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Entropy 2015, 17(7), 4485-4499; doi:10.3390/e17074485

On Monotone Embedding in Information Geometry

Department of Psychology and Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA
Received: 27 January 2015 / Revised: 28 February 2015 / Accepted: 17 March 2015 / Published: 25 June 2015
(This article belongs to the Special Issue Information, Entropy and Their Geometric Structures)
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Abstract

A paper was published (Harsha and Subrahamanian Moosath, 2014) in which the authors claimed to have discovered an extension to Amari's \(\alpha\)-geometry through a general monotone embedding function. It will be pointed out here that this so-called \((F, G)\)-geometry (which includes \(F\)-geometry as a special case) is identical to Zhang's (2004) extension to the \(\alpha\)-geometry, where the name of the pair of monotone embedding functions \(\rho\) and \(\tau\) were used instead of \(F\) and \(H\) used in Harsha and Subrahamanian Moosath (2014). Their weighting function \(G\) for the Riemannian metric appears cosmetically due to a rewrite of the score function in log-representation as opposed to \((\rho, \tau)\)-representation in Zhang (2004). It is further shown here that the resulting metric and \(\alpha\)-connections obtained by Zhang (2004) through arbitrary monotone embeddings is a unique extension of the \(\alpha\)-geometric structure. As a special case, Naudts' (2004) \(\phi\)-logarithm embedding (using the so-called \(\log_\phi\) function) is recovered with the identification \(\rho=\phi, \, \tau=\log_\phi\), with \(\phi\)-exponential \(\exp_\phi\) given by the associated convex function linking the two representations. View Full-Text
Keywords: α-embedding; monotone embedding; conjugate embedding; generalized Fisher–Rao metric; Amari–Chentsov tensor; deformed logarithm; representation duality; (ρ,τ)-geometry α-embedding; monotone embedding; conjugate embedding; generalized Fisher–Rao metric; Amari–Chentsov tensor; deformed logarithm; representation duality; (ρ,τ)-geometry
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Zhang, J. On Monotone Embedding in Information Geometry. Entropy 2015, 17, 4485-4499.

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