Pseudo‐marginal Metropolis‐Hastings sampling using averages of unbiased estimators

Summary We consider a pseudo‐marginal Metropolis‐Hastings kernel Symbol that is constructed using an average of m exchangeable random variables, and an analogous kernel Symbol 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 Symbol in terms of the asymptotic variance of the corresponding ergodic average associated with Symbol. 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 Symbol is never less than s/m times the variance under Symbol. 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. Symbol. No Caption available. Symbol. No Caption available. Symbol. No Caption available. Symbol. No Caption available. Symbol. No Caption available. Symbol. No Caption available.