  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.
