Parametrizations and reference priors for multinomial decomposable graphical models

Given a multinomial decomposable graphical model, we identify several alternative parametrizations; in particular we consider conditional probabilities of clique-residuals given separators, as well as generalized log-odds-ratios. For each such parametrization, we construct the corresponding reference prior for suitable groupings of the parameters. Each one of the reference priors we obtain is conjugate to the likelihood and is proper. Furthermore, all these priors are equivalent, in the sense that they can be deduced from each other by a change of variable. We also derive estimators of cell-probabilities based on the reference prior. Finally, we discuss in detail a parametrization associated to a collection of variables representing a cut for the statistical model, and derive the corresponding reference prior.