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Information Geometry: From Divergence Functions to Geometric Structures

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Information Geometry is the differential geometric study of the manifold of probability density functions. Divergence functions (such as KL divergence), as measure of proximity on this manifold, play an important role in machine learning, statistical inference, optimization, etc. This talk will review the various geometric structures induced from any divergence function. Most importantly, a Riemannian metric (Fisher information) with a family of torsion-free affine connections (alpha- connections) can be induced on the manifold, this is the so-called the “statistical structure” in Information Geometry. Divergence functions can induce other important structures/quantities, such as bi-orthogonal coordinates (namely expectation and natural parameters), parallel volume form (in modeling Bayesian priors), symplectic structure (for Hamiltonian systems).

This talk is part of the Statistics series.

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