**I** 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.

**T**he 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.)