Asymptotic Properties of Approximate Bayesian Computation
Christian Robert, Université Paris-Dauphine
12-1pm 8th May 2017
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 behavior 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. Using simple illustrative examples that have featured in the literature, we demonstrate that the required identification condition is far from guaranteed. The implications of the theoretical results for practitioners of ABC are also highlighted.
Christian P. Robert is Professor at Université Paris-Dauphine, PSL Research University since 2000. He is a senior member of the Institut Universitaire de France since 2010. He is part-time Professor at the University of Warwick since 2013. He has been co-editor of the Journal of the Royal Statistical Society, Series B, from 2006 till 2010, associate editor for the Annals of Statistics, Statistical Science, and JASA, and blog editor of the incoming Series B blog. He is a fellow of the Institute of Mathematical Statistics, of the American Statistical Association, and of the International Society for Bayesian Analysis (ISBA). Christian P. Robert’s research areas cover Bayesian statistics, with a focus on decision theory and model selection, numerical probability, with works centering on the application of Markov chain theory to simulation, and computational statistics, developing and evaluating new methodologies for the analysis of statistical models. He has written over 150 research papers in these areas. He has also written or co‑written ten books on Bayesian statistics and computational methods. He was awarded the 2004 DeGroot prize for the second edition of his book ‘The Bayesian Choice’.