Structural Properties of Entropic Vectors and Stability of the Ingleton Inequality

TL;DR

This paper investigates the stability of the Ingleton inequality in the context of entropic vectors, providing structural insights and bounds that extend understanding of entropy profiles under near-independence conditions.

Contribution

It introduces a structural lemma that captures mutual information effects and offers new bounds on the Ingleton inequality's stability under small violations.

Findings

Bounds that unify known inequalities by Matús and Dougherty et al.

New bounds on the stability of the Ingleton inequality.

Structural lemma simplifying the analysis of entropic vectors.

Abstract

We study constrained versions of the Ingleton inequality in the entropic setting and quantify its stability under small violations of conditional independence. Although the classical Ingleton inequality fails for general entropy profiles, it is known to hold under certain exact independence constraints. We focus on the regime where selected conditional mutual information terms are small (but not zero), and the inequality continues to hold up to controlled error terms. A central technical tool is a structural lemma that materializes part of the mutual information between two random variables, implicitly capturing the effect of infinitely many non-Shannon--type inequalities. This leads to conceptually transparent proofs without explicitly invoking such infinite families. Some of our bounds recover, in a unified way, what can also be deduced from the infinite families of inequalities of Mat\'u\v{s} (2007) and of Dougherty--Freiling--Zeger (2011), while others appear to be new.