Curvature of fluctuation geometry and its implications on Riemannian fluctuation theory

TL;DR

This paper explores the curvature of fluctuation geometry, linking it to irreducible statistical correlations and the validity of Gaussian approximations, with implications for statistical mechanics and inference theory.

Contribution

It introduces the notion of irreducible statistical correlations and connects the curvature tensor to these correlations, advancing the geometric understanding of statistical distributions.

Findings

Curvature tensor indicates the presence of irreducible statistical correlations.

Curvature scalar provides a criterion for Gaussian approximation validity.

Exact results enable derivation of invariant fluctuation theorems.

Abstract

Fluctuation geometry was recently proposed as a counterpart approach of Riemannian geometry of inference theory. This theory describes the geometric features of the statistical manifold $\mathcal{M}$ of random events that are described by a family of continuous distributions $dp(x|\theta)$. A main goal of this work is to clarify the statistical relevance of Levi-Civita curvature tensor $R_{ijkl}(x|\theta)$ of the statistical manifold $\mathcal{M}$. For this purpose, the notion of \emph{irreducible statistical correlations} is introduced. Specifically, a distribution $dp(x|\theta)$ exhibits irreducible statistical correlations if every distribution $dp(\check{x}|\theta)$ obtained from $dp(x|\theta)$ by considering a coordinate change $\check{x}=\phi(x)$ cannot be factorized into independent distributions as $dp(\check{x}|\theta)=\prod_{i}dp^{(i)}(\check{x}^{i}|\theta)$. It is shown that the curvature tensor $R_{ijkl}(x|\theta)$ arises as a direct indicator about the existence of irreducible statistical correlations. Moreover, the curvature scalar $R(x|\theta)$ allows to introduce a criterium for the applicability of the \emph{gaussian approximation} of a given distribution function. This type of asymptotic result is obtained in the framework of the second-order geometric expansion of the distributions family $dp(x|\theta)$, which appears as a counterpart development of the high-order asymptotic theory of statistical estimation. In physics, fluctuation geometry represents the mathematical apparatus of a Riemannian extension for Einstein's fluctuation theory of statistical mechanics. Some exact results of fluctuation geometry are now employed to derive the \emph{invariant fluctuation theorems}.