The Annals of Statistics

Parameter priors for directed acyclic graphical models and the characterization of several probability distributions

Dan Geiger and David Heckerman

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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 $n \times n$, $n \ge 3$, 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.

Article information

Ann. Statist. Volume 30, Number 5 (2002), 1412-1440.

First available: 28 October 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E10: Characterization and structure theory 60E05: Distributions: general theory
Secondary: 62A15 62C10: Bayesian problems; characterization of Bayes procedures 39B99: None of the above, but in this section

Bayesian network directed acyclic graphical model Dirichlet distribution Gaussian DAG model learning linear regression model normal distribution Wishart distribution


Geiger, Dan; Heckerman, David. Parameter priors for directed acyclic graphical models and the characterization of several probability distributions. The Annals of Statistics 30 (2002), no. 5, 1412--1440. doi:10.1214/aos/1035844981.

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