Abstract
Embedding or representing functions play important roles in order to produce various information geometric structure. This paper investigates them from a viewpoint of affine differential geometry [2]. By restricting affine immersions to a certain class, the probability simplex is realized to be 1-conformally flat [3] statistical manifolds immersed in \(\mathbf{R}^{n+1}\). Using this fact, we introduce a concept of conformal flattening of such manifolds to obtain dually flat statistical (Hessian) ones with conformal divergences, and show explicit forms of potential functions, dual coordinates. Finally, we demonstrate applications of the conformal flattening to nonextensive statistical physics and certain replicator equations on the probability simplex.
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Ohara, A. (2017). On Affine Immersions of the Probability Simplex and Their Conformal Flattening. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_29
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DOI: https://doi.org/10.1007/978-3-319-68445-1_29
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