Gaussoids are two-antecedental approximations of Gaussian conditional independence structures

TL;DR

This paper demonstrates that gaussoid axioms fully capture all two-antecedent inference rules valid for Gaussian conditional independence structures across any ground set, with realizations in rational positive-definite matrices.

Contribution

It proves the logical completeness of gaussoid axioms for two-antecedent rules in Gaussian CI structures over arbitrary ground sets and fields of characteristic zero.

Findings

Gaussoid axioms imply all two-antecedent inference rules for Gaussian CI.

Every minimal gaussoid extension has a rational positive-definite realization.

The results extend to algebraic and positive Gaussians over all fields of characteristic zero.

Abstract

The gaussoid axioms are conditional independence inference rules which characterize regular Gaussian CI structures over a three-element ground set. It is known that no finite set of inference rules completely describes regular Gaussian CI as the ground set grows. In this article we show that the gaussoid axioms logically imply every inference rule of at most two antecedents which is valid for regular Gaussians over any ground set. The proof is accomplished by exhibiting for each inclusion-minimal gaussoid extension of at most two CI statements a regular Gaussian realization. Moreover we prove that all those gaussoids have rational positive-definite realizations inside every $\varepsilon$-ball around the identity matrix. For the proof we introduce the concept of algebraic Gaussians over arbitrary fields and of positive Gaussians over ordered fields and obtain the same two-antecedental completeness of the gaussoid axioms for algebraic and positive Gaussians over all fields of characteristic zero as a byproduct.