Propriety of Intrinsic Priors in Invariant Testing Situations

The Theory of Intrinsic Priors, developed by Berger and Pericchi (1996a,b), is a general method of constructing objective priors for testing and model selection when proper priors are considered for the simpler null hypotheses. When this prior distribution is improper, as is typically the case for Objective Bayesian testing, they suggest approximating the (improper) prior by a sequence of proper priors on compacts. This “limiting procedure” was formalized by Moreno, Bertolino and Racugno (1998) who showed that the limiting Bayes factor is unique. Still, the natural question of whether some component of the intrinsic prior on parameters in the full model is proper in the limit (Sansó, 1997), remained to be answered. We develop a method here to partially answer this question for testing problems involving group invariance structures. We give a sufficient condition, which is easily verified, which implies that the conditional intrinsic prior under the full model is proper. This hitherto had to be verified by direct calculation. This paper also complements Berger, Pericchi and Varshavsky (1998), who develop methods, under group invariance situations, for non-nested models. For nested models, we give a class of initial non-informative priors and identify component parameters in the full model for which the default analysis results in a proper prior for these parameters. The proper prior is also identified as being the intrinsic prior arising from the use of the Arithmetic Intrinsic Bayes Factor (AIBF) methodology for the default analysis.