Diffeological Statistical Models, the Fisher Metric and Probabilistic Mappings

Hông Vân Lê
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Hông Vân Lê: Institute of Mathematics, Czech Academy of Sciences, Zitna 25, 11567 Praha 1, Czech Republic

Mathematics, 2020, vol. 8, issue 2, 1-13

Abstract: 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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