On Consistency of Approximate Bayesian Computation

TL;DR

This paper establishes conditions for the Bayesian consistency of Approximate Bayesian Computation (ABC) methods, highlighting the importance of parameter identification and proposing diagnostics to assess consistency in complex models.

Contribution

It provides the first general theoretical conditions for ABC consistency, linking summary statistic choice and model identification, and introduces diagnostics for practical assessment.

Findings

Consistency depends on parameter identification.

Many common summary statistics may not ensure consistency.

A diagnostic procedure can detect potential inconsistency issues.

Abstract

Approximate Bayesian computation (ABC) methods have become increasingly prevalent of late, facilitating as they do the analysis of intractable, or challenging, statistical problems. With the initial focus being primarily on the practical import of ABC, exploration of its formal statistical properties has begun to attract more attention. The aim of this paper is to establish general conditions under which ABC methods are Bayesian consistent, in the sense of producing draws that yield a degenerate posterior distribution at the true parameter (vector) asymptotically (in the sample size). We derive conditions under which arbitrary summary statistics yield consistent inference in the Bayesian sense, with these conditions linked to identification of the true parameters. Using simple illustrative examples that have featured in the literature, we demonstrate that identification, and hence consistency, is unlikely to be achieved in many cases, and propose a simple diagnostic procedure that can indicate the presence of this problem. We also formally explore the link between consistency and the use of auxiliary models within ABC, and illustrate the subsequent results in the Lotka-Volterra predator-prey model.