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

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.

Importance sampling, Pseudo-marginal Markov chain Monte Carlo