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Title: Discrete Bayesian DAG models with a restricted set of directions Authors:  Jacek Wesolowski - Warsaw University of Technology (Poland) [presenting]
Abstract: First, we develop a new family of conjugate prior distributions for the cell probabilities of discrete graphical models Markov with respect to a set $P$ of moral directed acyclic graphs (DAGs) with skeleton a given decomposable graph $G$. This family, called $P$-Dirichlet, is a generalization of the hyper Dirichlet: it keeps the directed strong hyper Markov property for every DAG in $P$ but increases the flexibility in the choice of its parameters. Second, we prove a characterization of the $P$-Dirichlet, which yields, as a corollary, a characterization of the hyper Dirichlet as well as the classical Dirichlet law. Like the characterization of the classical Dirichlet, our result is based on local and global independence of the probability parameters. We use also separability property explicitly defined but implicitly used by Geiger and Heckerman through their choice of two particular DAGs. Another advantage of our approach is that we need not make the assumption of the existence of a positive smooth density function. We use the method of moments for our proofs.