Sherlock, Christopher Gerrard and Thiery, Alex and Lee, Anthony (2017) Pseudo-marginal Metropolis-Hastings sampling using averages of unbiased estimators. Biometrika, 104 (3). pp. 727-734. ISSN 0006-3444
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Abstract
We consider a pseudo-marginal Metropolis--Hastings kernel Pm 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 Pm 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 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 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.