Using asymptotics to understand ABC

Approximate Bayesian computations are used typically when the model
is so complex that the likelihood is intractable but data can be generated
from the model. With the initial focus being primarily on the practical
import of this algorithm, exploration of its formal statistical properties has
begun to attract more attention. In this work we consider the asymptotic
behaviour of the posterior obtained by this method and the ensuing posterior
mean. We give general results on: (i) the rate of concentration of the
resulting posterior on sets containing the true parameter (vector); (ii) the
limiting shape of the posterior; and (iii) the asymptotic distribution of the
ensuing 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.
I will highlight what are the practical implications of these results on the
understanding of the behaviour of the algorithm.

(Joint work with David Frazier, Gael Martin and Christian Robert)