In my talk I shall first explain the concept of diffeological spaces introduced by Souriau. Then I shall explain how to use this concept to endow natural smooth structures on subsets of probability measures on an arbitrary measurable space. I shall discuss the concept of the diffeological Fisher metric and the resulting notion of the diffeological Hausdorff measure that are categorically natural, and meaningful for statistical estimations used in statistical physics and data analysis.
My talk is based on my paper https://doi.org/10.3390/math8020167 and my joint paper with Alexei Tuzhilin https://arxiv.org/abs/2011.13418
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