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.
Publié le : 2002-10-14
Classification:
Bayesian network,
directed acyclic graphical model,
Dirichlet distribution,
Gaussian DAG model,
learning,
linear regression model,
normal distribution,
Wishart distribution,
62E10,
60E05,
62A15,
62C10,
39B99
@article{1035844981,
author = {Geiger, Dan and Heckerman, David},
title = {Parameter priors for directed acyclic graphical models and the characterization of several probability distributions},
journal = {Ann. Statist.},
volume = {30},
number = {1},
year = {2002},
pages = { 1412-1440},
language = {en},
url = {http://dml.mathdoc.fr/item/1035844981}
}
Geiger, Dan; Heckerman, David. Parameter priors for directed acyclic graphical models and the characterization of several probability distributions. Ann. Statist., Tome 30 (2002) no. 1, pp. 1412-1440. http://gdmltest.u-ga.fr/item/1035844981/