Parametrized measure models

TL;DR

This paper introduces a broad framework for parametric measure models that do not require measures to share null sets, defining Fisher metrics and tensors in this general setting and exploring their invariance properties.

Contribution

It develops a new general notion of parametric measure models on arbitrary sample spaces, including definitions of roots, Fisher metric, and Amari-Chentsov tensor, with invariance properties.

Findings

Fisher metric and Amari-Chentsov tensor are well-defined in the general setting.

Invariance under sufficient statistics is preserved.

Monotonicity formulas hold in this broad framework.

Abstract

We develope a new and general notion of parametric measure models and statistical models on an arbitrary sample space $\Omega$ which does not assume that all measures of the model have the same null sets. This is given by a diffferentiable map from the parameter manifold $M$ into the set of finite measures or probability measures on $\Omega$, respectively, which is differentiable when regarded as a map into the Banach space of all signed measures on $\Omega$. Furthermore, we also give a rigorous definition of roots of measures and give a natural definition of the Fisher metric and the Amari-Chentsov tensor as the pullback of tensors defined on the space of roots of measures. We show that many features such as the preservation of this tensor under sufficient statistics and the monotonicity formula hold even in this very general set-up.