Diffeological statistical models, the Fisher metric and probabilistic mappings

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00521519" target="_blank" >RIV/67985840:_____/20:00521519 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.3390/math8020167" target="_blank" >https://doi.org/10.3390/math8020167</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.3390/math8020167" target="_blank" >10.3390/math8020167</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Diffeological statistical models, the Fisher metric and probabilistic mappings

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Diffeological statistical models, the Fisher metric and probabilistic mappings

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GC18-01953J" target="_blank" >GC18-01953J: Geometrické metody ve statistické teorie učení a aplikace</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Mathematics

ISSN

2227-7390

e-ISSN

—

Svazek periodika

8

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

13

Strana od-do

167

Kód UT WoS článku

000519234000022

EID výsledku v databázi Scopus

2-s2.0-85080125263