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.

翻译：在本文中,首先,我们用概率形态学的比较语言来调查地质渔业测量概念及其自然性,并略微扩展Lçée在\cite{Le2020}中的理论,将微弱的 $Cäk$-difficist 统计模型纳入其中。然后,我们引入了由此形成的地质渔业距离概念,即Difficist Hausdorff-Jeffrey测量法,并解释其在古典和巴伊西亚非参数估算统计问题中的作用。