adaptive copulas for ABC

A paper on ABC I read on my way back from Cambodia:  Yanzhi Chen and Michael Gutmann arXived an ABC [in Edinburgh] paper on learning the target via Gaussian copulas, to be presented at AISTATS this year (in Okinawa!). Linking post-processing (regression) ABC and sequential ABC. The drawback in the regression approach is that the correction often relies on an homogeneity assumption on the distribution of the noise or residual since this approach only applies a drift to the original simulated sample. Their method is based on two stages, a coarse-grained one where the posterior is approximated by ordinary linear regression ABC. And a fine-grained one, which uses the above coarse Gaussian version as a proposal and returns a Gaussian copula estimate of the posterior. This proposal is somewhat similar to the neural network approach of Papamakarios and Murray (2016). And to the Gaussian copula version of Li et al. (2017). The major difference being the presence of two stages. The new method is compared with other ABC proposals at a fixed simulation cost, which does not account for the construction costs, although they should be relatively negligible. To compare these ABC avatars, the authors use a symmetrised Kullback-Leibler divergence I had not met previously, requiring a massive numerical integration (although this is not an issue for the practical implementation of the method, which only calls for the construction of the neural network(s)). Note also that sequential ABC is only run for two iterations, and also that none of the importance sampling ABC versions of Fearnhead and Prangle (2012) and of Li and Fearnhead (2018) are considered, all versions relying on the same vector of summary statistics with a dimension much larger than the dimension of the parameter. Except in our MA(2) example, where regression does as well. I wonder at the impact of the dimension of the summary statistic on the performances of the neural network, i.e., whether or not it is able to manage the curse of dimensionality by ignoring all but essentially the data  statistics in the optimisation.

postprocessing for ABC

Two weeks ago, G.S. Rodrigues, Dennis Prangle and Scott Sisson have recently arXived a paper on recalibrating ABC output to make it correctly calibrated (in the frequentist sense). As in earlier papers, it takes advantage of the fact that the tail posterior probability should be uniformly distributed at the true value of the [simulated] parameter behind the [simulated] data. And as in Prangle et al. (2014), relies on a copula representation. The main notion is that marginals posteriors can be reasonably approximated by non-parametric kernel estimators, which means that an F⁰oF⁻¹ transform can be applied to an ABC reference table in a fully non-parametric extension of Beaumont et al.  (2002). Besides the issue that F is an approximation, I wonder about the computing cost of this approach, given that computing the post-processing transforms comes at a cost of O(pT²) when p is the dimension of the parameter and T the size of the ABC learning set… One question that came to me while discussing the paper with Jean-Michel Marin is why one would use F⁻¹(θ¹|s) instead of directly a uniform U(0,1) since in theory this should be a uniform U(0,1).