more than one, +then |X| = the sum of their sizes. +Then mu is a |Y|*|Q| vector, Sigma is a |Y|*|Y|*|Q| positive +semi-definite matrix, and W is a |Y|*|X|*|Q| regression (weight) +matrix. +<p> +We can create a Gaussian node with random parameters as follows. +<pre> +bnet.CPD{i} = gaussian_CPD(bnet, i); +</pre> +We can specify the value of one or more of the parameters as in the +following example, in which |Y|=2, and |Q|=1. +<pre> +bnet.CPD{i} = gaussian_CPD(bnet, i, 'mean', [0; 0], 'weights', randn(Y,X), 'cov', eye(Y)); +</pre> +<p> +We will see an example of conditional linear Gaussian nodes <a +href="#cg_model">below</a>. +<p> +When fitting a Gaussian using EM, you might encounter some numerical +problems. See <a +href="http://www.ai.mit.edu/~murphyk/Software/HMM/hmm.html#loglikpos">here</a> +for a discussion of this in the context of HMMs. +(Note: BNT does not currently support spherical covariances, although it +would be easy to add, since KPMstats/clg_Mstep supports this option; +you would just need to modify gaussian_CPD/update_ess to accumulate +weighted inner products.) + + +<h2><a name="nongauss">Other continuous distributions</h2> + +Currently BNT does not support any CPDs for continuous nodes other +than the Gaussian. +However, you can use a mixture of Gaussians to +approximate other continuous distributions. We will see some an example +of this with the IFA model <a href="#pca">below</a>. + + +<h2><a name="glm">Generalized linear model nodes</h2> + +In the future, we may incorporate some of the functionality of +<a href = +"http://www.sci.usq.edu.au/staff/dunn/glmlab/glmlab.html">glmlab</a> +into BNT. + + +<h2><a name="dtree">Classification/regression tree nodes</h2> + +We plan to add classification and regression trees to define CPDs for +discrete and continuous nodes, respectively. +Trees have many advantages: they are easy to interpret, they can do +feature selection, they can +handle discrete and continuous inputs, they do not make strong +assumptions about the form of the distribution, the number of +parameters can grow in a data-dependent way (i.e., they are +semi-parametric), they can handle missing data, etc. +However, they are not yet implemented. +<!-- +Yimin Zhang is currently (Feb '02) implementing this. +--> + + +<h2><a name="cpd_summary">Summary of CPD types</h2> + +We list all the different types of CPDs supported by BNT. +For each CPD, we specify if the child and parents can be discrete (D) or +continuous (C) (Binary (B) nodes are a special case). +We also specify which methods each class supports. +If a method is inherited, the name of the parent class is mentioned. +If a parent class calls a child method, this is mentioned. +<p> +The <tt>CPD_to_CPT</tt> method converts a CPD to a table; this +requires that the child and all parents are discrete. +The CPT might be exponentially big... +<tt>convert_to_table</tt> evaluates a CPD with evidence, and +represents the the resulting potential as an array. +This requires that the child is discrete, and any continuous parents +are observed. +<tt>convert_to_pot</tt> evaluates a CPD with evidence, and +represents the resulting potential as a dpot, gpot, cgpot or upot, as +requested. (d=discrete, g=Gaussian, cg = conditional Gaussian, u = +utility). + +<p> +When we sample a node, all the parents are observed. +When we compute the (log) probability of a node, all the parents and +the child are observed. +<p> +We also specify if the parameters are learnable. +For learning with EM, we require +the methods <tt>reset_ess</tt>, <tt>update_ess</tt> and +<tt>maximize_params</tt>. +For learning from fully observed data, we require +the method <tt>learn_params</tt>. +By default, all classes inherit this from generic_CPD, which simply +calls <tt>update_ess</tt> N times, once for each data case, followed +by <tt>maximize_params</tt>, i.e., it is like EM, without the E step. +Some classes implement a batch formula, which is quicker. +<p> +Bayesian learning means computing a posterior over the parameters +given fully observed data. +<p> +Pearl means we implement the methods <tt>compute_pi</tt> and +<tt>compute_lambda_msg</tt>, used by +<tt>pearl_inf_engine</tt>, which runs on directed graphs. +<tt>belprop_inf_engine</tt> only needs <tt>convert_to_pot</tt>.H +The pearl methods can exploit special properties of the CPDs for +computing the messages efficiently, whereas belprop does not. +<p> +The only method implemented by generic_CPD is <tt>adjustable_CPD</tt>, +which is not shown, since it is not very interesting. + + +<p> + + +<table> +<table border units = pixels><tr> +<td align=center>Name +<td align=center>Child +<td align=center>Parents +<td align=center>Comments +<td align=center>CPD_to_CPT +<td align=center>conv_to_table +<td align=center>conv_to_pot +<td align=center>sample +<td align=center>prob +<td align=center>learn +<td align=center>Bayes +<td align=center>Pearl + + +<tr> +<!-- Name--><td> +<!-- Child--><td> +<!-- Parents--><td> +<!-- Comments--><td> +<!-- CPD_to_CPT--><td> +<!-- conv_to_table--><td> +<!-- conv_to_pot--><td> +<!-- sample--><td> +<!-- prob--><td> +<!-- learn--><td> +<!-- Bayes--><td> +<!-- Pearl--><td> + +<tr> +<!-- Name--><td>boolean +<!-- Child--><td>B +<!-- Parents--><td>B +<!-- Comments--><td>Syntactic sugar for tabular +<!-- CPD_to_CPT--><td>- +<!-- conv_to_table--><td>- +<!-- conv_to_pot--><td>- +<!-- sample--><td>- +<!-- prob--><td>- +<!-- learn--><td>- +<!-- Bayes--><td>- +<!-- Pearl--><td>- + +<tr> +<!-- Name--><td>deterministic +<!-- Child--><td>D +<!-- Parents--><td>D +<!-- Comments--><td>Syntactic sugar for tabular +<!-- CPD_to_CPT--><td>- +<!-- conv_to_table--><td>- +<!-- conv_to_pot--><td>- +<!-- sample--><td>- +<!-- prob--><td>- +<!-- learn--><td>- +<!-- Bayes--><td>- +<!-- Pearl--><td>- + +<tr> +<!-- Name--><td>Discrete +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>Virtual class +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Calls CPD_to_CPT +<!-- conv_to_pot--><td>Calls conv_to_table +<!-- sample--><td>Calls conv_to_table +<!-- prob--><td>Calls conv_to_table +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + +<tr> +<!-- Name--><td>Gaussian +<!-- Child--><td>C +<!-- Parents--><td>C/D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>Y +<!-- prob--><td>Y +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + +<tr> +<!-- Name--><td>gmux +<!-- Child--><td>C +<!-- Parents--><td>C/D +<!-- Comments--><td>multiplexer +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>N +<!-- prob--><td>N +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>Y + + +<tr> +<!-- Name--><td>MLP +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>multi layer perceptron +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Y +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>noisy-or +<!-- Child--><td>B +<!-- Parents--><td>B +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>Y +<!-- conv_to_table--><td>Inherits from discrete +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>Y + + +<tr> +<!-- Name--><td>root +<!-- Child--><td>C/D +<!-- Parents--><td>none +<!-- Comments--><td>no params +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>Y +<!-- prob--><td>Y +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>softmax +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Y +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>generic +<!-- Child--><td>C/D +<!-- Parents--><td>C/D +<!-- Comments--><td>Virtual class +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>N +<!-- sample--><td>N +<!-- prob--><td>N +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>Tabular +<!-- Child--><td>D +<!-- Parents--><td>D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>Y +<!-- conv_to_table--><td>Inherits from discrete +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>Y +<!-- Pearl--><td>Y + +</table> + + + +<h1><a name="examples">Example models</h1> + + +<h2>Gaussian mixture models</h2> + +Richard W. DeVaul has made a detailed tutorial on how to fit mixtures +of Gaussians using BNT. Available +<a href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt">here</a>. + + +<h2><a name="pca">PCA, ICA, and all that </h2> + +In Figure (a) below, we show how Factor Analysis can be thought of as a +graphical model. Here, X has an N(0,I) prior, and +Y|X=x ~ N(mu + Wx, Psi), +where Psi is diagonal and W is called the "factor loading matrix". +Since the noise on both X and Y is diagonal, the components of these +vectors are uncorrelated, and hence can be represented as individual +scalar nodes, as we show in (b). +(This is useful if parts of the observations on the Y vector are occasionally missing.) +We usually take k=|X| << |Y|=D, so the model tries to explain +many observations using a low-dimensional subspace. + + +<center> +<table> +<tr> +<td><img src="Figures/fa.gif"> +<td><img src="Figures/fa_scalar.gif"> +<td><img src="Figures/mfa.gif"> +<td><img src="Figures/ifa.gif"> +<tr> +<td align=center> (a) +<td align=center> (b) +<td align=center> (c) +<td align=center> (d) +</table> +</center> + +<p> +We can create this model in BNT as follows. +<pre> +ns = [k D]; +dag = zeros(2,2); +dag(1,2) = 1; +bnet = mk_bnet(dag, ns, 'discrete', []); +bnet.CPD{1} = gaussian_CPD(bnet, 1, 'mean', zeros(k,1), 'cov', eye(k), ... + 'cov_type', 'diag', 'clamp_mean', 1, 'clamp_cov', 1); +bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(D,1), 'cov', diag(Psi0), 'weights', W0, ... + 'cov_type', 'diag', 'clamp_mean', 1); +</pre> + +The root node is clamped to the N(0,I) distribution, so that we will +not update these parameters during learning. +The mean of the leaf node is clamped to 0, +since we assume the data has been centered (had its mean subtracted +off); this is just for simplicity. +Finally, the covariance of the leaf node is constrained to be +diagonal. W0 and Psi0 are the initial parameter guesses. + +<p> +We can fit this model (i.e., estimate its parameters in a maximum +likelihood (ML) sense) using EM, as we +explain <a href="#em">below</a>. +Not surprisingly, the ML estimates for mu and Psi turn out to be +identical to the +sample mean and variance, which can be computed directly as +<pre> +mu_ML = mean(data); +Psi_ML = diag(cov(data)); +</pre> +Note that W can only be identified up to a rotation matrix, because of +the spherical symmetry of the source. + +<p> +If we restrict Psi to be spherical, i.e., Psi = sigma*I, +there is a closed-form solution for W as well, +i.e., we do not need to use EM. +In particular, W contains the first |X| eigenvectors of the sample covariance +matrix, with scalings determined by the eigenvalues and sigma. +Classical PCA can be obtained by taking the sigma->0 limit. +For details, see + +<ul> +<li> <a href="ftp://hope.caltech.edu/pub/roweis/Empca/empca.ps"> +"EM algorithms for PCA and SPCA"</a>, Sam Roweis, NIPS 97. +(<a href="ftp://hope.caltech.edu/pub/roweis/Code/empca.tar.gz"> +Matlab software</a>) + +<p> +<li> +<a +href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003> +"Mixtures of probabilistic principal component analyzers"</a>, +Tipping and Bishop, Neural Computation 11(2):443--482, 1999. +</ul> + +<p> +By adding a hidden discrete variable, we can create mixtures of FA +models, as shown in (c). +Now we can explain the data using a set of subspaces. +We can create this model in BNT as follows. +<pre> +ns = [M k D]; +dag = zeros(3); +dag(1,3) = 1; +dag(2,3) = 1; +bnet = mk_bnet(dag, ns, 'discrete', 1); +bnet.CPD{1} = tabular_CPD(bnet, 1, Pi0); +bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(k, 1), 'cov', eye(k), 'cov_type', 'diag', ... + 'clamp_mean', 1, 'clamp_cov', 1); +bnet.CPD{3} = gaussian_CPD(bnet, 3, 'mean', Mu0', 'cov', repmat(diag(Psi0), [1 1 M]), ... + 'weights', W0, 'cov_type', 'diag', 'tied_cov', 1); +</pre> +Notice how the covariance matrix for Y is the same for all values of +Q; that is, the noise level in each sub-space is assumed the same. +However, we allow the offset, mu, to vary. +For details, see +<ul> + +<LI> +<a HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/tr-96-1.ps.gz"> The EM +Algorithm for Mixtures of Factor Analyzers </A>, +Ghahramani, Z. and Hinton, G.E. (1996), +University of Toronto +Technical Report CRG-TR-96-1. +(<A HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/mfa.tar.gz">Matlab software</A>) + +<p> +<li> +<a +href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003> +"Mixtures of probabilistic principal component analyzers"</a>, +Tipping and Bishop, Neural Computation 11(2):443--482, 1999. +</ul> + +<p> +I have included Zoubin's specialized MFA code (with his permission) +with the toolbox, so you can check that BNT gives the same results: +see 'BNT/examples/static/mfa1.m'. + +<p> +Independent Factor Analysis (IFA) generalizes FA by allowing a +non-Gaussian prior on each component of X. +(Note that we can approximate a non-Gaussian prior using a mixture of +Gaussians.) +This means that the likelihood function is no longer rotationally +invariant, so we can uniquely identify W and the hidden +sources X. +IFA also allows a non-diagonal Psi (i.e. correlations between the components of Y). +We recover classical Independent Components Analysis (ICA) +in the Psi -> 0 limit, and by assuming that |X|=|Y|, so that the +weight matrix W is square and invertible. +For details, see +<ul> +<li> +<a href="http://www.gatsby.ucl.ac.uk/~hagai/ifa.ps">Independent Factor +Analysis</a>, H. Attias, Neural Computation 11: 803--851, 1998. +</ul> + + + +<h2><a name="mixexp">Mixtures of experts</h2> + +As an example of the use of the softmax function, +we introduce the Mixture of Experts model. +<!-- +We also show +the Hierarchical Mixture of Experts model, where the hierarchy has two +levels. +(This is essentially a probabilistic decision tree of height two.) +--> +As before, +circles denote continuous-valued nodes, +squares denote discrete nodes, clear +means hidden, and shaded means observed. +<p> +<center> +<table> +<tr> +<td><img src="Figures/mixexp.gif"> +<!-- +<td><img src="Figures/hme.gif"> +--> +</table> +</center> +<p> +X is the observed +input, Y is the output, and +the Q nodes are hidden "gating" nodes, which select the appropriate +set of parameters for Y. During training, Y is assumed observed, +but for testing, the goal is to predict Y given X. +Note that this is a <em>conditional</em> density model, so we don't +associate any parameters with X. +Hence X's CPD will be a root CPD, which is a way of modelling +exogenous nodes. +If the output is a continuous-valued quantity, +we assume the "experts" are linear-regression units, +and set Y's CPD to linear-Gaussian. +If the output is discrete, we set Y's CPD to a softmax function. +The Q CPDs will always be softmax functions. + +<p> +As a concrete example, consider the mixture of experts model where X and Y are +scalars, and Q is binary. +This is just piecewise linear regression, where +we have two line segments, i.e., +<P> +<IMG ALIGN=BOTTOM SRC="Eqns/lin_reg_eqn.gif"> +<P> +We can create this model with random parameters as follows. +(This code is bundled in BNT/examples/static/mixexp2.m.) +<PRE> +X = 1; +Q = 2; +Y = 3; +dag = zeros(3,3); +dag(X,[Q Y]) = 1 +dag(Q,Y) = 1; +ns = [1 2 1]; % make X and Y scalars, and have 2 experts +onodes = [1 3]; +bnet = mk_bnet(dag, ns, 'discrete', 2, 'observed', onodes); + +rand('state', 0); +randn('state', 0); +bnet.CPD{1} = root_CPD(bnet, 1); +bnet.CPD{2} = softmax_CPD(bnet, 2); +bnet.CPD{3} = gaussian_CPD(bnet, 3); +</PRE> +Now let us fit this model using <a href="#em">EM</a>. +First we <a href="#load_data">load the data</a> (1000 training cases) and plot them. +<P> +<PRE> +data = load('/examples/static/Misc/mixexp_data.txt', '-ascii'); +plot(data(:,1), data(:,2), '.'); +</PRE> +<p> +<center> +<IMG SRC="Figures/mixexp_data.gif"> +</center> +<p> +This is what the model looks like before training. +(Thanks to Thomas Hofman for writing this plotting routine.) +<p> +<center> +<IMG SRC="Figures/mixexp_before.gif"> +</center> +<p> +Now let's train the model, and plot the final performance. +(We will discuss how to train models in more detail <a href="#param_learning">below</a>.) +<P> +<PRE> +ncases = size(data, 1); % each row of data is a training case +cases = cell(3, ncases); +cases([1 3], :) = num2cell(data'); % each column of cases is a training case +engine = jtree_inf_engine(bnet); +max_iter = 20; +[bnet2, LLtrace] = learn_params_em(engine, cases, max_iter); +</PRE> +(We specify which nodes will be observed when we create the engine. +Hence BNT knows that the hidden nodes are all discrete. +For complex models, this can lead to a significant speedup.) +Below we show what the model looks like after 16 iterations of EM +(with 100 IRLS iterations per M step), when it converged +using the default convergence tolerance (that the +fractional change in the log-likelihood be less than 1e-3). +Before learning, the log-likelihood was +-322.927442; afterwards, it was -13.728778. +<p> +<center> +<IMG SRC="Figures/mixexp_after.gif"> +</center> +(See BNT/examples/static/mixexp2.m for details of the code.) + + + +<h2><a name="hme">Hierarchical mixtures of experts</h2> + +A hierarchical mixture of experts (HME) extends the mixture of experts +model by having more than one hidden node. A two-level example is shown below, along +with its more traditional representation as a neural network. +This is like a (balanced) probabilistic decision tree of height 2. +<p> +<center> +<IMG SRC="Figures/HMEforMatlab.jpg"> +</center> +<p> +<a href="mailto:pbrutti@stat.cmu.edu">Pierpaolo Brutti</a> +has written an extensive set of routines for HMEs, +which are bundled with BNT: see the examples/static/HME directory. +These routines allow you to choose the number of hidden (gating) +layers, and the form of the experts (softmax or MLP). +See the file hmemenu, which provides a demo. +For example, the figure below shows the decision boundaries learned +for a ternary classification problem, using a 2 level HME with softmax +gates and softmax experts; the training set is on the left, the +testing set on the right. +<p> +<center> +<!--<IMG SRC="Figures/hme_dec_boundary.gif">--> +<IMG SRC="Figures/hme_dec_boundary.png"> +</center> +<p> + + +<p> +For more details, see the following: +<ul> + +<li> <a href="http://www.cs.berkeley.edu/~jordan/papers/hierarchies.ps.Z"> +Hierarchical mixtures of experts and the EM algorithm</a> +M. I. Jordan and R. A. Jacobs. Neural Computation, 6, 181-214, 1994. + +<li> <a href = +"http://www.cs.berkeley.edu/~dmartin/software">David Martin's +matlab code for HME</a> + +<li> <a +href="http://www.cs.berkeley.edu/~jordan/papers/uai.ps.Z">Why the +logistic function? A tutorial discussion on +probabilities and neural networks.</a> M. I. Jordan. MIT Computational +Cognitive Science Report 9503, August 1995. + +<li> "Generalized Linear Models", McCullagh and Nelder, Chapman and +Halll, 1983. + +<li> +"Improved learning algorithms for mixtures of experts in multiclass +classification". +K. Chen, L. Xu, H. Chi. +Neural Networks (1999) 12: 1229-1252. + +<li> <a href="http://www.oigeeza.com/steve/"> +Classification Using Hierarchical Mixtures of Experts</a> +S.R. Waterhouse and A.J. Robinson. +In Proc. IEEE Workshop on Neural Network for Signal Processing IV (1994), pp. 177-186 + +<li> <a href="http://www.idiap.ch/~perry/"> +Localized mixtures of experts</a>, +P. Moerland, 1998. + +<li> "Nonlinear gated experts for time series", +A.S. Weigend and M. Mangeas, 1995. + +</ul> + + +<h2><a name="qmr">QMR</h2> + +Bayes nets originally arose out of an attempt to add probabilities to +expert systems, and this is still the most common use for BNs. +A famous example is +QMR-DT, a decision-theoretic reformulation of the Quick Medical +Reference (QMR) model. +<p> +<center> +<IMG ALIGN=BOTTOM SRC="Figures/qmr.gif"> +</center> +Here, the top layer represents hidden disease nodes, and the bottom +layer represents observed symptom nodes. +The goal is to infer the posterior probability of each disease given +all the symptoms (which can be present, absent or unknown). +Each node in the top layer has a Bernoulli prior (with a low prior +probability that the disease is present). +Since each node in the bottom layer has a high fan-in, we use a +noisy-OR parameterization; each disease has an independent chance of +causing each symptom. +The real QMR-DT model is copyright, but +we can create a random QMR-like model as follows. +<pre> +function bnet = mk_qmr_bnet(G, inhibit, leak, prior) +% MK_QMR_BNET Make a QMR model +% bnet = mk_qmr_bnet(G, inhibit, leak, prior) +% +% G(i,j) = 1 iff there is an arc from disease i to finding j +% inhibit(i,j) = inhibition probability on i->j arc +% leak(j) = inhibition prob. on leak->j arc +% prior(i) = prob. disease i is on + +[Ndiseases Nfindings] = size(inhibit); +N = Ndiseases + Nfindings; +finding_node = Ndiseases+1:N; +ns = 2*ones(1,N); +dag = zeros(N,N); +dag(1:Ndiseases, finding_node) = G; +bnet = mk_bnet(dag, ns, 'observed', finding_node); + +for d=1:Ndiseases + CPT = [1-prior(d) prior(d)]; + bnet.CPD{d} = tabular_CPD(bnet, d, CPT'); +end + +for i=1:Nfindings + fnode = finding_node(i); + ps = parents(G, i); + bnet.CPD{fnode} = noisyor_CPD(bnet, fnode, leak(i), inhibit(ps, i)); +end +</pre> +In the file BNT/examples/static/qmr1, we create a random bipartite +graph G, with 5 diseases and 10 findings, and random parameters. +(In general, to create a random dag, use 'mk_random_dag'.) +We can visualize the resulting graph structure using +the methods discussed <a href="#graphdraw">below</a>, with the +following results: +<p> +<img src="Figures/qmr.rnd.jpg"> + +<p> +Now let us put some random evidence on all the leaves except the very +first and very last, and compute the disease posteriors. +<pre> +pos = 2:floor(Nfindings/2); +neg = (pos(end)+1):(Nfindings-1); +onodes = myunion(pos, neg); +evidence = cell(1, N); +evidence(findings(pos)) = num2cell(repmat(2, 1, length(pos))); +evidence(findings(neg)) = num2cell(repmat(1, 1, length(neg))); + +engine = jtree_inf_engine(bnet); +[engine, ll] = enter_evidence(engine, evidence); +post = zeros(1, Ndiseases); +for i=diseases(:)' + m = marginal_nodes(engine, i); + post(i) = m.T(2); +end +</pre> +Junction tree can be quite slow on large QMR models. +Fortunately, it is possible to exploit properties of the noisy-OR +function to speed up exact inference using an algorithm called +<a href="#quickscore">quickscore</a>, discussed below. + + + + + +<h2><a name="cg_model">Conditional Gaussian models</h2> + +A conditional Gaussian model is one in which, conditioned on all the discrete +nodes, the distribution over the remaining (continuous) nodes is +multivariate Gaussian. This means we can have arcs from discrete (D) +to continuous (C) nodes, but not vice versa. +(We <em>are</em> allowed C->D arcs if the continuous nodes are observed, +as in the <a href="#mixexp">mixture of experts</a> model, +since this distribution can be represented with a discrete potential.) +<p> +We now give an example of a CG model, from +the paper "Propagation of Probabilities, Means amd +Variances in Mixed Graphical Association Models", Steffen Lauritzen, +JASA 87(420):1098--1108, 1992 (reprinted in the book "Probabilistic Networks and Expert +Systems", R. G. Cowell, A. P. Dawid, S. L. Lauritzen and +D. J. Spiegelhalter, Springer, 1999.) + +<h3>Specifying the graph</h3> + +Consider the model of waste emissions from an incinerator plant shown below. +We follow the standard convention that shaded nodes are observed, +clear nodes are hidden. +We also use the non-standard convention that +square nodes are discrete (tabular) and round nodes are +Gaussian. + +<p> +<center> +<IMG SRC="Figures/cg1.gif"> +</center> +<p> + +We can create this model as follows. +<pre> +F = 1; W = 2; E = 3; B = 4; C = 5; D = 6; Min = 7; Mout = 8; L = 9; +n = 9; + +dag = zeros(n); +dag(F,E)=1; +dag(W,[E Min D]) = 1; +dag(E,D)=1; +dag(B,[C D])=1; +dag(D,[L Mout])=1; +dag(Min,Mout)=1; + +% node sizes - all cts nodes are scalar, all discrete nodes are binary +ns = ones(1, n); +dnodes = [F W B]; +cnodes = mysetdiff(1:n, dnodes); +ns(dnodes) = 2; + +bnet = mk_bnet(dag, ns, 'discrete', dnodes); +</pre> +'dnodes' is a list of the discrete nodes; 'cnodes' is the continuous +nodes. 'mysetdiff' is a faster version of the built-in 'setdiff'. +<p> + + +<h3>Specifying the parameters</h3> + +The parameters of the discrete nodes can be specified as follows. +<pre> +bnet.CPD{B} = tabular_CPD(bnet, B, 'CPT', [0.85 0.15]); % 1=stable, 2=unstable +bnet.CPD{F} = tabular_CPD(bnet, F, 'CPT', [0.95 0.05]); % 1=intact, 2=defect +bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [2/7 5/7]); % 1=industrial, 2=household +</pre> + +<p> +The parameters of the continuous nodes can be specified as follows. +<pre> +bnet.CPD{E} = gaussian_CPD(bnet, E, 'mean', [-3.9 -0.4 -3.2 -0.5], ... + 'cov', [0.00002 0.0001 0.00002 0.0001]); +bnet.CPD{D} = gaussian_CPD(bnet, D, 'mean', [6.5 6.0 7.5 7.0], ... + 'cov', [0.03 0.04 0.1 0.1], 'weights', [1 1 1 1]); +bnet.CPD{C} = gaussian_CPD(bnet, C, 'mean', [-2 -1], 'cov', [0.1 0.3]); +bnet.CPD{L} = gaussian_CPD(bnet, L, 'mean', 3, 'cov', 0.25, 'weights', -0.5); +bnet.CPD{Min} = gaussian_CPD(bnet, Min, 'mean', [0.5 -0.5], 'cov', [0.01 0.005]); +bnet.CPD{Mout} = gaussian_CPD(bnet, Mout, 'mean', 0, 'cov', 0.002, 'weights', [1 1]); +</pre> + + +<h3><a name="cg_infer">Inference</h3> + +<!--Let us perform inference in the <a href="#cg_model">waste incinerator example</a>.--> +First we compute the unconditional marginals. +<pre> +engine = jtree_inf_engine(bnet); +evidence = cell(1,n); +[engine, ll] = enter_evidence(engine, evidence); +marg = marginal_nodes(engine, E); +</pre> +<!--(Of course, we could use <tt>cond_gauss_inf_engine</tt> instead of jtree.)--> +'marg' is a structure that contains the fields 'mu' and 'Sigma', which +contain the mean and (co)variance of the marginal on E. +In this case, they are both scalars. +Let us check they match the published figures (to 2 decimal places). +<!--(We can't expect +more precision than this in general because I have implemented the algorithm of +Lauritzen (1992), which can be numerically unstable.)--> +<pre> +tol = 1e-2; +assert(approxeq(marg.mu, -3.25, tol)); +assert(approxeq(sqrt(marg.Sigma), 0.709, tol)); +</pre> +We can compute the other posteriors similarly. +Now let us add some evidence. +<pre> +evidence = cell(1,n); +evidence{W} = 1; % industrial +evidence{L} = 1.1; +evidence{C} = -0.9; +[engine, ll] = enter_evidence(engine, evidence); +</pre> +Now we find +<pre> +marg = marginal_nodes(engine, E); +assert(approxeq(marg.mu, -3.8983, tol)); +assert(approxeq(sqrt(marg.Sigma), 0.0763, tol)); +</pre> + + +We can also compute the joint probability on a set of nodes. +For example, P(D, Mout | evidence) is a 2D Gaussian: +<pre> +marg = marginal_nodes(engine, [D Mout]) +marg = + domain: [6 8] + mu: [2x1 double] + Sigma: [2x2 double] + T: 1.0000 +</pre> +The mean is +<pre> +marg.mu +ans = + 3.6077 + 4.1077 +</pre> +and the covariance matrix is +<pre> +marg.Sigma +ans = + 0.1062 0.1062 + 0.1062 0.1182 +</pre> +It is easy to visualize this posterior using standard Matlab plotting +functions, e.g., +<pre> +gaussplot2d(marg.mu, marg.Sigma); +</pre> +produces the following picture. + +<p> +<center> +<IMG SRC="Figures/gaussplot.png"> +</center> +<p> + + +The T field indicates that the mixing weight of this Gaussian +component is 1.0. +If the joint contains discrete and continuous variables, the result +will be a mixture of Gaussians, e.g., +<pre> +marg = marginal_nodes(engine, [F E]) + domain: [1 3] + mu: [-3.9000 -0.4003] + Sigma: [1x1x2 double] + T: [0.9995 4.7373e-04] +</pre> +The interpretation is +Sigma(i,j,k) = Cov[ E(i) E(j) | F=k ]. +In this case, E is a scalar, so i=j=1; k specifies the mixture component. +<p> +We saw in the sprinkler network that BNT sets the effective size of +observed discrete nodes to 1, since they only have one legal value. +For continuous nodes, BNT sets their length to 0, +since they have been reduced to a point. +For example, +<pre> +marg = marginal_nodes(engine, [B C]) + domain: [4 5] + mu: [] + Sigma: [] + T: [0.0123 0.9877] +</pre> +It is simple to post-process the output of marginal_nodes. +For example, the file BNT/examples/static/cg1 sets the mu term of +observed nodes to their observed value, and the Sigma term to 0 (since +observed nodes have no variance). + +<p> +Note that the implemented version of the junction tree is numerically +unstable when using CG potentials +(which is why, in the example above, we only required our answers to agree with +the published ones to 2dp.) +This is why you might want to use <tt>stab_cond_gauss_inf_engine</tt>, +implemented by Shan Huang. This is described in + +<ul> +<li> "Stable Local Computation with Conditional Gaussian Distributions", +S. Lauritzen and F. Jensen, Tech Report R-99-2014, +Dept. Math. Sciences, Allborg Univ., 1999. +</ul> + +However, even the numerically stable version +can be computationally intractable if there are many hidden discrete +nodes, because the number of mixture components grows exponentially e.g., in a +<a href="usage_dbn.html#lds">switching linear dynamical system</a>. +In general, one must resort to approximate inference techniques: see +the discussion on <a href="#engines">inference engines</a> below. + + +<h2><a name="hybrid">Other hybrid models</h2> + +When we have C->D arcs, where C is hidden, we need to use +approximate inference. +One approach (not implemented in BNT) is described in +<ul> +<li> <a +href="http://www.cs.berkeley.edu/~murphyk/Papers/hybrid_uai99.ps.gz">A +Variational Approximation for Bayesian Networks with +Discrete and Continuous Latent Variables</a>, +K. Murphy, UAI 99. +</ul> +Of course, one can always use <a href="#sampling">sampling</a> methods +for approximate inference in such models. + + + +<h1><a name="param_learning">Parameter Learning</h1> + +The parameter estimation routines in BNT can be classified into 4 +types, depending on whether the goal is to compute +a full (Bayesian) posterior over the parameters or just a point +estimate (e.g., Maximum Likelihood or Maximum A Posteriori), +and whether all the variables are fully observed or there is missing +data/ hidden variables (partial observability). +<p> + +<TABLE BORDER> +<tr> + <TH></TH> + <th>Full obs</th> + <th>Partial obs</th> +</tr> +<tr> + <th>Point</th> + <td><tt>learn_params</tt></td> + <td><tt>learn_params_em</tt></td> +</tr> +<tr> + <th>Bayes</th> + <td><tt>bayes_update_params</tt></td> + <td>not yet supported</td> +</tr> +</table> + + +<h2><a name="load_data">Loading data from a file</h2> + +To load numeric data from an ASCII text file called 'dat.txt', where each row is a +case and columns are separated by white-space, such as +<pre> +011979 1626.5 0.0 +021979 1367.0 0.0 +... +</pre> +you can use +<pre> +data = load('dat.txt'); +</pre> +or +<pre> +load dat.txt -ascii +</pre> +In the latter case, the data is stored in a variable called 'dat' (the +filename minus the extension). +Alternatively, suppose the data is stored in a .csv file (has commas +separating the columns, and contains a header line), such as +<pre> +header info goes here +ORD,011979,1626.5,0.0 +DSM,021979,1367.0,0.0 +... +</pre> +You can load this using +<pre> +[a,b,c,d] = textread('dat.txt', '%s %d %f %f', 'delimiter', ',', 'headerlines', 1); +</pre> +If your file is not in either of these formats, you can either use Perl to convert +it to this format, or use the Matlab scanf command. +Type +<tt> +help iofun +</tt> +for more information on Matlab's file functions. +<!