Generalized Maximum Entropy: When and Why you need it

TL;DR

This paper explains when and why practitioners should use generalized maximum entropy methods instead of Shannon entropy, especially in systems with strong correlations, supported by theoretical review and practical applications.

Contribution

It provides a comprehensive guide on the conditions for abandoning Shannon entropy in favor of generalized entropies, with practical guidelines and illustrative applications.

Findings

Generalized entropies are necessary for strongly correlated systems.

The paper offers practical guidelines for selecting and reporting entropy measures.

Applications in economics and ecology demonstrate the advantages of generalized maximum entropy.

Abstract

The classical Maximum-Entropy Principle (MEP) based on Shannon entropy is widely used to construct least-biased probability distributions from partial information. However, the Shore-Johnson axioms that single out the Shannon functional hinge on strong system independence, an assumption often violated in real-world, strongly correlated systems. We provide a self-contained guide to when and why practitioners should abandon the Shannon form in favour of the one-parameter Uffink-Jizba-Korbel (UJK) family of generalized entropies. After reviewing the Shore and Johnson axioms from an applied perspective, we recall the most commonly used entropy functionals and locate them within the UJK family. The need for generalized entropies is made clear with two applications, one rooted in economics and the other in ecology. A simple mathematical model worked out in detail shows the power of generalized maximum entropy approaches in dealing with cases where strong system independence does not hold. We conclude with practical guidelines for choosing an entropy measure and reporting results so that analyses remain transparent and reproducible.