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

原文摘要

Summary We consider a pseudo-marginal Metropolis–Hastings kernel ${\mathbb{P}}_m$ that is constructed using an average of $m$ exchangeable random variables, and an analogous kernel ${\mathbb{P}}_s$ 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 ${\mathbb{P}}_m$ in terms of the asymptotic variance of the corresponding ergodic average associated with ${\mathbb{P}}_s$. 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 ${\mathbb{P}}_m$ is never less than $s/m$ times the variance under ${\mathbb{P}}_s$. 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.

摘要来源：Crossref