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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.