Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms

TL;DR

This paper analyzes the convergence properties of pseudo-marginal MCMC algorithms, showing how their spectral gaps and convergence rates relate to the marginal algorithms, with implications for their efficiency and accuracy.

Contribution

It establishes conditions under which pseudo-marginal algorithms inherit spectral gaps and convergence rates from marginal algorithms, including cases with unbounded weights.

Findings

Pseudo-marginal algorithms have asymptotic variance at least as large as marginal algorithms.

Spectral gap inheritance occurs when weights are bounded and the marginal chain has a spectral gap.

Convergence rates can be polynomial or geometric, depending on the target distribution and estimator accuracy.

Abstract

We study convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms (Andrieu and Roberts [Ann. Statist. 37 (2009) 697-725]). We find that the asymptotic variance of the pseudo-marginal algorithm is always at least as large as that of the marginal algorithm. We show that if the marginal chain admits a (right) spectral gap and the weights (normalised estimates of the target density) are uniformly bounded, then the pseudo-marginal chain has a spectral gap. In many cases, a similar result holds for the absolute spectral gap, which is equivalent to geometric ergodicity. We consider also unbounded weight distributions and recover polynomial convergence rates in more specific cases, when the marginal algorithm is uniformly ergodic or an independent Metropolis-Hastings or a random-walk Metropolis targeting a super-exponential density with regular contours. Our results on geometric and polynomial convergence rates imply central limit theorems. We also prove that under general conditions, the asymptotic variance of the pseudo-marginal algorithm converges to the asymptotic variance of the marginal algorithm if the accuracy of the estimators is increased.