  Markov chain Monte Carlo (MCMC) is the engine of modern Bayesian statistics,
being used to approximate the posterior and derived quantities of interest.
Despite this, the issue of how the output from a Markov chain is post-processed
and reported is often overlooked. Convergence diagnostics can be used to
control bias via burn-in removal, but these do not account for (common)
situations where a limited computational budget engenders a bias-variance
trade-off. The aim of this article is to review state-of-the-art techniques for
post-processing Markov chain output. Our review covers methods based on
discrepancy minimisation, which directly address the bias-variance trade-off,
as well as general-purpose control variate methods for approximating expected
quantities of interest.
