Establishing some order amongst exact approximations of MCMCs

Exact approximations of Markov chain Monte Carlo (MCMC) algorithms are a general emerging class of sampling algorithms. One of the main ideas behind exact approximations consists of replacing intractable quantities required to run standard MCMC algorithms, such as the target probability density in a Metropolis-Hastings algorithm, with estimators. Perhaps surprisingly, such approximations lead to powerful algorithms which are exact in the sense that they are guaranteed to have correct limiting distributions. In this paper we discover a general framework which allows one to compare, or order, performance measures of two implementations of such algorithms. In particular, we establish an order with respect to the mean acceptance probability, the first autocorrelation coefficient, the asymptotic variance and the right spectral gap. The key notion to guarantee the ordering is that of the convex order between estimators used to implement the algorithms. We believe that our convex order condition is close to optimal, and this is supported by a counter-example which shows that a weaker variance order is not sufficient. The convex order plays a central role by allowing us to construct a martingale coupling which enables the comparison of performance measures of Markov chain with differing invariant distributions, contrary to existing results. We detail applications of our result by identifying extremal distributions within given classes of approximations, by showing that averaging replicas improves performance in a monotonic fashion and that stratification is guaranteed to improve performance for the standard implementation of the Approximate Bayesian Computation (ABC) MCMC method.