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Fractional Conformable Approximation of Csiszar’s f-Divergence

  • George A. AnastassiouEmail author
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 886)

Abstract

Here are given tight probabilistic inequalities that give nearly best estimates for the Csiszar’s f-divergence. These use the right and left conformable fractional derivatives of the directing function f. Csiszar’s f-divergence or the so called Csiszar’s discrimination is used as a measure of dependence between two random variables which is a very important aspect of stochastics, we apply our results there. The Csiszar’s discrimination is the most essential and general measure for the comparison between two probability measures. See also [4].

References

  1. 1.
    T. Abdeljawad, On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    G.A. Anastassiou, Fractional and other approximation of Csiszar’s \(f\)-divergence, (Proc. FAAT 04-Ed. F. Altomare). Rend. Circ. Mat. Palermo, Serie II, Suppl. 76, 197–212 (2005)Google Scholar
  3. 3.
    G.A. Anastassiou, Nonlinearity: Ordinary and Fractional Approximations by Sublinear and Max-Product Operators. Studies in Systems, Decision and Control, vol. 147 (Springer, New York, 2018)Google Scholar
  4. 4.
    G.A. Anastassiou, Conformable Fractional Approximation of Csiszar’s \(f\)-Divergence, submitted for publication (2019)Google Scholar
  5. 5.
    N.S. Barnett, P. Gerone, S.S. Dragomir, A. Sofo, Approximating Csiszar’s \(f\)-divergence by the use of Taylor’s formula with integral remainder (paper #10, pp. 16), in, Inequalities for Csiszar\(f\)-Divergence in Information Theory, ed. by S.S. Dragomir (Victoria University, Melbourne, 2000), http://rgmia.vu.edu.au
  6. 6.
    I. Csiszar, Eine Informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. Magyar Trud. Akad. Mat. Kutato Int. Közl. 8, 85–108 (1963)Google Scholar
  7. 7.
    I. Csiszar, Information-type measures of difference of probability distributions and indirect observations. Stud. Sci. Math. Hung. 2, 299–318 (1967)MathSciNetzbMATHGoogle Scholar
  8. 8.
    S.S. Dragomir (ed.), Inequalities for Csiszar \(f\)-Divergence in Information Theory (Victoria University, Melbourne, 2000), http://rgmia.vu.edu.au
  9. 9.
    R. Khalil, M. Al. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S. Kullback, Information Theory and Statistics (Wiley, New York, 1959)Google Scholar
  11. 11.
    S. Kullback, R. Leibler, On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Rényi, On measures of dependence. Acta Math. Acad. Sci. Hung. 10, 441–451 (1959)MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. Rényi, On measures of entropy and information, in Proceedings of the 4th Berkeley Symposium on Mathematical Statistic and Probability, I, CA, Berkeley, 1960, pp. 547–561Google Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

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