On a Connection between Entropy, Extensive Measurement and Memoryless Characterization

TL;DR

This paper explores the relationship between entropy, measurement scales, and memoryless properties, deriving Tsallis entropy from a generalized distribution and emphasizing the importance of algebraic structures in analysis.

Contribution

It introduces a framework linking generalized entropies to measurement scales and algebraic structures, deriving Tsallis entropy from a specific distribution.

Findings

Generalized entropies correspond to alternative measurement scales.

An extensive measurement scale exhibits a memoryless property.

Tsallis entropy is derived from a generalized log-logistic distribution.

Abstract

We define an entropy based on a chosen governing probability distribution. If a certain kind of measurements follow such a distribution it also gives us a suitable scale to study it with. This scale will appear as a link function that is applied to the measurements. A link function can also be used to define an alternative structure on a set. We will see that generalized entropies are equivalent to using a different scale for the phenomenon that is studied compared to the scale the measurements arrive on. An extensive measurement scale is here a scale for which measurements fulfill a memoryless property. We conclude that the alternative algebraic structure defined by the link function must be used if we continue to work on the original scale. We derive Tsallis entropy by using a generalized log-logistic governing distribution. Typical applications of Tsallis entropy are related to phenomena with power-law behaviour.