Constructing sampling schemes via coupling: Markov semigroups and optimal transport
For researchers in computational statistics and Bayesian inference, it provides a theoretical foundation for designing efficient coupled MCMC samplers, though the results are largely theoretical with limited empirical validation.
This paper develops a general framework for constructing and analyzing coupled Markov chain Monte Carlo samplers, linking coupling efficiency to optimal transport theory. It proves a singularity theorem and derives a modified Poincaré inequality, with numerical experiments supporting the findings.
In this paper we develop a general framework for constructing and analysing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincaré inequality.