Pseudo-marginal Metropolis–Hastings 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, as well as an analogous kernel Ps that averages 15 s < m of these same random variables. Using an embedding technique to facilitate comparisons, we show that the asymptotic variances of ergodic averages associated with Pm are lower bounded in terms of those associated with Ps. We show that the bound provided 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, 20 however, hold when considering 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 pseudomarginal 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 25 the second there is a considerable start-up cost at each iteration