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

Abstract

In this paper, we first 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 given in [Ann. Statist. 21 (1993) 1272–1317]: it keeps the directed strong hyper Markov property for every DAG in $\mathcal{P}$ but increases the flexibility in the choice of its parameters, that is, the hyper parameters.

Our second contribution is a characterization of the $\mathcal{P}$-Dirichlet, which yields, as a corollary, a characterization of the hyper Dirichlet and a characterization of the Dirichlet also. Like the characterization of the Dirichlet given in [Ann. Statist. 25 (1997) 1344–1369], our characterization of the $\mathcal{P}$-Dirichlet is based on local and global independence of the probability parameters and also a separability property explicitly defined here but implicitly used in that paper through the choice of two particular DAGs. Another advantage of our approach is that we need not make the assumption of the existence of a positive density function. We use the method of moments for our proofs.