## kernel approximate Bayesian computation for population genetic inferences

**A** new posting about ABC on arXiv by Shigeki Nakagome, Kenji Fukumizu, and Shuhei Mano entitled *kernel approximate Bayesian computation for population genetic inferences* argues about an improvement brought by the use of reproducing kernel Hilbert space (RKHS) perspective in ABC methodology, when compared with more standard ABC relying on a rather arbitrary choice of summary statistics and metric. However, I feel that the paper does not substantially defend this point, only using a simulation experiment to compare mean square errors. In particular, the claim of consistency is unsubstantiated, as is the counterpoint that “*conventional ABC did not have consistency*” (page 14) [and several papers, including the just published Read Paper by Fearnhead and Prangle, claim the opposite]. Furthermore, a considerable amount of space is taken in the paper by the description of the existing ABC algorithms, while the complete version of the new kernel ABC-RKHS algorithm is missing. In particular, the coverage of kernel Bayes is too sketchy to be comprehensible [at least to me] without additional study. Actually, I do not get the notion of *kernel Bayes’ rule*, which seems defined only in terms of expectations

where the weights are the ridge-like matrix

where the parameter is generated from the prior, the data *s* is generated from the sampling distribution, and the matrix **G**_{S} is made of the *k(s _{i},s_{j})*‘s. The surrounding Hilbert space presentation does not seem particularly relevant, esp. in population genetics… I am also under the impression that the choice of the kernel function

*k(.,.)*is as important as the choice of the metric in regular ABC, although this is not discussed in the paper, since it implies [among other things] the choice of a metric. The implementation uses a Gaussian kernel and an Euclidean metric, which involves assumptions on the homogeneous nature of the components of the summary statistics or of the data. Similarly, the “regularization” parameter ε

_{n}needs to be calibrated and the paper is unclear about this, apparently picking the parameter that “showed the smallest MSEs” (page 10), which cannot be called a calibration.

*(There is a rather unimportant proposition about concentration of information on page 6 which proof relies on two densities being ordered, see top of page 7.)*

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