ABC methods for Bayesian model choice

Abstract

Approximate Bayesian computation (ABC), also known as likelihood-free methods, have become a standard tool for the analysis of complex models, primarily in population genetics. The development of new ABC methodology is undergoing a rapid increase in the past years, as shown by multiple publications, conferences and even software. While one valid interpretation of ABC based estimation is connected with nonparametrics, the setting is quite different for model choice issues. We examined in Grelaud et al. (2009) the use of ABC for Bayesian model choice in the specific of Gaussian random fields (GRF), relying on a sufficient property only enjoyed by GRFs to show that the approach was legitimate. Despite having previously suggested the use of ABC for model choice in a wider range of models in the DIYABC software (Cornuet et al., 2008), we present in Robert et al. (2011) theoretical evidence that the general use of ABC for model choice is fraught with danger in the sense that no amount of computation, however large, can guarantee a proper approximation of the posterior probabilities of the models under comparison. Finally, in Marin et al. (2011), we derive necessary and sufficient conditions on summary statistics for the corresponding Bayes factor to be convergent, namely to asymptotically select the true model. In this talk, we will present these different results.