Diffeological statistical models, the Fisher metric and probabilistic mappings

In this note 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 $P \subset {\cal P}({\cal X})$ is preserved under any probabilistic mapping $T: {\cal X}\leadsto {\cal Y}$ 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.

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