Conditional probabilities via line arrangements and point configurations

TL;DR

This paper explores the relationship between probability distributions with specific conditional independence constraints and geometric configurations of points and lines, using algebraic and combinatorial methods to characterize such distributions.

Contribution

It introduces a novel connection between CI constraints and incidence geometry, providing algebraic tools to analyze distributions with hidden variables.

Findings

Characterization of CI distributions via hypergraph decompositions

Extension of determinantal ideals to hypergraph ideals with geometric interpretations

Insight into the algebraic structure of distributions with hidden variables

Abstract

We study the connection between probability distributions satisfying certain conditional independence (CI) constraints, and point and line arrangements in incidence geometry. To a family of CI statements, we associate a polynomial ideal whose algebraic invariants are encoded in a hypergraph. The primary decompositions of these ideals give a characterisation of the distributions satisfying the original CI statements. Classically, these ideals are generated by 2-minors of a matrix of variables, however, in the presence of hidden variables, they contain higher degree minors. This leads to the study of the structure of determinantal hypergraph ideals whose decompositions can be understood in terms of point and line configurations in the projective space.