Nonparametric estimations and the diffeological Fisher metric

TL;DR

This paper surveys the diffeological Fisher metric, extends its theory to weakly smooth models, and explores its applications in classical and Bayesian nonparametric estimation.

Contribution

It introduces the diffeological Fisher distance and Hausdorff--Jeffrey measure, extending the theory to include weakly $C^k$-diffeological models and their role in statistical estimation.

Findings

Extended L extsuperscript{e}'s theory to weakly $C^k$-diffeological models.

Defined diffeological Fisher distance and Hausdorff--Jeffrey measure.

Applied concepts to classical and Bayesian nonparametric estimation.

Abstract

In this paper, first, we survey the concept of diffeological Fisher metric and its naturality, using functorial language of probability morphisms, and slightly extending L\^e's theory in \cite{Le2020} to include weakly $C^k$-diffeological statistical models. Then we introduce the resulting notions of the diffeological Fisher distance, the diffeological Hausdorff--Jeffrey measure and explain their role in classical and Bayesian nonparametric estimation problems in statistics.