Couplings play a central role in the analysis of Markov chain convergence and in the construction of new Markov chain Monte Carlo estimators, diagnostics, and variance reduction techniques. Tight bounds and efficient methods require an appropriate choice of coupling, which can be hard to find when the set of possible couplings is intractable. To address this challenge for the Metropolis--Hastings (MH) family of algorithms, we establish a simple characterization of the set of all MH transition kernel couplings. We then extend this result to describe the set of maximal couplings of the MH kernel, resolving an open question of O'Leary et al. [2021]. We conclude with a series of examples to build intuition. Our results represent an advance in understanding the MH kernel and a step forward for coupling this popular class of algorithms.

翻译：在分析Markov链条趋同和建造新的Markov链条Monte Carlo测算器、诊断和减少差异技术方面,串联在分析Markov链条趋同和建造新的Markov链条Monte Carlo测算器、诊断和减少差异技术方面发挥着核心作用。 紧凑的界限和有效方法要求适当选择结合,当一系列可能的联结难以解决时,很难找到这种结合。 要应对大都会-哈斯廷斯算法体系的这一挑战,我们就对所有MH过渡核心联结进行简单的定性。 然后,我们将这一结果扩展为描述MH内核的最大组合,解决O'Leary et al. [2021] 的开放问题。 我们以一系列例子来结束我们的工作,以建立直觉为例。 我们的结果代表了理解MH内核的进步,是将这一流行的算法组合向前迈出的一步。