A Bayesian Predictive Approach to Model Selection

This paper proposes a predictive approach to Bayesian model selection based on independent and identically distributed observations. In particular, we generalise the criterion of San Martini and Spezzaferri (J. Roy. Statist. Soc. B 46 (1984) 296–303) to take into account more realistic views as discussed by Bernardo and Smith (Bayesian Theory. Wiley, Chichester, 1994). The former authors only consider what the latter authors name the -closed view; that is, the assumption that one of the competing models is the true model. More realistic is the -open view in which it is believed that none of the competing models is the true model. Our new approach can encompass both of these views and moreover we introduce the -mixture view where the experimenter can express prior opinion concerning his/her belief as to whether one of the competing models is the true model or not. Essentially, we embed the -open view in a larger (nonparametric) -closed view.