  There has been an 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 (i) are often tailored to a
specific discrepancy, (ii) require, either explicitly or implicitly, regularity
conditions on the data generating process and the assumed statistical model,
and (iii) yield bounds depending on sequences of control functions that are not
made explicit. As such, there is the lack of a theoretical framework that (i)
is unified, (ii) facilitates the derivation of limiting properties that hold
uniformly, and (iii) relies on verifiable assumptions that provide
concentration bounds clarifying which factors govern the limiting behavior of
the ABC posterior. We address this gap via a novel theoretical framework that
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 informative asymptotic
results and uniform concentration bounds, even in settings not covered by
current studies. These advancements are obtained by relating the properties of
summary-free ABC posteriors to the behavior of the Rademacher complexity
associated with the chosen discrepancy within the family of integral
probability semimetrics. This family extends summary-based ABC, and includes
the Wasserstein distance and maximum mean discrepancy (MMD), among others. As
clarified through a focus on the MMD case and via illustrative simulations,
this perspective yields an improved understanding of summary-free ABC.
