Deformed Algebras and Generalizations of Independence on Deformed Exponential Families

Abstract

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.

Dualistic Hessian Structures Among the Thermodynamic Potentials in the κ-Thermostatistics

“…In addition, since the independence of an exponential pdf plays a fundamental role in the equilibrium statistical physics and thermodynamics, it is interesting to further study the generalization of the independence on a deformed pdf [20] based on the results obtained in this study.…”

A Sequence of Escort Distributions and Generalizations of Expectations on q-Exponential Family

“…In the previous works, the author showed that a deformed score function is unbiased with respect to the escort expectation [8,9]. This implies that a deformed score function is regarded as an estimating function on a deformed exponential family.…”

“…However, many results for the q-exponential family can be generalized for the χ-exponential family (cf. [6,8]). We remark that a q-exponential family and a χ-exponential family have further generalizations.…”

“…It is known that a q-exponential family naturally has at least three kinds of statistical manifold structures (cf. [6,8]). We reformulate these structures from the viewpoint of the sequence of escort distributions.…”

Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures

“…Up today, many papers have been written on the foundations and the theoretical consistency of κ-statistical mechanics [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48] (see also [49] and references therein). It has been applied in many research fields, such as statistical physics, thermostatistics, financial physics, social science, statistics and information theory [50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65].…”