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Information geometry is a branch of mathematics that applies the techniques of differential geometry to the field of probability theory. This is done by taking probability distributions for a statistical model as the points of a Riemannian manifold, forming a statistical manifold. The Fisher information metric provides the Riemannian metric.
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Comparison of Information and Wasserstein Topologies
Wasserstein distance has not good properties of invariances compared to Fisher Metric, that could be extended on convex cones by Jean-Louis Koszul tools (https://link.springer.com/chapter/10.1007/978- …