  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.
