Next Article in Journal
A Modified Hestenes-Stiefel-Type Derivative-Free Method for Large-Scale Nonlinear Monotone Equations
Previous Article in Journal
A Virus Infected Your Laptop. Let’s Play an Escape Game
Open AccessArticle

Diffeological Statistical Models, the Fisher Metric and Probabilistic Mappings

Institute of Mathematics, Czech Academy of Sciences, Zitna 25, 11567 Praha 1, Czech Republic
Mathematics 2020, 8(2), 167; https://doi.org/10.3390/math8020167
Received: 20 December 2019 / Revised: 19 January 2020 / Accepted: 21 January 2020 / Published: 30 January 2020
(This article belongs to the Special Issue Geometry and Topology in Statistics)
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 P ( X ) is preserved under any probabilistic mapping T : X 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.
Keywords: statistical model; diffeology; the Fisher metric; probabilistic mapping; Cramér-Rao inequality statistical model; diffeology; the Fisher metric; probabilistic mapping; Cramér-Rao inequality
MDPI and ACS Style

Lê, H.V. Diffeological Statistical Models, the Fisher Metric and Probabilistic Mappings. Mathematics 2020, 8, 167.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop