Prior specification across models with an application to Bayesian variable selection in linear models
Model determination and comparison is an important and active area of research especially from the Bayesian viewpoint. One critical issue emerges from the very beginning: the assignment of prior distributions on the parameter space of each model.
Often a model can be regarded as a submodel of a larger model, as for example in variable selection for linear models. Because of the potentially very high number of models under investigation, this suggests relating priors across models, in order to enhance the elicitation procedure.
Surprisingly relatively little is known in this area.
We discuss some issues concerning the interpretation of submodels, trying to clarify the appropriate notation, and show the implications on prior specifications.
In particular we focus on three strategies for prior assignments: marginalization, conditioning and Kullback-Leibler projection.
We exemplify our discussion with reference to the problem of variable selection in linear models.
This is joint work with Piero Veronese (Bocconi University, Milan).
A copy of the paper is available at