On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics

We give a new proof of the theorems on the maximum entropy principle in Tsallis statistics. That is, we show that the q -canonical distribution attains the maximum value of the Tsallis entropy, subject to the constraint on the q -expectation value, and the q -Gaussian distribution attains the maximu...

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Bibliographic Details
Published inJournal of mathematical physics Vol. 50; no. 1; pp. 013303 - 013303-12
Main Author Furuichi, Shigeru
Format Journal Article
LanguageEnglish
Published Melville, NY American Institute of Physics 01.01.2009
Subjects
Exact sciences and technology
Lagrange multiplier
Mathematical methods in physics
Mathematics
Maximum entropy method
Normal distribution
Physics
Sciences and techniques of general use
Theorems
Entropy
Mathematical physics
Fisher information
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ISSN0022-2488
1089-7658
DOI10.1063/1.3063640

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Summary:We give a new proof of the theorems on the maximum entropy principle in Tsallis statistics. That is, we show that the q -canonical distribution attains the maximum value of the Tsallis entropy, subject to the constraint on the q -expectation value, and the q -Gaussian distribution attains the maximum value of the Tsallis entropy, subject to the constraint on the q -variance, as applications of the non-negativity of the Tsallis relative entropy, without using the Lagrange multipliers method. In addition, we define a q -Fisher information and then prove a q -Cramér–Rao inequality that the q -Gaussian distribution with special q -variances attains the minimum value of the q -Fisher information.
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ISSN:0022-2488
1089-7658
DOI:10.1063/1.3063640