Conditional independence structures over four discrete random variables revisited: conditional Ingleton inequalities

TL;DR

This paper revisits conditional Ingleton inequalities for entropy functions of four discrete variables, proving a new inequality and simplifying existing proofs, to better characterize their conditional independence structures.

Contribution

It introduces a new conditional Ingleton inequality and provides simplified proofs for existing inequalities, enhancing understanding of independence structures among four variables.

Findings

Five conditional Ingleton inequalities characterized

New inequality proved and existing proofs simplified

Complete characterization of independence structures for four variables

Abstract

The paper deals with conditional linear information inequalities valid for entropy functions induced by discrete random variables. Specifically, the so-called conditional Ingleton inequalities are in the center of interest: these are valid under conditional independence assumptions on the inducing random variables. We discuss five inequalities of this particular type, four of which has appeared earlier in the literature. Besides the proof of the new fifth inequality, simpler proofs of (some of) former inequalities are presented. These five information inequalities are used to characterize all conditional independence structures induced by four discrete random variables.