[Paper Review] Parallel Markov Chain Monte Carlo

Douglas VanDerwerken, Scott C. Schmidler|arXiv (Cornell University)|Dec 28, 2013

Markov Chains and Monte Carlo Methods8 references18 citations

TL;DR

This paper introduces a general-purpose method for parallelizing Markov chain Monte Carlo (MCMC) by combining independent, asynchronous chains through partitioning and weight estimation. It achieves linear speedup proportional to the number of processors, even for unimodal targets, and exponential speedup in multimodal settings, enabling efficient sampling without sacrificing convergence rigor.

ABSTRACT

Markov chain Monte Carlo is an inherently serial algorithm. Although likelihood calculations for individual steps can sometimes be parallelized, the serial evolution of the process is widely viewed as incompatible with parallelization, offering no speedup for samplers which require large numbers of iterations to converge to equilibrium. We provide a methodology for parallelizing Markov chain Monte Carlo across large numbers of independent, asynchronous processors. Our approach uses a partitioning and weight estimation scheme to combine independent simulations run on separate processors into rigorous Monte Carlo estimates. The method is originally motivated by sampling multimodal target distributions, where we see an exponential speedup in running time. However we show that the approach is general-purpose and applicable to all Markov chain Monte Carlo simulations, and demonstrate speedups proportional to the number of available processors on slowly mixing chains with unimodal target distributions. The approach is simple and easy to implement, and suggests additional directions for further research.

Motivation & Objective

• To overcome the inherent serial nature of MCMC, which limits scalability despite advances in parallel computing.
• To enable efficient, rigorous estimation of equilibrium properties using multiple independent, asynchronous MCMC chains.
• To provide a general framework applicable to any MCMC algorithm, including complex samplers like parallel tempering and adaptive MCMC.
• To demonstrate significant speedups in both unimodal and multimodal target distributions through empirical validation.
• To offer a practical, easy-to-implement solution compatible with existing MCMC software such as BUGS.

Proposed method

• Partition the state space into disjoint regions using clustering (e.g., Voronoi partitioning) based on posterior samples from independent chains.
• Estimate the marginal probability of each partition using importance sampling with a proposal distribution tailored to each region.
• Use a weighted combination of samples from all chains, where weights are estimated via instrumental distributions and resampling techniques.
• Apply a two-stage estimation procedure: first estimate partition weights using a subset of draws, then combine full chains using these weights to produce unbiased Monte Carlo estimates.
• Utilize t-distributions with heavy tails as instrumental proposals to ensure robustness in weight estimation.
• Allow for asynchronous execution by decoupling chain simulation from weight estimation, enabling true parallelization.

Experimental results

Research questions

• RQ1 Can MCMC be effectively parallelized across independent, asynchronous processors without compromising the validity of posterior estimates?
• RQ2 Does the proposed method achieve linear speedup in the number of processors for unimodal target distributions?
• RQ3 Can the method significantly reduce convergence time in multimodal distributions, where traditional MCMC struggles?
• RQ4 How does the performance of the parallelized MCMC compare to serial MCMC in terms of accuracy and convergence speed?
• RQ5 Is the method general-purpose and compatible with existing MCMC software and complex sampling schemes?

Key findings

• The method achieves a speedup of over 20× in a unimodal Gibbs sampler, reducing convergence time from 1.2 million to under 60,000 effective iterations.
• In a high-dimensional regression model with 8 covariates, the parallelized Gibbs sampler converged 20× faster than a serial chain, using a convergence threshold of 0.10.
• For the loss of heterozygosity model, the parallel MCMC produced posterior means consistent with importance sampling (e.g., η = 0.816 vs. 0.814), validating accuracy.
• The method demonstrated exponential speedup in multimodal settings, with performance scaling favorably with problem dimension.
• Weight estimation using t-distributed instrumental proposals with 100 iterations and 10,000 draws per partition yielded stable and accurate posterior estimates.
• The approach is robust to slow mixing, as evidenced by successful estimation in chains with autocorrelation >0.999 for β₂, where serial MCMC would require vastly more iterations.