Concentration of discrepancy-based ABC via Rademacher complexity

There has been increasing interest on summary-free versions of approximate Bayesian computation (ABC), which replace distances among summaries with discrepancies between the empirical distributions of the observed data and the synthetic samples generated under the proposed parameter values. The success of these solutions has motivated theoretical studies on the limiting properties of the induced posteriors. However, current results are often specific to the discrepancy analyzed, and mostly rely on difficult-to-verify existence assumptions that are required to ease proofs at the expense of not-readily-interpretable bounds and a limited exploration of asymptotic properties in more complex settings. We address this gap via a novel theoretical framework which introduces the concept of Rademacher complexity in the analysis of the limiting properties for discrepancy-based ABC posteriors. This yields a unified theory that relies on constructive arguments and provides more interpretable asymptotic results and concentration bounds, even in challenging settings not considered by current theoretical studies. Such advancements are obtained by relating the limiting properties of summary-free ABC posteriors to the behavior of the Rademacher complexity associated with the chosen discrepancy within the broad family of integral probability semimetrics. This class extends summary-based ABC, and includes the widely-implemented Wasserstein distance and MMD, among others. As clarified through a focus of these results on the MMD case and via two illustrative simulations, this novel theoretical perspective yields an improved understanding of ABC and sets the premises to study the concentration of more general pseudo-posteriors, such as those induced by generalized Bayes.