Multi-Marginal Couplings for Metropolis-Hastings
Abstract
Convergence diagnosis for Markov chain Monte Carlo is a matter of fundamental importance in computational statistics: it determines the resources allocated to a particular sampling problem and influences the practitioner's view of the quality of estimates obtained from a Markov chain. Motivated by this, we contribute to the emerging class of coupling-based convergence diagnostic algorithms. Concretely, we study coupling multiple Metropolis-Hastings chains using multi-marginal coupling. We introduce a natural objective for this setting and establish lower and upper bounds by drawing connections to list-level distribution coupling and distributed pairwise-matching problems. This analysis ultimately leads to a shared-randomness Poisson Monte Carlo construction for coupling multiple Markov chains. In this process, we avoid a key dimension-dependent bottleneck in the runtime complexity of classical Poisson Monte Carlo by developing an adaptive rule for updating the point process, yielding significant gains in high-dimensional settings. Experiments on grand couplings of Markov chains show that our methods improve coalescence rates across dimensions, reducing meeting times by up to 50% compared with existing baselines.
Cite
@article{arxiv.2605.12807,
title = {Multi-Marginal Couplings for Metropolis-Hastings},
author = {Buu Phan and Gergely Flamich and Ashish Khisti and Shahab Asoodeh},
journal= {arXiv preprint arXiv:2605.12807},
year = {2026}
}