Foundations of Structural Statistics: Statistical Manifolds
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Upon a consistent topological statistical theory the application of structural statistics requires a quantification of the proximity structure of model spaces. An important tool to study these structures are (Pseudo-)Riemannian metrices, which in the category of statistical models are induced by statistical divergences. The present article is intended to extend the notation of topological statistical models by a differential structure to statistical manifolds and to introduce the differential geometric foundations to study specific families of probability distributions. In this purpose the article successively incorporates the structures of differential-, Riemannian- and symplectic geometry within an underlying topological statistical model. The last section addresses a specific structural category, termed a dually flat statistical manifold, which can be used to study the properties of exponential families, which are of particular importance in machine learning and deep learning.
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