AffineInvariantMCMC performs Bayesian sampling using Goodman & Weare's Affine Invariant Markov Chain Monte Carlo (MCMC) Ensemble sampler. AffineInvariantMCMC is a module of MADS. Goodman & Weare's algorithm implementation in Python is called Emcee.
Goodman, Jonathan, and Jonathan Weare. "Ensemble samplers with affine invariance." Communications in applied mathematics and computational science 5.1 (2010): 65-80. Link
import Pkg; Pkg.add("AffineInvariantMCMC")
import AffineInvariantMCMC numdims = 5 numwalkers = 100 thinning = 10 numsamples_perwalker = 1000 burnin = 100 const stds = exp(5 * randn(numdims)) const means = 1 + 5 * rand(numdims) llhood = x->begin retval = 0. for i in 1:length(x) retval -= .5 * ((x[i] - means[i]) / stds[i]) ^ 2 end return retval end x0 = rand(numdims, numwalkers) * 10 - 5 chain, llhoodvals = AffineInvariantMCMC.sample(llhood, numwalkers, x0, burnin, 1) chain, llhoodvals = AffineInvariantMCMC.sample(llhood, numwalkers, chain[:, :, end], numsamples_perwalker, thinning) flatchain, flatllhoodvals = AffineInvariantMCMC.flattenmcmcarray(chain, llhoodvals)
The figures below compare predicted marginal and joint posterior PDF's (probability density functions) using Classical vs. Affine Invariant MCMC for the same number of functional evaluations (in this case 1,000,000).
The synthetic problem tested below is designed to have a very complex structure. The Classical MCMC clearly fails to characterize sufficiently well the posterior PDF's.
- Classical MCMC
- Affine Invariant MCMC
The codes applied to perform these analyses are available here:
AffineInvariantMCMC can be executed efficiently in parallel using existing distributed network capabilities.
For more information, check out our Julia module RobustPmap.
AffineInvariantMCMC analyses can be performed utilizing extremely efficient restarts.
Typically, the AffineInvariantMCMC runs require large number of functional (model) evaluations which may take substantial computational time. Occasionally, the AffineInvariantMCMC runs may crash due to external issues (e.g., network/computer/disk failures). Furthermore, AffineInvariantMCMC runs may require more time than the allowed allocation time on existing HPC cluster queues. In all these cases, the AffineInvariantMCMC runs needs to be restarted. Our codes allow are very efficient restarts with very minimal overhead and without re-execution of completed functional (model) evaluations.
For more information, check out our Julia module ReusableFunctions.
All the available MADS modules and functions are described at madsjulia.github.io
AffineInvariantMCMC functions are documented at https://madsjulia.github.io/Mads.jl/Modules/AffineInvariantMCMC