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In this paper, we first develop a new family of conjugate prior distributionsfor the cell parameters of discrete graphical models Markov with respect to aset P of moral directed acyclic graphs with skeleton a given decomposable graphG. Such families arise when the set of conditional independences betweendiscrete variables is given and can be represented by a decomposable graph andadditionally, the direction of certain edges is imposed by the practitioner.This family, which we call the P-Dirichlet, is a generalization of the hyperDirichlet given in Dawid and Lauritzen (1993): it keeps the strong directedhyper Markov property for every DAG in P but increases the flexibility in thechoice of its parameters, i.e. the hyper parameters. Our second contribution isa characterization of the P-Dirichlet, which yields, as a corollary, acharacterization of the hyper Dirichlet and a characterization of the Dirichletalso. Like that given by Geiger and Heckerman (1997), our characterization ofthe Dirichlet is based on local and global independence of the probabilityparameters but we need not make the assumption of the existence of a positivedensity function. We use the method of moments for our proofs.

A new prior for the discrete DAG models with a restricted set of directions