A conjugate prior for discrete hierarchical loglinear models

In the Bayesian analysis of contingency table data, the selection of a prior distribution for either the loglinear parameters or the cell probability parameter is a major challenge. In this paper, we define a flexible family of conjugate priors for the wide class of discrete hierarchical loglinear models which includes the class of graphical models. These priors are first defined as the Diaconis- Ylvisaker conjugate priors on the loglinear parameters subject to "baseline constraints". We show that they have several desirable properties: the mathematical convenience of a conjugate prior, the flexibility of having as many hyperparameters as there are free cell probabilities in the model, the ability to easily reflect prior knowledge using the prior moments of cell probabilities and the strong hyper Markov property. We then obtain the priors induced on the cell probabilities parameter. For decomposable graphical models, we give the correspondance with the hyper Dirichlet as defined by Dawid and Lauritzen (1993). We also show that, in the decomposable case, the induced prior on the clique and separator marginal probability parametrization is standard conjugate while the induced prior on the joint cell probability parametrization is not so.