Deformed Algebras and Generalizations of Independence on Deformed Exponential Families
by
Hiroshi Matsuzoe 1,*
and Tatsuaki Wada 2
Hiroshi Matsuzoe 1,* and Tatsuaki Wada 2
1
Department of Computer Science and Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, 466-8555, Japan
2
Department of Electrical and Electronic Engineering, Ibaraki University, 4-12-1 Nakanarusawa, Hitachi, Ibaraki, 316-8511, Japan
*
Author to whom correspondence should be addressed.
Entropy 2015, 17(8), 5729-5751; https://doi.org/10.3390/e17085729
Received: 1 February 2015 / Revised: 1 February 2015 / Accepted: 4 August 2015 / Published: 10 August 2015
(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)
A deformed exponential family is a generalization of exponential families. Since the useful classes of power law tailed distributions are described by the deformed exponential families, they are important objects in the theory of complex systems. Though the deformed exponential families are defined by deformed exponential functions, these functions do not satisfy the law of exponents in general. The deformed algebras have been introduced based on the deformed exponential functions. In this paper, after summarizing such deformed algebraic structures, it is clarified how deformed algebras work on deformed exponential families. In fact, deformed algebras cause generalization of expectations. The three kinds of expectations for random variables are introduced in this paper, and it is discussed why these generalized expectations are natural from the viewpoint of information geometry. In addition, deformed algebras cause generalization of independences. Whereas it is difficult to check the well-definedness of deformed independence in general, the κ-independence is always well-defined on κ-exponential families. This is one of advantages of κ-exponential families in complex systems. Consequently, we can well generalize the maximum likelihood method for the κ-exponential family from the viewpoint of information geometry.
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Keywords:
deformed algebra; deformed exponential family; expectation functional; information geometry; statistical manifold; generalized maximum likelihood method
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MDPI and ACS Style
Matsuzoe, H.; Wada, T. Deformed Algebras and Generalizations of Independence on Deformed Exponential Families. Entropy 2015, 17, 5729-5751.
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