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.

翻译：近年来，无汇总统计的近似贝叶斯计算（ABC）方法日益受到关注，这类方法用观测数据经验分布与在提议参数值下生成的合成样本之间的差异，替代了汇总统计量之间的距离。这些解决方案的成功推动了关于诱导后验极限性质的理论研究。然而，现有结果通常局限于所分析的特定差异，且大多依赖于难以验证的存在性假设——这些假设虽便于证明，却导致界限难以解释，并限制了对更复杂设定下渐近性质的探索。我们通过一种新颖的理论框架解决了这一空白，该框架在分析基于差异的ABC后验极限性质时引入了Rademacher复杂度的概念。这构建了一个基于构造性论证的统一理论，在现有理论研究未考虑的挑战性设定中，也能提供更具可解释性的渐近结果和集中性界限。这些进展通过将无汇总统计ABC后验的极限性质与积分概率半度量族中选定差异对应的Rademacher复杂度行为建立联系而得以实现。该族方法扩展了基于汇总统计的ABC，并包含广泛使用的Wasserstein距离和最大均值差异（MMD）等。通过聚焦这些结果在MMD案例上的应用及两个示例性模拟，这一理论新视角深化了对ABC的理解，并为研究更广义伪后验（如广义贝叶斯方法诱导的后验）的集中性奠定了基础。