A large deviation principle for the empirical measures of Metropolis–Hastings chains

Abstract

To sample from a given target distribution, Markov chain Monte Carlo (MCMC) sampling relies on constructing an ergodic Markov chain with the target distribution as its invariant measure. For any MCMC method, an important question is how to evaluate its efficiency. One approach is to consider the associated empirical measure and how fast it converges to the stationary distribution of the underlying Markov process. Recently, this question has been considered from the perspective of large deviation theory, for different types of MCMC methods, including, e.g., non-reversible Metropolis–Hastings on a finite state space, non-reversible Langevin samplers, the zig-zag sampler, and parallel tempering. This approach, based on large deviations, has proven successful in analysing existing methods and designing new, efficient ones. However, for the Metropolis–Hastings algorithm on more general state spaces, the workhorse of MCMC sampling, the same techniques have not been available for analysing performance, as the underlying Markov chain dynamics violate the conditions used to prove existing large deviation results for empirical measures of a Markov chain. This also extends to methods built on the same idea as Metropolis–Hastings, such as the Metropolis-Adjusted Langevin Method or ABC-MCMC. In this paper, we take the first steps towards such a large-deviations based analysis of Metropolis–Hastings-like methods, by proving a large deviation principle for the empirical measures of Metropolis–Hastings chains. In addition, we also characterize the rate function and its properties in terms of the acceptance- and rejection-part of the Metropolis–Hastings dynamics.