Asymptotics of ABC when summaries converge at heterogeneous rates

We just posted a new arXival, jointly with Caroline Lawless, Judith Rousseau, and Robin Ryder. This is a significant component of Caroline’s PhD thesis in Oxford, on which we started working during the first COVID lockdown.  In this paper, we extend our results with David Frazier, Gael Martin, both with whom I’ll soon be reunited!, and Judith, published in Biometrika in 2018, to the more challenging case where different components of the summary statistic vector converge to their respective means at different rates, with some possibly not even converging at all. While this sounds impossible (!), we do prove consistency of the ABC posterior under such heterogeneous rates.

Wentao Li and Paul Fearnhead (also in Biometrika and in 2018)  reduce the curse of the dimension of the set of summary statistic by showing, in the specific case of asymptotically normal summary statistics concentrating at the same rate, that a local linear post-processing step leads to a significant improvement in the theoretical behaviour of the ABC posterior. However, due to this focus on reducing the impact of the dimension of the summary statistics, it is therefore important to study its efficiency in a context where the summary statistics are not as well behaved. Surprinsingly maybe, we show that the significant improvement due to local linear post-processing persists even when summary statistics have heterogeneous behaviour.  Most interestingly, the number of summary statistics which converge at the fast rate has no impact on the rate of posterior concentration nor on the shape of the ABC posterior (provided it exceeds the dimension of the parameter).