A Relationship Between Information Inequalities and Group Theory

This report presents the result recently published in [1] that establishes a one-to-one correspondence between information inequalities and group inequalities. Our aim in this report is to present this result in as concise a manner as possible whilst not excluding any steps in the derivation, thus making it suitable for brief perusal. In Sections 2–4, we introduce the notions of entropy functions, group-characterizable functions, information inequalities, and group inequalities. We confine most of the technical details to Section 5 — a section that may we skipped if one is interested only in the general idea of the method in which the result is demonstrated. We complete the demonstration in Section 6 and conclude in Section 7. 2 Entropy functions Let X1,X2,...,Xn be n discrete random variables whose sample spaces are X1,X2,...,Xn respectively. Let N = {1,..., n} and Ω be the collection of all non-empty subsets of N, so |Ω | = 2n−1. For α ∈ Ω, we define Xα to be the joint random variable (Xi)i∈α with sample space Xα