Log In Sign Up

Gaussoids are two-antecedental approximations of Gaussian conditional independence structures

by   Tobias Boege, et al.

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 ε-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.


page 1

page 2

page 3

page 4


Selfadhesivity in Gaussian conditional independence structures

Selfadhesivity is a property of entropic polymatroids which can be formu...

Constructing Faithful Homomorphisms over Fields of Finite Characteristic

We study the question of algebraic rank or transcendence degree preservi...

Semigraphoids Are Two-Antecedental Approximations of Stochastic Conditional Independence Models

The semigraphoid closure of every couple of CI-statements (GI=conditiona...

Incidence geometry in the projective plane via almost-principal minors of symmetric matrices

We present an encoding of a polynomial system into vanishing and non-van...

On the relative power of algebraic approximations of graph isomorphism

We compare the capabilities of two approaches to approximating graph iso...

Regular matroids have polynomial extension complexity

We prove that the extension complexity of the independence polytope of e...

A Coinductive Reformulation of Milner's Proof System for Regular Expressions Modulo Bisimilarity

Milner (1984) defined an operational semantics for regular expressions a...