On Normalization Functions and $$\varphi $$-Families of Probability Distributions

Andrade, Luiza Helena Félix de; Vieira, Francisca Leidmar Josué; Cavalcante, Charles C.

doi:10.1007/978-3-030-65459-7_2

Cited by 1 publication

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“…The work [ 6 ] unified the intermediary results in [ 10 , 11 , 31 ] and provided a general deformation framework that preserves the rigid interlocking of: (i) the functional form of entropy, cross-entropy, and relative entropy (divergence); (ii) the functional form of the deformed probability family with the corresponding normalization and potential and the duality between the natural and expectation parameterizations; (iii) the expressions of the Riemannian metric (Fisher–Rao metric in general and Hessian metric in particular) and of the conjugate connections. Some of these concepts have their correspondence in nonparametric probability families as well [ 32 , 33 , 34 ].…”

Section: The Standard Model Of Information Geometrymentioning

confidence: 99%

λ-Deformation: A Canonical Framework for Statistical Manifolds of Constant Curvature

Zhang

Wong

2022

Entropy

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This paper systematically presents the λ-deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have their correspondence in the λ-deformed case: λ-convexity, λ-conjugation, λ-biorthogonality, λ-logarithmic divergence, λ-exponential and λ-mixture families, etc. In particular, λ-deformation unifies Tsallis and Rényi deformations by relating them to two manifestations of an identical λ-exponential family, under subtractive or divisive probability normalization, respectively. Unlike the different Hessian geometries of the exponential and mixture families, the λ-exponential family, in turn, coincides with the λ-mixture family after a change of random variables. The resulting statistical manifolds, while still carrying a dualistic structure, replace the Hessian metric and a pair of dually flat conjugate affine connections with a conformal Hessian metric and a pair of projectively flat connections carrying constant (nonzero) curvature. Thus, λ-deformation is a canonical framework in generalizing the well-known dually flat Hessian structure of information geometry.

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