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

Matsuzoe, Hiroshi

doi:10.3390/e19010007

Cited by 12 publications

(6 citation statements)

References 26 publications

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“…The escort distributions are a useful tool in thermodynamics. Interested readers may refer to [15] for a concise introduction.…”

Section: Resultsmentioning

confidence: 99%

Generalized Mutual Information

Zhang

2020

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Mutual information is one of the essential building blocks of information theory. It is however only finitely defined for distributions in a subclass of the general class of all distributions on a joint alphabet. The unboundedness of mutual information prevents its potential utility from being extended to the general class. This is in fact a void in the foundation of information theory that needs to be filled. This article proposes a family of generalized mutual information whose members are indexed by a positive integer n, with the nth member being the mutual information of nth order. The mutual information of the first order coincides with Shannon’s, which may or may not be finite. It is however established (a) that each mutual information of an order greater than 1 is finitely defined for all distributions of two random elements on a joint countable alphabet, and (b) that each and every member of the family enjoys all the utilities of a finite Shannon’s mutual information.

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“…The escort distributions are a useful tool in thermodynamics. Interested readers may refer to [15] for a concise introduction.…”

Section: Resultsmentioning

confidence: 99%

Generalized Mutual Information

Zhang

2020

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“…For a = 1, the Riemannian metric g (q,1) and the cubic tensor C (q,1) correspond to the Fisher metric and the Amari-Čencov tensor, respectively. See [3] for further details.…”

Section: Remarkmentioning

confidence: 99%

Gauge Freedom of Entropies on q-Gaussian Measures

Matsuzoe

Takatsu

2021

Signals and Communication Technology

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A q-Gaussian measure is a generalization of a Gaussian measure. This generalization is obtained by replacing the exponential function with the power function of exponent 1/(1 − q) (q = 1). The limit case q = 1 recovers a Gaussian measure. On the set of all q-Gaussian densities over the real line with 1 ≤ q < 3, escort expectations determine information geometric structures such as an entropy and a relative entropy. The ordinary expectation of a random variable is the integral of the random variable with respect to its law. Escort expectations admit us to replace the law by any other measures. One of the most important escort expectations on the set of all q-Gaussian densities is the q-escort expectation since this escort expectation determines the Tsallis entropy and the Tsallis relative entropy. The phenomenon gauge freedom of entropies is that different escort expectations determine the same entropy, but different relative entropies. In this chapter, we first introduce a refinement of the q-logarithmic function. Then we demonstrate the phenomenon on an open set of all q-Gaussian densities over the real line by using the refined q-logarithmic functions. We write down the corresponding Riemannian metric. Keywords Information geometry • Gauge freedom of entropies • Refined q-logarithmic function • q-Gaussian measure

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“…In the field of nonextensive statistics, q -normal distributions and the generalization, q -exponential families, play an important role [ 1 , 2 , 3 ]. Since Ohara first pointed out the correspondence between the q -parameter of nonextensive statistics and the

-parameter of information geometry [ 4 , 5 ], the information geometric structure of q -exponential families has been investigated [ 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 ].…”

Section: Introductionmentioning

confidence: 99%

Extended Divergence on a Foliation by Deformed Probability Simplexes

Uohashi

2022

Entropy

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This study considers a new decomposition of an extended divergence on a foliation by deformed probability simplexes from the information geometry perspective. In particular, we treat the case where each deformed probability simplex corresponds to a set of q-escort distributions. For the foliation, different q-parameters and the corresponding α-parameters of dualistic structures are defined on each of the various leaves. We propose the divergence decomposition theorem that guides the proximity of q-escort distributions with different q-parameters and compare the new theorem to the previous theorem of the standard divergence on a Hessian manifold with a fixed α-parameter.

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

Cited by 12 publications

References 26 publications

Generalized Mutual Information

Generalized Mutual Information

Gauge Freedom of Entropies on q-Gaussian Measures

Extended Divergence on a Foliation by Deformed Probability Simplexes

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