Unbiased Markov chain Monte Carlo with couplings

TL;DR

This paper introduces a method to produce unbiased MCMC estimators using couplings and telescopic sums, enabling parallel computation and addressing bias issues in standard MCMC methods.

Contribution

It proposes a novel unbiased estimation technique for MCMC using couplings, with theoretical validation and practical algorithms for common MCMC methods.

Findings

Unbiased estimators can be computed independently and in parallel.

The method is effective on toy examples and complex models like Ising and high-dimensional problems.

Limitations include potential efficiency loss in certain scenarios.

Abstract

Markov chain Monte Carlo (MCMC) methods provide consistent of integrals as the number of iterations goes to infinity. MCMC estimators are generally biased after any fixed number of iterations. We propose to remove this bias by using couplings of Markov chains together with a telescopic sum argument of Glynn and Rhee (2014). The resulting unbiased estimators can be computed independently in parallel. We discuss practical couplings for popular MCMC algorithms. We establish the theoretical validity of the proposed estimators and study their efficiency relative to the underlying MCMC algorithms. Finally, we illustrate the performance and limitations of the method on toy examples, on an Ising model around its critical temperature, on a high-dimensional variable selection problem, and on an approximation of the cut distribution arising in Bayesian inference for models made of multiple modules.