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

TL;DR

This paper characterizes the unique parameter prior for complete Gaussian DAG models as the normal-Wishart distribution, based on a new property of the Wishart distribution, and discusses constructing priors for DAG models from regression models.

Contribution

It provides a novel characterization of the Wishart distribution and establishes the normal-Wishart as the unique prior satisfying key properties for Gaussian DAG models.

Findings

Normal-Wishart is the unique prior under specified conditions.

New characterization of Wishart distribution based on independence properties.

Method to construct priors for DAG models from regression model priors.

Abstract

We show that the only parameter prior for complete Gaussian DAG models that satisfies global parameter independence, complete model equivalence, and some weak regularity assumptions, is the normal-Wishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let W be an n x 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 as well. We also show how to construct a prior for every DAG model over X from the prior of a single regression model.