Metropolis-Hastings transition kernel couplings

TL;DR

This paper provides a simple characterization of couplings for Metropolis-Hastings transition kernels, including maximal couplings, advancing the understanding of these algorithms for convergence analysis and estimator development.

Contribution

It introduces a straightforward description of MH kernel couplings and resolves an open question about maximal couplings, enhancing theoretical understanding.

Findings

Characterization of MH transition kernel couplings

Extension to maximal couplings of MH kernels

Resolution of an open question by O'Leary et al.

Abstract

Couplings play a central role in the analysis of Markov chain convergence and in the construction of novel Markov chain Monte Carlo estimators, diagnostics, and variance reduction techniques. The set of possible couplings is often intractable, frustrating the search for tight bounds and efficient estimators. To address this challenge for algorithms in the Metropolis-Hastings (MH) family, we establish a simple characterization of the set of MH transition kernel couplings. We then extend this result to describe the set of maximal couplings of the MH kernel, resolving an open question of O'Leary et al.. Our results represent an advance in understanding the MH transition kernel and a step forward for coupling this popular class of algorithms.