This paper explores how and when to use common random number (CRN) simulation to evaluate MCMC convergence rates. We discuss how CRN simulation is closely related to theoretical convergence rate techniques such as one-shot coupling and coupling from the past. We present conditions under which the CRN technique generates an unbiased estimate of the Wasserstein distance between two random variables. We also discuss how unbiasedness of the Wasserstein distance between two Markov chains over a single iteration does not extend to unbiasedness over multiple iterations. We provide an upper bound on the Wasserstein distance of a Markov chain to its stationary distribution after $N$ steps in terms of averages over CRN simulations. We apply our result to a Bayesian regression Gibbs sampler.

翻译：本文探讨了如何以及何时使用通用随机数（CRN）模拟来评估马尔可夫链蒙特卡洛（MCMC）收敛速度。我们讨论了CRN模拟与理论收敛速率技术（如单次耦合和过去耦合）之间的密切联系。我们给出了CRN技术能够生成两个随机变量之间Wasserstein距离无偏估计的条件。同时，我们指出单次迭代中两条马尔可夫链之间Wasserstein距离的无偏性并不能推广至多次迭代。我们基于CRN模拟的平均值，提供了一个马尔可夫链经过$N$步后与其平稳分布之间Wasserstein距离的上界。我们将该结果应用于贝叶斯回归吉布斯采样器。