Probably approximate Bayesian computation: nonasymptotic convergence of ABC under misspecification

TL;DR

This paper provides theoretical bounds on the convergence of ABC methods under misspecification, demonstrating their robustness and proposing an improved SMC sampling algorithm for the pseudo-posterior.

Contribution

It develops nonasymptotic theoretical bounds for ABC under misspecification and introduces an enhanced SMC sampler for better posterior approximation.

Findings

ABC methods are inherently robust to misspecification.

Explicit bounds depend on parameter space dimension and number of statistics.

Proposed SMC improves sampling efficiency for the pseudo-posterior.

Abstract

Approximate Bayesian computation (ABC) is a widely used inference method in Bayesian statistics to bypass the point-wise computation of the likelihood. In this paper we develop theoretical bounds for the distance between the statistics used in ABC. We show that some versions of ABC are inherently robust to misspecification. The bounds are given in the form of oracle inequalities for a finite sample size. The dependence on the dimension of the parameter space and the number of statistics is made explicit. The results are shown to be amenable to oracle inequalities in parameter space. We apply our theoretical results to given prior distributions and data generating processes, including a non-parametric regression model. In a second part of the paper, we propose a sequential Monte Carlo (SMC) to sample from the pseudo-posterior, improving upon the state of the art samplers.