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

We consider a pseudo-marginal Metropolis–Hastings kernel Pm that is constructed using an average of m exchangeable random variables, and an analogous kernel Ps 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 Pm in terms of the asymptotic variance of the corresponding ergodic average associated with Ps . 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 Pm is never less than s/m times the variance under Ps . 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.