Integral equation solutions as prior distributions for Bayesian model selection

In many statistical problems we deal with more than one model. When the prior information on the parameters of the models is vague default priors are typically used. Unfortunately, these priors are usually improper provoking a calibration problem which precludes the comparison of the models. An attempt for solving this difficulty consists in using intrinsic priors, introduced in Berger and Pericchi (1996, The intrinsic Bayes factor for model selection and prediction. J Am Stat Assoc 91:109-122), instead of the original default priors; however, there are situations where the class of intrinsic priors is too large. Because of this we propose prior distributions for model selection that are solutions of a system of integral equations which is derived to calibrate the initial default priors. Under some assumptions our integral equations yield a unique solution. Some illustrative examples are provided.