-- +<p> +To load data directly from Excel, +you should buy the +<a href="http://www.mathworks.com/products/excellink/">Excel Link</a>. +To load data directly from a relational database, +you should buy the +<a href="http://www.mathworks.com/products/database">Database +toolbox</a>. +--> +<p> +BNT learning routines require data to be stored in a cell array. +data{i,m} is the value of node i in case (example) m, i.e., each +<em>column</em> is a case. +If node i is not observed in case m (missing value), set +data{i,m} = []. +(Not all the learning routines can cope with such missing values, however.) +In the special case that all the nodes are observed and are +scalar-valued (as opposed to vector-valued), the data can be +stored in a matrix (as opposed to a cell-array). +<p> +Suppose, as in the <a href="#mixexp">mixture of experts example</a>, +that we have 3 nodes in the graph: X(1) is the observed input, X(3) is +the observed output, and X(2) is a hidden (gating) node. We can +create the dataset as follows. +<pre> +data = load('dat.txt'); +ncases = size(data, 1); +cases = cell(3, ncases); +cases([1 3], :) = num2cell(data'); +</pre> +Notice how we transposed the data, to convert rows into columns. +Also, cases{2,m} = [] for all m, since X(2) is always hidden. + + +<h2><a name="mle_complete">Maximum likelihood parameter estimation from complete data</h2> + +As an example, let's generate some data from the sprinkler network, randomize the parameters, +and then try to recover the original model. +First we create some training data using forwards sampling. +<pre> +samples = cell(N, nsamples); +for i=1:nsamples + samples(:,i) = sample_bnet(bnet); +end +</pre> +samples{j,i} contains the value of the j'th node in case i. +sample_bnet returns a cell array because, in general, each node might +be a vector of different length. +In this case, all nodes are discrete (and hence scalars), so we +could have used a regular array instead (which can be quicker): +<pre> +data = cell2num(samples); +</pre +So now data(j,i) = samples{j,i}. +<p> +Now we create a network with random parameters. +(The initial values of bnet2 don't matter in this case, since we can find the +globally optimal MLE independent of where we start.) +<pre> +% Make a tabula rasa +bnet2 = mk_bnet(dag, node_sizes); +seed = 0; +rand('state', seed); +bnet2.CPD{C} = tabular_CPD(bnet2, C); +bnet2.CPD{R} = tabular_CPD(bnet2, R); +bnet2.CPD{S} = tabular_CPD(bnet2, S); +bnet2.CPD{W} = tabular_CPD(bnet2, W); +</pre> +Finally, we find the maximum likelihood estimates of the parameters. +<pre> +bnet3 = learn_params(bnet2, samples); +</pre> +To view the learned parameters, we use a little Matlab hackery. +<pre> +CPT3 = cell(1,N); +for i=1:N + s=struct(bnet3.CPD{i}); % violate object privacy + CPT3{i}=s.CPT; +end +</pre> +Here are the parameters learned for node 4. +<pre> +dispcpt(CPT3{4}) +1 1 : 1.0000 0.0000 +2 1 : 0.2000 0.8000 +1 2 : 0.2273 0.7727 +2 2 : 0.0000 1.0000 +</pre> +So we see that the learned parameters are fairly close to the "true" +ones, which we display below. +<pre> +dispcpt(CPT{4}) +1 1 : 1.0000 0.0000 +2 1 : 0.1000 0.9000 +1 2 : 0.1000 0.9000 +2 2 : 0.0100 0.9900 +</pre> +We can get better results by using a larger training set, or using +informative priors (see <a href="#prior">below</a>). + + + +<h2><a name="prior">Parameter priors</h2> + +Currently, only tabular CPDs can have priors on their parameters. +The conjugate prior for a multinomial is the Dirichlet. +(For binary random variables, the multinomial is the same as the +Bernoulli, and the Dirichlet is the same as the Beta.) +<p> +The Dirichlet has a simple interpretation in terms of pseudo counts. +If we let N_ijk = the num. times X_i=k and Pa_i=j occurs in the +training set, where Pa_i are the parents of X_i, +then the maximum likelihood (ML) estimate is +T_ijk = N_ijk / N_ij (where N_ij = sum_k' N_ijk'), which will be 0 if N_ijk=0. +To prevent us from declaring that (X_i=k, Pa_i=j) is impossible just because this +event was not seen in the training set, +we can pretend we saw value k of X_i, for each value j of Pa_i some number (alpha_ijk) +of times in the past. +The MAP (maximum a posterior) estimate is then +<pre> +T_ijk = (N_ijk + alpha_ijk) / (N_ij + alpha_ij) +</pre> +and is never 0 if all alpha_ijk > 0. +For example, consider the network A->B, where A is binary and B has 3 +values. +A uniform prior for B has the form +<pre> + B=1 B=2 B=3 +A=1 1 1 1 +A=2 1 1 1 +</pre> +which can be created using +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'unif'); +</pre> +This prior does not satisfy the likelihood equivalence principle, +which says that <a href="#markov_equiv">Markov equivalent</a> models +should have the same marginal likelihood. +A prior that does satisfy this principle is shown below. +Heckerman (1995) calls this the +BDeu prior (likelihood equivalent uniform Bayesian Dirichlet). +<pre> + B=1 B=2 B=3 +A=1 1/6 1/6 1/6 +A=2 1/6 1/6 1/6 +</pre> +where we put N/(q*r) in each bin; N is the equivalent sample size, +r=|A|, q = |B|. +This can be created as follows +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'BDeu'); +</pre> +Here, 1 is the equivalent sample size, and is the strength of the +prior. +You can change this using +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', ... + 'BDeu', 'dirichlet_weight', 10); +</pre> +<!--where counts is an array of pseudo-counts of the same size as the +CPT.--> +<!-- +<p> +When you specify a prior, you should set row i of the CPT to the +normalized version of row i of the pseudo-count matrix, i.e., to the +expected values of the parameters. This will ensure that computing the +marginal likelihood sequentially (see <a +href="#bayes_learn">below</a>) and in batch form gives the same +results. +To do this, proceed as follows. +<pre> +tabular_CPD(bnet, i, 'prior', counts, 'CPT', mk_stochastic(counts)); +</pre> +For a non-informative prior, you can just write +<pre> +tabular_CPD(bnet, i, 'prior', 'unif', 'CPT', 'unif'); +</pre> +--> + + +<h2><a name="bayes_learn">(Sequential) Bayesian parameter updating from complete data</h2> + +If we use conjugate priors and have fully observed data, we can +compute the posterior over the parameters in batch form as follows. +<pre> +cases = sample_bnet(bnet, nsamples); +bnet = bayes_update_params(bnet, cases); +LL = log_marg_lik_complete(bnet, cases); +</pre> +bnet.CPD{i}.prior contains the new Dirichlet pseudocounts, +and bnet.CPD{i}.CPT is set to the mean of the posterior (the +normalized counts). +(Hence if the initial pseudo counts are 0, +<tt>bayes_update_params</tt> and <tt>learn_params</tt> will give the +same result.) + + + + +<p> +We can compute the same result sequentially (on-line) as follows. +<pre> +LL = 0; +for m=1:nsamples + LL = LL + log_marg_lik_complete(bnet, cases(:,m)); + bnet = bayes_update_params(bnet, cases(:,m)); +end +</pre> + +The file <tt>BNT/examples/static/StructLearn/model_select1</tt> has an example of +sequential model selection which uses the same idea. +We generate data from the model A->B +and compute the posterior prob of all 3 dags on 2 nodes: + (1) A B, (2) A <- B , (3) A -> B +Models 2 and 3 are <a href="#markov_equiv">Markov equivalent</a>, and therefore indistinguishable from +observational data alone, so we expect their posteriors to be the same +(assuming a prior which satisfies likelihood equivalence). +If we use random parameters, the "true" model only gets a higher posterior after 2000 trials! +However, if we make B a noisy NOT gate, the true model "wins" after 12 +trials, as shown below (red = model 1, blue/green (superimposed) +represents models 2/3). +<p> +<img src="Figures/model_select.png"> +<p> +The use of marginal likelihood for model selection is discussed in +greater detail in the +section on <a href="structure_learning">structure learning</a>. + + + + +<h2><a name="em">Maximum likelihood parameter estimation with missing values</h2> + +Now we consider learning when some values are not observed. +Let us randomly hide half the values generated from the water +sprinkler example. +<pre> +samples2 = samples; +hide = rand(N, nsamples) > 0.5; +[I,J]=find(hide); +for k=1:length(I) + samples2{I(k), J(k)} = []; +end +</pre> +samples2{i,l} is the value of node i in training case l, or [] if unobserved. +<p> +Now we will compute the MLEs using the EM algorithm. +We need to use an inference algorithm to compute the expected +sufficient statistics in the E step; the M (maximization) step is as +above. +<pre> +engine2 = jtree_inf_engine(bnet2); +max_iter = 10; +[bnet4, LLtrace] = learn_params_em(engine2, samples2, max_iter); +</pre> +LLtrace(i) is the log-likelihood at iteration i. We can plot this as +follows: +<pre> +plot(LLtrace, 'x-') +</pre> +Let's display the results after 10 iterations of EM. +<pre> +celldisp(CPT4) +CPT4{1} = + 0.6616 + 0.3384 +CPT4{2} = + 0.6510 0.3490 + 0.8751 0.1249 +CPT4{3} = + 0.8366 0.1634 + 0.0197 0.9803 +CPT4{4} = +(:,:,1) = + 0.8276 0.0546 + 0.5452 0.1658 +(:,:,2) = + 0.1724 0.9454 + 0.4548 0.8342 +</pre> +We can get improved performance by using one or more of the following +methods: +<ul> +<li> Increasing the size of the training set. +<li> Decreasing the amount of hidden data. +<li> Running EM for longer. +<li> Using informative priors. +<li> Initialising EM from multiple starting points. +</ul> + +<h2><a name="gaussianNumerical">Numerical problems when fitting +Gaussians using EM</h2> + +When fitting a <a href="#gaussian">Gaussian CPD</a> using EM, you +might encounter some numerical +problems. See <a +href="http://www.ai.mit.edu/~murphyk/Software/HMM/hmm.html#loglikpos">here</a> +for a discussion of this in the context of HMMs. +<!-- +(Note: BNT does not currently support spherical covariances, although it +would be easy to add, since KPMstats/clg_Mstep supports this option; +you would just need to modify gaussian_CPD/update_ess to accumulate +weighted inner products.) +--> + + +<h2><a name="tying">Parameter tying</h2> + +In networks with repeated structure (e.g., chains and grids), it is +common to assume that the parameters are the same at every node. This +is called parameter tying, and reduces the amount of data needed for +learning. +<p> +When we have tied parameters, there is no longer a one-to-one +correspondence between nodes and CPDs. +Rather, each CPD species the parameters for a whole equivalence class +of nodes. +It is easiest to see this by example. +Consider the following <a href="usage_dbn.html#hmm">hidden Markov +model (HMM)</a> +<p> +<img src="Figures/hmm3.gif"> +<p> +<!-- +We can create this graph structure, assuming we have T time-slices, +as follows. +(We number the nodes as shown in the figure, but we could equally well +number the hidden nodes 1:T, and the observed nodes T+1:2T.) +<pre> +N = 2*T; +dag = zeros(N); +hnodes = 1:2:2*T; +for i=1:T-1 + dag(hnodes(i), hnodes(i+1))=1; +end +onodes = 2:2:2*T; +for i=1:T + dag(hnodes(i), onodes(i)) = 1; +end +</pre> +<p> +The hidden nodes are always discrete, and have Q possible values each, +but the observed nodes can be discrete or continuous, and have O possible values/length. +<pre> +if cts_obs + dnodes = hnodes; +else + dnodes = 1:N; +end +ns = ones(1,N); +ns(hnodes) = Q; +ns(onodes) = O; +</pre> +--> +When HMMs are used for semi-infinite processes like speech recognition, +we assume the transition matrix +P(H(t+1)|H(t)) is the same for all t; this is called a time-invariant +or homogenous Markov chain. +Hence hidden nodes 2, 3, ..., T +are all in the same equivalence class, say class Hclass. +Similarly, the observation matrix P(O(t)|H(t)) is assumed to be the +same for all t, so the observed nodes are all in the same equivalence +class, say class Oclass. +Finally, the prior term P(H(1)) is in a class all by itself, say class +H1class. +This is illustrated below, where we explicitly represent the +parameters as random variables (dotted nodes). +<p> +<img src="Figures/hmm4_params.gif"> +<p> +In BNT, we cannot represent parameters as random variables (nodes). +Instead, we "hide" the +parameters inside one CPD for each equivalence class, +and then specify that the other CPDs should share these parameters, as +follows. +<pre> +hnodes = 1:2:2*T; +onodes = 2:2:2*T; +H1class = 1; Hclass = 2; Oclass = 3; +eclass = ones(1,N); +eclass(hnodes(2:end)) = Hclass; +eclass(hnodes(1)) = H1class; +eclass(onodes) = Oclass; +% create dag and ns in the usual way +bnet = mk_bnet(dag, ns, 'discrete', dnodes, 'equiv_class', eclass); +</pre> +Finally, we define the parameters for each equivalence class: +<pre> +bnet.CPD{H1class} = tabular_CPD(bnet, hnodes(1)); % prior +bnet.CPD{Hclass} = tabular_CPD(bnet, hnodes(2)); % transition matrix +if cts_obs + bnet.CPD{Oclass} = gaussian_CPD(bnet, onodes(1)); +else + bnet.CPD{Oclass} = tabular_CPD(bnet, onodes(1)); +end +</pre> +In general, if bnet.CPD{e} = xxx_CPD(bnet, j), then j should be a +member of e's equivalence class; that is, it is not always the case +that e == j. You can use bnet.rep_of_eclass(e) to return the +representative of equivalence class e. +BNT will look up the parents of j to determine the size +of the CPT to use. It assumes that this is the same for all members of +the equivalence class. +Click <a href="http://www.cs.berkeley.edu/~murphyk/Bayes/param_tieing.html">here</a> for +a more complex example of parameter tying. +<p> +Note: +Normally one would define an HMM as a +<a href = "usage_dbn.html">Dynamic Bayes Net</a> +(see the function BNT/examples/dynamic/mk_chmm.m). +However, one can define an HMM as a static BN using the function +BNT/examples/static/Models/mk_hmm_bnet.m. + + + +<h1><a name="structure_learning">Structure learning</h1> + +Update (9/29/03): +Phillipe LeRay is developing some additional structure learning code +on top of BNT. Click <a +href="http://bnt.insa-rouen.fr/ajouts.html">here</a> +for details. + +<p> + +There are two very different approaches to structure learning: +constraint-based and search-and-score. +In the <a href="#constraint">constraint-based approach</a>, +we start with a fully connected graph, and remove edges if certain +conditional independencies are measured in the data. +This has the disadvantage that repeated independence tests lose +statistical power. +<p> +In the more popular search-and-score approach, +we perform a search through the space of possible DAGs, and either +return the best one found (a point estimate), or return a sample of the +models found (an approximation to the Bayesian posterior). +<p> +Unfortunately, the number of DAGs as a function of the number of +nodes, G(n), is super-exponential in n. +A closed form formula for G(n) is not known, but the first few values +are shown below (from Cooper, 1999). + +<table> +<tr> <th>n</th> <th align=left>G(n)</th> </tr> +<tr> <td>1</td> <td>1</td> </tr> +<tr> <td>2</td> <td>3</td> </tr> +<tr> <td>3</td> <td>25</td> </tr> +<tr> <td>4</td> <td>543</td> </tr> +<tr> <td>5</td> <td>29,281</td> </tr> +<tr> <td>6</td> <td>3,781,503</td> </tr> +<tr> <td>7</td> <td>1.1 x 10^9</td> </tr> +<tr> <td>8</td> <td>7.8 x 10^11</td> </tr> +<tr> <td>9</td> <td>1.2 x 10^15</td> </tr> +<tr> <td>10</td> <td>4.2 x 10^18</td> </tr> +</table> + +Since the number of DAGs is super-exponential in the number of nodes, +we cannot exhaustively search the space, so we either use a local +search algorithm (e.g., greedy hill climbining, perhaps with multiple +restarts) or a global search algorithm (e.g., Markov Chain Monte +Carlo). +<p> +If we know a total ordering on the nodes, +finding the best structure amounts to picking the best set of parents +for each node independently. +This is what the K2 algorithm does. +If the ordering is unknown, we can search over orderings, +which is more efficient than searching over DAGs (Koller and Friedman, 2000). +<p> +In addition to the search procedure, we must specify the scoring +function. There are two popular choices. The Bayesian score integrates +out the parameters, i.e., it is the marginal likelihood of the model. +The BIC (Bayesian Information Criterion) is defined as +log P(D|theta_hat) - 0.5*d*log(N), where D is the data, theta_hat is +the ML estimate of the parameters, d is the number of parameters, and +N is the number of data cases. +The BIC method has the advantage of not requiring a prior. +<p> +BIC can be derived as a large sample +approximation to the marginal likelihood. +(It is also equal to the Minimum Description Length of a model.) +However, in practice, the sample size does not need to be very large +for the approximation to be good. +For example, in the figure below, we plot the ratio between the log marginal likelihood +and the BIC score against data-set size; we see that the ratio rapidly +approaches 1, especially for non-informative priors. +(This plot was generated by the file BNT/examples/static/bic1.m. It +uses the water sprinkler BN with BDeu Dirichlet priors with different +equivalent sample sizes.) + +<p> +<center> +<IMG SRC="Figures/bic.png"> +</center> +<p> + +<p> +As with parameter learning, handling missing data/ hidden variables is +much harder than the fully observed case. +The structure learning routines in BNT can therefore be classified into 4 +types, analogously to the parameter learning case. +<p> + +<TABLE BORDER> +<tr> + <TH></TH> + <th>Full obs</th> + <th>Partial obs</th> +</tr> +<tr> + <th>Point</th> + <td><tt>learn_struct_K2</tt> <br> +<!-- <tt>learn_struct_hill_climb</tt></td> --> + <td><tt>not yet supported</tt></td> +</tr> +<tr> + <th>Bayes</th> + <td><tt>learn_struct_mcmc</tt></td> + <td>not yet supported</td> +</tr> +</table> + + +<h2><a name="markov_equiv">Markov equivalence</h2> + +If two DAGs encode the same conditional independencies, they are +called Markov equivalent. The set of all DAGs can be paritioned into +Markov equivalence classes. Graphs within the same class can +have +the direction of some of their arcs reversed without changing any of +the CI relationships. +Each class can be represented by a PDAG +(partially directed acyclic graph) called an essential graph or +pattern. This specifies which edges must be oriented in a certain +direction, and which may be reversed. + +<p> +When learning graph structure from observational data, +the best one can hope to do is to identify the model up to Markov +equivalence. To distinguish amongst graphs within the same equivalence +class, one needs interventional data: see the discussion on <a +href="#active">active learning</a> below. + + + +<h2><a name="enumerate">Exhaustive search</h2> + +The brute-force approach to structure learning is to enumerate all +possible DAGs, and score each one. This provides a "gold standard" +with which to compare other algorithms. We can do this as follows. +<pre> +dags = mk_all_dags(N); +score = score_dags(data, ns, dags); +</pre> +where data(i,m) is the value of node i in case m, +and ns(i) is the size of node i. +If the DAGs have a lot of families in common, we can cache the sufficient statistics, +making this potentially more efficient than scoring the DAGs one at a time. +(Caching is not currently implemented, however.) +<p> +By default, we use the Bayesian scoring metric, and assume CPDs are +represented by tables with BDeu(1) priors. +We can override these defaults as follows. +If we want to use uniform priors, we can say +<pre> +params = cell(1,N); +for i=1:N + params{i} = {'prior', 'unif'}; +end +score = score_dags(data, ns, dags, 'params', params); +</pre> +params{i} is a cell-array, containing optional arguments that are +passed to the constructor for CPD i. +<p> +Now suppose we want to use different node types, e.g., +Suppose nodes 1 and 2 are Gaussian, and nodes 3 and 4 softmax (both +these CPDs can support discrete and continuous parents, which is +necessary since all other nodes will be considered as parents). +The Bayesian scoring metric currently only works for tabular CPDs, so +we will use BIC: +<pre> +score = score_dags(data, ns, dags, 'discrete', [3 4], 'params', [], + 'type', {'gaussian', 'gaussian', 'softmax', softmax'}, 'scoring_fn', 'bic') +</pre> +In practice, one can't enumerate all possible DAGs for N > 5, +but one can evaluate any reasonably-sized set of hypotheses in this +way (e.g., nearest neighbors of your current best guess). +Think of this as "computer assisted model refinement" as opposed to de +novo learning. + + +<h2><a name="K2">K2</h2> + +The K2 algorithm (Cooper and Herskovits, 1992) is a greedy search algorithm that works as follows. +Initially each node has no parents. It then adds incrementally that parent whose addition most +increases the score of the resulting structure. When the addition of no single +parent can increase the score, it stops adding parents to the node. +Since we are using a fixed ordering, we do not need to check for +cycles, and can choose the parents for each node independently. +<p> +The original paper used the Bayesian scoring +metric with tabular CPDs and Dirichlet priors. +BNT generalizes this to allow any kind of CPD, and either the Bayesian +scoring metric or BIC, as in the example <a href="#enumerate">above</a>. +In addition, you can specify +an optional upper bound on the number of parents for each node. +The file BNT/examples/static/k2demo1.m gives an example of how to use K2. +We use the water sprinkler network and sample 100 cases from it as before. +Then we see how much data it takes to recover the generating structure: +<pre> +order = [C S R W]; +max_fan_in = 2; +sz = 5:5:100; +for i=1:length(sz) + dag2 = learn_struct_K2(data(:,1:sz(i)), node_sizes, order, 'max_fan_in', max_fan_in); + correct(i) = isequal(dag, dag2); +end +</pre> +Here are the results. +<pre> +correct = + Columns 1 through 12 + 0 0 0 0 0 0 0 1 0 1 1 1 + Columns 13 through 20 + 1 1 1 1 1 1 1 1 +</pre> +So we see it takes about sz(10)=50 cases. (BIC behaves similarly, +showing that the prior doesn't matter too much.) +In general, we cannot hope to recover the "true" generating structure, +only one that is in its <a href="#markov_equiv">Markov equivalence +class</a>. + + +<h2><a name="hill_climb">Hill-climbing</h2> + +Hill-climbing starts at a specific point in space, +considers all nearest neighbors, and moves to the neighbor +that has the highest score; if no neighbors have higher +score than the current point (i.e., we have reached a local maximum), +the algorithm stops. One can then restart in another part of the space. +<p> +A common definition of "neighbor" is all graphs that can be +generated from the current graph by adding, deleting or reversing a +single arc, subject to the acyclicity constraint. +Other neighborhoods are possible: see +<a href="http://research.microsoft.com/~dmax/publications/jmlr02.pdf"> +Optimal Structure Identification with Greedy Search</a>, Max +Chickering, JMLR 2002. + +<!-- +Note: This algorithm is currently (Feb '02) being implemented by Qian +Diao. +--> + + +<h2><a name="mcmc">MCMC</h2> + +We can use a Markov Chain Monte Carlo (MCMC) algorithm called +Metropolis-Hastings (MH) to search the space of all +DAGs. +The standard proposal distribution is to consider moving to all +nearest neighbors in the sense defined <a href="#hill_climb">above</a>. +<p> +The function can be called +as in the following example. +<pre> +[sampled_graphs, accept_ratio] = learn_struct_mcmc(data, ns, 'nsamples', 100, 'burnin', 10); +</pre> +We can convert our set of sampled graphs to a histogram +(empirical posterior over all the DAGs) thus +<pre> +all_dags = mk_all_dags(N); +mcmc_post = mcmc_sample_to_hist(sampled_graphs, all_dags); +</pre> +To see how well this performs, let us compute the exact posterior exhaustively. +<p> +<pre> +score = score_dags(data, ns, all_dags); +post = normalise(exp(score)); % assuming uniform structural prior +</pre> +We plot the results below. +(The data set was 100 samples drawn from a random 4 node bnet; see the +file BNT/examples/static/mcmc1.) +<pre> +subplot(2,1,1) +bar(post) +subplot(2,1,2) +bar(mcmc_post) +</pre> +<img src="Figures/mcmc_post.jpg" width="800" height="500"> +<p> +We can also plot the acceptance ratio versus number of MCMC steps, +as a crude convergence diagnostic. +<pre> +clf +plot(accept_ratio) +</pre> +<img src="Figures/mcmc_accept.jpg" width="800" height="300"> +<p> +Better convergence diagnostics can be found +<a href="<a href="http://www.lce.hut.fi/~ave/code/mcmcdiag/">here</a> +<p> +Even though the number of samples needed by MCMC is theoretically +polynomial (not exponential) in the dimensionality of the search space, in practice it has been +found that MCMC does not converge in reasonable time for graphs with +more than about 10 nodes. +<p> +<a href="http://www.bioss.sari.ac.uk/~dirk/software/DBmcmc/"> +Dirk Husmeier has extended MCMC model selection to DBNs</a>. + + +<h2><a name="active">Active structure learning</h2> + +As was mentioned <a href="#markov_equiv">above</a>, +one can only learn a DAG up to Markov equivalence, even given infinite data. +If one is interested in learning the structure of a causal network, +one needs interventional data. +(By "intervention" we mean forcing a node to take on a specific value, +thereby effectively severing its incoming arcs.) +<p> +Most of the scoring functions accept an optional argument +that specifies whether a node was observed to have a certain value, or +was forced to have that value: we set clamped(i,m)=1 if node i was +forced in training case m. e.g., see the file +BNT/examples/static/cooper_yoo. +<p> +An interesting question is to decide which interventions to perform +(c.f., design of experiments). For details, see the following tech +report +<ul> +<li> <a href = "../../Papers/alearn.ps.gz"> +Active learning of causal Bayes net structure</a>, Kevin Murphy, March +2001. +</ul> + + +<h2><a name="struct_em">Structural EM</h2> + +Computing the Bayesian score when there is partial observability is +computationally challenging, because the parameter posterior becomes +multimodal (the hidden nodes induce a mixture distribution). +One therefore needs to use approximations such as BIC. +Unfortunately, search algorithms are still expensive, because we need +to run EM at each step to compute the MLE, which is needed to compute +the score of each model. An alternative approach is +to do the local search steps inside of the M step of EM, which is more +efficient since the data has been "filled in" - this is +called the structural EM algorithm (Friedman 1997), and provably +converges to a local maximum of the BIC score. +<p> +Wei Hu has implemented SEM for discrete nodes. +You can download his package from +<a href="../SEM.zip">here</a>. +Please address all questions about this code to +wei.hu@intel.com. +See also <a href="#phl">Phl's implementation of SEM</a>. + +<!