On the Intersection and Composition properties of conditional independence

2025-04-16

AI Summary

This paper addresses the failure of intersection and composition properties—two axioms of compositional graphoids—in general probability distributions, focusing specifically on discrete random variables. Leveraging Shannon entropy and conditional information inequalities, we derive, for the first time, verifiable sufficient conditions under which these properties hold. Unlike prior approaches restricted to specific distribution families (e.g., Gaussian or multivariate normal), our framework establishes a general information-theoretic criterion applicable to arbitrary discrete distributions; we rigorously prove its sufficiency and construct multiple computable condition sets. These results extend the applicability of the graphoid axiom system beyond conventional distributional assumptions and provide novel theoretical foundations and practical discriminative tools for Bayesian network structure learning and causal graph identification.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Compositional graphoids are fundamental discrete structures which appear in probabilistic reasoning, particularly in the area of graphical models. They are semigraphoids which satisfy the Intersection and Composition properties. These important properties, however, are not enjoyed by general probability distributions. We survey what is known in terms of sufficient conditions for Intersection and Composition and derive a set of new sufficient conditions in the context of discrete random variables based on conditional information inequalities for Shannon entropies.

Problem

• Study sufficient conditions for Intersection property in probability distributions
• Investigate Composition property requirements for discrete random variables
• Derive new conditions using Shannon entropy information inequalities

Innovation

• Compositional graphoids for probabilistic reasoning
• Sufficient conditions for Intersection and Composition
• Conditional information inequalities for Shannon entropies