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Chris Sherlock, Alexandre H. Thiery, Anthony Lee, Pseudo-marginal Metropolis–Hastings sampling using averages of unbiased estimators, Biometrika, Volume 104, Issue 3, September 2017, Pages 727–734, https://doi.org/10.1093/biomet/asx031
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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.
