Bayes factor consistency

Good large sample performance is typically a minimum requirement of any model selection criterion. This article focuses on the consistency property of the Bayes factor, a commonly used model comparison tool, which has experienced a recent surge of attention in the literature. We thoroughly review existing results. As there exists such a wide variety of settings to be considered, e.g. parametric vs. nonparametric, nested vs. non-nested, etc., we adopt the view that a unified framework has didactic value. Using the basic marginal likelihood identity of Chib (1995), we study Bayes factor asymptotics by decomposing the natural logarithm of the ratio of marginal likelihoods into three components. These are, respectively, log ratios of likelihoods, prior densities, and posterior densities. This yields an interpretation of the log ratio of posteriors as a penalty term, and emphasizes that to understand Bayes factor consistency, the prior support conditions driving posterior consistency in each respective model under comparison should be contrasted in terms of the rates of posterior contraction they imply.