-- +<h2><a name="reveal">REVEAL algorithm</h2> + +A simple way to learn the structure of a fully observed, discrete, +factored DBN from a time series is described <a +href="usage_dbn.html#struct_learn">here</a>. +--> + + +<h2><a name="graphdraw">Visualizing the graph</h2> + +You can visualize an arbitrary graph (such as one learned using the +structure learning routines) with Matlab code contributed by +<a href="http://www.mbfys.kun.nl/~cemgil/matlab/layout.html">Ali +Taylan Cemgil</a> +from the University of Nijmegen. +For static BNs, call it as follows: +<pre> +draw_graph(bnet.dag); +</pre> +For example, this is the output produced on a +<a href="#qmr">random QMR-like model</a>: +<p> +<img src="Figures/qmr.rnd.jpg"> +<p> +If you install the excellent <a +href="http://www.research.att.com/sw/tools/graphviz">graphhviz</a>, an +open-source graph visualization package from AT&T, +you can create a much better visualization as follows +<pre> +graph_to_dot(bnet.dag) +</pre> +This works by converting the adjacency matrix to a file suitable +for input to graphviz (using the dot format), +then converting the output of graphviz to postscript, and displaying the results using +ghostview. +You can do each of these steps separately for more control, as shown +below. +<pre> +graph_to_dot(bnet.dag, 'filename', 'foo.dot'); +dot -Tps foo.dot -o foo.ps +ghostview foo.ps & +</pre> + +<h2><a name = "constraint">Constraint-based methods</h2> + +The IC algorithm (Pearl and Verma, 1991), +and the faster, but otherwise equivalent, PC algorithm (Spirtes, Glymour, and Scheines 1993), +computes many conditional independence tests, +and combines these constraints into a +PDAG to represent the whole +<a href="#markov_equiv">Markov equivalence class</a>. +<p> +IC*/FCI extend IC/PC to handle latent variables: see <a href="#ic_star">below</a>. +(IC stands for inductive causation; PC stands for Peter and Clark, +the first names of Spirtes and Glymour; FCI stands for fast causal +inference. +What we, following Pearl (2000), call IC* was called +IC in the original Pearl and Verma paper.) +For details, see +<ul> +<li> +<a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Causation, +Prediction, and Search</a>, Spirtes, Glymour and +Scheines (SGS), 2001 (2nd edition), MIT Press. +<li> +<a href="http://bayes.cs.ucla.edu/BOOK-2K/index.html">Causality: Models, Reasoning and Inference</a>, J. Pearl, +2000, Cambridge University Press. +</ul> + +<p> + +The PC algorithm takes as arguments a function f, the number of nodes N, +the maximum fan in K, and additional arguments A which are passed to f. +The function f(X,Y,S,A) returns 1 if X is conditionally independent of Y given S, and 0 +otherwise. +For example, suppose we cheat by +passing in a CI "oracle" which has access to the true DAG; the oracle +tests for d-separation in this DAG, i.e., +f(X,Y,S) calls dsep(X,Y,S,dag). We can to this as follows. +<pre> +pdag = learn_struct_pdag_pc('dsep', N, max_fan_in, dag); +</pre> +pdag(i,j) = -1 if there is definitely an i->j arc, +and pdag(i,j) = 1 if there is either an i->j or and i<-j arc. +<p> +Applied to the sprinkler network, this returns +<pre> +pdag = + 0 1 1 0 + 1 0 0 -1 + 1 0 0 -1 + 0 0 0 0 +</pre> +So as expected, we see that the V-structure at the W node is uniquely identified, +but the other arcs have ambiguous orientation. +<p> +We now give an example from p141 (1st edn) / p103 (2nd end) of the SGS +book. +This example concerns the female orgasm. +We are given a correlation matrix C between 7 measured factors (such +as subjective experiences of coital and masturbatory experiences), +derived from 281 samples, and want to learn a causal model of the +data. We will not discuss the merits of this type of work here, but +merely show how to reproduce the results in the SGS book. +Their program, +<a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Tetrad</a>, +makes use of the Fisher Z-test for conditional +independence, so we do the same: +<pre> +max_fan_in = 4; +nsamples = 281; +alpha = 0.05; +pdag = learn_struct_pdag_pc('cond_indep_fisher_z', n, max_fan_in, C, nsamples, alpha); +</pre> +In this case, the CI test is +<pre> +f(X,Y,S) = cond_indep_fisher_z(X,Y,S, C,nsamples,alpha) +</pre> +The results match those of Fig 12a of SGS apart from two edge +differences; presumably this is due to rounding error (although it +could be a bug, either in BNT or in Tetrad). +This example can be found in the file BNT/examples/static/pc2.m. + +<p> + +The IC* algorithm (Pearl and Verma, 1991), +and the faster FCI algorithm (Spirtes, Glymour, and Scheines 1993), +are like the IC/PC algorithm, except that they can detect the presence +of latent variables. +See the file <tt>learn_struct_pdag_ic_star</tt> written by Tamar +Kushnir. The output is a matrix P, defined as follows +(see Pearl (2000), p52 for details): +<pre> +% P(i,j) = -1 if there is either a latent variable L such that i <-L->j OR there is a directed edge from i->j. +% P(i,j) = -2 if there is a marked directed i-*>j edge. +% P(i,j) = P(j,i) = 1 if there is and undirected edge i--j +% P(i,j) = P(j,i) = 2 if there is a latent variable L such that i<-L->j. +</pre> + + +<h2><a name="phl">Philippe Leray's structure learning package</h2> + +Philippe Leray has written a +<a href="http://bnt.insa-rouen.fr/ajouts.html"> +structure learning package</a> that uses BNT. + +It currently (Juen 2003) has the following features: +<ul> +<li>PC with Chi2 statistical test +<li> MWST : Maximum weighted Spanning Tree +<li> Hill Climbing +<li> Greedy Search +<li> Structural EM +<li> hist_ic : optimal Histogram based on IC information criterion +<li> cpdag_to_dag +<li> dag_to_cpdag +<li> ... +</ul> + + +</a> + + +<!-- +<h2><a name="read_learning">Further reading on learning</h2> + +I recommend the following tutorials for more details on learning. +<ul> +<li> <a +href="http://www.cs.berkeley.edu/~murphyk/Papers/intel.ps.gz">My short +tutorial</a> on graphical models, which contains an overview of learning. + +<li> +<A HREF="ftp://ftp.research.microsoft.com/pub/tr/TR-95-06.PS"> +A tutorial on learning with Bayesian networks</a>, D. Heckerman, +Microsoft Research Tech Report, 1995. + +<li> <A HREF="http://www-cad.eecs.berkeley.edu/~wray/Mirror/lwgmja"> +Operations for Learning with Graphical Models</a>, +W. L. Buntine, JAIR'94, 159--225. +</ul> +<p> +--> + + + + + +<h1><a name="engines">Inference engines</h1> + +Up until now, we have used the junction tree algorithm for inference. +However, sometimes this is too slow, or not even applicable. +In general, there are many inference algorithms each of which make +different tradeoffs between speed, accuracy, complexity and +generality. Furthermore, there might be many implementations of the +same algorithm; for instance, a general purpose, readable version, +and a highly-optimized, specialized one. +To cope with this variety, we treat each inference algorithm as an +object, which we call an inference engine. + +<p> +An inference engine is an object that contains a bnet and supports the +'enter_evidence' and 'marginal_nodes' methods. The engine constructor +takes the bnet as argument and may do some model-specific processing. +When 'enter_evidence' is called, the engine may do some +evidence-specific processing. Finally, when 'marginal_nodes' is +called, the engine may do some query-specific processing. + +<p> +The amount of work done when each stage is specified -- structure, +parameters, evidence, and query -- depends on the engine. The cost of +work done early in this sequence can be amortized. On the other hand, +one can make better optimizations if one waits until later in the +sequence. +For example, the parameters might imply +conditional indpendencies that are not evident in the graph structure, +but can nevertheless be exploited; the evidence indicates which nodes +are observed and hence can effectively be disconnected from the +graph; and the query might indicate that large parts of the network +are d-separated from the query nodes. (Since it is not the actual +<em>values</em> of the evidence that matters, just which nodes are observed, +many engines allow you to specify which nodes will be observed when they are constructed, +i.e., before calling 'enter_evidence'. Some engines can still cope if +the actual pattern of evidence is different, e.g., if there is missing +data.) +<p> + +Although being maximally lazy (i.e., only doing work when a query is +issued) may seem desirable, +this is not always the most efficient. +For example, +when learning using EM, we need to call marginal_nodes N times, where N is the +number of nodes. <a href="varelim">Variable elimination</a> would end +up repeating a lot of work +each time marginal_nodes is called, making it inefficient for +learning. The junction tree algorithm, by contrast, uses dynamic +programming to avoid this redundant computation --- it calculates all +marginals in two passes during 'enter_evidence', so calling +'marginal_nodes' takes constant time. +<p> +We will discuss some of the inference algorithms implemented in BNT +below, and finish with a <a href="#engine_summary">summary</a> of all +of them. + + + + + + + +<h2><a name="varelim">Variable elimination</h2> + +The variable elimination algorithm, also known as bucket elimination +or peeling, is one of the simplest inference algorithms. +The basic idea is to "push sums inside of products"; this is explained +in more detail +<a +href="http://HTTP.CS.Berkeley.EDU/~murphyk/Bayes/bayes.html#infer">here</a>. +<p> +The principle of distributing sums over products can be generalized +greatly to apply to any commutative semiring. +This forms the basis of many common algorithms, such as Viterbi +decoding and the Fast Fourier Transform. For details, see + +<ul> +<li> R. McEliece and S. M. Aji, 2000. +<!--<a href="http://www.systems.caltech.edu/EE/Faculty/rjm/papers/GDL.ps">--> +<a href="GDL.pdf"> +The Generalized Distributive Law</a>, +IEEE Trans. Inform. Theory, vol. 46, no. 2 (March 2000), +pp. 325--343. + + +<li> +F. R. Kschischang, B. J. Frey and H.-A. Loeliger, 2001. +<a href="http://www.cs.toronto.edu/~frey/papers/fgspa.abs.html"> +Factor graphs and the sum-product algorithm</a> +IEEE Transactions on Information Theory, February, 2001. + +</ul> + +<p> +Choosing an order in which to sum out the variables so as to minimize +computational cost is known to be NP-hard. +The implementation of this algorithm in +<tt>var_elim_inf_engine</tt> makes no attempt to optimize this +ordering (in contrast, say, to <tt>jtree_inf_engine</tt>, which uses a +greedy search procedure to find a good ordering). +<p> +Note: unlike most algorithms, var_elim does all its computational work +inside of <tt>marginal_nodes</tt>, not inside of +<tt>enter_evidence</tt>. + + + + +<h2><a name="global">Global inference methods</h2> + +The simplest inference algorithm of all is to explicitely construct +the joint distribution over all the nodes, and then to marginalize it. +This is implemented in <tt>global_joint_inf_engine</tt>. +Since the size of the joint is exponential in the +number of discrete (hidden) nodes, this is not a very practical algorithm. +It is included merely for pedagogical and debugging purposes. +<p> +Three specialized versions of this algorithm have also been implemented, +corresponding to the cases where all the nodes are discrete (D), all +are Gaussian (G), and some are discrete and some Gaussian (CG). +They are called <tt>enumerative_inf_engine</tt>, +<tt>gaussian_inf_engine</tt>, +and <tt>cond_gauss_inf_engine</tt> respectively. +<p> +Note: unlike most algorithms, these global inference algorithms do all their computational work +inside of <tt>marginal_nodes</tt>, not inside of +<tt>enter_evidence</tt>. + + +<h2><a name="quickscore">Quickscore</h2> + +The junction tree algorithm is quite slow on the <a href="#qmr">QMR</a> network, +since the cliques are so big. +One simple trick we can use is to notice that hidden leaves do not +affect the posteriors on the roots, and hence do not need to be +included in the network. +A second trick is to notice that the negative findings can be +"absorbed" into the prior: +see the file +BNT/examples/static/mk_minimal_qmr_bnet for details. +<p> + +A much more significant speedup is obtained by exploiting special +properties of the noisy-or node, as done by the quickscore +algorithm. For details, see +<ul> +<li> Heckerman, "A tractable inference algorithm for diagnosing multiple diseases", UAI 89. +<li> Rish and Dechter, "On the impact of causal independence", UCI +tech report, 1998. +</ul> + +This has been implemented in BNT as a special-purpose inference +engine, which can be created and used as follows: +<pre> +engine = quickscore_inf_engine(inhibit, leak, prior); +engine = enter_evidence(engine, pos, neg); +m = marginal_nodes(engine, i); +</pre> + + +<h2><a name="belprop">Belief propagation</h2> + +Even using quickscore, exact inference takes time that is exponential +in the number of positive findings. +Hence for large networks we need to resort to approximate inference techniques. +See for example +<ul> +<li> T. Jaakkola and M. Jordan, "Variational probabilistic inference and the +QMR-DT network", JAIR 10, 1999. + +<li> K. Murphy, Y. Weiss and M. Jordan, "Loopy belief propagation for approximate inference: an empirical study", + UAI 99. +</ul> +The latter approximation +entails applying Pearl's belief propagation algorithm to a model even +if it has loops (hence the name loopy belief propagation). +Pearl's algorithm, implemented as <tt>pearl_inf_engine</tt>, gives +exact results when applied to singly-connected graphs +(a.k.a. polytrees, since +the underlying undirected topology is a tree, but a node may have +multiple parents). +To apply this algorithm to a graph with loops, +use <tt>pearl_inf_engine</tt>. +This can use a centralized or distributed message passing protocol. +You can use it as in the following example. +<pre> +engine = pearl_inf_engine(bnet, 'max_iter', 30); +engine = enter_evidence(engine, evidence); +m = marginal_nodes(engine, i); +</pre> +We found that this algorithm often converges, and when it does, often +is very accurate, but it depends on the precise setting of the +parameter values of the network. +(See the file BNT/examples/static/qmr1 to repeat the experiment for yourself.) +Understanding when and why belief propagation converges/ works +is a topic of ongoing research. +<p> +<tt>pearl_inf_engine</tt> can exploit special structure in noisy-or +and gmux nodes to compute messages efficiently. +<p> +<tt>belprop_inf_engine</tt> is like pearl, but uses potentials to +represent messages. Hence this is slower. +<p> +<tt>belprop_fg_inf_engine</tt> is like belprop, +but is designed for factor graphs. + + + +<h2><a name="sampling">Sampling</h2> + +BNT now (Mar '02) has two sampling (Monte Carlo) inference algorithms: +<ul> +<li> <tt>likelihood_weighting_inf_engine</tt> which does importance +sampling and can handle any node type. +<li> <tt>gibbs_sampling_inf_engine</tt>, written by Bhaskara Marthi. +Currently this can only handle tabular CPDs. +For a much faster and more powerful Gibbs sampling program, see +<a href="http://www.mrc-bsu.cam.ac.uk/bugs">BUGS</a>. +</ul> +Note: To generate samples from a network (which is not the same as inference!), +use <tt>sample_bnet</tt>. + + + +<h2><a name="engine_summary">Summary of inference engines</h2> + + +The inference engines differ in many ways. Here are +some of the major "axes": +<ul> +<li> Works for all topologies or makes restrictions? +<li> Works for all node types or makes restrictions? +<li> Exact or approximate inference? +</ul> + +<p> +In terms of topology, most engines handle any kind of DAG. +<tt>belprop_fg</tt> does approximate inference on factor graphs (FG), which +can be used to represent directed, undirected, and mixed (chain) +graphs. +(In the future, we plan to support exact inference on chain graphs.) +<tt>quickscore</tt> only works on QMR-like models. +<p> +In terms of node types: algorithms that use potentials can handle +discrete (D), Gaussian (G) or conditional Gaussian (CG) models. +Sampling algorithms can essentially handle any kind of node (distribution). +Other algorithms make more restrictive assumptions in exchange for +speed. +<p> +Finally, most algorithms are designed to give the exact answer. +The belief propagation algorithms are exact if applied to trees, and +in some other cases. +Sampling is considered approximate, even though, in the limit of an +infinite number of samples, it gives the exact answer. + +<p> + +Here is a summary of the properties +of all the engines in BNT which work on static networks. +<p> +<table> +<table border units = pixels><tr> +<td align=left width=0>Name +<td align=left width=0>Exact? +<td align=left width=0>Node type? +<td align=left width=0>topology +<tr> +<tr> +<td align=left> belprop +<td align=left> approx +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> belprop_fg +<td align=left> approx +<td align=left> D +<td align=left> factor graph +<tr> +<td align=left> cond_gauss +<td align=left> exact +<td align=left> CG +<td align=left> DAG +<tr> +<td align=left> enumerative +<td align=left> exact +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> gaussian +<td align=left> exact +<td align=left> G +<td align=left> DAG +<tr> +<td align=left> gibbs +<td align=left> approx +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> global_joint +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +<tr> +<td align=left> jtree +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +b<tr> +<td align=left> likelihood_weighting +<td align=left> approx +<td align=left> any +<td align=left> DAG +<tr> +<td align=left> pearl +<td align=left> approx +<td align=left> D,G +<td align=left> DAG +<tr> +<td align=left> pearl +<td align=left> exact +<td align=left> D,G +<td align=left> polytree +<tr> +<td align=left> quickscore +<td align=left> exact +<td align=left> noisy-or +<td align=left> QMR +<tr> +<td align=left> stab_cond_gauss +<td align=left> exact +<td align=left> CG +<td align=left> DAG +<tr> +<td align=left> var_elim +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +</table> + + + +<h1><a name="influence">Influence diagrams/ decision making</h1> + +BNT implements an exact algorithm for solving LIMIDs (limited memory +influence diagrams), described in +<ul> +<li> S. L. Lauritzen and D. Nilsson. +<a href="http://www.math.auc.dk/~steffen/papers/limids.pdf"> +Representing and solving decision problems with limited +information</a> +Management Science, 47, 1238 - 1251. September 2001. +</ul> +LIMIDs explicitely show all information arcs, rather than implicitely +assuming no forgetting. This allows them to model forgetful +controllers. +<p> +See the examples in <tt>BNT/examples/limids</tt> for details. + + + + +<h1>DBNs, HMMs, Kalman filters and all that</h1> + +Click <a href="usage_dbn.html">here</a> for documentation about how to +use BNT for dynamical systems and sequence data. + + +</BODY> diff --git a/sourcecodes/bnt-master/docs/usage_dbn.html b/sourcecodes/bnt-master/docs/usage_dbn.html new file mode 100644 index 00000000..7582a0cb --- /dev/null +++ b/sourcecodes/bnt-master/docs/usage_dbn.html @@ -0,0 +1,719 @@ +<HEAD> +<TITLE>How to use BNT for DBNs</TITLE> +</HEAD> + +<BODY BGCOLOR="#FFFFFF"> +<!-- white background is better for the pictures and equations --> + +Documentation last updated on 7 June 2004 + +<h1>How to use BNT for DBNs</h1> + +<p> +<ul> +<li> <a href="#spec">Model specification</a> +<ul> +<li> <a href="#hmm">HMMs</a> +<li> <a href="#lds">Kalman filters</a> +<li> <a href="#chmm">Coupled HMMs</a> +<li> <a href="#water">Water network</a> +<li> <a href="#bat">BAT network</a> +</ul> + +<li> <a href="#inf">Inference</a> +<ul> +<li> <a href="#discrete">Discrete hidden nodes</a> +<li> <a href="#cts">Continuous hidden nodes</a> +</ul> + +<li> <a href="#learn">Learning</a> +<ul> +<li> <a href="#param_learn">Parameter learning</a> +<li> <a href="#struct_learn">Structure learning</a> +</ul> + +</ul> + +Note: +you are recommended to read an introduction +to DBNs first, such as +<a href="http://www.ai.mit.edu/~murphyk/Papers/dbnchapter.pdf"> +this book chapter</a>. +<br> +You may also want to consider using +<a href=http://ssli.ee.washington.edu/~bilmes/gmtk/>GMTk</a>, which is +an excellent C++ package for DBNs. + + +<h1><a name="spec">Model specification</h1> + + +<!--<h1><a name="dbn_intro">Dynamic Bayesian Networks (DBNs)</h1>--> + +Dynamic Bayesian Networks (DBNs) are directed graphical models of stochastic +processes. +They generalise <a href="#hmm">hidden Markov models (HMMs)</a> +and <a href="#lds">linear dynamical systems (LDSs)</a> +by representing the hidden (and observed) state in terms of state +variables, which can have complex interdependencies. +The graphical structure provides an easy way to specify these +conditional independencies, and hence to provide a compact +parameterization of the model. +<p> +Note that "temporal Bayesian network" would be a better name than +"dynamic Bayesian network", since +it is assumed that the model structure does not change, but +the term DBN has become entrenched. +We also normally assume that the parameters do not +change, i.e., the model is time-invariant. +However, we can always add extra +hidden nodes to represent the current "regime", thereby creating +mixtures of models to capture periodic non-stationarities. +<p> +There are some cases where the size of the state space can change over +time, e.g., tracking a variable, but unknown, number of objects. +In this case, we need to change the model structure over time. +BNT does not support this. +<!-- +, but see the following paper for a +discussion of some of the issues: +<ul> +<li> <a href="ftp://ftp.cs.monash.edu.au/pub/annn/smc.ps"> +Dynamic belief networks for discrete monitoring</a>, +A. E. Nicholson and J. M. Brady. +IEEE Systems, Man and Cybernetics, 24(11):1593-1610, 1994. +</ul> +--> + + +<h2><a name="hmm">Hidden Markov Models (HMMs)</h2> + +The simplest kind of DBN is a Hidden Markov Model (HMM), which has +one discrete hidden node and one discrete or continuous +observed node per slice. We illustrate this below. +As before, circles denote continuous nodes, squares denote +discrete nodes, clear means hidden, shaded means observed. +<!-- +(The observed nodes can be +discrete or continuous; the crucial thing about an HMM is that the +hidden nodes are discrete, so the system can model arbitrary dynamics +-- providing, of course, that the hidden state space is large enough.) +--> +<p> +<img src="Figures/hmm3.gif"> +<p> +We have "unrolled" the model for three "time slices" -- the structure and parameters are +assumed to repeat as the model is unrolled further. +Hence to specify a DBN, we need to +define the intra-slice topology (within a slice), +the inter-slice topology (between two slices), +as well as the parameters for the first two slices. +(Such a two-slice temporal Bayes net is often called a 2TBN.) +<p> +We can specify the topology as follows. +<PRE> +intra = zeros(2); +intra(1,2) = 1; % node 1 in slice t connects to node 2 in slice t + +inter = zeros(2); +inter(1,1) = 1; % node 1 in slice t-1 connects to node 1 in slice t +</pre> +We can specify the parameters as follows, +where for simplicity we assume the observed node is discrete. +<pre> +Q = 2; % num hidden states +O = 2; % num observable symbols + +ns = [Q O]; +dnodes = 1:2; +bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes); +for i=1:4 + bnet.CPD{i} = tabular_CPD(bnet, i); +end +</pre> +<p> +We assume the distributions P(X(t) | X(t-1)) and +P(Y(t) | X(t)) are independent of t for t > 1. +Hence the CPD for nodes 5, 7, ... is the same as for node 3, so we say they +are in the same equivalence class, with node 3 being the "representative" +for this class. In other words, we have tied the parameters for nodes +3, 5, 7, ... +Similarly, nodes 4, 6, 8, ... are tied. +Note, however, that (the parameters for) nodes 1 and 2 are not tied to +subsequent slices. +<p> +Above we assumed the observation model P(Y(t) | X(t)) is independent of t for t>1, but +it is conventional to assume this is true for all t. +So we would like to put nodes 2, 4, 6, ... all in the same class. +We can do this by explicitely defining the equivalence classes, as +follows (see <a href="usage.html#tying">here</a> for more details on +parameter tying). +<p> +We define eclass1(i) to be the equivalence class that node i in slice +1 belongs to. +Similarly, we define eclass2(i) to be the equivalence class that node i in slice +2, 3, ..., belongs to. +For an HMM, we have +<pre> +eclass1 = [1 2]; +eclass2 = [3 2]; +eclass = [eclass1 eclass2]; +</pre> +This ties the observation model across slices, +since e.g., eclass(4) = eclass(2) = 2. +<p> +By default, +eclass1 = 1:ss, and eclass2 = (1:ss)+ss, where ss = slice size = the +number of nodes per slice. +<!--This will tie nodes in slices 3, 4, ... to the the nodes in slice 2, +but none of the nodes in slice 2 to any in slice 1.--> +But by using the above tieing pattern, +we now only have 3 CPDs to specify, instead of 4: +<pre> +bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes, 'eclass1', eclass1, 'eclass2', eclass2); +prior0 = normalise(rand(Q,1)); +transmat0 = mk_stochastic(rand(Q,Q)); +obsmat0 = mk_stochastic(rand(Q,O)); +bnet.CPD{1} = tabular_CPD(bnet, 1, prior0); +bnet.CPD{2} = tabular_CPD(bnet, 2, obsmat0); +bnet.CPD{3} = tabular_CPD(bnet, 3, transmat0); +</pre> +We discuss how to do <a href="#inf">inference</a> and <a href="#learn">learning</a> on this model +below. +(See also +my <a href="../HMM/hmm.html">HMM toolbox</a>, which is included with BNT.) + +<p> +Some common variants on HMMs are shown below. +BNT can handle all of these. +<p> +<center> +<table> +<tr> +<td><img src="Figures/hmm_gauss.gif"> +<td><img src="Figures/hmm_mixgauss.gif" +<td><img src="Figures/hmm_ar.gif"> +<tr> +<td><img src="Figures/hmm_factorial.gif"> +<td><img src="Figures/hmm_coupled.gif" +<td><img src="Figures/hmm_io.gif"> +<tr> +</table> +</center> + + + +<h2><a name="lds">Linear Dynamical Systems (LDSs) and Kalman filters</h2> + +A Linear Dynamical System (LDS) has the same topology as an HMM, but +all the nodes are assumed to have linear-Gaussian distributions, i.e., +<pre> + x(t+1) = A*x(t) + w(t), w ~ N(0, Q), x(0) ~ N(init_x, init_V) + y(t) = C*x(t) + v(t), v ~ N(0, R) +</pre> +Some simple variants are shown below. +<p> +<center> +<table> +<tr> +<td><img src="Figures/ar1.gif"> +<td><img src="Figures/sar.gif"> +<td><img src="Figures/kf.gif"> +<td><img src="Figures/skf.gif"> +</table> +</center> +<p> + +We can create a regular LDS in BNT as follows. +<pre> + +intra = zeros(2); +intra(1,2) = 1; +inter = zeros(2); +inter(1,1) = 1; +n = 2; + +X = 2; % size of hidden state +Y = 2; % size of observable state + +ns = [X Y]; +dnodes = []; +onodes = [2]; +eclass1 = [1 2]; +eclass2 = [3 2]; +bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes, 'eclass1', eclass1, 'eclass2', eclass2); + +x0 = rand(X,1); +V0 = eye(X); % must be positive semi definite! +C0 = rand(Y,X); +R0 = eye(Y); +A0 = rand(X,X); +Q0 = eye(X); + +bnet.CPD{1} = gaussian_CPD(bnet, 1, 'mean', x0, 'cov', V0, 'cov_prior_weight', 0); +bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(Y,1), 'cov', R0, 'weights', C0, ... + 'clamp_mean', 1, 'cov_prior_weight', 0); +bnet.CPD{3} = gaussian_CPD(bnet, 3, 'mean', zeros(X,1), 'cov', Q0, 'weights', A0, ... + 'clamp_mean', 1, 'cov_prior_weight', 0); +</pre> +We discuss how to do <a href="#inf">inference</a> and <a href="#learn">learning</a> on this model +below. +(See also +my <a href="../Kalman/kalman.html">Kalman filter toolbox</a>, which is included with BNT.) +<p> + + +<h2><a name="chmm">Coupled HMMs</h2> + +Here is an example of a coupled HMM with N=5 chains, unrolled for T=3 +slices. Each hidden discrete node has a private observed Gaussian +child. +<p> +<img src="Figures/chmm5.gif"> +<p> +We can make this using the function +<pre> +Q = 2; % binary hidden nodes +discrete_obs = 0; % cts observed nodes +Y = 1; % scalar observed nodes +bnet = mk_chmm(N, Q, Y, discrete_obs); +</pre> + +<!--We will use this model <a href="#pred">below</a> to illustrate online prediction.--> + + + +<h2><a name="water">Water network</h2> + +Consider the following model +of a water purification plant, developed +by Finn V. Jensen, Uffe Kjærulff, Kristian G. Olesen, and Jan +Pedersen. +<!-- +The clear nodes represent the hidden state of the system in +factored form, and the shaded nodes represent the observations in +factored form. +--> +<!-- +(Click <a +href="http://www-nt.cs.berkeley.edu/home/nir/public_html/Repository/water.htm">here</a> +for more details on this model. +Following Boyen and Koller, we have added discrete evidence nodes.) +--> +<!