Diffeological statistical models, the Fisher metric and probabilistic mappings

Abstract

In this note we introduce the notion of a $C^k$-diffeological statistical model, which allows us to apply the theory of diffeological spaces to (possibly singular) statistical models. In particular, we introduce a class of almost 2-integrable $C^k$-diffeological statistical models that encompasses all known statistical models for which the Fisher metric is defined. This class contains a statistical model which does not appear in the Ay-Jost-Lê-Schwachhöfer theory of parametrized measure models. Then we show that for any positive integer $k$ the class of almost 2-integrable $C^k$-diffeological statistical models is preserved under probabilistic mappings. Furthermore, the monotonicity theorem for the Fisher metric also holds for this class. As a consequence, the Fisher metric on an almost 2-integrable $C^k$-diffeological statistical model $P\subset \mathcal{P}(\mathcal{X})$ is preserved under any probabilistic mapping $T:\mathcal{X}⇝\mathcal{Y}$ that is sufficient w.r.t. $P$. Finally we extend the Cramér-Rao inequality to the class of 2-integrable $C^k$-diffeological statistical models.

16 p., final version, accepted to Journal Mathematics/MDPI

References in corpus (2)

• Learning Decentralized Controllers for Robot Swarms with Graph Neural Networks
• Probabilistic morphisms and Bayesian nonparametrics

Cited by in corpus (3)

• Probabilistic morphisms and Bayesian nonparametrics
• Natural differentiable structures on statistical models and the Fisher metric
• Nonparametric estimations and the diffeological Fisher metric