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

• 成果类型: Article
• 署名作者: Sherlock, Chris; Thiery, Alexandre H.; Lee, Anthony
• 署名单位: Lancaster University; National University of Singapore; University of Warwick
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asx031
• 发表日期: 2017
• 页码: 727734
• 关键词: approximate bayesian computation; algorithms; models

摘要

We consider a pseudo-marginal Metropolis-Hastings kernel P-m 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 P-m 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 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.