-- +We have "unrolled" the model for three "time slices" -- the structure and parameters are +assumed to repeat as the model is unrolled further. +Hence to specify a DBN, we need to +define the intra-slice topology (within a slice), +the inter-slice topology (between two slices), +as well as the parameters for the first two slices. +(Such a two-slice temporal Bayes net is often called a 2TBN.) +--> +<p> +<center> +<IMG SRC="Figures/water3_75.gif"> +</center> +We now show how to specify this model in BNT. +<pre> +ss = 12; % slice size +intra = zeros(ss); +intra(1,9) = 1; +intra(3,10) = 1; +intra(4,11) = 1; +intra(8,12) = 1; + +inter = zeros(ss); +inter(1, [1 3]) = 1; % node 1 in slice 1 connects to nodes 1 and 3 in slice 2 +inter(2, [2 3 7]) = 1; +inter(3, [3 4 5]) = 1; +inter(4, [3 4 6]) = 1; +inter(5, [3 5 6]) = 1; +inter(6, [4 5 6]) = 1; +inter(7, [7 8]) = 1; +inter(8, [6 7 8]) = 1; + +onodes = 9:12; % observed +dnodes = 1:ss; % discrete +ns = 2*ones(1,ss); % binary nodes +eclass1 = 1:12; +eclass2 = [13:20 9:12]; +eclass = [eclass1 eclass2]; +bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes, 'eclass1', eclass1, 'eclass2', eclass2); +for e=1:max(eclass) + bnet.CPD{e} = tabular_CPD(bnet, e); +end +</pre> +We have tied the observation parameters across all slices. +Click <a href="param_tieing.html">here</a> for a more complex example +of parameter tieing. + +<!-- +Let X(i,t) denote the i'th hidden node in slice t, +and Y(i,y) denote the i'th observed node in slice t. +We also use the notation Nj to refer to the j'th node in the +unrolled network, e.g., N25 = X(1,3), N33 = Y(1,3). +<p> +We assume the distributions P(X(i,t) | X(i,t-1)) and +P(Y(i,t) | X(i,t)) are independent of t for t > 1 and for all i. +Hence the CPD for N25, N37, ... is the same as for N13, so we say they +are in the same equivalence class, with N13 being the "representative" +for this class. In other words, we have tied the parameters for nodes +N13, N25, N37, ... +Note, however, that the parameters for the nodes in the first slice +are not tied, so each equivalence class for nodes 1..12 contains a +single node. +<p> +Above we assumed P(Y(i,t) | X(i,t)) is independent of t for t>1, but +it is conventional to assume this is true for all t. +So we would like to put N9, N21, N33, ... all in the same class, and +similarly for the other observed nodes. +We can do this by explicitely defining the equivalence classes, as +follows. +<p> +We define eclass1(i) to be the equivalence class that node i in slice +1 belongs to. +Similarly, we define eclass2(i) to be the equivalence class that node i in slice +2, 3, ..., belongs to. +For the water model, we have +<pre> +</pre> +This ties the observation model across slices, +since e.g., eclass(9) = eclass(21) = 9, so Y(1,1) and Y(1,2) belong to the +same class. +<p> +By default, +eclass1 = 1:ss, and eclass2 = (1:ss)+ss, where ss = slice size = the +number of nodes per slice. +This will tie nodes in slices 3, 4, ... to the the nodes in slice 2, +but none of the nodes in slice 2 to any in slice 1. +By using the above tieing pattern, +we now only have 20 CPDs to specify, instead of 24: +<pre> +bnet = mk_dbn(intra, inter, ns, dnodes, eclass1, eclass2); +for e=1:max(eclass) + bnet.CPD{e} = tabular_CPD(bnet, e); +end +</pre> +--> + + + +<h2><a name="bat">BATnet</h2> + +As an example of a more complicated DBN, consider the following +example, +which is a model of a car's high level state, as might be used by +an automated car. +(The model is from Forbes, Huang, Kanazawa and Russell, "The BATmobile: Towards a +Bayesian Automated Taxi", IJCAI 95. The figure is from +Boyen and Koller, "Tractable Inference for Complex Stochastic +Processes", UAI98. +For simplicity, we only show the observed nodes for slice 2.) +<p> +<center> +<IMG SRC="Figures/batnet.gif"> +</center> +<p> +Since this topology is so complicated, +it is useful to be able to refer to the nodes by name, instead of +number. +<pre> +names = {'LeftClr', 'RightClr', 'LatAct', ... 'Bclr', 'BYdotDiff'}; +ss = length(names); +</pre> +We can specify the intra-slice topology using a cell array as follows, +where each row specifies a connection between two named nodes: +<pre> +intrac = {... + 'LeftClr', 'LeftClrSens'; + 'RightClr', 'RightClrSens'; + ... + 'BYdotDiff', 'BcloseFast'}; +</pre> +Finally, we can convert this cell array to an adjacency matrix using +the following function: +<pre> +[intra, names] = mk_adj_mat(intrac, names, 1); +</pre> +This function also permutes the names so that they are in topological +order. +Given this ordering of the names, we can make the inter-slice +connectivity matrix as follows: +<pre> +interc = {... + 'LeftClr', 'LeftClr'; + 'LeftClr', 'LatAct'; + ... + 'FBStatus', 'LatAct'}; + +inter = mk_adj_mat(interc, names, 0); +</pre> + +To refer to a node, we must know its number, which can be computed as +in the following example: +<pre> +obs = {'LeftClrSens', 'RightClrSens', 'TurnSignalSens', 'XdotSens', 'YdotSens', 'FYdotDiffSens', ... + 'FclrSens', 'BXdotSens', 'BclrSens', 'BYdotDiffSens'}; +for i=1:length(obs) + onodes(i) = strmatch(obs{i}, names); +end +onodes = sort(onodes); +</pre> +(We sort the onodes since most BNT routines assume that set-valued +arguments are in sorted order.) +We can now make the DBN: +<pre> +dnodes = 1:ss; +ns = 2*ones(1,ss); % binary nodes +bnet = mk_dbn(intra, inter, ns, 'iscrete', dnodes); +</pre> +To specify the parameters, we must know the order of the parents. +See the function BNT/general/mk_named_CPT for a way to do this in the +case of tabular nodes. For simplicity, we just generate random +parameters: +<pre> +for i=1:2*ss + bnet.CPD{i} = tabular_CPD(bnet, i); +end +</pre> +A complete version of this example is available in BNT/examples/dynamic/bat1.m. + + + + +<h1><a name="inf">Inference</h1> + + +The general inference problem for DBNs is to compute +P(X(i,t0) | Y(:, t1:t2)), where X(i,t) represents the i'th hidden +variable at time t and Y(:,t1:t2) represents all the evidence +between times t1 and t2. +There are several special cases of interest, illustrated below. +The arrow indicates t0: it is X(t0) that we are trying to estimate. +The shaded region denotes t1:t2, the available data. +<p> + +<img src="Figures/filter.gif"> + +<p> +BNT can currently only handle offline smoothing. +(The HMM engine handles filtering and, to a limited extent, prediction.) +The usage is similar to static +inference engines, except now the evidence is a 2D cell array of +size ss*T, where ss is the number of nodes per slice (ss = slice sizee) and T is the +number of slices. +Also, 'marginal_nodes' takes two arguments, the nodes and the time-slice. +For example, to compute P(X(i,t) | y(:,1:T)), we proceed as follows +(where onodes are the indices of the observedd nodes in each slice, +which correspond to y): +<pre> +ev = sample_dbn(bnet, T); +evidence = cell(ss,T); +evidence(onodes,:) = ev(onodes, :); % all cells besides onodes are empty +[engine, ll] = enter_evidence(engine, evidence); +marg = marginal_nodes(engine, i, t); +</pre> + + +<h2><a name="discrete">Discrete hidden nodes</h2> + +If all the hidden nodes are discrete, +we can use the junction tree algorithm to perform inference. +The simplest approach, +<tt>jtree_unrolled_dbn_inf_engine</tt>, +unrolls the DBN into a static network and applies jtree; however, for +long sequences, this +can be very slow and can result in numerical underflow. +A better approach is to apply the jtree algorithm to pairs of +neighboring slices at a time; this is implemented in +<tt>jtree_dbn_inf_engine</tt>. + +<p> +A DBN can be converted to an HMM if all the hidden nodes are discrete. +In this case, you can use +<tt>hmm_inf_engine</tt>. This is faster than jtree for small models +because the constant factors of the algorithm are lower, but can be +exponentially slower for models with many variables +(e.g., > 6 binary hidden nodes). + +<p> +The use of both +<tt>jtree_dbn_inf_engine</tt> +and +<tt>hmm_inf_engine</tt> +is deprecated. +A better approach is to construct a smoother engine out of lower-level +engines, which implement forward/backward operators. +You can create these engines as follows. +<pre> +engine = smoother_engine(hmm_2TBN_inf_engine(bnet)); +or +engine = smoother_engine(jtree_2TBN_inf_engine(bnet)); +</pre> +You then call them in the usual way: +<pre> +engine = enter_evidence(engine, evidence); +m = marginal_nodes(engine, nodes, t); +</pre> +Note: you must declare the observed nodes in the bnet before using +hmm_2TBN_inf_engine. + + +<p> +Unfortunately, when all the hiddden nodes are discrete, +exact inference takes O(2^n) time, where n is the number of hidden +nodes per slice, +even if the model is sparse. +The basic reason for this is that two nodes become correlated, even if +there is no direct connection between them in the 2TBN, +by virtue of sharing common ancestors in the past. +Hence we need to use approximations. +<p> +A popular approximate inference algorithm for discrete DBNs, known as BK, is described in +<ul> +<li> +<A HREF="http://robotics.Stanford.EDU/~xb/uai98/index.html"> +Tractable inference for complex stochastic processes </A>, +Boyen and Koller, UAI 1998 +<li> +<A HREF="http://robotics.Stanford.EDU/~xb/nips98/index.html"> +Approximate learning of dynamic models</a>, Boyen and Koller, NIPS +1998. +</ul> +This approximates the belief state with a product of +marginals on a specified set of clusters. For example, +in the water network, we might use the following clusters: +<pre> +engine = bk_inf_engine(bnet, { [1 2], [3 4 5 6], [7 8] }); +</pre> +This engine can now be used just like the jtree engine. +Two special cases of the BK algorithm are supported: 'ff' (fully +factored) means each node has its own cluster, and 'exact' means there +is 1 cluster that contains the whole slice. These can be created as +follows: +<pre> +engine = bk_inf_engine(bnet, 'ff'); +engine = bk_inf_engine(bnet, 'exact'); +</pre> +For pedagogical purposes, an implementation of BK-FF that uses an HMM +instead of junction tree is available at +<tt>bk_ff_hmm_inf_engine</tt>. + + + +<h2><a name="cts">Continuous hidden nodes</h2> + +If all the hidden nodes are linear-Gaussian, <em>and</em> the observed nodes are +linear-Gaussian, +the model is a <a href="http://www.cs.berkeley.edu/~murphyk/Bayes/kalman.html"> +linear dynamical system</a> (LDS). +A DBN can be converted to an LDS if all the hidden nodes are linear-Gaussian +and if they are all persistent. In this case, you can use +<tt>kalman_inf_engine</tt>. +For more general linear-gaussian models, you can use +<tt>jtree_dbn_inf_engine</tt> or <tt>jtree_unrolled_dbn_inf_engine</tt>. + +<p> +For nonlinear systems with Gaussian noise, the unscented Kalman filter (UKF), +due to Julier and Uhlmann, is far superior to the well-known extended Kalman +filter (EKF), both in theory and practice. +<!-- +See +<A HREF="http://phoebe.robots.ox.ac.uk/default.html">"A General Method for +Approximating Nonlinear Transformations of +Probability Distributions"</A>. +(If the above link is down, +try <a href="http://www.ece.ogi.edu/~ericwan/pubs.html">Eric Wan's</a> +page, who has done a lot of work on the UKF.) +<p> +--> +The key idea of the UKF is that it is easier to estimate a Gaussian distribution +from a set of points than to approximate an arbitrary non-linear +function. +We start with points that are plus/minus sigma away from the mean along +each dimension, and then pipe them through the nonlinearity, and +then fit a Gaussian to the transformed points. +(No need to compute Jacobians, unlike the EKF!) + +<p> +For systems with non-Gaussian noise, I recommend +<a href="http://www.cs.berkeley.edu/~jfgf/smc/">Particle +filtering</a> (PF), which is a popular sequential Monte Carlo technique. + +<p> +The EKF can be used as a proposal distribution for a PF. +This method is better than either one alone. +See <a href="http://www.cs.berkeley.edu/~jfgf/upf.ps.gz">The Unscented Particle Filter</a>, +by R van der Merwe, A Doucet, JFG de Freitas and E Wan, May 2000. +<a href="http://www.cs.berkeley.edu/~jfgf/software.html">Matlab +software</a> for the UPF is also available. +<p> +Note: none of this software is part of BNT. + + + +<h1><a name="learn">Learning</h1> + +Learning in DBNs can be done online or offline. +Currently only offline learning is implemented in BNT. + + +<h2><a name="param_learn">Parameter learning</h2> + +Offline parameter learning is very similar to learning in static networks, +except now the training data is a cell-array of 2D cell-arrays. +For example, +cases{l}{i,t} is the value of node i in slice t in sequence l, or [] +if unobserved. +Each sequence can be a different length, and may have missing values +in arbitrary locations. +Here is a typical code fragment for using EM. +<pre> +ncases = 2; +cases = cell(1, ncases); +for i=1:ncases + ev = sample_dbn(bnet, T); + cases{i} = cell(ss,T); + cases{i}(onodes,:) = ev(onodes, :); +end +[bnet2, LLtrace] = learn_params_dbn_em(engine, cases, 'max_iter', 10); +</pre> +If the observed node is vector-valued and stored in an OxT array, you +need to assign each vector to a single cell, as in the following +example. +<pre> +data = [xpos(:)'; ypos(:)']; +ncases = 1; +cases = cell(1, ncases); +onodes = bnet.observed; +for i=1:ncases + cases{i} = cell(ss,T); + cases{i}(onodes,:) = num2cell(data(:,1:T), 1); +end +</pre> +<p> +For a complete code listing of how to do EM in a simple DBN, click +<a href="dbn_hmm_demo.m">here</a>. + +<h2><a name="struct_learn">Structure learning</h2> + +There is currently only one structure learning algorithm for DBNs. +This assumes all nodes are tabular and observed, and that there are +no intra-slice connections. Hence we can find the optimal set of +parents for each node separately, without worrying about directed +cycles or node orderings. +The function is called as follows +<pre> +inter = learn_struct_dbn_reveal(cases, ns, max_fan_in, penalty) +</pre> +A full example is given in BNT/examples/dynamic/reveal1.m. +Setting the penalty term to 0 gives the maximum likelihood model; this +is equivalent to maximizing the mutual information between parents and +child (in the bioinformatics community, this is known as the REVEAL +algorithm). A non-zero penalty invokes the BIC criterion, which +lessens the chance of overfitting. +<p> +<a href="http://www.bioss.sari.ac.uk/~dirk/software/DBmcmc/"> +Dirk Husmeier has extended MCMC model selection to DBNs</a>. + +</BODY> diff --git a/sourcecodes/bnt-master/docs/usage_dbn_02nov13.html b/sourcecodes/bnt-master/docs/usage_dbn_02nov13.html new file mode 100644 index 00000000..a1cd8f3a --- /dev/null +++ b/sourcecodes/bnt-master/docs/usage_dbn_02nov13.html @@ -0,0 +1,715 @@ +<HEAD> +<TITLE>How to use BNT for DBNs</TITLE> +</HEAD> + +<BODY BGCOLOR="#FFFFFF"> +<!-- white background is better for the pictures and equations --> + +Documentation last updated on 13 November 2002 + +<h1>How to use BNT for DBNs</h1> + +<p> +<ul> +<li> <a href="#spec">Model specification</a> +<ul> +<li> <a href="#hmm">HMMs</a> +<li> <a href="#lds">Kalman filters</a> +<li> <a href="#chmm">Coupled HMMs</a> +<li> <a href="#water">Water network</a> +<li> <a href="#bat">BAT network</a> +</ul> + +<li> <a href="#inf">Inference</a> +<ul> +<li> <a href="#discrete">Discrete hidden nodes</a> +<li> <a href="#cts">Continuous hidden nodes</a> +</ul> + +<li> <a href="#learn">Learning</a> +<ul> +<li> <a href="#param_learn">Parameter learning</a> +<li> <a href="#struct_learn">Structure learning</a> +</ul> + +</ul> + +Note: +you are recommended to read an introduction +to DBNs first, such as +<a href="http://www.ai.mit.edu/~murphyk/Papers/dbnchapter.pdf"> +this book chapter</a>. + + +<h1><a name="spec">Model specification</h1> + + +<!--<h1><a name="dbn_intro">Dynamic Bayesian Networks (DBNs)</h1>--> + +Dynamic Bayesian Networks (DBNs) are directed graphical models of stochastic +processes. +They generalise <a href="#hmm">hidden Markov models (HMMs)</a> +and <a href="#lds">linear dynamical systems (LDSs)</a> +by representing the hidden (and observed) state in terms of state +variables, which can have complex interdependencies. +The graphical structure provides an easy way to specify these +conditional independencies, and hence to provide a compact +parameterization of the model. +<p> +Note that "temporal Bayesian network" would be a better name than +"dynamic Bayesian network", since +it is assumed that the model structure does not change, but +the term DBN has become entrenched. +We also normally assume that the parameters do not +change, i.e., the model is time-invariant. +However, we can always add extra +hidden nodes to represent the current "regime", thereby creating +mixtures of models to capture periodic non-stationarities. +<p> +There are some cases where the size of the state space can change over +time, e.g., tracking a variable, but unknown, number of objects. +In this case, we need to change the model structure over time. +BNT does not support this. +<!-- +, but see the following paper for a +discussion of some of the issues: +<ul> +<li> <a href="ftp://ftp.cs.monash.edu.au/pub/annn/smc.ps"> +Dynamic belief networks for discrete monitoring</a>, +A. E. Nicholson and J. M. Brady. +IEEE Systems, Man and Cybernetics, 24(11):1593-1610, 1994. +</ul> +--> + + +<h2><a name="hmm">Hidden Markov Models (HMMs)</h2> + +The simplest kind of DBN is a Hidden Markov Model (HMM), which has +one discrete hidden node and one discrete or continuous +observed node per slice. We illustrate this below. +As before, circles denote continuous nodes, squares denote +discrete nodes, clear means hidden, shaded means observed. +<!-- +(The observed nodes can be +discrete or continuous; the crucial thing about an HMM is that the +hidden nodes are discrete, so the system can model arbitrary dynamics +-- providing, of course, that the hidden state space is large enough.) +--> +<p> +<img src="Figures/hmm3.gif"> +<p> +We have "unrolled" the model for three "time slices" -- the structure and parameters are +assumed to repeat as the model is unrolled further. +Hence to specify a DBN, we need to +define the intra-slice topology (within a slice), +the inter-slice topology (between two slices), +as well as the parameters for the first two slices. +(Such a two-slice temporal Bayes net is often called a 2TBN.) +<p> +We can specify the topology as follows. +<PRE> +intra = zeros(2); +intra(1,2) = 1; % node 1 in slice t connects to node 2 in slice t + +inter = zeros(2); +inter(1,1) = 1; % node 1 in slice t-1 connects to node 1 in slice t +</pre> +We can specify the parameters as follows, +where for simplicity we assume the observed node is discrete. +<pre> +Q = 2; % num hidden states +O = 2; % num observable symbols + +ns = [Q O]; +dnodes = 1:2; +bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes); +for i=1:4 + bnet.CPD{i} = tabular_CPD(bnet, i); +end +</pre> +<p> +We assume the distributions P(X(t) | X(t-1)) and +P(Y(t) | X(t)) are independent of t for t > 1. +Hence the CPD for nodes 5, 7, ... is the same as for node 3, so we say they +are in the same equivalence class, with node 3 being the "representative" +for this class. In other words, we have tied the parameters for nodes +3, 5, 7, ... +Similarly, nodes 4, 6, 8, ... are tied. +Note, however, that (the parameters for) nodes 1 and 2 are not tied to +subsequent slices. +<p> +Above we assumed the observation model P(Y(t) | X(t)) is independent of t for t>1, but +it is conventional to assume this is true for all t. +So we would like to put nodes 2, 4, 6, ... all in the same class. +We can do this by explicitely defining the equivalence classes, as +follows (see <a href="usage.html#tying">here</a> for more details on +parameter tying). +<p> +We define eclass1(i) to be the equivalence class that node i in slice +1 belongs to. +Similarly, we define eclass2(i) to be the equivalence class that node i in slice +2, 3, ..., belongs to. +For an HMM, we have +<pre> +eclass1 = [1 2]; +eclass2 = [3 2]; +eclass = [eclass1 eclass2]; +</pre> +This ties the observation model across slices, +since e.g., eclass(4) = eclass(2) = 2. +<p> +By default, +eclass1 = 1:ss, and eclass2 = (1:ss)+ss, where ss = slice size = the +number of nodes per slice. +<!--This will tie nodes in slices 3, 4, ... to the the nodes in slice 2, +but none of the nodes in slice 2 to any in slice 1.--> +But by using the above tieing pattern, +we now only have 3 CPDs to specify, instead of 4: +<pre> +bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes, 'eclass1', eclass1, 'eclass2', eclass2); +prior0 = normalise(rand(Q,1)); +transmat0 = mk_stochastic(rand(Q,Q)); +obsmat0 = mk_stochastic(rand(Q,O)); +bnet.CPD{1} = tabular_CPD(bnet, 1, prior0); +bnet.CPD{2} = tabular_CPD(bnet, 2, obsmat0); +bnet.CPD{3} = tabular_CPD(bnet, 3, transmat0); +</pre> +We discuss how to do <a href="#inf">inference</a> and <a href="#learn">learning</a> on this model +below. +(See also +my <a href="../HMM/hmm.html">HMM toolbox</a>, which is included with BNT.) + +<p> +Some common variants on HMMs are shown below. +BNT can handle all of these. +<p> +<center> +<table> +<tr> +<td><img src="Figures/hmm_gauss.gif"> +<td><img src="Figures/hmm_mixgauss.gif" +<td><img src="Figures/hmm_ar.gif"> +<tr> +<td><img src="Figures/hmm_factorial.gif"> +<td><img src="Figures/hmm_coupled.gif" +<td><img src="Figures/hmm_io.gif"> +<tr> +</table> +</center> + + + +<h2><a name="lds">Linear Dynamical Systems (LDSs) and Kalman filters</h2> + +A Linear Dynamical System (LDS) has the same topology as an HMM, but +all the nodes are assumed to have linear-Gaussian distributions, i.e., +<pre> + x(t+1) = A*x(t) + w(t), w ~ N(0, Q), x(0) ~ N(init_x, init_V) + y(t) = C*x(t) + v(t), v ~ N(0, R) +</pre> +Some simple variants are shown below. +<p> +<center> +<table> +<tr> +<td><img src="Figures/ar1.gif"> +<td><img src="Figures/sar.gif"> +<td><img src="Figures/kf.gif"> +<td><img src="Figures/skf.gif"> +</table> +</center> +<p> + +We can create a regular LDS in BNT as follows. +<pre> + +intra = zeros(2); +intra(1,2) = 1; +inter = zeros(2); +inter(1,1) = 1; +n = 2; + +X = 2; % size of hidden state +Y = 2; % size of observable state + +ns = [X Y]; +dnodes = []; +onodes = [2]; +eclass1 = [1 2]; +eclass2 = [3 2]; +bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes, 'eclass1', eclass1, 'eclass2', eclass2); + +x0 = rand(X,1); +V0 = eye(X); % must be positive semi definite! +C0 = rand(Y,X); +R0 = eye(Y); +A0 = rand(X,X); +Q0 = eye(X); + +bnet.CPD{1} = gaussian_CPD(bnet, 1, 'mean', x0, 'cov', V0, 'cov_prior_weight', 0); +bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(Y,1), 'cov', R0, 'weights', C0, ... + 'clamp_mean', 1, 'cov_prior_weight', 0); +bnet.CPD{3} = gaussian_CPD(bnet, 3, 'mean', zeros(X,1), 'cov', Q0, 'weights', A0, ... + 'clamp_mean', 1, 'cov_prior_weight', 0); +</pre> +We discuss how to do <a href="#inf">inference</a> and <a href="#learn">learning</a> on this model +below. +(See also +my <a href="../Kalman/kalman.html">Kalman filter toolbox</a>, which is included with BNT.) +<p> + + +<h2><a name="chmm">Coupled HMMs</h2> + +Here is an example of a coupled HMM with N=5 chains, unrolled for T=3 +slices. Each hidden discrete node has a private observed Gaussian +child. +<p> +<img src="Figures/chmm5.gif"> +<p> +We can make this using the function +<pre> +Q = 2; % binary hidden nodes +discrete_obs = 0; % cts observed nodes +Y = 1; % scalar observed nodes +bnet = mk_chmm(N, Q, Y, discrete_obs); +</pre> + +<!--We will use this model <a href="#pred">below</a> to illustrate online prediction.--> + + + +<h2><a name="water">Water network</h2> + +Consider the following model +of a water purification plant, developed +by Finn V. Jensen, Uffe Kjærulff, Kristian G. Olesen, and Jan +Pedersen. +<!-- +The clear nodes represent the hidden state of the system in +factored form, and the shaded nodes represent the observations in +factored form. +--> +<!-- +(Click <a +href="http://www-nt.cs.berkeley.edu/home/nir/public_html/Repository/water.htm">here</a> +for more details on this model. +Following Boyen and Koller, we have added discrete evidence nodes.) +--> +<!-- +We have "unrolled" the model for three "time slices" -- the structure and parameters are +assumed to repeat as the model is unrolled further. +Hence to specify a DBN, we need to +define the intra-slice topology (within a slice), +the inter-slice topology (between two slices), +as well as the parameters for the first two slices. +(Such a two-slice temporal Bayes net is often called a 2TBN.) +--> +<p> +<center> +<IMG SRC="Figures/water3_75.gif"> +</center> +We now show how to specify this model in BNT. +<pre> +ss = 12; % slice size +intra = zeros(ss); +intra(1,9) = 1; +intra(3,10) = 1; +intra(4,11) = 1; +intra(8,12) = 1; + +inter = zeros(ss); +inter(1, [1 3]) = 1; % node 1 in slice 1 connects to nodes 1 and 3 in slice 2 +inter(2, [2 3 7]) = 1; +inter(3, [3 4 5]) = 1; +inter(4, [3 4 6]) = 1; +inter(5, [3 5 6]) = 1; +inter(6, [4 5 6]) = 1; +inter(7, [7 8]) = 1; +inter(8, [6 7 8]) = 1; + +onodes = 9:12; % observed +dnodes = 1:ss; % discrete +ns = 2*ones(1,ss); % binary nodes +eclass1 = 1:12; +eclass2 = [13:20 9:12]; +eclass = [eclass1 eclass2]; +bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes, 'eclass1', eclass1, 'eclass2', eclass2); +for e=1:max(eclass) + bnet.CPD{e} = tabular_CPD(bnet, e); +end +</pre> +We have tied the observation parameters across all slices. +Click <a href="param_tieing.html">here</a> for a more complex example +of parameter tieing. + +<!-- +Let X(i,t) denote the i'th hidden node in slice t, +and Y(i,y) denote the i'th observed node in slice t. +We also use the notation Nj to refer to the j'th node in the +unrolled network, e.g., N25 = X(1,3), N33 = Y(1,3). +<p> +We assume the distributions P(X(i,t) | X(i,t-1)) and +P(Y(i,t) | X(i,t)) are independent of t for t > 1 and for all i. +Hence the CPD for N25, N37, ... is the same as for N13, so we say they +are in the same equivalence class, with N13 being the "representative" +for this class. In other words, we have tied the parameters for nodes +N13, N25, N37, ... +Note, however, that the parameters for the nodes in the first slice +are not tied, so each equivalence class for nodes 1..12 contains a +single node. +<p> +Above we assumed P(Y(i,t) | X(i,t)) is independent of t for t>1, but +it is conventional to assume this is true for all t. +So we would like to put N9, N21, N33, ... all in the same class, and +similarly for the other observed nodes. +We can do this by explicitely defining the equivalence classes, as +follows. +<p> +We define eclass1(i) to be the equivalence class that node i in slice +1 belongs to. +Similarly, we define eclass2(i) to be the equivalence class that node i in slice +2, 3, ..., belongs to. +For the water model, we have +<pre> +</pre> +This ties the observation model across slices, +since e.g., eclass(9) = eclass(21) = 9, so Y(1,1) and Y(1,2) belong to the +same class. +<p> +By default, +eclass1 = 1:ss, and eclass2 = (1:ss)+ss, where ss = slice size = the +number of nodes per slice. +This will tie nodes in slices 3, 4, ... to the the nodes in slice 2, +but none of the nodes in slice 2 to any in slice 1. +By using the above tieing pattern, +we now only have 20 CPDs to specify, instead of 24: +<pre> +bnet = mk_dbn(intra, inter, ns, dnodes, eclass1, eclass2); +for e=1:max(eclass) + bnet.CPD{e} = tabular_CPD(bnet, e); +end +</pre> +--> + + + +<h2><a name="bat">BATnet</h2> + +As an example of a more complicated DBN, consider the following +example, +which is a model of a car's high level state, as might be used by +an automated car. +(The model is from Forbes, Huang, Kanazawa and Russell, "The BATmobile: Towards a +Bayesian Automated Taxi", IJCAI 95. The figure is from +Boyen and Koller, "Tractable Inference for Complex Stochastic +Processes", UAI98. +For simplicity, we only show the observed nodes for slice 2.) +<p> +<center> +<IMG SRC="Figures/batnet.gif"> +</center> +<p> +Since this topology is so complicated, +it is useful to be able to refer to the nodes by name, instead of +number. +<pre> +names = {'LeftClr', 'RightClr', 'LatAct', ... 