Bayes Factors and Marginal Distributions in Invariant Situations

James O. Berger, Luis R. Pericchi, and Julia Varshavsky

In Bayesian analysis with a ``minimal'' data set and common noninformative priors, the (formal) marginal density of the data is surprisingly often independent of the error distribution. This results in great simplifications in certain model selection methodologies; for instance, the Intrinsic Bayes Factor for models with this property reduces simply to the Bayes factor with respect to the noninformative priors. The basic result holds for comparison of models which are invariant with respect to the same group structure. Indeed the condition reduces to a condition on the distributions of the common maximal invariant. In these situations, the marginal density of a ``minimal'' data set is typically available in closed form, regardless of the error distribution. This provides very useful expressions for computation of Intrinsic Bayes Factors in more general settings. The conditions for the results to hold are explored in some detail for nonnormal linear models and various transformations thereof. Postscript File (526kB)