The copula ABC approximation for the banana distribution holds actually for p=250 dimensions in the paper (not p=50), but of course, by construction it will hold for p=\infty, computing time and storage requirements notwithstanding!

The “incoherent” model (with different summary statistics for different univariate and bivariate margins) is a reasonable point, although it is simple enough to check that using a low-dimensional subset/function of the full vector of summary statistics gives a more precise marginal posterior estimate compared to the full vector in each case. (And if it doesn’t then don’t use it!) So in this sense, even any “incoherence” is actually an improvement when trading off against the otherwise poor vanilla ABC approximation.

In addition, there is some work in the density estimation literature (discussed in our previous paper on high-dimensional ABC in JCGS, 2014) that involves using different datasets to estimate different marginals for various parts of the posterior. So whatever inherent problems these techniques have, we inherit in this approach. Although to date, we have only experienced improvements over standard ABC using these approaches.

Scott

]]>