Bayesian Computation and Model Selection Without Likelihoods

Until recently, the use of Bayesian inference was limited to a few cases because for many realistic probability models the likelihood function cannot be calculated analytically. The situation changed with the advent of likelihood-free inference algorithms, often subsumed under the term approximate Bayesian computation (ABC). A key innovation was the use of a postsampling regression adjustment, allowing larger tolerance values and as such shifting computation time to realistic orders of magnitude. Here we propose a reformulation of the regression adjustment in terms of a general linear model (GLM). This allows the integration into the sound theoretical framework of Bayesian statistics and the use of its methods, including model selection via Bayes factors. We then apply the proposed methodology to the question of population subdivision among western chimpanzees, Pan troglodytes verus.