  Approximate Bayesian computation (ABC) is a class of algorithmic methods in
Bayesian inference using statistical summaries and computer simulations. ABC
has become popular in evolutionary genetics and in other branches of biology.
However model selection under ABC algorithms has been a subject of intense
debate during the recent years. Here we propose novel approaches to model
selection based on posterior predictive distributions and approximations of the
deviance. We argue that this framework can settle some contradictions between
the computation of model probabilities and posterior predictive checks using
ABC posterior distributions. A simulation study and an analysis of a
resequencing data set of human DNA show that the deviance criteria lead to
sensible results in a number of model choice problems of interest to population
geneticists.
