Properties of intrinsic Bayes factors
Abstract
The Bayes factor (BF) is commonly used in parametric Bayesian model selection or hypothesis testing problems. In the simplest situation the Bayes factor is the ratio of the marginal distributions of the observed data under two competing models. The BF cannot be used directly if at least one of the (conditional) priors is improper, if, as seems reasonable, we view such an improper prior as determined only up to a multiplicative constant. Berger and Pericchi (5) proposed the use of the Intrinsic Bayes Factor (IBF) for determining a reasonable value for the unknown multiplicative constant(s). Part of the sample is used to obtain a proper (integrable) posterity, and this posterity is treated as a proper prior for the remaining data. In this way the BF conditioned on a subsample is obtained. Then in order to get the IBF such conditional BFs are averaged over all possible subsamples. The goal of our investigations was to look for properties of this new method which would allow better understanding of the IBF. Technically the IBF is a Bayes factor (BF) multiplied by a data-dependent "correction term". The correction term is just the Bayes factor calculated for every subsample and averaged over all subsamples of specified size. We describe the asymptotic behavior of both the BF and the correction term in the iid case using Wald's (34) consistency-of-the-MLE method and U-statistics theory. Another motivation for using the IBF method is the existence of intrinsic priors. Intrinsic priors are priors for which the resulting BF is asymptotically equivalent to the IBF. Asymptotic considerations lead to one (nested case) or two (non-nested case) functional equations. These intrinsic equations involve Kullback-Leibler projections. A geometrical interpretation and the general form of the solutions for the nested case are presented and illustrated through examples. For the non-nested case the discussion is restricted to multiplicate exponential families with $\Theta\sb0$ and $\Theta\sb1$ being two disjoint regions in the natural parameter space. Sufficient conditions for the existence of Kullback-Leibler projections are stated. Solutions of intrinsic equations do not always exist. Existence usually depends on the resolution of conflicts on the boundaries of both regions. The most important conflict is caused by cycles of Kullback-Leibler projections. The existence of such cycles is investigated and illustrated through examples.
Degree
Ph.D.
Advisors
Sellke, Purdue University.
Subject Area
Statistics
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