Approximate Bayesian computation via sufficient dimension reduction

“One of our contribution comes from the mathematical analysis of the consequence of conditioning the parameters of interest on consistent statistics and intrinsically inconsistent statistics”

Xiaolong Zhong and Malay Ghosh have just arXived an ABC paper focussing on the convergence of the method. And on the use of sufficient dimension reduction techniques for the construction of summary statistics. I had not heard of this approach before so read the paper with interest. I however regret that the paper does not link with the recent consistency results of Liu and Fearnhead and of Daniel Frazier, Gael Martin, Judith Rousseau and myself. When conditioning upon the MLE [or the posterior mean] as the summary statistic, Theorem 1 states that the Bernstein-von Mises theorem holds, missing a limit in the tolerance ε. And apparently missing conditions on the speed of convergence of this tolerance to zero although the conditioning event involves the true value of the parameter. This makes me wonder at the relevance of the result. The part about partial posteriors and the characterisation of limiting posterior distributions stats with the natural remark that the mean of the summary statistic must identify the whole parameter θ to achieve consistency, a point central to our 2014 JRSS B paper. The authors suggest using a support vector machine to derive the summary statistics, an idea already exploited by Heiko Strathmann et al.. There is no consistency result of relevance for ABC in that second and final part, which ends up rather abruptly. Overall, while the paper contributes to the current reflection on the convergence properties of ABC, the lack of scaling of the tolerance ε calls for further investigations.