Parameter Priors for Directed Acyclic Graphical Models and the Characterization of Several Probability Distributions
This work addresses a methodological challenge in Bayesian statistics for researchers using DAG models, offering incremental improvements in prior specification and computational efficiency.
The paper tackles the problem of constructing parameter priors for model choice among Directed Acyclic Graphical (DAG) models, introducing assumptions that simplify prior construction and presenting a method for computing marginal likelihoods, with results showing that the normal-Wishart distribution is the only prior satisfying these assumptions for complete Gaussian DAG models.
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.