Pseudo-marginal Metropolis–Hastings sampling using averages of unbiased estimators

Abstract

We consider a pseudo-marginal Metropolis–Hastings kernel P m that is constructed using an average of m exchangeable random variables, and an analogous kernel P s that averages s < m of these same random variables. Using an embedding technique to facilitate comparisons, we provide a lower bound for the asymptotic variance of any ergodic average associated with P m in terms of the asymptotic variance of the corresponding ergodic average associated with P s ⁠ . We show that the bound is tight and disprove a conjecture that when the random variables to be averaged are independent, the asymptotic variance under P m is never less than s / m times the variance under P s ⁠ . The conjecture does, however, hold for continuous-time Markov chains. These results imply that if the computational cost of the algorithm is proportional to m ⁠ , it is often better to set m = 1 ⁠ . We provide intuition as to why these findings differ so markedly from recent results for pseudo-marginal kernels employing particle filter approximations. Our results are exemplified through two simulation studies; in the first the computational cost is effectively proportional to m and in the second there is a considerable start-up cost at each iteration.