**O**n the flight back from Warwick, I read a fairly recently arXived paper by Umberto Picchini and Julie Forman entitled “Accelerating inference for diffusions observed with measurement error and large sample sizes using Approximate Bayesian Computation: A case study” that relates to earlier ABC works (and the MATLAB abc-sde package) by the first author (earlier works I missed). Among other things, the authors propose an acceleration device for ABC-MCMC: when simulating from the proposal, the Metropolis-Hastings acceptance probability can be computed and compared with a uniform rv *prior to* simulating pseudo-data. In case of rejection, the pseudo-data does not need to be simulated. In case of acceptance, it is compared with the observed data as usual. This is interesting for two reasons: first it always speeds up the algorithm. Second, it shows the strict limitations of ABC-MCMC, since the rejection takes place without incorporating the information contained in the data. (Even when the proposal incorporates this information, the comparison with the prior does not go this way.) This also relates to one of my open problems, namely how to simulate directly summary statistics without simulating the whole pseudo-dataset.

**A**nother thing (related with acceleration) is that the authors use a simulated subsample rather than the simulated sample in order to gain time: this worries me somehow as the statistics corresponding to the observed data is based on the *whole* observed data. I thus wonder how both statistics could be compared, since they have different distributions and variabilities, even when using *the same* parameter value. Or is this a sort of pluggin/bootstrap principle, the true parameter being replaced with its estimator based on *the whole* data? Maybe this does not matter in the end (when compared with the several levels of approximation)…