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

TL;DR

This paper introduces methods for constructing parameter priors in DAG models, especially Gaussian DAGs, and characterizes the Wishart distribution through a new independence property, enabling efficient model comparison.

Contribution

It presents a novel approach to deriving parameter priors for DAG models from minimal assessments and characterizes the Wishart distribution via a new independence criterion.

Findings

The normal-Wishart distribution is the unique prior satisfying the assumptions for Gaussian DAGs.

A new characterization of the Wishart distribution based on independence properties.

Method for computing marginal likelihoods for DAG models with no missing data.

Abstract

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.