Estimating Convergence of Markov chains with L-Lag Couplings

Abstract

Markov chain Monte Carlo (MCMC) methods generate samples that are asymptotically distributed from a target distribution of interest as the number of iterations goes to infinity. Various theoretical results provide upper bounds on the distance between the target and marginal distribution after a fixed number of iterations. These upper bounds are on a case by case basis and typically involve intractable quantities, which limits their use for practitioners. We introduce L-lag couplings to generate computable, non-asymptotic upper bound estimates for the total variation or the Wasserstein distance of general Markov chains. We apply L-lag couplings to the tasks of (i) determining MCMC burn-in, (ii) comparing different MCMC algorithms with the same target, and (iii) comparing exact and approximate MCMC. Lastly, we (iv) assess the bias of sequential Monte Carlo and self-normalized importance samplers.

Unbiased Markov chain Monte Carlo with couplings

“…), and we can approximate E[max(0, (τ − k − 1))] by Monte Carlo for a range of k values. This is pursued in Biswas and Jacob [2019], where the proposed construction is extended to allow for arbitrary time lags between the coupled chains.…”

“…Therefore, the cost of H A variant of (2.1) can be obtained by considering a time lag greater than one between the two chains (X t ) t≥0 and (Y t ) t≥0 , with the meeting time defined as the first time t for which {X t = Y t−lag } occurs. This introduces another tuning parameter but is found to be fruitful in Biswas and Jacob [2019].…”