Transition kernel couplings of the Metropolis-Hastings algorithm

Couplings play a central role in the analysis of Markov chain convergence to stationarity and in the construction of novel Markov chain Monte Carlo diagnostics, estimators, and variance reduction techniques. The quality of the resulting bounds or methods typically depends on how quickly the coupling induces meeting between chains, a property sometimes referred to as its efficiency. The design of efficient Markovian couplings remains a difficult open question, especially for discrete time processes. In pursuit of this goal, in this paper we fully characterize the couplings of the Metropolis-Hastings (MH) transition kernel, providing necessary and sufficient conditions in terms of the underlying proposal and acceptance distributions. We apply these results to characterize the set of maximal couplings of the MH kernel, resolving open questions posed in \citet{OLeary2020} on the structure and properties of these couplings. These results represent an advance in the understanding of the MH kernel and a step toward the formulation of efficient couplings for this popular family of algorithms.