🤖 AI Summary
This study investigates the stability of the Ingleton inequality under approximate conditional independence—specifically, when the conditional mutual information is small but nonzero—and quantifies the extent to which the inequality may be violated. To this end, we introduce a novel analytical framework that avoids tropicalization and dispenses with the intricate machinery of tropical probability spaces, thereby substantially simplifying the stability analysis. Within this framework, we derive explicit error bounds that improve upon existing estimates, resolve an open problem concerning the stability of the sum of two Ingleton expressions, and consequently construct a new family of entropy inequalities that confirm the stability of this sum.
📝 Abstract
The classical Ingleton inequality is known to hold for entropic points under specific exact conditional independence constraints. Recently, Matveev and Romashchenko (2026) investigated the stability of these implications, quantifying the extent to which the Ingleton inequality can be violated when a group of conditional mutual information terms is small but non-zero. While their proofs relied fundamentally on the complex framework of tropical probability spaces, we revisit these stability results using a completely tropicalization-free approach. By developing an alternative framework, we significantly streamline the underlying concepts and proofs, derive explicit error terms, and improve some estimates. Furthermore, we resolve an open problem posed in prior work by exhibiting a new infinite family of entropy inequalities that establishes the stability of the sum of two Ingleton expressions.