'Bclr', 'BYdotDiff'}; +ss = length(names); +</pre> +We can specify the intra-slice topology using a cell array as follows, +where each row specifies a connection between two named nodes: +<pre> +intrac = {... + 'LeftClr', 'LeftClrSens'; + 'RightClr', 'RightClrSens'; + ... + 'BYdotDiff', 'BcloseFast'}; +</pre> +Finally, we can convert this cell array to an adjacency matrix using +the following function: +<pre> +[intra, names] = mk_adj_mat(intrac, names, 1); +</pre> +This function also permutes the names so that they are in topological +order. +Given this ordering of the names, we can make the inter-slice +connectivity matrix as follows: +<pre> +interc = {... + 'LeftClr', 'LeftClr'; + 'LeftClr', 'LatAct'; + ... + 'FBStatus', 'LatAct'}; + +inter = mk_adj_mat(interc, names, 0); +</pre> + +To refer to a node, we must know its number, which can be computed as +in the following example: +<pre> +obs = {'LeftClrSens', 'RightClrSens', 'TurnSignalSens', 'XdotSens', 'YdotSens', 'FYdotDiffSens', ... + 'FclrSens', 'BXdotSens', 'BclrSens', 'BYdotDiffSens'}; +for i=1:length(obs) + onodes(i) = stringmatch(obs{i}, names); +end +onodes = sort(onodes); +</pre> +(We sort the onodes since most BNT routines assume that set-valued +arguments are in sorted order.) +We can now make the DBN: +<pre> +dnodes = 1:ss; +ns = 2*ones(1,ss); % binary nodes +bnet = mk_dbn(intra, inter, ns, 'iscrete', dnodes); +</pre> +To specify the parameters, we must know the order of the parents. +See the function BNT/general/mk_named_CPT for a way to do this in the +case of tabular nodes. For simplicity, we just generate random +parameters: +<pre> +for i=1:2*ss + bnet.CPD{i} = tabular_CPD(bnet, i); +end +</pre> +A complete version of this example is available in BNT/examples/dynamic/bat1.m. + + + + +<h1><a name="inf">Inference</h1> + + +The general inference problem for DBNs is to compute +P(X(i,t0) | Y(:, t1:t2)), where X(i,t) represents the i'th hidden +variable at time t and Y(:,t1:t2) represents all the evidence +between times t1 and t2. +There are several special cases of interest, illustrated below. +The arrow indicates t0: it is X(t0) that we are trying to estimate. +The shaded region denotes t1:t2, the available data. +<p> + +<img src="Figures/filter.gif"> + +<p> +BNT can currently only handle offline smoothing. +(The HMM engine handles filtering and, to a limited extent, prediction.) +The usage is similar to static +inference engines, except now the evidence is a 2D cell array of +size ss*T, where ss is the number of nodes per slice (ss = slice sizee) and T is the +number of slices. +Also, 'marginal_nodes' takes two arguments, the nodes and the time-slice. +For example, to compute P(X(i,t) | y(:,1:T)), we proceed as follows +(where onodes are the indices of the observedd nodes in each slice, +which correspond to y): +<pre> +ev = sample_dbn(bnet, T); +evidence = cell(ss,T); +evidence(onodes,:) = ev(onodes, :); % all cells besides onodes are empty +[engine, ll] = enter_evidence(engine, evidence); +marg = marginal_nodes(engine, i, t); +</pre> + + +<h2><a name="discrete">Discrete hidden nodes</h2> + +If all the hidden nodes are discrete, +we can use the junction tree algorithm to perform inference. +The simplest approach, +<tt>jtree_unrolled_dbn_inf_engine</tt>, +unrolls the DBN into a static network and applies jtree; however, for +long sequences, this +can be very slow and can result in numerical underflow. +A better approach is to apply the jtree algorithm to pairs of +neighboring slices at a time; this is implemented in +<tt>jtree_dbn_inf_engine</tt>. + +<p> +A DBN can be converted to an HMM if all the hidden nodes are discrete. +In this case, you can use +<tt>hmm_inf_engine</tt>. This is faster than jtree for small models +because the constant factors of the algorithm are lower, but can be +exponentially slower for models with many variables +(e.g., > 6 binary hidden nodes). + +<p> +The use of both +<tt>jtree_dbn_inf_engine</tt> +and +<tt>hmm_inf_engine</tt> +is deprecated. +A better approach is to construct a smoother engine out of lower-level +engines, which implement forward/backward operators. +You can create these engines as follows. +<pre> +engine = smoother_engine(hmm_2TBN_inf_engine(bnet)); +or +engine = smoother_engine(jtree_2TBN_inf_engine(bnet)); +</pre> +You then call them in the usual way: +<pre> +engine = enter_evidence(engine, evidence); +m = marginal_nodes(engine, nodes, t); +</pre> +Note: you must declare the observed nodes in the bnet before using +hmm_2TBN_inf_engine. + + +<p> +Unfortunately, when all the hiddden nodes are discrete, +exact inference takes O(2^n) time, where n is the number of hidden +nodes per slice, +even if the model is sparse. +The basic reason for this is that two nodes become correlated, even if +there is no direct connection between them in the 2TBN, +by virtue of sharing common ancestors in the past. +Hence we need to use approximations. +<p> +A popular approximate inference algorithm for discrete DBNs, known as BK, is described in +<ul> +<li> +<A HREF="http://robotics.Stanford.EDU/~xb/uai98/index.html"> +Tractable inference for complex stochastic processes </A>, +Boyen and Koller, UAI 1998 +<li> +<A HREF="http://robotics.Stanford.EDU/~xb/nips98/index.html"> +Approximate learning of dynamic models</a>, Boyen and Koller, NIPS +1998. +</ul> +This approximates the belief state with a product of +marginals on a specified set of clusters. For example, +in the water network, we might use the following clusters: +<pre> +engine = bk_inf_engine(bnet, { [1 2], [3 4 5 6], [7 8] }); +</pre> +This engine can now be used just like the jtree engine. +Two special cases of the BK algorithm are supported: 'ff' (fully +factored) means each node has its own cluster, and 'exact' means there +is 1 cluster that contains the whole slice. These can be created as +follows: +<pre> +engine = bk_inf_engine(bnet, 'ff'); +engine = bk_inf_engine(bnet, 'exact'); +</pre> +For pedagogical purposes, an implementation of BK-FF that uses an HMM +instead of junction tree is available at +<tt>bk_ff_hmm_inf_engine</tt>. + + + +<h2><a name="cts">Continuous hidden nodes</h2> + +If all the hidden nodes are linear-Gaussian, <em>and</em> the observed nodes are +linear-Gaussian, +the model is a <a href="http://www.cs.berkeley.edu/~murphyk/Bayes/kalman.html"> +linear dynamical system</a> (LDS). +A DBN can be converted to an LDS if all the hidden nodes are linear-Gaussian +and if they are all persistent. In this case, you can use +<tt>kalman_inf_engine</tt>. +For more general linear-gaussian models, you can use +<tt>jtree_dbn_inf_engine</tt> or <tt>jtree_unrolled_dbn_inf_engine</tt>. + +<p> +For nonlinear systems with Gaussian noise, the unscented Kalman filter (UKF), +due to Julier and Uhlmann, is far superior to the well-known extended Kalman +filter (EKF), both in theory and practice. +<!-- +See +<A HREF="http://phoebe.robots.ox.ac.uk/default.html">"A General Method for +Approximating Nonlinear Transformations of +Probability Distributions"</A>. +(If the above link is down, +try <a href="http://www.ece.ogi.edu/~ericwan/pubs.html">Eric Wan's</a> +page, who has done a lot of work on the UKF.) +<p> +--> +The key idea of the UKF is that it is easier to estimate a Gaussian distribution +from a set of points than to approximate an arbitrary non-linear +function. +We start with points that are plus/minus sigma away from the mean along +each dimension, and then pipe them through the nonlinearity, and +then fit a Gaussian to the transformed points. +(No need to compute Jacobians, unlike the EKF!) + +<p> +For systems with non-Gaussian noise, I recommend +<a href="http://www.cs.berkeley.edu/~jfgf/smc/">Particle +filtering</a> (PF), which is a popular sequential Monte Carlo technique. + +<p> +The EKF can be used as a proposal distribution for a PF. +This method is better than either one alone. +See <a href="http://www.cs.berkeley.edu/~jfgf/upf.ps.gz">The Unscented Particle Filter</a>, +by R van der Merwe, A Doucet, JFG de Freitas and E Wan, May 2000. +<a href="http://www.cs.berkeley.edu/~jfgf/software.html">Matlab +software</a> for the UPF is also available. +<p> +Note: none of this software is part of BNT. + + + +<h1><a name="learn">Learning</h1> + +Learning in DBNs can be done online or offline. +Currently only offline learning is implemented in BNT. + + +<h2><a name="param_learn">Parameter learning</h2> + +Offline parameter learning is very similar to learning in static networks, +except now the training data is a cell-array of 2D cell-arrays. +For example, +cases{l}{i,t} is the value of node i in slice t in sequence l, or [] +if unobserved. +Each sequence can be a different length, and may have missing values +in arbitrary locations. +Here is a typical code fragment for using EM. +<pre> +ncases = 2; +cases = cell(1, ncases); +for i=1:ncases + ev = sample_dbn(bnet, T); + cases{i} = cell(ss,T); + cases{i}(onodes,:) = ev(onodes, :); +end +[bnet2, LLtrace] = learn_params_dbn_em(engine, cases, 'max_iter', 10); +</pre> +If the observed node is vector-valued and stored in an OxT array, you +need to assign each vector to a single cell, as in the following +example. +<pre> +data = [xpos(:)'; ypos(:)']; +ncases = 1; +cases = cell(1, ncases); +onodes = bnet.observed; +for i=1:ncases + cases{i} = cell(ss,T); + cases{i}(onodes,:) = num2cell(data(:,1:T), 1); +end +</pre> +<p> +For a complete code listing of how to do EM in a simple DBN, click +<a href="dbn_hmm_demo.m">here</a>. + +<h2><a name="struct_learn">Structure learning</h2> + +There is currently only one structure learning algorithm for DBNs. +This assumes all nodes are tabular and observed, and that there are +no intra-slice connections. Hence we can find the optimal set of +parents for each node separately, without worrying about directed +cycles or node orderings. +The function is called as follows +<pre> +inter = learn_struct_dbn_reveal(cases, ns, max_fan_in, penalty) +</pre> +A full example is given in BNT/examples/dynamic/reveal1.m. +Setting the penalty term to 0 gives the maximum likelihood model; this +is equivalent to maximizing the mutual information between parents and +child (in the bioinformatics community, this is known as the REVEAL +algorithm). A non-zero penalty invokes the BIC criterion, which +lessens the chance of overfitting. +<p> +<a href="http://www.bioss.sari.ac.uk/~dirk/software/DBmcmc/"> +Dirk Husmeier has extended MCMC model selection to DBNs</a>. + +</BODY> diff --git a/sourcecodes/bnt-master/docs/usage_sf.html b/sourcecodes/bnt-master/docs/usage_sf.html new file mode 100644 index 00000000..7fc3d2c5 --- /dev/null +++ b/sourcecodes/bnt-master/docs/usage_sf.html @@ -0,0 +1,3242 @@ +<HEAD> +<TITLE>How to use the Bayes Net Toolbox</TITLE> +</HEAD> + +<BODY BGCOLOR="#FFFFFF"> +<!-- white background is better for the pictures and equations --> + +<h1>How to use the Bayes Net Toolbox</h1> + +This documentation was last updated on 7 June 2004. +<br> +Click <a href="changelog.html">here</a> for a list of changes made to +BNT. +<br> +Click +<a href="http://bnt.insa-rouen.fr/">here</a> +for a French version of this documentation (which might not +be up-to-date). +<br> +Update 23 May 2005: +Philippe LeRay has written +a +<a href="http://banquiseasi.insa-rouen.fr/projects/bnt-editor/"> +BNT GUI</a> +and +<a href="http://banquiseasi.insa-rouen.fr/projects/bnt-slp/"> +BNT Structure Learning Package</a>. + +<p> + +<ul> +<li> <a href="#install">Installation</a> +<ul> +<li> <a href="#install">Installing the Matlab code</a> +<li> <a href="#installC">Installing the C code</a> +<li> <a href="../matlab_tips.html">Useful Matlab tips</a>. +</ul> + +<li> <a href="#basics">Creating your first Bayes net</a> + <ul> + <li> <a href="#basics">Creating a model by hand</a> + <li> <a href="#file">Loading a model from a file</a> + <li> <a href="http://bnt.insa-rouen.fr/ajouts.html">Creating a model using a GUI</a> + </ul> + +<li> <a href="#inference">Inference</a> + <ul> + <li> <a href="#marginal">Computing marginal distributions</a> + <li> <a href="#joint">Computing joint distributions</a> + <li> <a href="#soft">Soft/virtual evidence</a> + <li> <a href="#mpe">Most probable explanation</a> + </ul> + +<li> <a href="#cpd">Conditional Probability Distributions</a> + <ul> + <li> <a href="#tabular">Tabular (multinomial) nodes</a> + <li> <a href="#noisyor">Noisy-or nodes</a> + <li> <a href="#deterministic">Other (noisy) deterministic nodes</a> + <li> <a href="#softmax">Softmax (multinomial logit) nodes</a> + <li> <a href="#mlp">Neural network nodes</a> + <li> <a href="#root">Root nodes</a> + <li> <a href="#gaussian">Gaussian nodes</a> + <li> <a href="#glm">Generalized linear model nodes</a> + <li> <a href="#dtree">Classification/regression tree nodes</a> + <li> <a href="#nongauss">Other continuous distributions</a> + <li> <a href="#cpd_summary">Summary of CPD types</a> + </ul> + +<li> <a href="#examples">Example models</a> + <ul> + <li> <a + href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt"> +Gaussian mixture models</a> + <li> <a href="#pca">PCA, ICA, and all that</a> + <li> <a href="#mixep">Mixtures of experts</a> + <li> <a href="#hme">Hierarchical mixtures of experts</a> + <li> <a href="#qmr">QMR</a> + <li> <a href="#cg_model">Conditional Gaussian models</a> + <li> <a href="#hybrid">Other hybrid models</a> + </ul> + +<li> <a href="#param_learning">Parameter learning</a> + <ul> + <li> <a href="#load_data">Loading data from a file</a> + <li> <a href="#mle_complete">Maximum likelihood parameter estimation from complete data</a> + <li> <a href="#prior">Parameter priors</a> + <li> <a href="#bayes_learn">(Sequential) Bayesian parameter updating from complete data</a> + <li> <a href="#em">Maximum likelihood parameter estimation with missing values (EM)</a> + <li> <a href="#tying">Parameter tying</a> + </ul> + +<li> <a href="#structure_learning">Structure learning</a> + <ul> + <li> <a href="#enumerate">Exhaustive search</a> + <li> <a href="#K2">K2</a> + <li> <a href="#hill_climb">Hill-climbing</a> + <li> <a href="#mcmc">MCMC</a> + <li> <a href="#active">Active learning</a> + <li> <a href="#struct_em">Structural EM</a> + <li> <a href="#graphdraw">Visualizing the learned graph structure</a> + <li> <a href="#constraint">Constraint-based methods</a> + </ul> + + +<li> <a href="#engines">Inference engines</a> + <ul> + <li> <a href="#jtree">Junction tree</a> + <li> <a href="#varelim">Variable elimination</a> + <li> <a href="#global">Global inference methods</a> + <li> <a href="#quickscore">Quickscore</a> + <li> <a href="#belprop">Belief propagation</a> + <li> <a href="#sampling">Sampling (Monte Carlo)</a> + <li> <a href="#engine_summary">Summary of inference engines</a> + </ul> + + +<li> <a href="#influence">Influence diagrams/ decision making</a> + + +<li> <a href="usage_dbn.html">DBNs, HMMs, Kalman filters and all that</a> +</ul> + +</ul> + + + + +<h1><a name="install">Installation</h1> + +<h2><a name="installM">Installing the Matlab code</h2> + +<ul> +<li> <a href="bnt_download.html">Download</a> the FullBNT.zip file. + +<p> +<li> Unpack the file. In Unix, type +<!--"tar xvf BNT.tar".--> +"unzip FullBNT.zip". +In Windows, use +a program like <a href="http://www.winzip.com">Winzip</a>. This will +create a directory called FullBNT, which contains BNT and other libraries. +(Files ending in ~ or # are emacs backup files, and can be ignored.) + +<p> +<li> Read the file <tt>BNT/README.txt</tt> to make sure the date +matches the one on the top of <a href=bnt.html>the BNT home page</a>. +If not, you may need to press 'refresh' on your browser, and download +again, to get the most recent version. + +<p> +<li> <b>Edit the file "FullBNT/BNT/add_BNT_to_path.m"</b> so it contains the correct +pathname. +For example, in Windows, +I download FullBNT.zip into C:\kmurphy\matlab, and +then ensure the second lines reads +<pre> +BNT_HOME = 'C:\kmurphy\matlab\FullBNT'; +</pre> + +<p> +<li> Start up Matlab. + +<p> +<li> Type "ver" at the Matlab prompt (">>"). +<b>You need Matlab version 5.2 or newer to run BNT</b>. +(Versions 5.0 and 5.1 have a memory leak which seems to sometimes +crash BNT.) +<b>BNT will not run on Octave</b>. + +<p> +<li> Move to the BNT directory. +For example, in Windows, I type +<pre> +>> cd C:\kpmurphy\matlab\FullBNT\BNT +</pre> + +<p> +<li> Type "add_BNT_to_path". +This executes the command +<tt>addpath(genpath(BNT_HOME))</tt>, +which adds all directories below FullBNT to the matlab path. + +<p> +<li> Type "test_BNT". + +If all goes well, this will produce a bunch of numbers and maybe some +warning messages (which you can ignore), but no error messages. +(The warnings should only be of the form +"Warning: Maximum number of iterations has been exceeded", and are +produced by Netlab.) + +<p> +<li> Problems? Did you remember to +<b>Edit the file "FullBNT/BNT/add_BNT_to_path.m"</b> so it contains +the right path?? + +<p> +<li> <a href="http://groups.yahoo.com/group/BayesNetToolbox/join"> +Join the BNT email list</a> + +<p> +<li>Read +<a href="../matlab_tips.html">some useful Matlab tips</a>. +<!-- +For instance, this explains how to create a startup file, which can be +used to set your path variable automatically, so you can avoid having +to type the above commands every time. +--> + +</ul> + + + +<h2><a name="installC">Installing the C code</h2> + +Some BNT functions also have C implementations. +<b>It is not necessary to install the C code</b>, but it can result in a speedup +of a factor of 2-5. +To install all the C code, +edit installC_BNT.m so it contains the right path, +then type <tt>installC_BNT</tt>. +(Ignore warnings of the form 'invalid white space character in directive'.) +To uninstall all the C code, +edit uninstallC_BNT.m so it contains the right path, +then type <tt>uninstallC_BNT</tt>. +For an up-to-date list of the files which have C implementations, see +BNT/installC_BNT.m. + +<p> +mex is a script that lets you call C code from Matlab - it does not compile matlab to +C (see mcc below). +If your C/C++ compiler is set up correctly, mex should work out of +the box. +If not, you might need to type +<p> +<tt> mex -setup</tt> +<p> +before calling installC. +<p> +To make mex call gcc on Windows, +you must install <a +href="http://www.mrc-cbu.cam.ac.uk/Imaging/gnumex20.html">gnumex</a>. +You can use the <a href="http://www.mingw.org/">minimalist GNU for +Windows</a> version of gcc, or +the <a href="http://sources.redhat.com/cygwin/">cygwin</a> version. +<p> +In general, typing +'mex foo.c' from inside Matlab creates a file called +'foo.mexglx' or 'foo.dll' (the exact file +extension is system dependent - on Linux it is 'mexglx', on Windows it is '.dll'). +The resulting file will hide the original 'foo.m' (if it existed), i.e., +typing 'foo' at the prompt will call the compiled C version. +To reveal the original matlab version, just delete foo.mexglx (this is +what uninstallC does). +<p> +Sometimes it takes time for Matlab to realize that the file has +changed from matlab to C or vice versa; try typing 'clear all' or +restarting Matlab to refresh it. +To find out which version of a file you are running, type +'which foo'. +<p> +<a href="http://www.mathworks.com/products/compiler">mcc</a>, the +Matlab to C compiler, is a separate product, +and is quite different from mex. It does not yet support +objects/classes, which is why we can't compile all of BNT to C automatically. +Also, hand-written C code is usually much +better than the C code generated by mcc. + + +<p> +Acknowledgements: +Most of the C code (e.g., for jtree and dpot) was written by Wei Hu; +the triangulation C code was written by Ilya Shpitser; +the Gibbs sampling C code (for discrete nodes) was written by Bhaskara +Marthi. + + + +<h1><a name="basics">Creating your first Bayes net</h1> + +To define a Bayes net, you must specify the graph structure and then +the parameters. We look at each in turn, using a simple example +(adapted from Russell and +Norvig, "Artificial Intelligence: a Modern Approach", Prentice Hall, +1995, p454). + + +<h2>Graph structure</h2> + + +Consider the following network. + +<p> +<center> +<IMG SRC="Figures/sprinkler.gif"> +</center> +<p> + +<P> +To specify this directed acyclic graph (dag), we create an adjacency matrix: +<PRE> +N = 4; +dag = zeros(N,N); +C = 1; S = 2; R = 3; W = 4; +dag(C,[R S]) = 1; +dag(R,W) = 1; +dag(S,W)=1; +</PRE> +<P> +We have numbered the nodes as follows: +Cloudy = 1, Sprinkler = 2, Rain = 3, WetGrass = 4. +<b>The nodes must always be numbered in topological order, i.e., +ancestors before descendants.</b> +For a more complicated graph, this is a little inconvenient: we will +see how to get around this <a href="usage_dbn.html#bat">below</a>. +<p> +In Matlab 6, you can use logical arrays instead of double arrays, +which are 4 times smaller: +<pre> +dag = false(N,N); +dag(C,[R S]) = true; +... +</pre> +However, <b>some graph functions (eg acyclic) do not work on +logical arrays</b>! +<p> +A preliminary attempt to make a <b>GUI</b> +has been writte by Philippe LeRay and can be downloaded +from <a href="http://bnt.insa-rouen.fr/ajouts.html">here</a>. +<p> +You can visualize the resulting graph structure using +the methods discussed <a href="#graphdraw">below</a>. + +<h2>Creating the Bayes net shell</h2> + +In addition to specifying the graph structure, +we must specify the size and type of each node. +If a node is discrete, its size is the +number of possible values +each node can take on; if a node is continuous, +it can be a vector, and its size is the length of this vector. +In this case, we will assume all nodes are discrete and binary. +<PRE> +discrete_nodes = 1:N; +node_sizes = 2*ones(1,N); +</pre> +If the nodes were not binary, you could type e.g., +<pre> +node_sizes = [4 2 3 5]; +</pre> +meaning that Cloudy has 4 possible values, +Sprinkler has 2 possible values, etc. +Note that these are cardinal values, not ordinal, i.e., +they are not ordered in any way, like 'low', 'medium', 'high'. +<p> +We are now ready to make the Bayes net: +<pre> +bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes); +</PRE> +By default, all nodes are assumed to be discrete, so we can also just +write +<pre> +bnet = mk_bnet(dag, node_sizes); +</PRE> +You may also specify which nodes will be observed. +If you don't know, or if this not fixed in advance, +just use the empty list (the default). +<pre> +onodes = []; +bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes, 'observed', onodes); +</PRE> +Note that optional arguments are specified using a name/value syntax. +This is common for many BNT functions. +In general, to find out more about a function (e.g., which optional +arguments it takes), please see its +documentation string by typing +<pre> +help mk_bnet +</pre> +See also other <a href="matlab_tips.html">useful Matlab tips</a>. +<p> +It is possible to associate names with nodes, as follows: +<pre> +bnet = mk_bnet(dag, node_sizes, 'names', {'cloudy','S','R','W'}, 'discrete', 1:4); +</pre> +You can then refer to a node by its name: +<pre> +C = bnet.names('cloudy'); % bnet.names is an associative array +bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]); +</pre> +This feature uses my own associative array class. + + +<h2><a name="cpt">Parameters</h2> + +A model consists of the graph structure and the parameters. +The parameters are represented by CPD objects (CPD = Conditional +Probability Distribution), which define the probability distribution +of a node given its parents. +(We will use the terms "node" and "random variable" interchangeably.) +The simplest kind of CPD is a table (multi-dimensional array), which +is suitable when all the nodes are discrete-valued. Note that the discrete +values are not assumed to be ordered in any way; that is, they +represent categorical quantities, like male and female, rather than +ordinal quantities, like low, medium and high. +(We will discuss CPDs in more detail <a href="#cpd">below</a>.) +<p> +Tabular CPDs, also called CPTs (conditional probability tables), +are stored as multidimensional arrays, where the dimensions +are arranged in the same order as the nodes, e.g., the CPT for node 4 +(WetGrass) is indexed by Sprinkler (2), Rain (3) and then WetGrass (4) itself. +Hence the child is always the last dimension. +If a node has no parents, its CPT is a column vector representing its +prior. +Note that in Matlab (unlike C), arrays are indexed +from 1, and are layed out in memory such that the first index toggles +fastest, e.g., the CPT for node 4 (WetGrass) is as follows +<P> +<P><IMG ALIGN=BOTTOM SRC="Figures/CPTgrass.gif"><P> +<P> +where we have used the convention that false==1, true==2. +We can create this CPT in Matlab as follows +<PRE> +CPT = zeros(2,2,2); +CPT(1,1,1) = 1.0; +CPT(2,1,1) = 0.1; +... +</PRE> +Here is an easier way: +<PRE> +CPT = reshape([1 0.1 0.1 0.01 0 0.9 0.9 0.99], [2 2 2]); +</PRE> +In fact, we don't need to reshape the array, since the CPD constructor +will do that for us. So we can just write +<pre> +bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [1 0.1 0.1 0.01 0 0.9 0.9 0.99]); +</pre> +The other nodes are created similarly (using the old syntax for +optional parameters) +<PRE> +bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]); +bnet.CPD{R} = tabular_CPD(bnet, R, [0.8 0.2 0.2 0.8]); +bnet.CPD{S} = tabular_CPD(bnet, S, [0.5 0.9 0.5 0.1]); +bnet.CPD{W} = tabular_CPD(bnet, W, [1 0.1 0.1 0.01 0 0.9 0.9 0.99]); +</PRE> + + +<h2><a name="rnd_cpt">Random Parameters</h2> + +If we do not specify the CPT, random parameters will be +created, i.e., each "row" of the CPT will be drawn from the uniform distribution. +To ensure repeatable results, use +<pre> +rand('state', seed); +randn('state', seed); +</pre> +To control the degree of randomness (entropy), +you can sample each row of the CPT from a Dirichlet(p,p,...) distribution. +If p << 1, this encourages "deterministic" CPTs (one entry near 1, the rest near 0). +If p = 1, each entry is drawn from U[0,1]. +If p >> 1, the entries will all be near 1/k, where k is the arity of +this node, i.e., each row will be nearly uniform. +You can do this as follows, assuming this node +is number i, and ns is the node_sizes. +<pre> +k = ns(i); +ps = parents(dag, i); +psz = prod(ns(ps)); +CPT = sample_dirichlet(p*ones(1,k), psz); +bnet.CPD{i} = tabular_CPD(bnet, i, 'CPT', CPT); +</pre> + + +<h2><a name="file">Loading a network from a file</h2> + +If you already have a Bayes net represented in the XML-based +<a href="http://www.cs.cmu.edu/afs/cs/user/fgcozman/www/Research/InterchangeFormat/"> +Bayes Net Interchange Format (BNIF)</a> (e.g., downloaded from the +<a +href="http://www.cs.huji.ac.il/labs/compbio/Repository"> +Bayes Net repository</a>), +you can convert it to BNT format using +the +<a href="http://www.digitas.harvard.edu/~ken/bif2bnt/">BIF-BNT Java +program</a> written by Ken Shan. +(This is not necessarily up-to-date.) +<p> +<b>It is currently not possible to save/load a BNT matlab object to +file</b>, but this is easily fixed if you modify all the constructors +for all the classes (see matlab documentation). + +<h2>Creating a model using a GUI</h2> + +Click <a href="http://bnt.insa-rouen.fr/ajouts.html">here</a>. + + + +<h1><a name="inference">Inference</h1> + +Having created the BN, we can now use it for inference. +There are many different algorithms for doing inference in Bayes nets, +that make different tradeoffs between speed, +complexity, generality, and accuracy. +BNT therefore offers a variety of different inference +"engines". We will discuss these +in more detail <a href="#engines">below</a>. +For now, we will use the junction tree +engine, which is the mother of all exact inference algorithms. +This can be created as follows. +<pre> +engine = jtree_inf_engine(bnet); +</pre> +The other engines have similar constructors, but might take +additional, algorithm-specific parameters. +All engines are used in the same way, once they have been created. +We illustrate this in the following sections. + + +<h2><a name="marginal">Computing marginal distributions</h2> + +Suppose we want to compute the probability that the sprinker was on +given that the grass is wet. +The evidence consists of the fact that W=2. All the other nodes +are hidden (unobserved). We can specify this as follows. +<pre> +evidence = cell(1,N); +evidence{W} = 2; +</pre> +We use a 1D cell array instead of a vector to +cope with the fact that nodes can be vectors of different lengths. +In addition, the value [] can be used +to denote 'no evidence', instead of having to specify the observation +pattern as a separate argument. +(Click <a href="cellarray.html">here</a> for a quick tutorial on cell +arrays in matlab.) +<p> +We are now ready to add the evidence to the engine. +<pre> +[engine, loglik] = enter_evidence(engine, evidence); +</pre> +The behavior of this function is algorithm-specific, and is discussed +in more detail <a href="#engines">below</a>. +In the case of the jtree engine, +enter_evidence implements a two-pass message-passing scheme. +The first return argument contains the modified engine, which +incorporates the evidence. The second return argument contains the +log-likelihood of the evidence. (Not all engines are capable of +computing the log-likelihood.) +<p> +Finally, we can compute p=P(S=2|W=2) as follows. +<PRE> +marg = marginal_nodes(engine, S); +marg.T +ans = + 0.57024 + 0.42976 +p = marg.T(2); +</PRE> +We see that p = 0.4298. +<p> +Now let us add the evidence that it was raining, and see what +difference it makes. +<PRE> +evidence{R} = 2; +[engine, loglik] = enter_evidence(engine, evidence); +marg = marginal_nodes(engine, S); +p = marg.T(2); +</PRE> +We find that p = P(S=2|W=2,R=2) = 0.1945, +which is lower than +before, because the rain can ``explain away'' the +fact that the grass is wet. +<p> +You can plot a marginal distribution over a discrete variable +as a barchart using the built 'bar' function: +<pre> +bar(marg.T) +</pre> +This is what it looks like + +<p> +<center> +<IMG SRC="Figures/sprinkler_bar.gif"> +</center> +<p> + +<h2><a name="observed">Observed nodes</h2> + +What happens if we ask for the marginal on an observed node, e.g. P(W|W=2)? +An observed discrete node effectively only has 1 value (the observed + one) --- all other values would result in 0 probability. +For efficiency, BNT treats observed (discrete) nodes as if they were + set to 1, as we see below: +<pre> +evidence = cell(1,N); +evidence{W} = 2; +engine = enter_evidence(engine, evidence); +m = marginal_nodes(engine, W); +m.T +ans = + 1 +</pre> +This can get a little confusing, since we assigned W=2. +So we can ask BNT to add the evidence back in by passing in an optional argument: +<pre> +m = marginal_nodes(engine, W, 1); +m.T +ans = + 0 + 1 +</pre> +This shows that P(W=1|W=2) = 0 and P(W=2|W=2) = 1. + + + +<h2><a name="joint">Computing joint distributions</h2> + +We can compute the joint probability on a set of nodes