Unbiased Markov chain Monte Carlo with couplings

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.

Final version, accepted as a JRSS discussion paper; includes supplementary material as appendices; 12 figures, 48 pages

References in corpus (7)

• Geometric integrators and the Hamiltonian Monte Carlo method
• Piecewise-Deterministic Markov Chain Monte Carlo
• Rapid Mixing of Hamiltonian Monte Carlo on Strongly Log-Concave Distributions
• Better together? Statistical learning in models made of modules
• Circularly-Coupled Markov Chain Sampling
• Unbiased estimation of log normalizing constants with applications to Bayesian cross-validation
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