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On the Maximum \(f\)-Divergence of Probability Distributions Given the Value of Their Coupling

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Abstract

The paper is a supplement to the author’s paper [1]. Here we present explicit upper bounds (which are optimal in some cases) on the maximum value of the \(f\)-divergence \(D_f(P\,\|\, Q)\) of discrete probability distributions \(P\) and \(Q\) provided that the distribution \(Q\) (or its minimal component \(q_{\min}\)) and the value of the coupling of \(P\) and \(Q\) are fixed. We also obtain an explicit expression for the maximum value of the divergence \(D_f(P\,\|\, Q)\) provided that only the value of the coupling of \(P\) and \(Q\) is given. Results of [1] concerning the Kullback–Leibler divergence and \(\chi^2\)-divergence are particular cases of the results proved in the present paper.

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References

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Funding

The research was supported in part by the Russian Foundation for Basic Research, project no. 19-01-00364.

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Authors and Affiliations

  1. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

    V. V. Prelov

Authors
  1. V. V. Prelov
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Additional information

Translated from Problemy Peredachi Informatsii, 2021, Vol. 57, No. 4, pp. 24–33 https://doi.org/10.31857/S0555292321040021.

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Prelov, V. On the Maximum \(f\)-Divergence of Probability Distributions Given the Value of Their Coupling. Probl Inf Transm 57, 321–330 (2021). https://doi.org/10.1134/S0032946021040025

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  • DOI: https://doi.org/10.1134/S0032946021040025

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