Natural differentiable structures on statistical models and the Fisher metric

TL;DR

This paper explores the relationship between Fisher metrics and differentiability in statistical models, comparing various smoothness concepts across Information Geometry, measure theory, and statistics.

Contribution

It provides a comprehensive comparison of different notions of differentiability and smoothness in statistical models, clarifying their interrelations and implications.

Findings

Identifies connections between Fisher metric and differentiability concepts

Clarifies the role of smooth statistical manifolds in information geometry

Discusses open problems in differentiable measure models

Abstract

In this paper I discuss the relation between the concept of the Fisher metric and the concept of differentiability of a family of probability measures. I compare the concepts of smooth statistical manifolds, differentiable families of measures, $k$-integrable parameterized measure models, diffeological statistical models, differentiable measures, which arise in Information Geometry, mathematical statistics and measure theory, and discuss some related problems.