Bayes Factors for Linear Models and Improper Priors

A new method is suggested to evaluate the Bayes factor for choosing between two nested models using improper priors for the model parameters. Within the above framework it is shown that commonly used vague priors always lead to an infinite Bayes factor. For normal linear model we identify a class of improper priors consistent within a finite Bayes factor. Furthermore we single out a subclass of such priors under which the Bayes factor is a function of the standard F-statistics. A numerical illustration is provided in the one-way analysis of variance setup.