- The paper presents a framework for parameter prior specification in Gaussian DAG models using a regression-based approach that ensures global parameter independence.
- It uniquely characterizes the Wishart, normal, and normal-Wishart distributions by establishing strict independence and regularity conditions.
- The results streamline Bayesian model selection and inference, reducing computational complexity in high-dimensional graphical models.
Parameter Priors for Directed Acyclic Graphical Models and the Characterization of Several Probability Distributions
This paper explores the specification of parameter priors for Gaussian Directed Acyclic Graphical Models (DAGs) and introduces a characterization of related probability distributions. In the context of complete Gaussian DAG models, the authors demonstrate that the only parameter prior meeting the criteria of global parameter independence, complete model equivalence, and weak regularity assumptions is the normal-Wishart distribution. The characterization of the Wishart distribution is central to this analysis, with specific conditions outlined that define its unique independence properties. Similar characterizations are developed for normal and normal-Wishart distributions.
The primary contribution of the paper is the development of a systematic approach to parameter prior specification for Gaussian DAG models, enabling the construction of priors for all potential DAG models using a single prior from a regression model. This framework utilizes assumptions of complete model equivalence, regularity, likelihood modularity, prior modularity, and global parameter independence. The consolidation of these assumptions allows the derivation of a parameter prior for any DAG model from the prior of a single complete model.
Notably, the paper provides formal proofs for Theorems characterizing the Wishart, normal, and normal-Wishart distributions, which hinge on global parameter independence. The authors argue that local parameter independence is a redundant assumption for characterizing these distributions when the number of variables is three or more, further proposing that the same redundancy may apply to Dirichlet distribution characterizations for discrete models.
The implications of this work are significant in Bayesian network learning, as it presents an efficient way to manage the computational complexity associated with exploring the vast space of DAG models. By reducing the need for exhaustive specification across multiple models, this approach potentially enhances the feasibility of Bayesian inference in high-dimensional domains.
The theoretical results are particularly applicable to Gaussian models with continuous variables. The derivation of parameter priors and marginal likelihood calculations within this framework could simplify Bayesian model selection and averaging procedures, as demonstrated by the closed-form marginal likelihood expression provided in the paper.
Looking towards future advancements in AI, these findings could impact the design and implementation of more efficient probabilistic models in areas requiring intricate dependency mappings, such as causal inference or decision analysis. Additionally, while the paper explicitly assumes complete model equivalence, exploring relaxations of this assumption could yield further practical methodologies for incomplete or hierarchical model structures.
In summary, the insights offered by Geiger and Heckerman contribute to a deeper understanding of the foundational structures underpinning probabilistic graphical models, raising pertinent questions regarding assumptions traditionally held in statistical modeling and encouraging further exploration into relaxed conditions that preserve computational tractability.