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We elaborate the notion of a Ricci curvature lower bound for parametrizedstatistical models. Following the seminal ideas of Lott-Strum-Villani, wedefine this notion based on the geodesic convexity of the Kullback-Leiblerdivergence in a Wasserstein statistical manifold, that is, a manifold ofprobability distributions endowed with a Wasserstein metric tensor structure.Within these definitions, the Ricci curvature is related to both, informationgeometry and Wasserstein geometry. These definitions allow us to formulatebounds on the convergence rate of Wasserstein gradient flows and informationfunctional inequalities in parameter space. We discuss examples of Riccicurvature lower bounds and convergence rates in exponential family models.

We elaborate the notion of a Ricci curvature lower bound for parametrizedstatistical models. Following the seminal ideas of Lott-Strum-Villani, wedefine this notion based on the geodesic convexity of the Kullback-Leiblerdivergence in a Wasserstein statistical manifold, that is, a manifold ofprobability distributions endowed with a Wasserstein metric tensor structure.Within these definitions, the Ricci curvature is related to both, informationgeometry and Wasserstein geometry. These definitions allow us to formulatebounds on the convergence rate of Wasserstein gradient flows and informationfunctional inequalities in parameter space. We discuss examples of Riccicurvature lower bounds and convergence rates in exponential family models.