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Tsallis and Rényi Deformations Linked via a New λ-Duality.
- Source :
-
IEEE Transactions on Information Theory . Aug2022, Vol. 68 Issue 8, p5353-5373. 21p. - Publication Year :
- 2022
-
Abstract
- Tsallis and Rényi entropies, which are monotone transformations of each other, are deformations of the celebrated Shannon entropy. Maximization of these deformed entropies, under suitable constraints, leads to the $q$ -exponential family which has applications in non-extensive statistical physics, information theory and statistics. In previous information-geometric studies, the $q$ -exponential family was analyzed using classical convex duality and Bregman divergence. In this paper, we show that a generalized $\lambda $ -duality, where $\lambda = 1 - q$ is to be interpreted as the constant information-geometric curvature, leads to a generalized exponential family which is essentially equivalent to the $q$ -exponential family and has deep connections with Rényi entropy and optimal transport. Using this generalized convex duality and its associated logarithmic divergence, we show that our $\lambda $ -exponential family satisfies properties that parallel and generalize those of the exponential family. Under our framework, the Rényi entropy and divergence arise naturally, and we give a new proof of the Tsallis/Rényi entropy maximizing property of the $q$ -exponential family. We also introduce a $\lambda $ -mixture family which may be regarded as the dual of the $\lambda $ -exponential family, and connect it with other mixture-type families. Finally, we discuss a duality between the $\lambda $ -exponential family and the $\lambda $ -logarithmic divergence, and study its statistical consequences. [ABSTRACT FROM AUTHOR]
- Subjects :
- *MAXIMUM entropy method
*RENYI'S entropy
*INFORMATION theory
*STATISTICAL physics
Subjects
Details
- Language :
- English
- ISSN :
- 00189448
- Volume :
- 68
- Issue :
- 8
- Database :
- Academic Search Index
- Journal :
- IEEE Transactions on Information Theory
- Publication Type :
- Academic Journal
- Accession number :
- 157958013
- Full Text :
- https://doi.org/10.1109/TIT.2022.3159385