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