Title: Concentration and robustness of discrepancy-based ABC through Rademacher complexity
Authors: Sirio Legramanti - University of Bergamo (Italy) [presenting]
Daniele Durante - Bocconi University (Italy)
Pierre Alquier - CREST, ENSAE ParisTech (France)
Abstract: Approximate Bayesian Computation (ABC) typically employs summary statistics to measure the discrepancy among the observed data and synthetic data generated from each proposed parameter value. However, finding good summary statistics (that are close to sufficiency) is non-trivial for most of the models for which ABC is needed. This motivated summary-free versions of ABC based on discrepancies between the empirical distributions of observed and synthetic data. The studies on the properties of the corresponding ABC posteriors are quite uneven, ranging from empirical assessments to more theoretical investigations. Even when available, the existing theory is often limited to a single discrepancy or relies on hypotheses that are difficult to verify. We propose a unifying view through Rademacher complexity over a general class of discrepancies known as integral probability semimetrics, which include the maximum mean discrepancy and the total variation, Kolmogorov-Smirnov, and Wasserstein distances. For rejection ABC based on this class of semimetrics, we prove results on both posterior concentration and robustness. Such results connect the properties of the ABC posterior to the Rademacher complexity of the class of test functions that characterizes each integral probability semimetric. This provides a new understanding of why some discrepancies work well with ABC and others do not.