Diffeological statistical models, the Fisher metric and probabilistic mappings

Abstract: In this note we introduce the notion of a $C<sup>k$-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 $C<sup>k$-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-L^e-Schwachh\"ofer theory of parametrized measure models. Then we show that for any positive integer $k$ the class of almost 2-integrable $C<sup>k$-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 $C<sup>k$-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 Cram\'er-Rao inequality to the class of 2-integrable $C<sup>k$-diffeological statistical models.