A generalization of the maximum entropy principle for curved statistical manifolds

TL;DR

This paper extends the maximum entropy principle by incorporating information geometry and curved statistical manifolds, enabling the use of Rényi entropy for modeling complex systems beyond traditional Boltzmann-Gibbs distributions.

Contribution

It introduces a geometric generalization of the MEP based on Rényi entropy, connecting non-Euclidean geometry with entropy maximization.

Findings

Provides a theoretical foundation linking curved manifolds and Rényi entropy

Enables modeling of complex systems with non-Boltzmann distributions

Lays groundwork for new analytical methods in complex systems

Abstract

The maximum entropy principle (MEP) is one of the most prominent methods to investigate and model complex systems. Despite its popularity, the standard form of the MEP can only generate Boltzmann-Gibbs distributions, which are ill-suited for many scenarios of interest. As a principled approach to extend the reach of the MEP, this paper revisits its foundations in information geometry and shows how the geometry of curved statistical manifolds naturally leads to a generalization of the MEP based on the R\'enyi entropy. By establishing a bridge between non-Euclidean geometry and the MEP, our proposal sets a solid foundation for the numerous applications of the R\'enyi entropy, and enables a range of novel methods for complex systems analysis.