#### 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.
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