Foundations of Structural Statistics: Statistical Manifolds

TL;DR

This paper develops a differential geometric framework for statistical models, extending topological models to manifolds and analyzing their structure using Riemannian and symplectic geometry.

Contribution

It introduces a differential geometric foundation for statistical models, integrating Riemannian and symplectic structures into the study of distribution families.

Findings

Defines statistical manifolds with differential structures

Extends divergence-induced metrics to a geometric setting

Provides tools for analyzing model proximity via geometry

Abstract

Upon a consistent topological statistical theory the application of structural statistics requires a quantification of the proximity structure of model spaces. An important tool to study these structures are Pseudo-Riemannian metrices, which in the category of statistical models are induced by statistical divergences. The present article extends the notation of topological statistical models by a differential structure to statistical manifolds and introduces the differential geometric foundations to study distribution families by their differential-, Riemannian- and symplectic geometry.