  Couplings play a central role in the analysis of Markov chain Monte Carlo
algorithms and appear increasingly often in the algorithms themselves, e.g. in
convergence diagnostics, parallelization, and variance reduction techniques.
Existing couplings of the Metropolis-Hastings algorithm handle the proposal and
acceptance steps separately and fall short of the upper bound on one-step
meeting probabilities given by the coupling inequality. This paper introduces
maximal couplings which achieve this bound while retaining the practical
advantages of current methods. We consider the properties of these couplings
and examine their behavior on a selection of numerical examples.
