Maximal couplings of the Metropolis-Hastings algorithm

TL;DR

This paper introduces maximal couplings for the Metropolis-Hastings algorithm that achieve the theoretical upper bound on meeting probabilities, enhancing convergence analysis and practical applications.

Contribution

It develops a new class of couplings that attain the coupling inequality bound while maintaining practical usability in MCMC algorithms.

Findings

Maximal couplings improve meeting probabilities in MCMC.

Numerical examples demonstrate enhanced convergence properties.

Theoretical analysis confirms optimality of the proposed couplings.

Abstract

Couplings play a central role in the analysis of Markov chain Monte Carlo algorithms and appear increasingly often in the algorithms themselves, e.g. in convergence diagnostics, parallelization, and variance reduction techniques. Existing couplings of the Metropolis-Hastings algorithm handle the proposal and acceptance steps separately and fall short of the upper bound on one-step meeting probabilities given by the coupling inequality. This paper introduces maximal couplings which achieve this bound while retaining the practical advantages of current methods. We consider the properties of these couplings and examine their behavior on a selection of numerical examples.