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A note on bounded entropies

  • Amblard, Pierre-Olivier
  • Vignat, Christophe
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    The aim of the paper is to study the link between non-additivity of some entropies and their boundedness. We propose an axiomatic construction of the entropy relying on the fact that entropy belongs to a group isomorphic to the usual additive group. This allows to show that the entropies that are additive with respect to the addition of the group for independent random variables are nonlinear transforms of the Rényi entropies, including the particular case of the Shannon entropy. As a particular example, we study as a group a bounded interval in which the addition is a generalization of the addition of velocities in special relativity. We show that Tsallis–Havrda–Charvat entropy is included in the family of entropies we define. Finally, a link is made between the approach developed in the paper and the theory of deformed logarithms.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0378437106000409
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    Article provided by Elsevier in its journal Physica A: Statistical Mechanics and its Applications.

    Volume (Year): 365 (2006)
    Issue (Month): 1 ()
    Pages: 50-56

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    Handle: RePEc:eee:phsmap:v:365:y:2006:i:1:p:50-56
    Contact details of provider: Web page: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/

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    1. Kaniadakis, G. & Lissia, M. & Scarfone, A.M., 2004. "Deformed logarithms and entropies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 41-49.
    2. Naudts, Jan, 2002. "Deformed exponentials and logarithms in generalized thermostatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 316(1), pages 323-334.
    3. Naudts, Jan, 2004. "Generalized thermostatistics based on deformed exponential and logarithmic functions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 32-40.
    4. Kaniadakis, G., 2001. "Non-linear kinetics underlying generalized statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 296(3), pages 405-425.
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