Seminar Details

Seminar Details


Wednesday

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.