Statistical manifolds are family of density functions with respect to some measure endowed to geometric structures used to model information, their field of study belonging to Information Geometry, a relatively recent branch of mathematics, that uses tools of differential geometry to study statistical inference, information loss, and estimation. Our approach first considers a statistical model seen as a riemaniann differentiable manifold. This technique shows the importance of differential geometry in modeling in the sense that it uses a Riemannian metric on a statistical manifold, linear connection, parallel transport maps, etc.

In this lecture, we firstly recall the importance of differential geometry through the work of some researchers. Moreover we study the geometrical structure of parametrical statistical model. We study some geometric properties (see [1]) of statistical manifold equipped with the Riemannian Otto metric which is related to the Wasserstein distance. We construct alpha- connections on such manifold and we prove that the proposed connections are torsion-free and coincide with the Levi-Civita connection when alpha = 0. In addition, the exponentialy families and the mixture families are shown to be respectively (1)-flat and (−1)-flat. As an application, we study the Wasserstein Riemannian Gamma manifold (see [2]) which is a space of Gamma probability density functions endowed with the Riemannian Otto metric. We compute in this case, the coefficients of alpha-connections and the sectional curvature of those manifolds.

[1] Ogouyandjou, C., & Wadagni, N. (2020). Wasserstein Riemannian Geometry on Statistical Manifold. International Electronic
Journal of Geometry, 13(2), 144-151.

[2] Ogouyandjou, C., & Wadagni, N. (2020). Wasserstein Riemannian geometry of Gamma densities. Computer Science, 15(4), 1253-
1270.
https://indico.math.cnrs.fr/event/7086/