Asymptotic Properties of Approximate Bayesian Computation

TL;DR

This paper studies the long-term behavior of approximate Bayesian computation posteriors, providing theoretical insights into their concentration, shape, and distribution under certain conditions, aiding practitioners in understanding its reliability.

Contribution

It offers the first comprehensive theoretical analysis of the asymptotic properties of ABC posteriors, including convergence rates and limiting distributions.

Findings

Posterior distribution concentrates at a quantifiable rate.

Limiting shape of the ABC posterior is characterized.

Asymptotic distribution of the posterior mean is derived.

Abstract

Approximate Bayesian computation allows for statistical analysis in models with intractable likelihoods. In this paper we consider the asymptotic behaviour of the posterior distribution obtained by this method. We give general results on the rate at which the posterior distribution concentrates on sets containing the true parameter, its limiting shape, and the asymptotic distribution of the posterior mean. These results hold under given rates for the tolerance used within the method, mild regularity conditions on the summary statistics, and a condition linked to identification of the true parameters. Implications for practitioners are discussed.