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We develop simple methods for constructing parameter priors for model choiceamong Directed Acyclic Graphical (DAG) models. In particular, we introduceseveral assumptions that permit the construction of parameter priors for alarge number of DAG models from a small set of assessments. We then present amethod for directly computing the marginal likelihood of every DAG model givena random sample with no missing observations. We apply this methodology toGaussian DAG models which consist of a recursive set of linear regressionmodels. We show that the only parameter prior for complete Gaussian DAG modelsthat satisfies our assumptions is the normal-Wishart distribution. Our analysisis based on the following new characterization of the Wishart distribution: let$W$ be an $n \times n$, $n \ge 3$, positive-definite symmetric matrix of randomvariables and $f(W)$ be a pdf of $W$. Then, f$(W)$ is a Wishart distribution ifand only if $W_{11} - W_{12} W_{22}^{-1} W'_{12}$ is independent of$\{W_{12},W_{22}\}$ for every block partitioning $W_{11},W_{12}, W'_{12},W_{22}$ of $W$. Similar characterizations of the normal and normal-Wishartdistributions are provided as well.

Parameter Priors for Directed Acyclic Graphical Models and the Characterization of Several Probability Distributions