Composition of q-entropies and hyperbolic orthogonality

TL;DR

This paper reveals a deep geometric analogy between q-entropy composition and hyperbolic geometry, linking entropy, orthogonality, and independence through the hyperboloid model and the Dvoretzky-Rogers lemma.

Contribution

It establishes a formal connection between q-entropy composition and hyperbolic geometry, providing a geometric interpretation of entropy and independence.

Findings

q-entropy composition mirrors the hyperbolic Pythagorean theorem

Hyperbolic geometry underpins the structure of q-entropy

Orthogonality relates to probability independence via the Dvoretzky-Rogers lemma

Abstract

We point out that the q-entropy composition for independent events has exactly the same form as the Pythagorean theorem in hyperbolic geometry. We justify the formal relation of hyperbolic geometry with the q-entropy through the use of the $\kappa$-entropy, which is directly related to the hyperboloid model of hyperbolic space. We comment on the relation between orthogonality in this form of the Pythagorean theorem and the independence of the probability distributions appearing in the q-entropy composition through the use of the Dvoretzky-Rogers lemma.