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|
<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.
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