## postprocessing for ABC

**T**wo 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).

June 12, 2017 at 11:44 am

Hi Christian, thanks for your comments (and sorry for the slow response!)

The idea behind this paper is to try to correct for F_ABC(theta|s) being non-uniform. Assuming it is uniform in our algorithm would simply return the original ABC posterior. But perhaps I’m missing the intention of your remarks…

Regarding the cost, a pure ABC implementation can be expensive. But Scott and Guilherme had a nice idea of speeding up the process using faster estimators for some of the steps – see Section 2.3 of the paper for example. This produces a hybrid method that uses ABC to correct another approximate Bayesian method.

June 15, 2017 at 2:20 pm

Hi Dennis, thanks for taking the time to read and address my insubstantial remarks!!! I understand the fact that using directly a uniform gets back to an earlier stage, but wonder at some potential alternative where the uniforms would be correlated by a copula or something like that, in order to avoid the extra noise due to the cdf estimation, since in theory it should really be a (dependent) uniform.