Nonparametric estimations and the diffeological Fisher metric

In this paper, first, we survey the concept of diffeological Fisher metric and its naturality, using functorial language of probability morphisms, and slightly extending L\^e's theory in \cite{Le2020} to include weakly $C^k$-diffeological statistical models. Then we introduce the resulting notions of the diffeological Fisher distance, the diffeological Hausdorff--Jeffrey measure and explain their role in classical and Bayesian nonparametric estimation problems in statistics.