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Asymptotic Properties of Approximate Bayesian Computation

Frazier, David T.; Martin, Gaël; Robert, Christian P.; Rousseau, Judith (2018), Asymptotic Properties of Approximate Bayesian Computation, Biometrika, 105, 3, p. 593-607. 10.1093/biomet/asy027

Type
Article accepté pour publication ou publié
External document link
https://arxiv.org/abs/1607.06903
Date
2018
Journal name
Biometrika
Volume
105
Number
3
Publisher
Oxford University Press
Pages
593-607
Publication identifier
Metadata
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Author(s)
Frazier, David T.

Martin, Gaël

Robert, Christian P.
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Rousseau, Judith
Abstract (EN)
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.
Subjects / Keywords
asymptotic properties; Bayesian consistency; Bernstein-von Mises theorem; likelihood-free methods