Unique additive information measures - Boltzmann-Gibbs-Shannon, Fisher and beyond

TL;DR

This paper proves that the only additive, isotropic information measures depending on probability distributions and their first derivatives are linear combinations of Boltzmann-Gibbs-Shannon and Fisher measures, revealing their role in power law distributions.

Contribution

It establishes a unique characterization of additive information measures involving derivatives, extending the understanding of information measures beyond classical forms.

Findings

Power law equilibrium distributions arise from the interaction of Shannon and Fisher measures.

A second order derivative dependent information measure is also derived.

Theoretical framework for additive information measures involving derivatives is developed.

Abstract

It is proved that the only additive and isotropic information measure that can depend on the probability distribution and also on its first derivative is a linear combination of the Boltzmann-Gibbs-Shannon and Fisher information measures. Power law equilibrium distributions are found as a result of the interaction of the two terms. The case of second order derivative dependence is investigated and a corresponding additive information measure is given.