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.

翻译：马尔可夫链蒙特卡洛的收敛诊断是计算统计学中的一个基础性问题：它决定了特定采样问题的资源分配，并影响从业者对马尔可夫链估计质量的评估。基于此，我们针对新兴的基于耦合的收敛诊断算法做出贡献。具体而言，我们研究了利用多边际耦合耦合多条Metropolis-Hastings链的方法。我们为此场景提出一个自然目标，并通过建立与列表级分布耦合和分布式配对匹配问题的联系，确定了下界与上界。这一分析最终导出了用于耦合多条马尔可夫链的共享随机性泊松蒙特卡洛构造。在此过程中，我们通过开发一种更新点过程的自适应规则，避免了经典泊松蒙特卡洛运行时复杂度中关键维度依赖的瓶颈，在高维场景下实现了显著性能提升。对马尔可夫链的大规模耦合实验表明，我们的方法在多个维度上改善了合并率，相较于现有基线，相遇时间减少了高达50%。