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In this paper a new algorithm combining population-based MCMC methods with ABC requirements is proposed, using an analogy with the Parallel Tempering algorithm (Geyer, 1991).“

**A**nother of those arXiv papers that had sat on my to-read pile for way too long: *Likelihood-free parallel tempering* by Meïli Baragtti, Agnès Grimaud, and Denys Pommeret, from Luminy, Marseilles. The paper mentions our population Monte Carlo (PMC) algorithm (Beaumont et al., 2009) and other ABC-SMC algorithms, but opts instead for an ABC-MCMC basis. The purpose is to build a parallel tempering method. Tolerances and temperatures evolve simultaneously. I however fail to see where the tempering occurs in the algorithm (page 7): there is a set of temperatures *T*_{1},….,*T*_{N}, but they do not appear within the algorithm. My first idea of a tempering mechanism in a likelihood-free setting was to replicate our SAME algorithm (Doucet, Godsill, and Robert, 2004), by creating *T*_{j} copies of the [pseudo-]observations to mimic the likelihood taken to the power *T*_{j}. But this is annealing, not tempering, and I cannot think of the opposite of copies of the data. Unless of course a power of the likelihood can be simulated (and even then, what would the equivalent be for the data…?) Maybe a natural solution would be to operate some kind of data-attrition, e.g. by subsampling the original vector of observations.

**D**iscussing the issue with Jean-Michel Marin, during a visit to Montpellier today, I realised that the true tempering came from the tolerances ε_{i}, while the temperatures *T*_{j} were there to calibrate the proposal distributions. And that the major innovation contained in the thesis (if not so clearly in the paper) was to boost exchanges between different tolerances, improving upon the regular ABC-MCMC sampler by an equi-energy move.