Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms

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 is geometrically ergodic and the weights (normalised estimates of the target density) are uniformly bounded, then the pseudo-marginal chain is geometric. We consider also unbounded weight distributions and recover polynomial convergence rates in more specific cases, when the marginal algorithm is uniformly ergodic, 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.