as in the +following example. +<pre> +evidence = cell(1,N); +[engine, ll] = enter_evidence(engine, evidence); +m = marginal_nodes(engine, [S R W]); +</pre> +m is a structure. The 'T' field is a multi-dimensional array (in +this case, 3-dimensional) that contains the joint probability +distribution on the specified nodes. +<pre> +>> m.T +ans(:,:,1) = + 0.2900 0.0410 + 0.0210 0.0009 +ans(:,:,2) = + 0 0.3690 + 0.1890 0.0891 +</pre> +We see that P(S=1,R=1,W=2) = 0, since it is impossible for the grass +to be wet if both the rain and sprinkler are off. +<p> +Let us now add some evidence to R. +<pre> +evidence{R} = 2; +[engine, ll] = enter_evidence(engine, evidence); +m = marginal_nodes(engine, [S R W]) +m = + domain: [2 3 4] + T: [2x1x2 double] +>> m.T +m.T +ans(:,:,1) = + 0.0820 + 0.0018 +ans(:,:,2) = + 0.7380 + 0.1782 +</pre> +The joint T(i,j,k) = P(S=i,R=j,W=k|evidence) +should have T(i,1,k) = 0 for all i,k, since R=1 is incompatible +with the evidence that R=2. +Instead of creating large tables with many 0s, BNT sets the effective +size of observed (discrete) nodes to 1, as explained above. +This is why m.T has size 2x1x2. +To get a 2x2x2 table, type +<pre> +m = marginal_nodes(engine, [S R W], 1) +m = + domain: [2 3 4] + T: [2x2x2 double] +>> m.T +m.T +ans(:,:,1) = + 0 0.082 + 0 0.0018 +ans(:,:,2) = + 0 0.738 + 0 0.1782 +</pre> + +<p> +Note: It is not always possible to compute the joint on arbitrary +sets of nodes: it depends on which inference engine you use, as discussed +in more detail <a href="#engines">below</a>. + + +<h2><a name="soft">Soft/virtual evidence</h2> + +Sometimes a node is not observed, but we have some distribution over +its possible values; this is often called "soft" or "virtual" +evidence. +One can use this as follows +<pre> +[engine, loglik] = enter_evidence(engine, evidence, 'soft', soft_evidence); +</pre> +where soft_evidence{i} is either [] (if node i has no soft evidence) +or is a vector representing the probability distribution over i's +possible values. +For example, if we don't know i's exact value, but we know its +likelihood ratio is 60/40, we can write evidence{i} = [] and +soft_evidence{i} = [0.6 0.4]. +<p> +Currently only jtree_inf_engine supports this option. +It assumes that all hidden nodes, and all nodes for +which we have soft evidence, are discrete. +For a longer example, see BNT/examples/static/softev1.m. + + +<h2><a name="mpe">Most probable explanation</h2> + +To compute the most probable explanation (MPE) of the evidence (i.e., +the most probable assignment, or a mode of the joint), use +<pre> +[mpe, ll] = calc_mpe(engine, evidence); +</pre> +mpe{i} is the most likely value of node i. +This calls enter_evidence with the 'maximize' flag set to 1, which +causes the engine to do max-product instead of sum-product. +The resulting max-marginals are then thresholded. +If there is more than one maximum probability assignment, we must take + care to break ties in a consistent manner (thresholding the + max-marginals may give the wrong result). To force this behavior, + type +<pre> +[mpe, ll] = calc_mpe(engine, evidence, 1); +</pre> +Note that computing the MPE is someties called abductive reasoning. + +<p> +You can also use <tt>calc_mpe_bucket</tt> written by Ron Zohar, +that does a forwards max-product pass, and then a backwards traceback +pass, which is how Viterbi is traditionally implemented. + + + +<h1><a name="cpd">Conditional Probability Distributions</h1> + +A Conditional Probability Distributions (CPD) +defines P(X(i) | X(Pa(i))), where X(i) is the i'th node, and X(Pa(i)) +are the parents of node i. There are many ways to represent this +distribution, which depend in part on whether X(i) and X(Pa(i)) are +discrete, continuous, or a combination. +We will discuss various representations below. + + +<h2><a name="tabular">Tabular nodes</h2> + +If the CPD is represented as a table (i.e., if it is a multinomial +distribution), it has a number of parameters that is exponential in +the number of parents. See the example <a href="#cpt">above</a>. + + +<h2><a name="noisyor">Noisy-or nodes</h2> + +A noisy-OR node is like a regular logical OR gate except that +sometimes the effects of parents that are on get inhibited. +Let the prob. that parent i gets inhibited be q(i). +Then a node, C, with 2 parents, A and B, has the following CPD, where +we use F and T to represent off and on (1 and 2 in BNT). +<pre> +A B P(C=off) P(C=on) +--------------------------- +F F 1.0 0.0 +T F q(A) 1-q(A) +F T q(B) 1-q(B) +T T q(A)q(B) q-q(A)q(B) +</pre> +Thus we see that the causes get inhibited independently. +It is common to associate a "leak" node with a noisy-or CPD, which is +like a parent that is always on. This can account for all other unmodelled +causes which might turn the node on. +<p> +The noisy-or distribution is similar to the logistic distribution. +To see this, let the nodes, S(i), have values in {0,1}, and let q(i,j) +be the prob. that j inhibits i. Then +<pre> +Pr(S(i)=1 | parents(S(i))) = 1 - prod_{j} q(i,j)^S(j) +</pre> +Now define w(i,j) = -ln q(i,j) and rho(x) = 1-exp(-x). Then +<pre> +Pr(S(i)=1 | parents(S(i))) = rho(sum_j w(i,j) S(j)) +</pre> +For a sigmoid node, we have +<pre> +Pr(S(i)=1 | parents(S(i))) = sigma(-sum_j w(i,j) S(j)) +</pre> +where sigma(x) = 1/(1+exp(-x)). Hence they differ in the choice of +the activation function (although both are monotonically increasing). +In addition, in the case of a noisy-or, the weights are constrained to be +positive, since they derive from probabilities q(i,j). +In both cases, the number of parameters is <em>linear</em> in the +number of parents, unlike the case of a multinomial distribution, +where the number of parameters is exponential in the number of parents. +We will see an example of noisy-OR nodes <a href="#qmr">below</a>. + + +<h2><a name="deterministic">Other (noisy) deterministic nodes</h2> + +Deterministic CPDs for discrete random variables can be created using +the deterministic_CPD class. It is also possible to 'flip' the output +of the function with some probability, to simulate noise. +The boolean_CPD class is just a special case of a +deterministic CPD, where the parents and child are all binary. +<p> +Both of these classes are just "syntactic sugar" for the tabular_CPD +class. + + + +<h2><a name="softmax">Softmax nodes</h2> + +If we have a discrete node with a continuous parent, +we can define its CPD using a softmax function +(also known as the multinomial logit function). +This acts like a soft thresholding operator, and is defined as follows: +<pre> + exp(w(:,i)'*x + b(i)) +Pr(Q=i | X=x) = ----------------------------- + sum_j exp(w(:,j)'*x + b(j)) + +</pre> +The parameters of a softmax node, w(:,i) and b(i), i=1..|Q|, have the +following interpretation: w(:,i)-w(:,j) is the normal vector to the +decision boundary between classes i and j, +and b(i)-b(j) is its offset (bias). For example, suppose +X is a 2-vector, and Q is binary. Then +<pre> +w = [1 -1; + 0 0]; + +b = [0 0]; +</pre> +means class 1 are points in the 2D plane with positive x coordinate, +and class 2 are points in the 2D plane with negative x coordinate. +If w has large magnitude, the decision boundary is sharp, otherwise it +is soft. +In the special case that Q is binary (0/1), the softmax function reduces to the logistic +(sigmoid) function. +<p> +Fitting a softmax function can be done using the iteratively reweighted +least squares (IRLS) algorithm. +We use the implementation from +<a href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>. +Note that since +the softmax distribution is not in the exponential family, it does not +have finite sufficient statistics, and hence we must store all the +training data in uncompressed form. +If this takes too much space, one should use online (stochastic) gradient +descent (not implemented in BNT). +<p> +If a softmax node also has discrete parents, +we use a different set of w/b parameters for each combination of +parent values, as in the <a href="#gaussian">conditional linear +Gaussian CPD</a>. +This feature was implemented by Pierpaolo Brutti. +He is currently extending it so that discrete parents can be treated +as if they were continuous, by adding indicator variables to the X +vector. +<p> +We will see an example of softmax nodes <a href="#mixexp">below</a>. + + +<h2><a name="mlp">Neural network nodes</h2> + +Pierpaolo Brutti has implemented the mlp_CPD class, which uses a multi layer perceptron +to implement a mapping from continuous parents to discrete children, +similar to the softmax function. +(If there are also discrete parents, it creates a mixture of MLPs.) +It uses code from <a +href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>. +This is work in progress. + +<h2><a name="root">Root nodes</h2> + +A root node has no parents and no parameters; it can be used to model +an observed, exogeneous input variable, i.e., one which is "outside" +the model. +This is useful for conditional density models. +We will see an example of root nodes <a href="#mixexp">below</a>. + + +<h2><a name="gaussian">Gaussian nodes</h2> + +We now consider a distribution suitable for the continuous-valued nodes. +Suppose the node is called Y, its continuous parents (if any) are +called X, and its discrete parents (if any) are called Q. +The distribution on Y is defined as follows: +<pre> +- no parents: Y ~ N(mu, Sigma) +- cts parents : Y|X=x ~ N(mu + W x, Sigma) +- discrete parents: Y|Q=i ~ N(mu(:,i), Sigma(:,:,i)) +- cts and discrete parents: Y|X=x,Q=i ~ N(mu(:,i) + W(:,:,i) * x, Sigma(:,:,i)) +</pre> +where N(mu, Sigma) denotes a Normal distribution with mean mu and +covariance Sigma. Let |X|, |Y| and |Q| denote the sizes of X, Y and Q +respectively. +If there are no discrete parents, |Q|=1; if there is +more than one, then |Q| = a vector of the sizes of each discrete parent. +If there are no continuous parents, |X|=0; if there is more than one, +then |X| = the sum of their sizes. +Then mu is a |Y|*|Q| vector, Sigma is a |Y|*|Y|*|Q| positive +semi-definite matrix, and W is a |Y|*|X|*|Q| regression (weight) +matrix. +<p> +We can create a Gaussian node with random parameters as follows. +<pre> +bnet.CPD{i} = gaussian_CPD(bnet, i); +</pre> +We can specify the value of one or more of the parameters as in the +following example, in which |Y|=2, and |Q|=1. +<pre> +bnet.CPD{i} = gaussian_CPD(bnet, i, 'mean', [0; 0], 'weights', randn(Y,X), 'cov', eye(Y)); +</pre> +<p> +We will see an example of conditional linear Gaussian nodes <a +href="#cg_model">below</a>. +<p> +<b>When learning Gaussians from data</b>, it is helpful to ensure the +data has a small magnitde +(see e.g., KPMstats/standardize) to prevent numerical problems. +Unless you have a lot of data, it is also a very good idea to use +diagonal instead of full covariance matrices. +(BNT does not currently support spherical covariances, although it +would be easy to add, since KPMstats/clg_Mstep supports this option; +you would just need to modify gaussian_CPD/update_ess to accumulate +weighted inner products.) + + + +<h2><a name="nongauss">Other continuous distributions</h2> + +Currently BNT does not support any CPDs for continuous nodes other +than the Gaussian. +However, you can use a mixture of Gaussians to +approximate other continuous distributions. We will see some an example +of this with the IFA model <a href="#pca">below</a>. + + +<h2><a name="glm">Generalized linear model nodes</h2> + +In the future, we may incorporate some of the functionality of +<a href = +"http://www.sci.usq.edu.au/staff/dunn/glmlab/glmlab.html">glmlab</a> +into BNT. + + +<h2><a name="dtree">Classification/regression tree nodes</h2> + +We plan to add classification and regression trees to define CPDs for +discrete and continuous nodes, respectively. +Trees have many advantages: they are easy to interpret, they can do +feature selection, they can +handle discrete and continuous inputs, they do not make strong +assumptions about the form of the distribution, the number of +parameters can grow in a data-dependent way (i.e., they are +semi-parametric), they can handle missing data, etc. +However, they are not yet implemented. +<!-- +Yimin Zhang is currently (Feb '02) implementing this. +--> + + +<h2><a name="cpd_summary">Summary of CPD types</h2> + +We list all the different types of CPDs supported by BNT. +For each CPD, we specify if the child and parents can be discrete (D) or +continuous (C) (Binary (B) nodes are a special case). +We also specify which methods each class supports. +If a method is inherited, the name of the parent class is mentioned. +If a parent class calls a child method, this is mentioned. +<p> +The <tt>CPD_to_CPT</tt> method converts a CPD to a table; this +requires that the child and all parents are discrete. +The CPT might be exponentially big... +<tt>convert_to_table</tt> evaluates a CPD with evidence, and +represents the the resulting potential as an array. +This requires that the child is discrete, and any continuous parents +are observed. +<tt>convert_to_pot</tt> evaluates a CPD with evidence, and +represents the resulting potential as a dpot, gpot, cgpot or upot, as +requested. (d=discrete, g=Gaussian, cg = conditional Gaussian, u = +utility). + +<p> +When we sample a node, all the parents are observed. +When we compute the (log) probability of a node, all the parents and +the child are observed. +<p> +We also specify if the parameters are learnable. +For learning with EM, we require +the methods <tt>reset_ess</tt>, <tt>update_ess</tt> and +<tt>maximize_params</tt>. +For learning from fully observed data, we require +the method <tt>learn_params</tt>. +By default, all classes inherit this from generic_CPD, which simply +calls <tt>update_ess</tt> N times, once for each data case, followed +by <tt>maximize_params</tt>, i.e., it is like EM, without the E step. +Some classes implement a batch formula, which is quicker. +<p> +Bayesian learning means computing a posterior over the parameters +given fully observed data. +<p> +Pearl means we implement the methods <tt>compute_pi</tt> and +<tt>compute_lambda_msg</tt>, used by +<tt>pearl_inf_engine</tt>, which runs on directed graphs. +<tt>belprop_inf_engine</tt> only needs <tt>convert_to_pot</tt>.H +The pearl methods can exploit special properties of the CPDs for +computing the messages efficiently, whereas belprop does not. +<p> +The only method implemented by generic_CPD is <tt>adjustable_CPD</tt>, +which is not shown, since it is not very interesting. + + +<p> + + +<table> +<table border units = pixels><tr> +<td align=center>Name +<td align=center>Child +<td align=center>Parents +<td align=center>Comments +<td align=center>CPD_to_CPT +<td align=center>conv_to_table +<td align=center>conv_to_pot +<td align=center>sample +<td align=center>prob +<td align=center>learn +<td align=center>Bayes +<td align=center>Pearl + + +<tr> +<!-- Name--><td> +<!-- Child--><td> +<!-- Parents--><td> +<!-- Comments--><td> +<!-- CPD_to_CPT--><td> +<!-- conv_to_table--><td> +<!-- conv_to_pot--><td> +<!-- sample--><td> +<!-- prob--><td> +<!-- learn--><td> +<!-- Bayes--><td> +<!-- Pearl--><td> + +<tr> +<!-- Name--><td>boolean +<!-- Child--><td>B +<!-- Parents--><td>B +<!-- Comments--><td>Syntactic sugar for tabular +<!-- CPD_to_CPT--><td>- +<!-- conv_to_table--><td>- +<!-- conv_to_pot--><td>- +<!-- sample--><td>- +<!-- prob--><td>- +<!-- learn--><td>- +<!-- Bayes--><td>- +<!-- Pearl--><td>- + +<tr> +<!-- Name--><td>deterministic +<!-- Child--><td>D +<!-- Parents--><td>D +<!-- Comments--><td>Syntactic sugar for tabular +<!-- CPD_to_CPT--><td>- +<!-- conv_to_table--><td>- +<!-- conv_to_pot--><td>- +<!-- sample--><td>- +<!-- prob--><td>- +<!-- learn--><td>- +<!-- Bayes--><td>- +<!-- Pearl--><td>- + +<tr> +<!-- Name--><td>Discrete +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>Virtual class +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Calls CPD_to_CPT +<!-- conv_to_pot--><td>Calls conv_to_table +<!-- sample--><td>Calls conv_to_table +<!-- prob--><td>Calls conv_to_table +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + +<tr> +<!-- Name--><td>Gaussian +<!-- Child--><td>C +<!-- Parents--><td>C/D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>Y +<!-- prob--><td>Y +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + +<tr> +<!-- Name--><td>gmux +<!-- Child--><td>C +<!-- Parents--><td>C/D +<!-- Comments--><td>multiplexer +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>N +<!-- prob--><td>N +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>Y + + +<tr> +<!-- Name--><td>MLP +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>multi layer perceptron +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Y +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>noisy-or +<!-- Child--><td>B +<!-- Parents--><td>B +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>Y +<!-- conv_to_table--><td>Inherits from discrete +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>Y + + +<tr> +<!-- Name--><td>root +<!-- Child--><td>C/D +<!-- Parents--><td>none +<!-- Comments--><td>no params +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>Y +<!-- prob--><td>Y +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>softmax +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Y +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>generic +<!-- Child--><td>C/D +<!-- Parents--><td>C/D +<!-- Comments--><td>Virtual class +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>N +<!-- sample--><td>N +<!-- prob--><td>N +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>Tabular +<!-- Child--><td>D +<!-- Parents--><td>D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>Y +<!-- conv_to_table--><td>Inherits from discrete +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>Y +<!-- Pearl--><td>Y + +</table> + + + +<h1><a name="examples">Example models</h1> + + +<h2>Gaussian mixture models</h2> + +Richard W. DeVaul has made a detailed tutorial on how to fit mixtures +of Gaussians using BNT. Available +<a href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt">here</a>. + + +<h2><a name="pca">PCA, ICA, and all that </h2> + +In Figure (a) below, we show how Factor Analysis can be thought of as a +graphical model. Here, X has an N(0,I) prior, and +Y|X=x ~ N(mu + Wx, Psi), +where Psi is diagonal and W is called the "factor loading matrix". +Since the noise on both X and Y is diagonal, the components of these +vectors are uncorrelated, and hence can be represented as individual +scalar nodes, as we show in (b). +(This is useful if parts of the observations on the Y vector are occasionally missing.) +We usually take k=|X| << |Y|=D, so the model tries to explain +many observations using a low-dimensional subspace. + + +<center> +<table> +<tr> +<td><img src="Figures/fa.gif"> +<td><img src="Figures/fa_scalar.gif"> +<td><img src="Figures/mfa.gif"> +<td><img src="Figures/ifa.gif"> +<tr> +<td align=center> (a) +<td align=center> (b) +<td align=center> (c) +<td align=center> (d) +</table> +</center> + +<p> +We can create this model in BNT as follows. +<pre> +ns = [k D]; +dag = zeros(2,2); +dag(1,2) = 1; +bnet = mk_bnet(dag, ns, 'discrete', []); +bnet.CPD{1} = gaussian_CPD(bnet, 1, 'mean', zeros(k,1), 'cov', eye(k), ... + 'cov_type', 'diag', 'clamp_mean', 1, 'clamp_cov', 1); +bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(D,1), 'cov', diag(Psi0), 'weights', W0, ... + 'cov_type', 'diag', 'clamp_mean', 1); +</pre> + +The root node is clamped to the N(0,I) distribution, so that we will +not update these parameters during learning. +The mean of the leaf node is clamped to 0, +since we assume the data has been centered (had its mean subtracted +off); this is just for simplicity. +Finally, the covariance of the leaf node is constrained to be +diagonal. W0 and Psi0 are the initial parameter guesses. + +<p> +We can fit this model (i.e., estimate its parameters in a maximum +likelihood (ML) sense) using EM, as we +explain <a href="#em">below</a>. +Not surprisingly, the ML estimates for mu and Psi turn out to be +identical to the +sample mean and variance, which can be computed directly as +<pre> +mu_ML = mean(data); +Psi_ML = diag(cov(data)); +</pre> +Note that W can only be identified up to a rotation matrix, because of +the spherical symmetry of the source. + +<p> +If we restrict Psi to be spherical, i.e., Psi = sigma*I, +there is a closed-form solution for W as well, +i.e., we do not need to use EM. +In particular, W contains the first |X| eigenvectors of the sample covariance +matrix, with scalings determined by the eigenvalues and sigma. +Classical PCA can be obtained by taking the sigma->0 limit. +For details, see + +<ul> +<li> <a href="ftp://hope.caltech.edu/pub/roweis/Empca/empca.ps"> +"EM algorithms for PCA and SPCA"</a>, Sam Roweis, NIPS 97. +(<a href="ftp://hope.caltech.edu/pub/roweis/Code/empca.tar.gz"> +Matlab software</a>) + +<p> +<li> +<a +href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003> +"Mixtures of probabilistic principal component analyzers"</a>, +Tipping and Bishop, Neural Computation 11(2):443--482, 1999. +</ul> + +<p> +By adding a hidden discrete variable, we can create mixtures of FA +models, as shown in (c). +Now we can explain the data using a set of subspaces. +We can create this model in BNT as follows. +<pre> +ns = [M k D]; +dag = zeros(3); +dag(1,3) = 1; +dag(2,3) = 1; +bnet = mk_bnet(dag, ns, 'discrete', 1); +bnet.CPD{1} = tabular_CPD(bnet, 1, Pi0); +bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(k, 1), 'cov', eye(k), 'cov_type', 'diag', ... + 'clamp_mean', 1, 'clamp_cov', 1); +bnet.CPD{3} = gaussian_CPD(bnet, 3, 'mean', Mu0', 'cov', repmat(diag(Psi0), [1 1 M]), ... + 'weights', W0, 'cov_type', 'diag', 'tied_cov', 1); +</pre> +Notice how the covariance matrix for Y is the same for all values of +Q; that is, the noise level in each sub-space is assumed the same. +However, we allow the offset, mu, to vary. +For details, see +<ul> + +<LI> +<a HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/tr-96-1.ps.gz"> The EM +Algorithm for Mixtures of Factor Analyzers </A>, +Ghahramani, Z. and Hinton, G.E. (1996), +University of Toronto +Technical Report CRG-TR-96-1. +(<A HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/mfa.tar.gz">Matlab software</A>) + +<p> +<li> +<a +href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003> +"Mixtures of probabilistic principal component analyzers"</a>, +Tipping and Bishop, Neural Computation 11(2):443--482, 1999. +</ul> + +<p> +I have included Zoubin's specialized MFA code (with his permission) +with the toolbox, so you can check that BNT gives the same results: +see 'BNT/examples/static/mfa1.m'. + +<p> +Independent Factor Analysis (IFA) generalizes FA by allowing a +non-Gaussian prior on each component of X. +(Note that we can approximate a non-Gaussian prior using a mixture of +Gaussians.) +This means that the likelihood function is no longer rotationally +invariant, so we can uniquely identify W and the hidden +sources X. +IFA also allows a non-diagonal Psi (i.e. correlations between the components of Y). +We recover classical Independent Components Analysis (ICA) +in the Psi -> 0 limit, and by assuming that |X|=|Y|, so that the +weight matrix W is square and invertible. +For details, see +<ul> +<li> +<a href="http://www.gatsby.ucl.ac.uk/~hagai/ifa.ps">Independent Factor +Analysis</a>, H. Attias, Neural Computation 11: 803--851, 1998. +</ul> + + + +<h2><a name="mixexp">Mixtures of experts</h2> + +As an example of the use of the softmax function, +we introduce the Mixture of Experts model. +<!-- +We also show +the Hierarchical Mixture of Experts model, where the hierarchy has two +levels. +(This is essentially a probabilistic decision tree of height two.) +--> +As before, +circles denote continuous-valued nodes, +squares denote discrete nodes, clear +means hidden, and shaded means observed. +<p> +<center> +<table> +<tr> +<td><img src="Figures/mixexp.gif"> +<!-- +<td><img src="Figures/hme.gif"> +--> +</table> +</center> +<p> +X is the observed +input, Y is the output, and +the Q nodes are hidden "gating" nodes, which select the appropriate +set of parameters for Y. During training, Y is assumed observed, +but for testing, the goal is to predict Y given X. +Note that this is a <em>conditional</em> density model, so we don't +associate any parameters with X. +Hence X's CPD will be a root CPD, which is a way of modelling +exogenous nodes. +If the output is a continuous-valued quantity, +we assume the "experts" are linear-regression units, +and set Y's CPD to linear-Gaussian. +If the output is discrete, we set Y's CPD to a softmax function. +The Q CPDs will always be softmax functions. + +<p> +As a concrete example, consider the mixture of experts model where X and Y are +scalars, and Q is binary. +This is just piecewise linear regression, where +we have two line segments, i.e., +<P> +<IMG ALIGN=BOTTOM SRC="Eqns/lin_reg_eqn.gif"> +<P> +We can create this model with random parameters as follows. +(This code is bundled in BNT/examples/static/mixexp2.m.) +<PRE> +X = 1; +Q = 2; +Y = 3; +dag = zeros(3,3); +dag(X,[Q Y]) = 1 +dag(Q,Y) = 1; +ns = [1 2 1]; % make X and Y scalars, and have 2 experts +onodes = [1 3]; +bnet = mk_bnet(dag, ns, 'discrete', 2, 'observed', onodes); + +rand('state', 0); +randn('state', 0); +bnet.CPD{1} = root_CPD(bnet, 1); +bnet.CPD{2} = softmax_CPD(bnet, 2); +bnet.CPD{3} = gaussian_CPD(bnet, 3); +</PRE> +Now let us fit this model using <a href="#em">EM</a>. +First we <a href="#load_data">load the data</a> (1000 training cases) and plot them. +<P> +<PRE> +data = load('/examples/static/Misc/mixexp_data.txt', '-ascii'); +plot(data(:,1), data(:,2), '.'); +</PRE> +<p> +<center> +<IMG SRC="Figures/mixexp_data.gif"> +</center> +<p> +This is what the model looks like before training. +(Thanks to Thomas Hofman for writing this plotting routine.) +<p> +<center> +<IMG SRC="Figures/mixexp_before.gif"> +</center> +<p> +Now let's train the model, and plot the final performance. +(We will discuss how to train models in more detail <a href="#param_learning">below</a>.) +<P> +<PRE> +ncases = size(data, 1); % each row of data is a training case +cases = cell(3, ncases); +cases([1 3], :) = num2cell(data'); % each column of cases is a training case +engine = jtree_inf_engine(bnet); +max_iter = 20; +[bnet2, LLtrace] = learn_params_em(engine, cases, max_iter); +</PRE> +(We specify which nodes will be observed when we create the engine. +Hence BNT knows that the hidden nodes are all discrete. +For complex models, this can lead to a significant speedup.) +Below we show what the model looks like after 16 iterations of EM +(with 100 IRLS iterations per M step), when it converged +using the default convergence tolerance (that the +fractional change in the log-likelihood be less than 1e-3). +Before learning, the log-likelihood was +-322.927442; afterwards, it was -13.728778. +<p> +<center> +<IMG SRC="Figures/mixexp_after.gif"> +</center> +(See BNT/examples/static/mixexp2.m for details of the code.) + + + +<h2><a name="hme">Hierarchical mixtures of experts</h2> + +A hierarchical mixture of experts (HME) extends the mixture of experts +model by having more than one hidden node. A two-level example is shown below, along +with its more traditional representation as a neural network. +This is like a (balanced) probabilistic decision tree of height 2. +<p> +<center> +<IMG SRC="Figures/HMEforMatlab.jpg"> +</center> +<p> +<a href="mailto:pbrutti@stat.cmu.edu">Pierpaolo Brutti</a> +has written an extensive set of routines for HMEs, +which are bundled with BNT: see the examples/static/HME directory. +These routines allow you to choose the number of hidden (gating) +layers, and the form of the experts (softmax or MLP). +See the file hmemenu, which provides a demo. +For example, the figure below shows the decision boundaries learned +for a ternary classification problem, using a 2 level HME with softmax +gates and softmax experts; the training set is on the left, the +testing set on the right. +<p> +<center> +<!--<IMG SRC="Figures/hme_dec_boundary.gif">--> +<IMG SRC="Figures/hme_dec_boundary.png"> +</center> +<p> + + +<p> +For more details, see the following: +<ul> + +<li> <a href="http://www.cs.berkeley.edu/~jordan/papers/hierarchies.ps.Z"> +Hierarchical mixtures of experts and the EM algorithm</a> +M. I. Jordan and R. A. Jacobs. Neural Computation, 6, 181-214, 1994. + +<li> <a href = +"http://www.cs.berkeley.edu/~dmartin/software">David Martin's +matlab code for HME</a> + +<li> <a +href="http://www.cs.berkeley.edu/~jordan/papers/uai.ps.Z">Why the +logistic function? A tutorial discussion on +probabilities and neural networks.