Dec 2

4:00 pm

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

### Dan Geiger

Seminar

Microsoft Research

**Multivariate Analysis & Graphical Models of Association (MAGMA 4) Workshop**

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 by 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 and a related characterization of the Dirichlet distribution is discussed as well.

This work was done at Microsoft research jointly with David Heckerman.