Structural Properties of Entropic Vectors and Stability of the Ingleton Inequality
Rostislav Matveev, Andrei Romashchenko
TL;DR
The paper tackles the stability of the Ingleton inequality in entropic settings when certain conditional independencies are only approximately satisfied. It introduces a structural lemma that materializes parts of I(X:Y) through auxiliary variables, avoiding explicit reliance on infinite non-Shannon inequalities by leveraging tropicalization, the Copy Lemma, and AK reductions. Through a main construction that builds a sequence of W_k variables, the authors derive sharp bounds: additive errors of order √ε when two CMI terms are small, and order ε log(1/ε) when four CMIs are small. The results unify and extend known non-Shannon-type inequality families while offering a conceptually transparent framework amenable to computer-assisted proof strategies and potential discovery of new conditional information inequalities.
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 Matúš (2007) and of Dougherty--Freiling--Zeger (2011), while others appear to be new.