论文标题
高斯条件独立性结构的两种抗态度近似
Gaussoids are two-antecedental approximations of Gaussian conditional independence structures
论文作者
论文摘要
Gaussoid公理是有条件的独立推理规则，它表征了三元素地面集合的常规高斯CI结构。众所周知，没有有限的推理规则完全描述了随着地面套件的增长，常规的高斯CI。在本文中，我们显示Gaussoid公理在逻辑上暗示着最多两个先例的所有推论规则，这对于任何地面组合都对常规高斯有效。该证明是通过为最多两个CI陈述的每个纳入最小的gaussoid扩展来实现的证明，这是常规的高斯实现。此外，我们证明所有这些Gaussoid在身份矩阵周围的每个$ \ varepsilon $ -Ball中都具有合理的正定实现。为了证明，我们介绍了代数高斯在任意领域和积极的高斯在有序领域上的概念，并在代数和积极的高斯在所有特征性零领域的高质体公理的两种抗态度完整性作为副产品。
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.
