TY - JOUR

T1 - Pseudo-marginal Metropolis-Hastings sampling using averages of unbiased estimators

AU - Sherlock, Christopher Gerrard

AU - Thiery, Alex

AU - Lee, Anthony

N1 - This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Biometrika following peer review. The definitive publisher-authenticated version Chris Sherlock, Alexandre H. Thiery, Anthony Lee; Pseudo-marginal Metropolis–Hastings sampling using averages of unbiased estimators, Biometrika, Volume 104, Issue 3, 1 September 2017, Pages 727–734, https://doi.org/10.1093/biomet/asx031

PY - 2017/9

Y1 - 2017/9

N2 - 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.

AB - 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.

KW - Importance sampling

KW - Pseudo-marginal Markov chain Monte Carlo

U2 - 10.1093/biomet/asx031

DO - 10.1093/biomet/asx031

M3 - Journal article

VL - 104

SP - 727

EP - 734

JO - Biometrika

JF - Biometrika

SN - 0006-3444

IS - 3

ER -