Parameter Priors for Directed Acyclic Graphical Models and the Characterization of Several Probability Distributions
This work addresses theoretical foundations for Bayesian inference in graphical models, providing a rigorous characterization that is incremental but clarifies prior assumptions for statisticians and machine learning researchers.
The paper proves that the normal-Wishart distribution is the unique parameter prior for complete Gaussian directed acyclic graphical models under conditions like global parameter independence and complete model equivalence, based on a new characterization of the Wishart distribution involving independence properties of matrix partitions.
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.