Diffeological statistical models, the Fisher metric and probabilistic mappings

12/04/2019

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by   Hông Vân Lê, et al.

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In this note I 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. Then I 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 I extend the Cramér-Rao inequality to the class of 2-integrable C^k-diffeological statistical models.