Stability of noisy Metropolis–Hastings

Pseudo-marginal Markov chain Monte Carlo methods for sampling from intractable distributions have gained recent interest and have been theoretically studied in considerable depth. Their main appeal is that they are exact, in the sense that they target marginally the correct invariant distribution. However, the pseudo-marginal Markov chain can exhibit poor mixing and slow convergence towards its target. As an alternative, a subtly different Markov chain can be simulated, where better mixing is possible but the exactness property is sacrificed. This is the noisy algorithm, initially conceptualised as Monte Carlo within Metropolis, which has also been studied but to a lesser extent. The present article provides a further characterisation of the noisy algorithm, with a focus on fundamental stability properties like positive recurrence and geometric ergodicity. Sufficient conditions for inheriting geometric ergodicity from a standard Metropolis–Hastings chain are given, as well as convergence of the invariant distribution towards the true target distribution.