Bayesian model comparison based on expected posterior priors for discrete decomposable graphical models

The implementation of the Bayesian paradigm to model comparison can be problematic. In particular, prior distributions on the parameter space of each candidate model require special care. While it is well known that improper priors cannot be routinely used for Bayesian model comparison, we claim that also the use of proper conventional priors under each model should be regarded as suspicious, especially when comparing models having different dimensions. The basic idea is that priors should not be assigned separately under each model; rather they should be related across models, in order to acquire some degree of compatibility, and thus allow fairer and more robust comparisons. In this connection, the intrinsic prior as well as the expected posterior prior (EPP) methodology represent a useful tool. In this paper we develop a procedure based on EPP to perform Bayesian model comparison for discrete undirected decomposable graphical models, although our method could be adapted to deal also with directed acyclic graph models. We present two possible approaches. One based on imaginary data, and one which makes use of a limited number of actual data. The methodology is illustrated through the analysis of a 2 × 3 × 4 contingency table.