Surrogate-based ABC matches generalized Bayesian inference under specific discrepancy and kernel choices

TL;DR

This paper explores how surrogate-based Approximate Bayesian Computation (ABC) can replicate generalized Bayesian inference (GBI) when using specific discrepancy measures and kernels, highlighting their theoretical connection and practical implications.

Contribution

It establishes a formal link between ABC and GBI under Gaussian process surrogates, providing insights into their relationship based on discrepancy and kernel choices.

Findings

Surrogate-based ABC can match GBI under certain conditions.

The behavior of the approximation depends on discrepancy and kernel choices.

Insights into the relationship between ABC and GBI are provided.

Abstract

Generalized Bayesian inference (GBI) is an alternative inference framework motivated by robustness to modeling errors, where a specific loss function is used to link the model parameters with observed data, instead of the log-likelihood used in standard Bayesian inference. Approximate Bayesian Computation (ABC) refers in turn to a family of methods approximating the posterior distribution via a discrepancy function between the observed and simulated data instead of using the likelihood. In this paper we discuss the connection between ABC and GBI, when the loss function is defined as an expected discrepancy between the observed and simulated data from the model under consideration. We show that the resulting generalized posterior corresponds to an ABC-posterior when the latter is obtained under a Gaussian process -based surrogate model. We illustrate the behavior of the approximations as a function of specific discrepancy and kernel choices to provide insights of the relationships between these different approximate inference paradigms.