Multi-Marginal Couplings for Metropolis-Hastings

2026-05-12

AI Summary

This work addresses the challenge of diagnosing convergence in Markov chain Monte Carlo (MCMC) methods by proposing an efficient diagnostic framework based on multi-marginal coupling. By introducing shared randomness across multiple Metropolis–Hastings chains, the authors construct a Poisson Monte Carlo estimator and develop an adaptive point-process update rule alongside a distributed matching algorithm, substantially alleviating computational bottlenecks in high-dimensional settings. The approach establishes theoretical connections to list-level distribution coupling and distributed matching problems, yielding a natural optimization objective tailored for multi-chain coupling. Experimental results demonstrate that, across various dimensional configurations, the proposed method reduces coupling time by up to 50% compared to existing baselines, achieving significantly improved diagnostic efficiency.

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.

Problem

• Markov chain Monte Carlo
• convergence diagnosis
• Metropolis-Hastings
• multi-marginal coupling
• coalescence

Innovation

• multi-marginal coupling
• Metropolis-Hastings
• convergence diagnosis
• Poisson Monte Carlo
• high-dimensional efficiency