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function [samples, energies, diagn] = metrop(f, x, options, gradf, varargin)
%METROP Markov Chain Monte Carlo sampling with Metropolis algorithm.
%
% Description
% SAMPLES = METROP(F, X, OPTIONS) uses the Metropolis algorithm to
% sample from the distribution P ~ EXP(-F), where F is the first
% argument to METROP. The Markov chain starts at the point X and each
% candidate state is picked from a Gaussian proposal distribution and
% accepted or rejected according to the Metropolis criterion.
%
% SAMPLES = METROP(F, X, OPTIONS, [], P1, P2, ...) allows additional
% arguments to be passed to F(). The fourth argument is ignored, but
% is included for compatibility with HMC and the optimisers.
%
% [SAMPLES, ENERGIES, DIAGN] = METROP(F, X, OPTIONS) also returns a log
% of the energy values (i.e. negative log probabilities) for the
% samples in ENERGIES and DIAGN, a structure containing diagnostic
% information (position and acceptance threshold) for each step of the
% chain in DIAGN.POS and DIAGN.ACC respectively. All candidate states
% (including rejected ones) are stored in DIAGN.POS.
%
% S = METROP('STATE') returns a state structure that contains the state
% of the two random number generators RAND and RANDN. These are
% contained in fields randstate, randnstate.
%
% METROP('STATE', S) resets the state to S. If S is an integer, then
% it is passed to RAND and RANDN. If S is a structure returned by
% METROP('STATE') then it resets the generator to exactly the same
% state.
%
% The optional parameters in the OPTIONS vector have the following
% interpretations.
%
% OPTIONS(1) is set to 1 to display the energy values and rejection
% threshold at each step of the Markov chain. If the value is 2, then
% the position vectors at each step are also displayed.
%
% OPTIONS(14) is the number of samples retained from the Markov chain;
% default 100.
%
% OPTIONS(15) is the number of samples omitted from the start of the
% chain; default 0.
%
% OPTIONS(18) is the variance of the proposal distribution; default 1.
%
% See also
% HMC
%
% Copyright (c) Ian T Nabney (1996-2001)
if nargin <= 2
if ~strcmp(f, 'state')
error('Unknown argument to metrop');
end
switch nargin
case 1
% Return state of sampler
samples = get_state(f); % Function defined in this module
return;
case 2
% Set the state of the sampler
set_state(f, x); % Function defined in this module
return;
end
end
if 0
seed = 42;
randn('state', seed);
rand('state', seed)
end
display = options(1);
if options(14) > 0
nsamples = options(14);
else
nsamples = 100;
end
if options(15) >= 0
nomit = options(15);
else
nomit = 0;
end
if options(18) > 0.0
std_dev = sqrt(options(18));
else
std_dev = 1.0; % default
end
nparams = length(x);
% Set up string for evaluating potential function.
f = fcnchk(f, length(varargin));
samples = zeros(nsamples, nparams); % Matrix of returned samples.
if nargout >= 2
en_save = 1;
energies = zeros(nsamples, 1);
else
en_save = 0;
end
if nargout >= 3
diagnostics = 1;
diagn_pos = zeros(nsamples, nparams);
diagn_acc = zeros(nsamples, 1);
else
diagnostics = 0;
end
% Main loop.
n = - nomit + 1;
Eold = feval(f, x, varargin{:}); % Evaluate starting energy.
nreject = 0; % Initialise count of rejected states.
while n <= nsamples
xold = x;
% Sample a new point from the proposal distribution
x = xold + randn(1, nparams)*std_dev;
%fprintf('netlab propose: xold = %5.3f,%5.3f, xnew = %5.3f,%5.3f\n',...
% xold(1), xold(2), x(1), x(2));
% Now apply Metropolis algorithm.
Enew = feval(f, x, varargin{:}); % Evaluate new energy.
a = exp(Eold - Enew); % Acceptance threshold.
if (diagnostics & n > 0)
diagn_pos(n,:) = x;
diagn_acc(n,:) = a;
end
if (display > 1)
fprintf(1, 'New position is\n');
disp(x);
end
r = rand(1);
%fprintf('netlab: n=%d, a=%f/%f=%5.3f (%5.3f), r=%5.3f\n',...
% n, exp(-Enew), exp(-Eold), a, exp(-Enew)/exp(-Eold), r);
if a > r % Accept the new state.
Eold = Enew;
if (display > 0)
fprintf(1, 'Finished step %4d Threshold: %g\n', n, a);
end
else % Reject the new state
if n > 0
nreject = nreject + 1;
end
x = xold; % Reset position
if (display > 0)
fprintf(1, ' Sample rejected %4d. Threshold: %g\n', n, a);
end
end
if n > 0
samples(n,:) = x; % Store sample.
if en_save
energies(n) = Eold; % Store energy.
end
end
n = n + 1;
end
if (display > 0)
fprintf(1, '\nFraction of samples rejected: %g\n', ...
nreject/(nsamples));
end
if diagnostics
diagn.pos = diagn_pos;
diagn.acc = diagn_acc;
end
% Return complete state of the sampler.
function state = get_state(f)
state.randstate = rand('state');
state.randnstate = randn('state');
return
% Set state of sampler, either from full state, or with an integer
function set_state(f, x)
if isnumeric(x)
rand('state', x);
randn('state', x);
else
if ~isstruct(x)
error('Second argument to metrop must be number or state structure');
end
if (~isfield(x, 'randstate') | ~isfield(x, 'randnstate'))
error('Second argument to metrop must contain correct fields')
end
rand('state', x.randstate);
randn('state', x.randnstate);
end
return
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