Some Bayesian predictive approaches to model selection

• Published in: Statistics & probability letters Vol. 73; no. 4; pp. 369 - 379
• Main Authors: Mukhopadhyay, Nitai, Ghosh, Jayanta K., Berger, James O.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.07.2005; Elsevier
• Series: Statistics & Probability Letters
• Subjects: Bayes factor; Bayes factor Training sample; Decision theory; Exact sciences and technology; Mathematics; Parametric inference; Probability and statistics; Sciences and techniques of general use; Statistics; Training sample; Training sample; Bayes factor; Statistical method; Prior distribution; Model selection; Probability theory; Predictive distribution; Prediction theory; Bootstrap; Cross validation; Mean estimation; Empirical Bayes estimation
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/j.spl.2005.04.012

A variety of pseudo-Bayes factors have been proposed, based on using part of the data to update an improper prior, and using the remainder of the data to compute the Bayes factor. A number of these approaches are of a bootstrap or cross-validation nature, with some type of average being taken over the data used for updating. Asymptotic characteristics of a number of these pseudo-Bayes factors are discussed, and it is shown how many behave quite differently from ordinary Bayes factors. It is also shown that arguments of predictive optimality, based on simply inserting the empirical distribution in place of the ‘true predictive distribution’, can be misleading; the particular example of this that is studied is the argument given in Bernardo and Smith [1994. Bayesian Theory. Wiley, Chichester] to the effect that the geometric intrinsic Bayes factor has an optimal predictive property.