Bayesian Inference for Model Choice

Two nested or non-nested candidate sampling models for an observed data set may be compared by consideration of summaries of a probability plot, which contrasts the posterior quantiles of the log-likelihoods under the two models. The procedures address both preference inference and refutation inference, and extensions to DIC and alternatives to AIC are developed. Preference inference favors models with more parameters, perhaps on a tentative basis when further data are anticipated, while refutation inference emphasizes parameter parsimony. A characterization relating to an α-profile motivates the comparison of the posterior medians of the log-likelihoods, when considering simple model preference. For nested models, a stronger omega-preference procedure is developed via a Bayes-frequency compromise. The Bayes-frequency performances of the different preference and refutation inference procedures are investigated when the models are nested. While attention is primarily confined to model inference within the linear paradigm, most of the methods are approximately applicable in a range of non-linear cases. A Gamma approximation to an Upsilon distribution facilitates a general approach, for the linear model with unknown variance. A data set for 71 hypertensive diabetic patients is analyzed, and a symptom of high blood pressure is related to four out of the eight explanatory variables available. of Statistics at the Universities of Wisconsin-Madison and Edinburgh, is retired and lives at 4/3 Hopetoun Crescent, Edinburgh, Scotland, EH7 4AY. The authors would like to thank Angelika van der Linde for her helpful advice on the literature.