</a> M. I. Jordan. MIT Computational +Cognitive Science Report 9503, August 1995. + +<li> "Generalized Linear Models", McCullagh and Nelder, Chapman and +Halll, 1983. + +<li> +"Improved learning algorithms for mixtures of experts in multiclass +classification". +K. Chen, L. Xu, H. Chi. +Neural Networks (1999) 12: 1229-1252. + +<li> <a href="http://www.oigeeza.com/steve/"> +Classification Using Hierarchical Mixtures of Experts</a> +S.R. Waterhouse and A.J. Robinson. +In Proc. IEEE Workshop on Neural Network for Signal Processing IV (1994), pp. 177-186 + +<li> <a href="http://www.idiap.ch/~perry/"> +Localized mixtures of experts</a>, +P. Moerland, 1998. + +<li> "Nonlinear gated experts for time series", +A.S. Weigend and M. Mangeas, 1995. + +</ul> + + +<h2><a name="qmr">QMR</h2> + +Bayes nets originally arose out of an attempt to add probabilities to +expert systems, and this is still the most common use for BNs. +A famous example is +QMR-DT, a decision-theoretic reformulation of the Quick Medical +Reference (QMR) model. +<p> +<center> +<IMG ALIGN=BOTTOM SRC="Figures/qmr.gif"> +</center> +Here, the top layer represents hidden disease nodes, and the bottom +layer represents observed symptom nodes. +The goal is to infer the posterior probability of each disease given +all the symptoms (which can be present, absent or unknown). +Each node in the top layer has a Bernoulli prior (with a low prior +probability that the disease is present). +Since each node in the bottom layer has a high fan-in, we use a +noisy-OR parameterization; each disease has an independent chance of +causing each symptom. +The real QMR-DT model is copyright, but +we can create a random QMR-like model as follows. +<pre> +function bnet = mk_qmr_bnet(G, inhibit, leak, prior) +% MK_QMR_BNET Make a QMR model +% bnet = mk_qmr_bnet(G, inhibit, leak, prior) +% +% G(i,j) = 1 iff there is an arc from disease i to finding j +% inhibit(i,j) = inhibition probability on i->j arc +% leak(j) = inhibition prob. on leak->j arc +% prior(i) = prob. disease i is on + +[Ndiseases Nfindings] = size(inhibit); +N = Ndiseases + Nfindings; +finding_node = Ndiseases+1:N; +ns = 2*ones(1,N); +dag = zeros(N,N); +dag(1:Ndiseases, finding_node) = G; +bnet = mk_bnet(dag, ns, 'observed', finding_node); + +for d=1:Ndiseases + CPT = [1-prior(d) prior(d)]; + bnet.CPD{d} = tabular_CPD(bnet, d, CPT'); +end + +for i=1:Nfindings + fnode = finding_node(i); + ps = parents(G, i); + bnet.CPD{fnode} = noisyor_CPD(bnet, fnode, leak(i), inhibit(ps, i)); +end +</pre> +In the file BNT/examples/static/qmr1, we create a random bipartite +graph G, with 5 diseases and 10 findings, and random parameters. +(In general, to create a random dag, use 'mk_random_dag'.) +We can visualize the resulting graph structure using +the methods discussed <a href="#graphdraw">below</a>, with the +following results: +<p> +<img src="Figures/qmr.rnd.jpg"> + +<p> +Now let us put some random evidence on all the leaves except the very +first and very last, and compute the disease posteriors. +<pre> +pos = 2:floor(Nfindings/2); +neg = (pos(end)+1):(Nfindings-1); +onodes = myunion(pos, neg); +evidence = cell(1, N); +evidence(findings(pos)) = num2cell(repmat(2, 1, length(pos))); +evidence(findings(neg)) = num2cell(repmat(1, 1, length(neg))); + +engine = jtree_inf_engine(bnet); +[engine, ll] = enter_evidence(engine, evidence); +post = zeros(1, Ndiseases); +for i=diseases(:)' + m = marginal_nodes(engine, i); + post(i) = m.T(2); +end +</pre> +Junction tree can be quite slow on large QMR models. +Fortunately, it is possible to exploit properties of the noisy-OR +function to speed up exact inference using an algorithm called +<a href="#quickscore">quickscore</a>, discussed below. + + + + + +<h2><a name="cg_model">Conditional Gaussian models</h2> + +A conditional Gaussian model is one in which, conditioned on all the discrete +nodes, the distribution over the remaining (continuous) nodes is +multivariate Gaussian. This means we can have arcs from discrete (D) +to continuous (C) nodes, but not vice versa. +(We <em>are</em> allowed C->D arcs if the continuous nodes are observed, +as in the <a href="#mixexp">mixture of experts</a> model, +since this distribution can be represented with a discrete potential.) +<p> +We now give an example of a CG model, from +the paper "Propagation of Probabilities, Means amd +Variances in Mixed Graphical Association Models", Steffen Lauritzen, +JASA 87(420):1098--1108, 1992 (reprinted in the book "Probabilistic Networks and Expert +Systems", R. G. Cowell, A. P. Dawid, S. L. Lauritzen and +D. J. Spiegelhalter, Springer, 1999.) + +<h3>Specifying the graph</h3> + +Consider the model of waste emissions from an incinerator plant shown below. +We follow the standard convention that shaded nodes are observed, +clear nodes are hidden. +We also use the non-standard convention that +square nodes are discrete (tabular) and round nodes are +Gaussian. + +<p> +<center> +<IMG SRC="Figures/cg1.gif"> +</center> +<p> + +We can create this model as follows. +<pre> +F = 1; W = 2; E = 3; B = 4; C = 5; D = 6; Min = 7; Mout = 8; L = 9; +n = 9; + +dag = zeros(n); +dag(F,E)=1; +dag(W,[E Min D]) = 1; +dag(E,D)=1; +dag(B,[C D])=1; +dag(D,[L Mout])=1; +dag(Min,Mout)=1; + +% node sizes - all cts nodes are scalar, all discrete nodes are binary +ns = ones(1, n); +dnodes = [F W B]; +cnodes = mysetdiff(1:n, dnodes); +ns(dnodes) = 2; + +bnet = mk_bnet(dag, ns, 'discrete', dnodes); +</pre> +'dnodes' is a list of the discrete nodes; 'cnodes' is the continuous +nodes. 'mysetdiff' is a faster version of the built-in 'setdiff'. +<p> + + +<h3>Specifying the parameters</h3> + +The parameters of the discrete nodes can be specified as follows. +<pre> +bnet.CPD{B} = tabular_CPD(bnet, B, 'CPT', [0.85 0.15]); % 1=stable, 2=unstable +bnet.CPD{F} = tabular_CPD(bnet, F, 'CPT', [0.95 0.05]); % 1=intact, 2=defect +bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [2/7 5/7]); % 1=industrial, 2=household +</pre> + +<p> +The parameters of the continuous nodes can be specified as follows. +<pre> +bnet.CPD{E} = gaussian_CPD(bnet, E, 'mean', [-3.9 -0.4 -3.2 -0.5], ... + 'cov', [0.00002 0.0001 0.00002 0.0001]); +bnet.CPD{D} = gaussian_CPD(bnet, D, 'mean', [6.5 6.0 7.5 7.0], ... + 'cov', [0.03 0.04 0.1 0.1], 'weights', [1 1 1 1]); +bnet.CPD{C} = gaussian_CPD(bnet, C, 'mean', [-2 -1], 'cov', [0.1 0.3]); +bnet.CPD{L} = gaussian_CPD(bnet, L, 'mean', 3, 'cov', 0.25, 'weights', -0.5); +bnet.CPD{Min} = gaussian_CPD(bnet, Min, 'mean', [0.5 -0.5], 'cov', [0.01 0.005]); +bnet.CPD{Mout} = gaussian_CPD(bnet, Mout, 'mean', 0, 'cov', 0.002, 'weights', [1 1]); +</pre> + + +<h3><a name="cg_infer">Inference</h3> + +<!--Let us perform inference in the <a href="#cg_model">waste incinerator example</a>.--> +First we compute the unconditional marginals. +<pre> +engine = jtree_inf_engine(bnet); +evidence = cell(1,n); +[engine, ll] = enter_evidence(engine, evidence); +marg = marginal_nodes(engine, E); +</pre> +<!--(Of course, we could use <tt>cond_gauss_inf_engine</tt> instead of jtree.)--> +'marg' is a structure that contains the fields 'mu' and 'Sigma', which +contain the mean and (co)variance of the marginal on E. +In this case, they are both scalars. +Let us check they match the published figures (to 2 decimal places). +<!--(We can't expect +more precision than this in general because I have implemented the algorithm of +Lauritzen (1992), which can be numerically unstable.)--> +<pre> +tol = 1e-2; +assert(approxeq(marg.mu, -3.25, tol)); +assert(approxeq(sqrt(marg.Sigma), 0.709, tol)); +</pre> +We can compute the other posteriors similarly. +Now let us add some evidence. +<pre> +evidence = cell(1,n); +evidence{W} = 1; % industrial +evidence{L} = 1.1; +evidence{C} = -0.9; +[engine, ll] = enter_evidence(engine, evidence); +</pre> +Now we find +<pre> +marg = marginal_nodes(engine, E); +assert(approxeq(marg.mu, -3.8983, tol)); +assert(approxeq(sqrt(marg.Sigma), 0.0763, tol)); +</pre> + + +We can also compute the joint probability on a set of nodes. +For example, P(D, Mout | evidence) is a 2D Gaussian: +<pre> +marg = marginal_nodes(engine, [D Mout]) +marg = + domain: [6 8] + mu: [2x1 double] + Sigma: [2x2 double] + T: 1.0000 +</pre> +The mean is +<pre> +marg.mu +ans = + 3.6077 + 4.1077 +</pre> +and the covariance matrix is +<pre> +marg.Sigma +ans = + 0.1062 0.1062 + 0.1062 0.1182 +</pre> +It is easy to visualize this posterior using standard Matlab plotting +functions, e.g., +<pre> +gaussplot2d(marg.mu, marg.Sigma); +</pre> +produces the following picture. + +<p> +<center> +<IMG SRC="Figures/gaussplot.png"> +</center> +<p> + + +The T field indicates that the mixing weight of this Gaussian +component is 1.0. +If the joint contains discrete and continuous variables, the result +will be a mixture of Gaussians, e.g., +<pre> +marg = marginal_nodes(engine, [F E]) + domain: [1 3] + mu: [-3.9000 -0.4003] + Sigma: [1x1x2 double] + T: [0.9995 4.7373e-04] +</pre> +The interpretation is +Sigma(i,j,k) = Cov[ E(i) E(j) | F=k ]. +In this case, E is a scalar, so i=j=1; k specifies the mixture component. +<p> +We saw in the sprinkler network that BNT sets the effective size of +observed discrete nodes to 1, since they only have one legal value. +For continuous nodes, BNT sets their length to 0, +since they have been reduced to a point. +For example, +<pre> +marg = marginal_nodes(engine, [B C]) + domain: [4 5] + mu: [] + Sigma: [] + T: [0.0123 0.9877] +</pre> +It is simple to post-process the output of marginal_nodes. +For example, the file BNT/examples/static/cg1 sets the mu term of +observed nodes to their observed value, and the Sigma term to 0 (since +observed nodes have no variance). + +<p> +Note that the implemented version of the junction tree is numerically +unstable when using CG potentials +(which is why, in the example above, we only required our answers to agree with +the published ones to 2dp.) +This is why you might want to use <tt>stab_cond_gauss_inf_engine</tt>, +implemented by Shan Huang. This is described in + +<ul> +<li> "Stable Local Computation with Conditional Gaussian Distributions", +S. Lauritzen and F. Jensen, Tech Report R-99-2014, +Dept. Math. Sciences, Allborg Univ., 1999. +</ul> + +However, even the numerically stable version +can be computationally intractable if there are many hidden discrete +nodes, because the number of mixture components grows exponentially e.g., in a +<a href="usage_dbn.html#lds">switching linear dynamical system</a>. +In general, one must resort to approximate inference techniques: see +the discussion on <a href="#engines">inference engines</a> below. + + +<h2><a name="hybrid">Other hybrid models</h2> + +When we have C->D arcs, where C is hidden, we need to use +approximate inference. +One approach (not implemented in BNT) is described in +<ul> +<li> <a +href="http://www.cs.berkeley.edu/~murphyk/Papers/hybrid_uai99.ps.gz">A +Variational Approximation for Bayesian Networks with +Discrete and Continuous Latent Variables</a>, +K. Murphy, UAI 99. +</ul> +Of course, one can always use <a href="#sampling">sampling</a> methods +for approximate inference in such models. + + + +<h1><a name="param_learning">Parameter Learning</h1> + +The parameter estimation routines in BNT can be classified into 4 +types, depending on whether the goal is to compute +a full (Bayesian) posterior over the parameters or just a point +estimate (e.g., Maximum Likelihood or Maximum A Posteriori), +and whether all the variables are fully observed or there is missing +data/ hidden variables (partial observability). +<p> + +<TABLE BORDER> +<tr> + <TH></TH> + <th>Full obs</th> + <th>Partial obs</th> +</tr> +<tr> + <th>Point</th> + <td><tt>learn_params</tt></td> + <td><tt>learn_params_em</tt></td> +</tr> +<tr> + <th>Bayes</th> + <td><tt>bayes_update_params</tt></td> + <td>not yet supported</td> +</tr> +</table> + + +<h2><a name="load_data">Loading data from a file</h2> + +To load numeric data from an ASCII text file called 'dat.txt', where each row is a +case and columns are separated by white-space, such as +<pre> +011979 1626.5 0.0 +021979 1367.0 0.0 +... +</pre> +you can use +<pre> +data = load('dat.txt'); +</pre> +or +<pre> +load dat.txt -ascii +</pre> +In the latter case, the data is stored in a variable called 'dat' (the +filename minus the extension). +Alternatively, suppose the data is stored in a .csv file (has commas +separating the columns, and contains a header line), such as +<pre> +header info goes here +ORD,011979,1626.5,0.0 +DSM,021979,1367.0,0.0 +... +</pre> +You can load this using +<pre> +[a,b,c,d] = textread('dat.txt', '%s %d %f %f', 'delimiter', ',', 'headerlines', 1); +</pre> +If your file is not in either of these formats, you can either use Perl to convert +it to this format, or use the Matlab scanf command. +Type +<tt> +help iofun +</tt> +for more information on Matlab's file functions. +<!-- +<p> +To load data directly from Excel, +you should buy the +<a href="http://www.mathworks.com/products/excellink/">Excel Link</a>. +To load data directly from a relational database, +you should buy the +<a href="http://www.mathworks.com/products/database">Database +toolbox</a>. +--> +<p> +BNT learning routines require data to be stored in a cell array. +data{i,m} is the value of node i in case (example) m, i.e., each +<em>column</em> is a case. +If node i is not observed in case m (missing value), set +data{i,m} = []. +(Not all the learning routines can cope with such missing values, however.) +In the special case that all the nodes are observed and are +scalar-valued (as opposed to vector-valued), the data can be +stored in a matrix (as opposed to a cell-array). +<p> +Suppose, as in the <a href="#mixexp">mixture of experts example</a>, +that we have 3 nodes in the graph: X(1) is the observed input, X(3) is +the observed output, and X(2) is a hidden (gating) node. We can +create the dataset as follows. +<pre> +data = load('dat.txt'); +ncases = size(data, 1); +cases = cell(3, ncases); +cases([1 3], :) = num2cell(data'); +</pre> +Notice how we transposed the data, to convert rows into columns. +Also, cases{2,m} = [] for all m, since X(2) is always hidden. + + +<h2><a name="mle_complete">Maximum likelihood parameter estimation from complete data</h2> + +As an example, let's generate some data from the sprinkler network, randomize the parameters, +and then try to recover the original model. +First we create some training data using forwards sampling. +<pre> +samples = cell(N, nsamples); +for i=1:nsamples + samples(:,i) = sample_bnet(bnet); +end +</pre> +samples{j,i} contains the value of the j'th node in case i. +sample_bnet returns a cell array because, in general, each node might +be a vector of different length. +In this case, all nodes are discrete (and hence scalars), so we +could have used a regular array instead (which can be quicker): +<pre> +data = cell2num(samples); +</pre +So now data(j,i) = samples{j,i}. +<p> +Now we create a network with random parameters. +(The initial values of bnet2 don't matter in this case, since we can find the +globally optimal MLE independent of where we start.) +<pre> +% Make a tabula rasa +bnet2 = mk_bnet(dag, node_sizes); +seed = 0; +rand('state', seed); +bnet2.CPD{C} = tabular_CPD(bnet2, C); +bnet2.CPD{R} = tabular_CPD(bnet2, R); +bnet2.CPD{S} = tabular_CPD(bnet2, S); +bnet2.CPD{W} = tabular_CPD(bnet2, W); +</pre> +Finally, we find the maximum likelihood estimates of the parameters. +<pre> +bnet3 = learn_params(bnet2, samples); +</pre> +To view the learned parameters, we use a little Matlab hackery. +<pre> +CPT3 = cell(1,N); +for i=1:N + s=struct(bnet3.CPD{i}); % violate object privacy + CPT3{i}=s.CPT; +end +</pre> +Here are the parameters learned for node 4. +<pre> +dispcpt(CPT3{4}) +1 1 : 1.0000 0.0000 +2 1 : 0.2000 0.8000 +1 2 : 0.2273 0.7727 +2 2 : 0.0000 1.0000 +</pre> +So we see that the learned parameters are fairly close to the "true" +ones, which we display below. +<pre> +dispcpt(CPT{4}) +1 1 : 1.0000 0.0000 +2 1 : 0.1000 0.9000 +1 2 : 0.1000 0.9000 +2 2 : 0.0100 0.9900 +</pre> +We can get better results by using a larger training set, or using +informative priors (see <a href="#prior">below</a>). + + + +<h2><a name="prior">Parameter priors</h2> + +Currently, only tabular CPDs can have priors on their parameters. +The conjugate prior for a multinomial is the Dirichlet. +(For binary random variables, the multinomial is the same as the +Bernoulli, and the Dirichlet is the same as the Beta.) +<p> +The Dirichlet has a simple interpretation in terms of pseudo counts. +If we let N_ijk = the num. times X_i=k and Pa_i=j occurs in the +training set, where Pa_i are the parents of X_i, +then the maximum likelihood (ML) estimate is +T_ijk = N_ijk / N_ij (where N_ij = sum_k' N_ijk'), which will be 0 if N_ijk=0. +To prevent us from declaring that (X_i=k, Pa_i=j) is impossible just because this +event was not seen in the training set, +we can pretend we saw value k of X_i, for each value j of Pa_i some number (alpha_ijk) +of times in the past. +The MAP (maximum a posterior) estimate is then +<pre> +T_ijk = (N_ijk + alpha_ijk) / (N_ij + alpha_ij) +</pre> +and is never 0 if all alpha_ijk > 0. +For example, consider the network A->B, where A is binary and B has 3 +values. +A uniform prior for B has the form +<pre> + B=1 B=2 B=3 +A=1 1 1 1 +A=2 1 1 1 +</pre> +which can be created using +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'unif'); +</pre> +This prior does not satisfy the likelihood equivalence principle, +which says that <a href="#markov_equiv">Markov equivalent</a> models +should have the same marginal likelihood. +A prior that does satisfy this principle is shown below. +Heckerman (1995) calls this the +BDeu prior (likelihood equivalent uniform Bayesian Dirichlet). +<pre> + B=1 B=2 B=3 +A=1 1/6 1/6 1/6 +A=2 1/6 1/6 1/6 +</pre> +where we put N/(q*r) in each bin; N is the equivalent sample size, +r=|A|, q = |B|. +This can be created as follows +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'BDeu'); +</pre> +Here, 1 is the equivalent sample size, and is the strength of the +prior. +You can change this using +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', ... + 'BDeu', 'dirichlet_weight', 10); +</pre> +<!--where counts is an array of pseudo-counts of the same size as the +CPT.--> +<!-- +<p> +When you specify a prior, you should set row i of the CPT to the +normalized version of row i of the pseudo-count matrix, i.e., to the +expected values of the parameters. This will ensure that computing the +marginal likelihood sequentially (see <a +href="#bayes_learn">below</a>) and in batch form gives the same +results. +To do this, proceed as follows. +<pre> +tabular_CPD(bnet, i, 'prior', counts, 'CPT', mk_stochastic(counts)); +</pre> +For a non-informative prior, you can just write +<pre> +tabular_CPD(bnet, i, 'prior', 'unif', 'CPT', 'unif'); +</pre> +--> + + +<h2><a name="bayes_learn">(Sequential) Bayesian parameter updating from complete data</h2> + +If we use conjugate priors and have fully observed data, we can +compute the posterior over the parameters in batch form as follows. +<pre> +cases = sample_bnet(bnet, nsamples); +bnet = bayes_update_params(bnet, cases); +LL = log_marg_lik_complete(bnet, cases); +</pre> +bnet.CPD{i}.prior contains the new Dirichlet pseudocounts, +and bnet.CPD{i}.CPT is set to the mean of the posterior (the +normalized counts). +(Hence if the initial pseudo counts are 0, +<tt>bayes_update_params</tt> and <tt>learn_params</tt> will give the +same result.) + + + + +<p> +We can compute the same result sequentially (on-line) as follows. +<pre> +LL = 0; +for m=1:nsamples + LL = LL + log_marg_lik_complete(bnet, cases(:,m)); + bnet = bayes_update_params(bnet, cases(:,m)); +end +</pre> + +The file <tt>BNT/examples/static/StructLearn/model_select1</tt> has an example of +sequential model selection which uses the same idea. +We generate data from the model A->B +and compute the posterior prob of all 3 dags on 2 nodes: + (1) A B, (2) A <- B , (3) A -> B +Models 2 and 3 are <a href="#markov_equiv">Markov equivalent</a>, and therefore indistinguishable from +observational data alone, so we expect their posteriors to be the same +(assuming a prior which satisfies likelihood equivalence). +If we use random parameters, the "true" model only gets a higher posterior after 2000 trials! +However, if we make B a noisy NOT gate, the true model "wins" after 12 +trials, as shown below (red = model 1, blue/green (superimposed) +represents models 2/3). +<p> +<img src="Figures/model_select.png"> +<p> +The use of marginal likelihood for model selection is discussed in +greater detail in the +section on <a href="structure_learning">structure learning</a>. + + + + +<h2><a name="em">Maximum likelihood parameter estimation with missing values</h2> + +Now we consider learning when some values are not observed. +Let us randomly hide half the values generated from the water +sprinkler example. +<pre> +samples2 = samples; +hide = rand(N, nsamples) > 0.5; +[I,J]=find(hide); +for k=1:length(I) + samples2{I(k), J(k)} = []; +end +</pre> +samples2{i,l} is the value of node i in training case l, or [] if unobserved. +<p> +Now we will compute the MLEs using the EM algorithm. +We need to use an inference algorithm to compute the expected +sufficient statistics in the E step; the M (maximization) step is as +above. +<pre> +engine2 = jtree_inf_engine(bnet2); +max_iter = 10; +[bnet4, LLtrace] = learn_params_em(engine2, samples2, max_iter); +</pre> +LLtrace(i) is the log-likelihood at iteration i. We can plot this as +follows: +<pre> +plot(LLtrace, 'x-') +</pre> +Let's display the results after 10 iterations of EM. +<pre> +celldisp(CPT4) +CPT4{1} = + 0.6616 + 0.3384 +CPT4{2} = + 0.6510 0.3490 + 0.8751 0.1249 +CPT4{3} = + 0.8366 0.1634 + 0.0197 0.9803 +CPT4{4} = +(:,:,1) = + 0.8276 0.0546 + 0.5452 0.1658 +(:,:,2) = + 0.1724 0.9454 + 0.4548 0.8342 +</pre> +We can get improved performance by using one or more of the following +methods: +<ul> +<li> Increasing the size of the training set. +<li> Decreasing the amount of hidden data. +<li> Running EM for longer. +<li> Using informative priors. +<li> Initialising EM from multiple starting points. +</ul> + +Click <a href="#gaussian">here</a> for a discussion of learning +Gaussians, which can cause numerical problems. +<p> +For a more complete example of learning with EM, +see the script BNT/examples/static/learn1.m. + +<h2><a name="tying">Parameter tying</h2> + +In networks with repeated structure (e.g., chains and grids), it is +common to assume that the parameters are the same at every node. This +is called parameter tying, and reduces the amount of data needed for +learning. +<p> +When we have tied parameters, there is no longer a one-to-one +correspondence between nodes and CPDs. +Rather, each CPD species the parameters for a whole equivalence class +of nodes. +It is easiest to see this by example. +Consider the following <a href="usage_dbn.html#hmm">hidden Markov +model (HMM)</a> +<p> +<img src="Figures/hmm3.gif"> +<p> +<!-- +We can create this graph structure, assuming we have T time-slices, +as follows. +(We number the nodes as shown in the figure, but we could equally well +number the hidden nodes 1:T, and the observed nodes T+1:2T.) +<pre> +N = 2*T; +dag = zeros(N); +hnodes = 1:2:2*T; +for i=1:T-1 + dag(hnodes(i), hnodes(i+1))=1; +end +onodes = 2:2:2*T; +for i=1:T + dag(hnodes(i), onodes(i)) = 1; +end +</pre> +<p> +The hidden nodes are always discrete, and have Q possible values each, +but the observed nodes can be discrete or continuous, and have O possible values/length. +<pre> +if cts_obs + dnodes = hnodes; +else + dnodes = 1:N; +end +ns = ones(1,N); +ns(hnodes) = Q; +ns(onodes) = O; +</pre> +--> +When HMMs are used for semi-infinite processes like speech recognition, +we assume the transition matrix +P(H(t+1)|H(t)) is the same for all t; this is called a time-invariant +or homogenous Markov chain. +Hence hidden nodes 2, 3, ..., T +are all in the same equivalence class, say class Hclass. +Similarly, the observation matrix P(O(t)|H(t)) is assumed to be the +same for all t, so the observed nodes are all in the same equivalence +class, say class Oclass. +Finally, the prior term P(H(1)) is in a class all by itself, say class +H1class. +This is illustrated below, where we explicitly represent the +parameters as random variables (dotted nodes). +<p> +<img src="Figures/hmm4_params.gif"> +<p> +In BNT, we cannot represent parameters as random variables (nodes). +Instead, we "hide" the +parameters inside one CPD for each equivalence class, +and then specify that the other CPDs should share these parameters, as +follows. +<pre> +hnodes = 1:2:2*T; +onodes = 2:2:2*T; +H1class = 1; Hclass = 2; Oclass = 3; +eclass = ones(1,N); +eclass(hnodes(2:end)) = Hclass; +eclass(hnodes(1)) = H1class; +eclass(onodes) = Oclass; +% create dag and ns in the usual way +bnet = mk_bnet(dag, ns, 'discrete', dnodes, 'equiv_class', eclass); +</pre> +Finally, we define the parameters for each equivalence class: +<pre> +bnet.CPD{H1class} = tabular_CPD(bnet, hnodes(1)); % prior +bnet.CPD{Hclass} = tabular_CPD(bnet, hnodes(2)); % transition matrix +if cts_obs + bnet.CPD{Oclass} = gaussian_CPD(bnet, onodes(1)); +else + bnet.CPD{Oclass} = tabular_CPD(bnet, onodes(1)); +end +</pre> +In general, if bnet.CPD{e} = xxx_CPD(bnet, j), then j should be a +member of e's equivalence class; that is, it is not always the case +that e == j. You can use bnet.rep_of_eclass(e) to return the +representative of equivalence class e. +BNT will look up the parents of j to determine the size +of the CPT to use. It assumes that this is the same for all members of +the equivalence class. +Click <a href="param_tieing.html">here</a> for +a more complex example of parameter tying. +<p> +Note: +Normally one would define an HMM as a +<a href = "usage_dbn.html">Dynamic Bayes Net</a> +(see the function BNT/examples/dynamic/mk_chmm.m). +However, one can define an HMM as a static BN using the function +BNT/examples/static/Models/mk_hmm_bnet.m. + + + +<h1><a name="structure_learning">Structure learning</h1> + +Update (9/29/03): +Phillipe LeRay is developing some additional structure learning code +on top of BNT. Click <a +href="http://bnt.insa-rouen.fr/ajouts.html">here</a> +for details. + +<p> + +There are two very different approaches to structure learning: +constraint-based and search-and-score. +In the <a href="#constraint">constraint-based approach</a>, +we start with a fully connected graph, and remove edges if certain +conditional independencies are measured in the data. +This has the disadvantage that repeated independence tests lose +statistical power. +<p> +In the more popular search-and-score approach, +we perform a search through the space of possible DAGs, and either +return the best one found (a point estimate), or return a sample of the +models found (an approximation to the Bayesian posterior). +<p> +Unfortunately, the number of DAGs as a function of the number of +nodes, G(n), is super-exponential in n. +A closed form formula for G(n) is not known, but the first few values +are shown below (from Cooper, 1999). + +<table> +<tr> <th>n</th> <th align=left>G(n)</th> </tr> +<tr> <td>1</td> <td>1</td> </tr> +<tr> <td>2</td> <td>3</td> </tr> +<tr> <td>3</td> <td>25</td> </tr> +<tr> <td>4</td> <td>543</td> </tr> +<tr> <td>5</td> <td>29,281</td> </tr> +<tr> <td>6</td> <td>3,781,503</td> </tr> +<tr> <td>7</td> <td>1.1 x 10^9</td> </tr> +<tr> <td>8</td> <td>7.8 x 10^11</td> </tr> +<tr> <td>9</td> <td>1.2 x 10^15</td> </tr> +<tr> <td>10</td> <td>4.2 x 10^18</td> </tr> +</table> + +Since the number of DAGs is super-exponential in the number of nodes, +we cannot exhaustively search the space, so we either use a local +search algorithm (e.g., greedy hill climbining, perhaps with multiple +restarts) or a global search algorithm (e.g., Markov Chain Monte +Carlo). +<p> +If we know a total ordering on the nodes, +finding the best structure amounts to picking the best set of parents +for each node independently. +This is what the K2 algorithm does. +If the ordering is unknown, we can search over orderings, +which is more efficient than searching over DAGs (Koller and Friedman, 2000). +<p> +In addition to the search procedure, we must specify the scoring +function. There are two popular choices. The Bayesian score integrates +out the parameters, i.e., it is the marginal likelihood of the model. +The BIC (Bayesian Information Criterion) is defined as +log P(D|theta_hat) - 0.5*d*log(N), where D is the data, theta_hat is +the ML estimate of the parameters, d is the number of parameters, and +N is the number of data cases. +The BIC method has the advantage of not requiring a prior. +<p> +BIC can be derived as a large sample +approximation to the marginal likelihood. +(It is also equal to the Minimum Description Length of a model.) +However, in practice, the sample size does not need to be very large +for the approximation to be good. +For example, in the figure below, we plot the ratio between the log marginal likelihood +and the BIC score against data-set size; we see that the ratio rapidly +approaches 1, especially for non-informative priors. +(This plot was generated by the file BNT/examples/static/bic1.m. It +uses the water sprinkler BN with BDeu Dirichlet priors with different +equivalent sample sizes.) + +<p> +<center> +<IMG SRC="Figures/bic.png"> +</center> +<p> + +<p> +As with parameter learning, handling missing data/ hidden variables is +much harder than the fully observed case. +The structure learning routines in BNT can therefore be classified into 4 +types, analogously to the parameter learning case. +<p> + +<TABLE BORDER> +<tr> + <TH></TH> + <th>Full obs</th> + <th>Partial obs</th> +</tr> +<tr> + <th>Point</th> + <td><tt>learn_struct_K2</tt> <br> +<!