Bounding and estimating MCMC convergence rates using common random number simulations

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.