Title
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Rho-tau embedding and gauge freedom in information geometry
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Author
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Abstract
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The standard model of information geometry, expressed as FisherRao metric and Amari-Chensov tensor, reflects an embedding of probability density by log log -transform. The present paper studies parametrized statistical models and the induced geometry using arbitrary embedding functions, comparing single-function approaches (Eguchis U-embedding and Naudts deformed-log or phi-embedding) and a two-function embedding approach (Zhangs conjugate rho-tau embedding). In terms of geometry, the rho-tau embedding of a parametric statistical model defines both a Riemannian metric, called rho-tau metric, and an alpha-family of rho-tau connections, with the former controlled by a single function and the latter by both embedding functions ρ ρ and τ τ in general. We identify conditions under which the rho-tau metric becomes Hessian and hence the ±1 ±1 rho-tau connections are dually flat. For any choice of rho and tau there exist models belonging to the phi-deformed exponential family for which the rho-tau metric is Hessian. In other cases the rhotau metric may be only conformally equivalent with a Hessian metric. Finally, we show a formulation of the maximum entropy framework which yields the phi-exponential family as the solution. |
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Language
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English
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Source (journal)
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Information geometry. - Singapore, 2018, currens
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Publication
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Singapore
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Springer
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2018
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ISSN
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2511-249X
[online]
2511-2481
[print]
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DOI
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10.1007/S41884-018-0004-6
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Volume/pages
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(2018)
, p. 1-37
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Full text (Publisher's DOI)
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Full text (open access)
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