Diffeological statistical models, the Fisher metric and probabilistic mappings

0521519 - MÚ 2021 RIV CH eng J - Journal Article
Le, Hong-Van
Diffeological statistical models, the Fisher metric and probabilistic mappings.
Mathematics. Roč. 8, č. 2 (2020), č. článku 167. E-ISSN 2227-7390
R&D Projects: GA ČR(CZ) GC18-01953J
Institutional support: RVO:67985840
Keywords : statistical model * diffeology * the Fisher metric * probabilistic mapping
OECD category: Pure mathematics
Impact factor: 2.258, year: 2020
Method of publishing: Open access
https://doi.org/10.3390/math8020167

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.
Permanent Link: http://hdl.handle.net/11104/0306118