The Geometry of Gaussoids

TL;DR

This paper develops the geometric theory of gaussoids, combinatorial structures encoding statistical independence, connecting them to advanced geometric objects like the Lagrangian Grassmannian and exploring their realizability.

Contribution

It introduces the geometric framework for gaussoids, including oriented and valuated variants, and classifies small realizable and non-realizable cases, linking to tropical and real geometry.

Findings

Gaussoids are connected to the Lagrangian Grassmannian and quadratic relations.

Classification of small realizable and non-realizable gaussoids.

Positive gaussoids are all realizable via graphical models.

Abstract

A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. The gaussoid axioms of Lnenicka and Mat\'us are equivalent to compatibility with certain quadratic relations among principal and almost-principal minors of a symmetric matrix. We develop the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. We introduce oriented gaussoids and valuated gaussoids, thus connecting to real and tropical geometry. We classify small realizable and non-realizable gaussoids. Positive gaussoids are as nice as positroids: they are all realizable via graphical models.