Ensemble MCMC sampler

An affine invariant ensemble Markov Chain Monte Carlo sampler

Cascaded affine invariant ensemble MCMC sampler. "The MCMC hammer"

gwmcmc is an implementation of the Goodman and Weare 2010 Affine
invariant ensemble Markov Chain Monte Carlo (MCMC) sampler. MCMC sampling
enables bayesian inference. The problem with many traditional MCMC samplers
is that they can have slow convergence for badly scaled problems, and that
it is difficult to optimize the random walk for high-dimensional problems.
This is where the GW-algorithm really excels as it is affine invariant. It
can achieve much better convergence on badly scaled problems. It is much
simpler to get to work straight out of the box, and for that reason it
truly deserves to be called the MCMC hammer.

(This code uses a cascaded variant of the Goodman and Weare algorithm).

USAGE:
[models,logP]=gwmcmc(minit,logPfuns,mccount,[Parameter,Value,Parameter,Value]);

INPUTS:
minit: an MxW matrix of initial values for each of the walkers in the
ensemble. (M:number of model params. W: number of walkers). W
should be atleast 2xM. (see e.g. mvnrnd).
logPfuns: a cell of function handles returning the log probality of a
proposed set of model parameters. Typically this cell will
contain two function handles: one to the logprior and another
to the loglikelihood. E.g. {@(m)logprior(m) @(m)loglike(m)}
mccount: What is the desired total number of monte carlo proposals.
This is the total number, -NOT the number per chain.

Named Parameter-Value pairs:
'StepSize': unit-less stepsize (default=2.5).
'ThinChain': Thin all the chains by only storing every N'th step (default=10)
'ProgressBar': Show a text progress bar (default=true)
'Parallel': Run in ensemble of walkers in parallel. (default=false)

OUTPUTS:
models: A MxWxT matrix with the thinned markov chains (with T samples
per walker). T=~mccount/p.ThinChain/W.
logP: A PxWxT matrix of log probabilities for each model in the
models. here P is the number of functions in logPfuns.

Note on cascaded evaluation of log probabilities:
The logPfuns-argument can be specifed as a cell-array to allow a cascaded
evaluation of the probabilities. The computationally cheapest function should be
placed first in the cell (this will typically the prior). This allows the
routine to avoid calculating the likelihood, if the proposed model can be
rejected based on the prior alone.
logPfuns={logprior loglike} is faster but equivalent to
logPfuns={@(m)logprior(m)+loglike(m)}

TIP: if you aim to analyze the entire set of ensemble members as a single
sample from the distribution then you may collapse output models-matrix
thus: models=models(:,:); This will reshape the MxWxT matrix into a
Mx(W*T)-matrix while preserving the order.


EXAMPLE: Here we sample a multivariate normal distribution.

%define problem:
mu = [5;-3;6];
C = [.5 -.4 0;-.4 .5 0; 0 0 1];
iC=pinv(C);
logPfuns={@(m)-0.5*sum((m-mu)'*iC*(m-mu))}

%make a set of starting points for the entire ensemble of walkers
minit=randn(length(mu),length(mu)*2);

%Apply the MCMC hammer
[models,logP]=gwmcmc(minit,logPfuns,100000);
models(:,:,1:floor(size(models,3)*.2))=[]; %remove 20% as burn-in
models=models(:,:)'; %reshape matrix to collapse the ensemble member dimension
scatter(models(:,1),models(:,2))
prctile(models,[5 50 95])


References:
Goodman & Weare (2010), Ensemble Samplers With Affine Invariance, Comm. App. Math. Comp. Sci., Vol. 5, No. 1, 65–80
Foreman-Mackey, Hogg, Lang, Goodman (2013), emcee: The MCMC Hammer, arXiv:1202.3665

WebPage: https://github.com/grinsted/gwmcmc

-Aslak Grinsted 2015