# Statistical Meaning of Mean Functions

## TL;DR

This paper explores the statistical interpretation of mean functions using Fisher information, introducing an informational mean that generalizes classical means for non-commuting matrices and extends inequalities.

## Contribution

It introduces a new informational mean derived from Fisher information, applicable to non-commuting matrices, and generalizes classical mean inequalities.

## Key findings

- The informational mean lies between the arithmetic and harmonic means.
- Fisher information properties lead to a new inequality generalizing classical mean inequalities.
- The informational mean provides a statistical interpretation of mean functions.

## Abstract

The basic properties of the Fisher information allow to reveal the statistical meaning of classical inequalities between mean functions. The properties applied to scale mixtures of Gaussian distributions lead to a new mean function of purely statistical origin, unrelated to the classical arithmetic, geometric, and harmonic means. We call it the informational mean and show that when the arguments of the mean functions are Hermitian positive definite matrices, not necessarily commuting, the informational mean lies between the arithmetic and harmonic means, playing, in a sense, the role of the geometric mean that cannot be correctly defined in case of non-commuting matrices.\\ Surprisingly the monotonicity and additivity properties of the Fisher information lead to a new generalization of the classical inequality between the arithmetic and harmonic means.