An Intrinsic Limiting Procedure for Model Selection and Hypotheses Testing

Improper priors typically arise in default Bayesian estimation problems. In the Bayesian approach to model selection or hypothesis testing, the main tool is the Bayes factor. When improper priors for the parameters appearing in the models are used, the Bayes factor is not well defined. The intrinsic Bayes factor introduced by Berger and Pericchi is an interesting method for overcoming that difficulty. That method is of particular interest as a means for generating proper prior distributions (intrinsic priors) for model comparison from the improper priors typically used in estimation. The goal of this article is to develop a limiting procedure that provides a solid justification for the use of Bayes factor with intrinsic priors. The procedure is formalized and discussed for nested and nonnested models. Illustrations and comparisons with other approximations to Bayes factors, such as the Bayesian information criterion of Schwarz and the fractional Bayes factor of O'Hagan are provided.