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

MSR-TR-98-67 |

We develop simple methods for constructing parameter priors for model choice among Directed Acyclic Graphical (DAG) models. In particular, we introduce several assumptions that permit the construction of parameter priors for a large number of DAG models from a small set of assessments. We then present a method for directly computing the marginal likelihood of every DAG model given a random sample with no missing observations. We apply this methodology to Gaussian DAG models which consist of a recursive set of linear regression models. We show that the only parameter prior for complete Gaussian DAG models that satisfies our assumptions is the normal-Wishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let W be an n x n , n > 3, positive-definite symmetric matrix of random variables and f ( W ) be a pdf of W . Then, f( W ) is a Wishart distribution if and 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-Wishart distributions are provided as well.