Entropies and Ingleton's inequality

The Ingleton inequality is a necessary condition for a matroid to be linearly representable and it comes in the form of a linear inequality in its rank function. In a probability-theoretic reinterpretation of the inequality, linear subspaces are replaced by discrete random variables and ranks by Shannon entropies. In this setting, the Ingleton inequality no longer holds universally for representable rank functions but only if additional linear constraints are assumed.

In this talk, I give an overview of Milan Studený's recent systematic work on these so-called conditional Ingleton inequalities, their historical roots and my own contribution to finishing their classification for four discrete random variables.