efficient implementation of MCMC when using an unbiased likelihood estimator

bhamI read this paper by Arnaud Doucet, Mike Pitt, George Deligiannidis and Robert Kohn, re-arXived last month, when travelling to Warwick this morning. In a very pleasant weather, both sides of the Channel. (Little was I aware then that it was a public (“bank”) holiday in the UK and hence that the department here would be empty of people.) Actually, Mike had already talked with me about it during my previous visit to Warwick, as the proof in the paper is making use of our vanilla Rao-Blackwellisation paper, by considering the jump kernels associated with the original kernels.

The purpose of the paper is to determine the precision of (i.e., the number of terms N in) an unbiased estimation of the likelihood function in order to minimise the asymptotic variance of the corresponding Metropolis-Hastings estimate. For a given total number of simulations. While this is a very pertinent issue with pseudo-marginal and particle MCMC algorithms, I would overall deem the paper to be more theoretical than methodological in that it relies on special assumptions like a known parametric family for the distribution of the noise in the approximation of the log-likelihood and independence (of this distribution) from the parameter value. The central result of the paper is that the number of terms N should be such that the variance of the log-likelihood estimator is around 1. Definitely a manageable target. (The above assumptions are used to break the Metropolis-Hastings acceptance probability in two independent parts and to run two separate acceptance checks. Ending up with an upper bound on the asymptotic variance.)

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