-- <tt>learn_struct_hill_climb</tt></td> --> + <td><tt>not yet supported</tt></td> +</tr> +<tr> + <th>Bayes</th> + <td><tt>learn_struct_mcmc</tt></td> + <td>not yet supported</td> +</tr> +</table> + + +<h2><a name="markov_equiv">Markov equivalence</h2> + +If two DAGs encode the same conditional independencies, they are +called Markov equivalent. The set of all DAGs can be paritioned into +Markov equivalence classes. Graphs within the same class can +have +the direction of some of their arcs reversed without changing any of +the CI relationships. +Each class can be represented by a PDAG +(partially directed acyclic graph) called an essential graph or +pattern. This specifies which edges must be oriented in a certain +direction, and which may be reversed. + +<p> +When learning graph structure from observational data, +the best one can hope to do is to identify the model up to Markov +equivalence. To distinguish amongst graphs within the same equivalence +class, one needs interventional data: see the discussion on <a +href="#active">active learning</a> below. + + + +<h2><a name="enumerate">Exhaustive search</h2> + +The brute-force approach to structure learning is to enumerate all +possible DAGs, and score each one. This provides a "gold standard" +with which to compare other algorithms. We can do this as follows. +<pre> +dags = mk_all_dags(N); +score = score_dags(data, ns, dags); +</pre> +where data(i,m) is the value of node i in case m, +and ns(i) is the size of node i. +If the DAGs have a lot of families in common, we can cache the sufficient statistics, +making this potentially more efficient than scoring the DAGs one at a time. +(Caching is not currently implemented, however.) +<p> +By default, we use the Bayesian scoring metric, and assume CPDs are +represented by tables with BDeu(1) priors. +We can override these defaults as follows. +If we want to use uniform priors, we can say +<pre> +params = cell(1,N); +for i=1:N + params{i} = {'prior', 'unif'}; +end +score = score_dags(data, ns, dags, 'params', params); +</pre> +params{i} is a cell-array, containing optional arguments that are +passed to the constructor for CPD i. +<p> +Now suppose we want to use different node types, e.g., +Suppose nodes 1 and 2 are Gaussian, and nodes 3 and 4 softmax (both +these CPDs can support discrete and continuous parents, which is +necessary since all other nodes will be considered as parents). +The Bayesian scoring metric currently only works for tabular CPDs, so +we will use BIC: +<pre> +score = score_dags(data, ns, dags, 'discrete', [3 4], 'params', [], + 'type', {'gaussian', 'gaussian', 'softmax', softmax'}, 'scoring_fn', 'bic') +</pre> +In practice, one can't enumerate all possible DAGs for N > 5, +but one can evaluate any reasonably-sized set of hypotheses in this +way (e.g., nearest neighbors of your current best guess). +Think of this as "computer assisted model refinement" as opposed to de +novo learning. + + +<h2><a name="K2">K2</h2> + +The K2 algorithm (Cooper and Herskovits, 1992) is a greedy search algorithm that works as follows. +Initially each node has no parents. It then adds incrementally that parent whose addition most +increases the score of the resulting structure. When the addition of no single +parent can increase the score, it stops adding parents to the node. +Since we are using a fixed ordering, we do not need to check for +cycles, and can choose the parents for each node independently. +<p> +The original paper used the Bayesian scoring +metric with tabular CPDs and Dirichlet priors. +BNT generalizes this to allow any kind of CPD, and either the Bayesian +scoring metric or BIC, as in the example <a href="#enumerate">above</a>. +In addition, you can specify +an optional upper bound on the number of parents for each node. +The file BNT/examples/static/k2demo1.m gives an example of how to use K2. +We use the water sprinkler network and sample 100 cases from it as before. +Then we see how much data it takes to recover the generating structure: +<pre> +order = [C S R W]; +max_fan_in = 2; +sz = 5:5:100; +for i=1:length(sz) + dag2 = learn_struct_K2(data(:,1:sz(i)), node_sizes, order, 'max_fan_in', max_fan_in); + correct(i) = isequal(dag, dag2); +end +</pre> +Here are the results. +<pre> +correct = + Columns 1 through 12 + 0 0 0 0 0 0 0 1 0 1 1 1 + Columns 13 through 20 + 1 1 1 1 1 1 1 1 +</pre> +So we see it takes about sz(10)=50 cases. (BIC behaves similarly, +showing that the prior doesn't matter too much.) +In general, we cannot hope to recover the "true" generating structure, +only one that is in its <a href="#markov_equiv">Markov equivalence +class</a>. + + +<h2><a name="hill_climb">Hill-climbing</h2> + +Hill-climbing starts at a specific point in space, +considers all nearest neighbors, and moves to the neighbor +that has the highest score; if no neighbors have higher +score than the current point (i.e., we have reached a local maximum), +the algorithm stops. One can then restart in another part of the space. +<p> +A common definition of "neighbor" is all graphs that can be +generated from the current graph by adding, deleting or reversing a +single arc, subject to the acyclicity constraint. +Other neighborhoods are possible: see +<a href="http://research.microsoft.com/~dmax/publications/jmlr02.pdf"> +Optimal Structure Identification with Greedy Search</a>, Max +Chickering, JMLR 2002. + +<!-- +Note: This algorithm is currently (Feb '02) being implemented by Qian +Diao. +--> + + +<h2><a name="mcmc">MCMC</h2> + +We can use a Markov Chain Monte Carlo (MCMC) algorithm called +Metropolis-Hastings (MH) to search the space of all +DAGs. +The standard proposal distribution is to consider moving to all +nearest neighbors in the sense defined <a href="#hill_climb">above</a>. +<p> +The function can be called +as in the following example. +<pre> +[sampled_graphs, accept_ratio] = learn_struct_mcmc(data, ns, 'nsamples', 100, 'burnin', 10); +</pre> +We can convert our set of sampled graphs to a histogram +(empirical posterior over all the DAGs) thus +<pre> +all_dags = mk_all_dags(N); +mcmc_post = mcmc_sample_to_hist(sampled_graphs, all_dags); +</pre> +To see how well this performs, let us compute the exact posterior exhaustively. +<p> +<pre> +score = score_dags(data, ns, all_dags); +post = normalise(exp(score)); % assuming uniform structural prior +</pre> +We plot the results below. +(The data set was 100 samples drawn from a random 4 node bnet; see the +file BNT/examples/static/mcmc1.) +<pre> +subplot(2,1,1) +bar(post) +subplot(2,1,2) +bar(mcmc_post) +</pre> +<img src="Figures/mcmc_post.jpg" width="800" height="500"> +<p> +We can also plot the acceptance ratio versus number of MCMC steps, +as a crude convergence diagnostic. +<pre> +clf +plot(accept_ratio) +</pre> +<img src="Figures/mcmc_accept.jpg" width="800" height="300"> +<p> +Even though the number of samples needed by MCMC is theoretically +polynomial (not exponential) in the dimensionality of the search space, in practice it has been +found that MCMC does not converge in reasonable time for graphs with +more than about 10 nodes. + + + + +<h2><a name="active">Active structure learning</h2> + +As was mentioned <a href="#markov_equiv">above</a>, +one can only learn a DAG up to Markov equivalence, even given infinite data. +If one is interested in learning the structure of a causal network, +one needs interventional data. +(By "intervention" we mean forcing a node to take on a specific value, +thereby effectively severing its incoming arcs.) +<p> +Most of the scoring functions accept an optional argument +that specifies whether a node was observed to have a certain value, or +was forced to have that value: we set clamped(i,m)=1 if node i was +forced in training case m. e.g., see the file +BNT/examples/static/cooper_yoo. +<p> +An interesting question is to decide which interventions to perform +(c.f., design of experiments). For details, see the following tech +report +<ul> +<li> <a href = "../../Papers/alearn.ps.gz"> +Active learning of causal Bayes net structure</a>, Kevin Murphy, March +2001. +</ul> + + +<h2><a name="struct_em">Structural EM</h2> + +Computing the Bayesian score when there is partial observability is +computationally challenging, because the parameter posterior becomes +multimodal (the hidden nodes induce a mixture distribution). +One therefore needs to use approximations such as BIC. +Unfortunately, search algorithms are still expensive, because we need +to run EM at each step to compute the MLE, which is needed to compute +the score of each model. An alternative approach is +to do the local search steps inside of the M step of EM, which is more +efficient since the data has been "filled in" - this is +called the structural EM algorithm (Friedman 1997), and provably +converges to a local maximum of the BIC score. +<p> +Wei Hu has implemented SEM for discrete nodes. +You can download his package from +<a href="../SEM.zip">here</a>. +Please address all questions about this code to +wei.hu@intel.com. +See also <a href="#phl">Phl's implementation of SEM</a>. + +<!-- +<h2><a name="reveal">REVEAL algorithm</h2> + +A simple way to learn the structure of a fully observed, discrete, +factored DBN from a time series is described <a +href="usage_dbn.html#struct_learn">here</a>. +--> + + +<h2><a name="graphdraw">Visualizing the graph</h2> + +You can visualize an arbitrary graph (such as one learned using the +structure learning routines) with Matlab code contributed by +<a href="http://www.mbfys.kun.nl/~cemgil/matlab/layout.html">Ali +Taylan Cemgil</a> +from the University of Nijmegen. +For static BNs, call it as follows: +<pre> +draw_graph(bnet.dag); +</pre> +For example, this is the output produced on a +<a href="#qmr">random QMR-like model</a>: +<p> +<img src="Figures/qmr.rnd.jpg"> +<p> +If you install the excellent <a +href="http://www.research.att.com/sw/tools/graphviz">graphhviz</a>, an +open-source graph visualization package from AT&T, +you can create a much better visualization as follows +<pre> +graph_to_dot(bnet.dag) +</pre> +This works by converting the adjacency matrix to a file suitable +for input to graphviz (using the dot format), +then converting the output of graphviz to postscript, and displaying the results using +ghostview. +You can do each of these steps separately for more control, as shown +below. +<pre> +graph_to_dot(bnet.dag, 'filename', 'foo.dot'); +dot -Tps foo.dot -o foo.ps +ghostview foo.ps & +</pre> + +<h2><a name = "constraint">Constraint-based methods</h2> + +The IC algorithm (Pearl and Verma, 1991), +and the faster, but otherwise equivalent, PC algorithm (Spirtes, Glymour, and Scheines 1993), +computes many conditional independence tests, +and combines these constraints into a +PDAG to represent the whole +<a href="#markov_equiv">Markov equivalence class</a>. +<p> +IC*/FCI extend IC/PC to handle latent variables: see <a href="#ic_star">below</a>. +(IC stands for inductive causation; PC stands for Peter and Clark, +the first names of Spirtes and Glymour; FCI stands for fast causal +inference. +What we, following Pearl (2000), call IC* was called +IC in the original Pearl and Verma paper.) +For details, see +<ul> +<li> +<a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Causation, +Prediction, and Search</a>, Spirtes, Glymour and +Scheines (SGS), 2001 (2nd edition), MIT Press. +<li> +<a href="http://bayes.cs.ucla.edu/BOOK-2K/index.html">Causality: Models, Reasoning and Inference</a>, J. Pearl, +2000, Cambridge University Press. +</ul> + +<p> + +The PC algorithm takes as arguments a function f, the number of nodes N, +the maximum fan in K, and additional arguments A which are passed to f. +The function f(X,Y,S,A) returns 1 if X is conditionally independent of Y given S, and 0 +otherwise. +For example, suppose we cheat by +passing in a CI "oracle" which has access to the true DAG; the oracle +tests for d-separation in this DAG, i.e., +f(X,Y,S) calls dsep(X,Y,S,dag). We can to this as follows. +<pre> +pdag = learn_struct_pdag_pc('dsep', N, max_fan_in, dag); +</pre> +pdag(i,j) = -1 if there is definitely an i->j arc, +and pdag(i,j) = 1 if there is either an i->j or and i<-j arc. +<p> +Applied to the sprinkler network, this returns +<pre> +pdag = + 0 1 1 0 + 1 0 0 -1 + 1 0 0 -1 + 0 0 0 0 +</pre> +So as expected, we see that the V-structure at the W node is uniquely identified, +but the other arcs have ambiguous orientation. +<p> +We now give an example from p141 (1st edn) / p103 (2nd end) of the SGS +book. +This example concerns the female orgasm. +We are given a correlation matrix C between 7 measured factors (such +as subjective experiences of coital and masturbatory experiences), +derived from 281 samples, and want to learn a causal model of the +data. We will not discuss the merits of this type of work here, but +merely show how to reproduce the results in the SGS book. +Their program, +<a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Tetrad</a>, +makes use of the Fisher Z-test for conditional +independence, so we do the same: +<pre> +max_fan_in = 4; +nsamples = 281; +alpha = 0.05; +pdag = learn_struct_pdag_pc('cond_indep_fisher_z', n, max_fan_in, C, nsamples, alpha); +</pre> +In this case, the CI test is +<pre> +f(X,Y,S) = cond_indep_fisher_z(X,Y,S, C,nsamples,alpha) +</pre> +The results match those of Fig 12a of SGS apart from two edge +differences; presumably this is due to rounding error (although it +could be a bug, either in BNT or in Tetrad). +This example can be found in the file BNT/examples/static/pc2.m. + +<p> + +The IC* algorithm (Pearl and Verma, 1991), +and the faster FCI algorithm (Spirtes, Glymour, and Scheines 1993), +are like the IC/PC algorithm, except that they can detect the presence +of latent variables. +See the file <tt>learn_struct_pdag_ic_star</tt> written by Tamar +Kushnir. The output is a matrix P, defined as follows +(see Pearl (2000), p52 for details): +<pre> +% P(i,j) = -1 if there is either a latent variable L such that i <-L->j OR there is a directed edge from i->j. +% P(i,j) = -2 if there is a marked directed i-*>j edge. +% P(i,j) = P(j,i) = 1 if there is and undirected edge i--j +% P(i,j) = P(j,i) = 2 if there is a latent variable L such that i<-L->j. +</pre> + + +<h2><a name="phl">Philippe Leray's structure learning package</h2> + +Philippe Leray has written a +<a href="http://bnt.insa-rouen.fr/ajouts.html"> +structure learning package</a> that uses BNT. + +It currently (Juen 2003) has the following features: +<ul> +<li>PC with Chi2 statistical test +<li> MWST : Maximum weighted Spanning Tree +<li> Hill Climbing +<li> Greedy Search +<li> Structural EM +<li> hist_ic : optimal Histogram based on IC information criterion +<li> cpdag_to_dag +<li> dag_to_cpdag +<li> ... +</ul> + + +</a> + + +<!-- +<h2><a name="read_learning">Further reading on learning</h2> + +I recommend the following tutorials for more details on learning. +<ul> +<li> <a +href="http://www.cs.berkeley.edu/~murphyk/Papers/intel.ps.gz">My short +tutorial</a> on graphical models, which contains an overview of learning. + +<li> +<A HREF="ftp://ftp.research.microsoft.com/pub/tr/TR-95-06.PS"> +A tutorial on learning with Bayesian networks</a>, D. Heckerman, +Microsoft Research Tech Report, 1995. + +<li> <A HREF="http://www-cad.eecs.berkeley.edu/~wray/Mirror/lwgmja"> +Operations for Learning with Graphical Models</a>, +W. L. Buntine, JAIR'94, 159--225. +</ul> +<p> +--> + + + + + +<h1><a name="engines">Inference engines</h1> + +Up until now, we have used the junction tree algorithm for inference. +However, sometimes this is too slow, or not even applicable. +In general, there are many inference algorithms each of which make +different tradeoffs between speed, accuracy, complexity and +generality. Furthermore, there might be many implementations of the +same algorithm; for instance, a general purpose, readable version, +and a highly-optimized, specialized one. +To cope with this variety, we treat each inference algorithm as an +object, which we call an inference engine. + +<p> +An inference engine is an object that contains a bnet and supports the +'enter_evidence' and 'marginal_nodes' methods. The engine constructor +takes the bnet as argument and may do some model-specific processing. +When 'enter_evidence' is called, the engine may do some +evidence-specific processing. Finally, when 'marginal_nodes' is +called, the engine may do some query-specific processing. + +<p> +The amount of work done when each stage is specified -- structure, +parameters, evidence, and query -- depends on the engine. The cost of +work done early in this sequence can be amortized. On the other hand, +one can make better optimizations if one waits until later in the +sequence. +For example, the parameters might imply +conditional indpendencies that are not evident in the graph structure, +but can nevertheless be exploited; the evidence indicates which nodes +are observed and hence can effectively be disconnected from the +graph; and the query might indicate that large parts of the network +are d-separated from the query nodes. (Since it is not the actual +<em>values</em> of the evidence that matters, just which nodes are observed, +many engines allow you to specify which nodes will be observed when they are constructed, +i.e., before calling 'enter_evidence'. Some engines can still cope if +the actual pattern of evidence is different, e.g., if there is missing +data.) +<p> + +Although being maximally lazy (i.e., only doing work when a query is +issued) may seem desirable, +this is not always the most efficient. +For example, +when learning using EM, we need to call marginal_nodes N times, where N is the +number of nodes. <a href="varelim">Variable elimination</a> would end +up repeating a lot of work +each time marginal_nodes is called, making it inefficient for +learning. The junction tree algorithm, by contrast, uses dynamic +programming to avoid this redundant computation --- it calculates all +marginals in two passes during 'enter_evidence', so calling +'marginal_nodes' takes constant time. +<p> +We will discuss some of the inference algorithms implemented in BNT +below, and finish with a <a href="#engine_summary">summary</a> of all +of them. + + + + + + + +<h2><a name="varelim">Variable elimination</h2> + +The variable elimination algorithm, also known as bucket elimination +or peeling, is one of the simplest inference algorithms. +The basic idea is to "push sums inside of products"; this is explained +in more detail +<a +href="http://HTTP.CS.Berkeley.EDU/~murphyk/Bayes/bayes.html#infer">here</a>. +<p> +The principle of distributing sums over products can be generalized +greatly to apply to any commutative semiring. +This forms the basis of many common algorithms, such as Viterbi +decoding and the Fast Fourier Transform. For details, see + +<ul> +<li> R. McEliece and S. M. Aji, 2000. +<!--<a href="http://www.systems.caltech.edu/EE/Faculty/rjm/papers/GDL.ps">--> +<a href="GDL.pdf"> +The Generalized Distributive Law</a>, +IEEE Trans. Inform. Theory, vol. 46, no. 2 (March 2000), +pp. 325--343. + + +<li> +F. R. Kschischang, B. J. Frey and H.-A. Loeliger, 2001. +<a href="http://www.cs.toronto.edu/~frey/papers/fgspa.abs.html"> +Factor graphs and the sum-product algorithm</a> +IEEE Transactions on Information Theory, February, 2001. + +</ul> + +<p> +Choosing an order in which to sum out the variables so as to minimize +computational cost is known to be NP-hard. +The implementation of this algorithm in +<tt>var_elim_inf_engine</tt> makes no attempt to optimize this +ordering (in contrast, say, to <tt>jtree_inf_engine</tt>, which uses a +greedy search procedure to find a good ordering). +<p> +Note: unlike most algorithms, var_elim does all its computational work +inside of <tt>marginal_nodes</tt>, not inside of +<tt>enter_evidence</tt>. + + + + +<h2><a name="global">Global inference methods</h2> + +The simplest inference algorithm of all is to explicitely construct +the joint distribution over all the nodes, and then to marginalize it. +This is implemented in <tt>global_joint_inf_engine</tt>. +Since the size of the joint is exponential in the +number of discrete (hidden) nodes, this is not a very practical algorithm. +It is included merely for pedagogical and debugging purposes. +<p> +Three specialized versions of this algorithm have also been implemented, +corresponding to the cases where all the nodes are discrete (D), all +are Gaussian (G), and some are discrete and some Gaussian (CG). +They are called <tt>enumerative_inf_engine</tt>, +<tt>gaussian_inf_engine</tt>, +and <tt>cond_gauss_inf_engine</tt> respectively. +<p> +Note: unlike most algorithms, these global inference algorithms do all their computational work +inside of <tt>marginal_nodes</tt>, not inside of +<tt>enter_evidence</tt>. + + +<h2><a name="quickscore">Quickscore</h2> + +The junction tree algorithm is quite slow on the <a href="#qmr">QMR</a> network, +since the cliques are so big. +One simple trick we can use is to notice that hidden leaves do not +affect the posteriors on the roots, and hence do not need to be +included in the network. +A second trick is to notice that the negative findings can be +"absorbed" into the prior: +see the file +BNT/examples/static/mk_minimal_qmr_bnet for details. +<p> + +A much more significant speedup is obtained by exploiting special +properties of the noisy-or node, as done by the quickscore +algorithm. For details, see +<ul> +<li> Heckerman, "A tractable inference algorithm for diagnosing multiple diseases", UAI 89. +<li> Rish and Dechter, "On the impact of causal independence", UCI +tech report, 1998. +</ul> + +This has been implemented in BNT as a special-purpose inference +engine, which can be created and used as follows: +<pre> +engine = quickscore_inf_engine(inhibit, leak, prior); +engine = enter_evidence(engine, pos, neg); +m = marginal_nodes(engine, i); +</pre> + + +<h2><a name="belprop">Belief propagation</h2> + +Even using quickscore, exact inference takes time that is exponential +in the number of positive findings. +Hence for large networks we need to resort to approximate inference techniques. +See for example +<ul> +<li> T. Jaakkola and M. Jordan, "Variational probabilistic inference and the +QMR-DT network", JAIR 10, 1999. + +<li> K. Murphy, Y. Weiss and M. Jordan, "Loopy belief propagation for approximate inference: an empirical study", + UAI 99. +</ul> +The latter approximation +entails applying Pearl's belief propagation algorithm to a model even +if it has loops (hence the name loopy belief propagation). +Pearl's algorithm, implemented as <tt>pearl_inf_engine</tt>, gives +exact results when applied to singly-connected graphs +(a.k.a. polytrees, since +the underlying undirected topology is a tree, but a node may have +multiple parents). +To apply this algorithm to a graph with loops, +use <tt>pearl_inf_engine</tt>. +This can use a centralized or distributed message passing protocol. +You can use it as in the following example. +<pre> +engine = pearl_inf_engine(bnet, 'max_iter', 30); +engine = enter_evidence(engine, evidence); +m = marginal_nodes(engine, i); +</pre> +We found that this algorithm often converges, and when it does, often +is very accurate, but it depends on the precise setting of the +parameter values of the network. +(See the file BNT/examples/static/qmr1 to repeat the experiment for yourself.) +Understanding when and why belief propagation converges/ works +is a topic of ongoing research. +<p> +<tt>pearl_inf_engine</tt> can exploit special structure in noisy-or +and gmux nodes to compute messages efficiently. +<p> +<tt>belprop_inf_engine</tt> is like pearl, but uses potentials to +represent messages. Hence this is slower. +<p> +<tt>belprop_fg_inf_engine</tt> is like belprop, +but is designed for factor graphs. + + + +<h2><a name="sampling">Sampling</h2> + +BNT now (Mar '02) has two sampling (Monte Carlo) inference algorithms: +<ul> +<li> <tt>likelihood_weighting_inf_engine</tt> which does importance +sampling and can handle any node type. +<li> <tt>gibbs_sampling_inf_engine</tt>, written by Bhaskara Marthi. +Currently this can only handle tabular CPDs. +For a much faster and more powerful Gibbs sampling program, see +<a href="http://www.mrc-bsu.cam.ac.uk/bugs">BUGS</a>. +</ul> +Note: To generate samples from a network (which is not the same as inference!), +use <tt>sample_bnet</tt>. + + + +<h2><a name="engine_summary">Summary of inference engines</h2> + + +The inference engines differ in many ways. Here are +some of the major "axes": +<ul> +<li> Works for all topologies or makes restrictions? +<li> Works for all node types or makes restrictions? +<li> Exact or approximate inference? +</ul> + +<p> +In terms of topology, most engines handle any kind of DAG. +<tt>belprop_fg</tt> does approximate inference on factor graphs (FG), which +can be used to represent directed, undirected, and mixed (chain) +graphs. +(In the future, we plan to support exact inference on chain graphs.) +<tt>quickscore</tt> only works on QMR-like models. +<p> +In terms of node types: algorithms that use potentials can handle +discrete (D), Gaussian (G) or conditional Gaussian (CG) models. +Sampling algorithms can essentially handle any kind of node (distribution). +Other algorithms make more restrictive assumptions in exchange for +speed. +<p> +Finally, most algorithms are designed to give the exact answer. +The belief propagation algorithms are exact if applied to trees, and +in some other cases. +Sampling is considered approximate, even though, in the limit of an +infinite number of samples, it gives the exact answer. + +<p> + +Here is a summary of the properties +of all the engines in BNT which work on static networks. +<p> +<table> +<table border units = pixels><tr> +<td align=left width=0>Name +<td align=left width=0>Exact? +<td align=left width=0>Node type? +<td align=left width=0>topology +<tr> +<tr> +<td align=left> belprop +<td align=left> approx +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> belprop_fg +<td align=left> approx +<td align=left> D +<td align=left> factor graph +<tr> +<td align=left> cond_gauss +<td align=left> exact +<td align=left> CG +<td align=left> DAG +<tr> +<td align=left> enumerative +<td align=left> exact +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> gaussian +<td align=left> exact +<td align=left> G +<td align=left> DAG +<tr> +<td align=left> gibbs +<td align=left> approx +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> global_joint +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +<tr> +<td align=left> jtree +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +b<tr> +<td align=left> likelihood_weighting +<td align=left> approx +<td align=left> any +<td align=left> DAG +<tr> +<td align=left> pearl +<td align=left> approx +<td align=left> D,G +<td align=left> DAG +<tr> +<td align=left> pearl +<td align=left> exact +<td align=left> D,G +<td align=left> polytree +<tr> +<td align=left> quickscore +<td align=left> exact +<td align=left> noisy-or +<td align=left> QMR +<tr> +<td align=left> stab_cond_gauss +<td align=left> exact +<td align=left> CG +<td align=left> DAG +<tr> +<td align=left> var_elim +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +</table> + + + +<h1><a name="influence">Influence diagrams/ decision making</h1> + +BNT implements an exact algorithm for solving LIMIDs (limited memory +influence diagrams), described in +<ul> +<li> S. L. Lauritzen and D. Nilsson. +<a href="http://www.math.auc.dk/~steffen/papers/limids.pdf"> +Representing and solving decision problems with limited +information</a> +Management Science, 47, 1238 - 1251. September 2001. +</ul> +LIMIDs explicitely show all information arcs, rather than implicitely +assuming no forgetting. This allows them to model forgetful +controllers. +<p> +See the examples in <tt>BNT/examples/limids</tt> for details. + + + + +<h1>DBNs, HMMs, Kalman filters and all that</h1> + +Click <a href="usage_dbn.html">here</a> for documentation about how to +use BNT for dynamical systems and sequence data. + + +</BODY> diff --git a/sourcecodes/bnt-master/docs/whyNotSourceforge.html b/sourcecodes/bnt-master/docs/whyNotSourceforge.html new file mode 100644 index 00000000..7c17e425 --- /dev/null +++ b/sourcecodes/bnt-master/docs/whyNotSourceforge.html @@ -0,0 +1,17 @@ +On 4 October 2007, I decided to move BNT back from +<a href="http://bnt.sourceforge.net/">sourceforge</a> to my +own website, where I could modify it more easily. I have no plans to +add new functionality, but I at least want it to ensure it runs +correctly on new versions of matlab. Despite new users complaining on +the Yahoo group that BNT would +not work on the latest version of Matlab, nobody in the sourceforce +community did anything about it. As an opensource project, it was basically +a failure (although I would like +to thank Nicholas Saunier and Tom Murray (yozhik) for +making code updates, +as well as Hiroaki Ogawa and mayliszt, +and Bob Welch for answering many questions on the newsgroup.) +For historical purposes, +v1.0.3 (as of 4 Oct 07) has been cached +<a href="http://www.cs.ubc.ca/~murphyk/Software/BNT/FullBNT-1.0.3.zip">here</a>. + |
