Asymptotic Properties of Approximate Bayesian Computation

Approximate Bayesian computation (ABC) is becoming an accepted tool for statistical analysis in models with intractable likelihoods. With the initial focus being primarily on the practical import of ABC, exploration of its formal statistical properties has begun to attract more attention. In this paper we consider the asymptotic behaviour of the posterior obtained from ABC and the ensuing posterior mean. We give general results on: (i) the rate of concentration of the ABC posterior on sets containing the true parameter (vector); (ii) the limiting shape of the posterior; and (iii) the asymptotic distribution of the ABC posterior mean. These results hold under given rates for the tolerance used within ABC, mild regularity conditions on the summary statistics, and a condition linked to identification of the true parameters. Important implication of the theoretical results for practitioners of ABC are highlighted.