No eleventh conditional Ingleton inequality

TL;DR

This paper constructs a specific probability distribution on four binary variables that satisfies certain conditional independencies but violates the Ingleton inequality, resolving a recent open question and classifying minimal sets of assumptions related to this inequality.

Contribution

It provides a counterexample distribution demonstrating the violation of Ingleton inequality under specified conditions and classifies minimal conditional independence sets that uphold it.

Findings

Constructed a distribution violating Ingleton inequality with given independencies

Classified minimal sets of assumptions making Ingleton inequality hold

Resolved a recent open question by Studený about conditional independence and Ingleton inequality

Abstract

A rational probability distribution on four binary random variables $X, Y, Z, U$ is constructed which satisfies the conditional independence relations $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Y]$, $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Z \mid U]$, $[Y \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid Z]$ and $[Z \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid XY]$ but whose entropy vector violates the Ingleton inequality. This settles a recent question of Studen\'y (IEEE Trans. Inf. Theory vol. 67, no. 11) and shows that there are, up to symmetry, precisely ten inclusion-minimal sets of conditional independence assumptions on four discrete random variables which make the Ingleton inequality hold. The last case in the classification of which of these inequalities are essentially conditional is also settled.