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
Aslak Grinsted (2019). Ensemble MCMC sampler (https://www.github.com/grinsted/gwmcmc), GitHub. Retrieved .
1.8.0.0 | changed title |
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1.8.0.0 | included: ecornerplot, eacorr, ess calculation |
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1.7.0.0 | Adding FEX description yet again. |
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1.6.0.0 | Fixed important bug in inputparsing |
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1.6.0.0 | accidently removed description |
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1.6.0.0 | Updated title, and |
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1.5.0.0 | updated published html examples, and FEX description |
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1.3.0.0 | Changed title |
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1.2.0.0 | Moved to github |
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1.1.0.0 | updated help description and typo in example. |
Inspired by: 2D Bandwidth Estimator for KDE, Markov Chain Monte Carlo sampling of posterior distribution, Cornerplot
Inspired: Markov Chain Monte Carlo sampling of posterior distribution
Create scripts with code, output, and formatted text in a single executable document.
Sanjay Manohar (view profile)
Sanjay Manohar (view profile)
Note- the parallel implementation generates one sample in each worker, then needs to coalesce before running the next sample. So only useful with very slow objective functions... Needs to be improved using spmd? if anyone knows how to do this that would be great!
name1 name2 (view profile)
hi what is the difference between your algorithm and the rjmcmc of green?
Gilberto Gonzalez-Parra (view profile)
Can I get 95% confidence intervals for each parameter ?
Aslak Grinsted (view profile)
The algorithm can only explore the dimensions of space spanned by the starting points of the walkers. If all the walkers start in the same point then they will never move.
Romain Fiévet (view profile)
works great otherwise, nice good!
Romain Fiévet (view profile)
I am having a problem describing a simple Gaussian prior with this code. The chain only has constant value! I tried the line_fit example and it works, but the moment I remove the randomness from the initial ensemble of walkers, it also contains only constant chains. Is this normal behavior ? Shouldnt the random walk move away from the initial position ?
Kusa Amu (view profile)
Is it possible to apply hierarchical bayes? If yes, could you provide an example to determine hyper-parameters of the slope from the linefit example?
That would be very helpful.
Liguang Jiang (view profile)
No boundary handing when new candidate is proposed!
AMRANE youcef (view profile)
can help me
Nathan Bowman (view profile)
Aslak Grinsted (view profile)
@Yalei: Good question. I guess this website shows uncompressed size. But it is out of my control.
Yalei Ding (view profile)
File Size: 1.35 MB
File ID: #49820
Version: 1.8
why the zip i downloaded was 391kb?
Aslak Grinsted (view profile)
@Arpad Rozsas: thank you. I have fixed the typo in the example
Arpad Rozsas (view profile)
Thanks for this great code!
I think there is a minor error in the example.
logP={@(m)-0.5*sum((m-mu)'*iC*(m-mu))}
should be:
logPfuns={@(m)-0.5*sum((m-mu)'*iC*(m-mu))}
Luis O'Shea (view profile)