Structural Properties of Entropic Vectors and Stability of the Ingleton Inequality
This work addresses a theoretical problem in information theory by providing new bounds and transparent proofs for the stability of the Ingleton inequality, which is incremental but offers conceptual clarity.
The paper tackles the stability of the Ingleton inequality under small violations of conditional independence in entropic vectors, showing that the inequality holds up to controlled error terms when selected conditional mutual information terms are small but not zero.
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 Matúš (2007) and of Dougherty--Freiling--Zeger (2011), while others appear to be new.