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Open AccessArticle

Information Geometric Duality of ϕ-Deformed Exponential Families

by Jan Korbel 1,2, Rudolf Hanel 1,2 and Stefan Thurner 1,2,3,4,*
1
Section for Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, A-1090 Vienna, Austria
2
Complexity Science Hub Vienna, Josefstädter Strasse 39, A-1080 Vienna, Austria
3
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
4
IIASA, Schlossplatz 1, A-2361 Laxenburg, Austria
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(2), 112; https://doi.org/10.3390/e21020112
Received: 24 December 2018 / Revised: 11 January 2019 / Accepted: 16 January 2019 / Published: 24 January 2019
(This article belongs to the Special Issue Information Theory in Complex Systems)
In the world of generalized entropies—which, for example, play a role in physical systems with sub- and super-exponential phase space growth per degree of freedom—there are two ways for implementing constraints in the maximum entropy principle: linear and escort constraints. Both appear naturally in different contexts. Linear constraints appear, e.g., in physical systems, when additional information about the system is available through higher moments. Escort distributions appear naturally in the context of multifractals and information geometry. It was shown recently that there exists a fundamental duality that relates both approaches on the basis of the corresponding deformed logarithms (deformed-log duality). Here, we show that there exists another duality that arises in the context of information geometry, relating the Fisher information of ϕ -deformed exponential families that correspond to linear constraints (as studied by J.Naudts) to those that are based on escort constraints (as studied by S.-I. Amari). We explicitly demonstrate this information geometric duality for the case of ( c , d ) -entropy, which covers all situations that are compatible with the first three Shannon–Khinchin axioms and that include Shannon, Tsallis, Anteneodo–Plastino entropy, and many more as special cases. Finally, we discuss the relation between the deformed-log duality and the information geometric duality and mention that the escort distributions arising in these two dualities are generally different and only coincide for the case of the Tsallis deformation. View Full-Text
Keywords: generalized entropy; ϕ-deformed family; Fisher information; information geometry; Cramér–Rao bound; (c,d)-entropy generalized entropy; ϕ-deformed family; Fisher information; information geometry; Cramér–Rao bound; (c,d)-entropy
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Korbel, J.; Hanel, R.; Thurner, S. Information Geometric Duality of ϕ-Deformed Exponential Families. Entropy 2019, 21, 112.

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