Generalized measurement of uncertainty and the maximizable entropy

TL;DR

This paper introduces a generalized framework for measuring uncertainty and deriving maximizable entropies for various probability distributions, encompassing both extensive and nonextensive statistics.

Contribution

It proposes a new variational relationship for uncertainty measurement and a generalized expectation definition to find concrete forms of maximizable entropies.

Findings

Derived explicit forms of maximizable entropies for different distributions.

Unified approach applicable to extensive and nonextensive statistics.

Provides a practical physical interpretation of uncertainty measurement.

Abstract

For a random variable we can define a variational relationship with practical physical meaning as dI=dbar(x)-bar(dx), where I is called as uncertainty measurement. With the help of a generalized definition of expectation, bar(x)=sum_(i)g(p_i)x_i, and the expression of dI, we can find the concrete forms of the maximizable entropies for any given probability distribution function, where g(p_i) may have different forms for different statistics which includes the extensive and nonextensive statistics.