Bounding Wasserstein distance with couplings

Markov chain Monte Carlo (MCMC) provides asymptotically consistent estimates of intractable posterior expectations as the number of iterations tends to infinity. However, in large data applications, MCMC can be computationally expensive per iteration. This has catalyzed interest in approximating MCMC in a manner that improves computational speed per iteration but does not produce asymptotically consistent estimates. In this article, we propose estimators based on couplings of Markov chains to assess the quality of such asymptotically biased sampling methods. The estimators give empirical upper bounds of the Wasserstein distance between the limiting distribution of the asymptotically biased sampling method and the original target distribution of interest. We establish theoretical guarantees for our upper bounds and show that our estimators can remain effective in high dimensions. We apply our quality measures to stochastic gradient MCMC, variational Bayes, and Laplace approximations for tall data and to approximate MCMC for Bayesian logistic regression in 4500 dimensions and Bayesian linear regression in 50000 dimensions.