Gaussoids are two-antecedental approximations of Gaussian conditional independence structures

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.Comment: 24 pages; v3: added Preliminaries section; corrected miscalculations in examples and added source code lin

Semigraphoids are Two-Antecedental Approximations of Stochastic Conditional Independence Models

The semigraphoid closure of every couple of CI-statements (CI=conditional independence) is a stochastic CI-model. As a consequence of this result it is shown that every probabilistically sound inference rule for CI-models, having at most two antecedents, is derivable from the semigraphoid inference rules. This justifies the use of semigraphoids as approximations of stochastic CI-models in probabilistic reasoning. The list of all 19 potential dominant elements of the mentioned semigraphoid closure is given as a byproduct. 1 INTRODUCTION Many reasoning tasks arising in AI can be considerably simplified if a suitable concept of relevance or irrelevance of variables is utilized. The conditional irrelevance in probabilistic reasoning is modelled by means of the concept of stochastic conditional independence (CI) -- details about the probabilistic approach to uncertainty handling in AI are in Pearl's book (1988). The fact that every CI-statement can be interpreted as certain qualitative re..