In the present literature, prior distributions for discrete directed acyclic graph (DAG) models are either derived from the Dirichlet distribution on the complete model or consists of a set of different Dirichlet priors for each vertex of the DAG and each configuration of its parents.

In this talk, we develop a new family of conjugate prior distributions for the cell probability parameters of discrete graphical models Markov with respect to a set $\mathcal{P}$ of moral directed acyclic graphs with skeleton a given decomposable graph $G$. This family, which we call the $\mathcal{P}$-Dirichlet, is a generalization of the hyper Dirichlet: it keeps the directed strong hyper Markov property of the hyper Dirichlet for every DAG in $\mathcal{P}$ but increases the flexibility in the choice of its parameters, i.e. the hyper parameters.

We also give a characterization of this $\mathcal{P}$-Dirichlet which yields, as corollaries, a characterization of the hyper Dirichlet and a characterization of the Dirichlet.