Some $f$-Divergence Measures Related to Jensen's One

Year 2023, Volume: 6 Issue: 4, 140 - 154, 18.12.2023

Sever Dragomır

https://doi.org/10.32323/ujma.1362709

Abstract

In this paper, we introduce some $f$-divergence measures that are related to the Jensen's divergence introduced by Burbea and Rao in 1982. We establish their joint convexity and provide some inequalities between these measures and a combination of Csisz\'{a}r's $f$-divergence, $f$-midpoint divergence and $f$-integral divergence measures.

Keywords

$f$-divergence measures, $\chi ^{2}$-divergence, HH $f$-divergence measures, Jensen divergence

References

• [1] I. Csiszar, Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen Ketten, (German), Magyar Tud. Akad. Mat. Kutato Int. Kozl., 8 (1963), 85–108.
• [2] P. Cerone, S. S. Dragomir, F. Osterreicher, Bounds on extended f -divergences for a variety of classes, Kybernetika (Prague), 40(6) (2004), 745–756.
• [3] P. Kafka, F. Osterreicher, I. Vincze, On powers of f -divergence defining a distance, Studia Sci. Math. Hungar., 26 (1991), 415–422.
• [4] F. Osterreicher, I. Vajda, A new class of metric divergences on probability spaces and its applicability in statistics, Ann. Inst. Statist. Math., 55(3) (2003), 639–653.
• [5] F. Liese, I. Vajda, Convex Statistical Distances, Teubuer–Texte zur Mathematik, Band, 95, Leipzig, (1987).
• [6] P. Cerone, S. S. Dragomir, Approximation of the integral mean divergence and f -divergence via mean results, Math. Comput. Modelling, 42(1-2) (2005), 207–219.
• [7] S. S. Dragomir, Some inequalities for (m;M)-convex mappings and applications for the Csisz´ar F-divergence in information theory, Math. J. Ibaraki Univ., 33 (2001), 35–50.
• [8] S. S. Dragomir, Some inequalities for two Csisz´ar divergences and applications, Mat. Bilten, 25 (2001), 73–90.
• [9] S. S. Dragomir, An upper bound for the Csiszar f-divergence in terms of the variational distance and applications, Panamer. Math. J. 12 (2002), no. 4, 43–54.
• [10] S. S. Dragomir, An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure and Appl. Math., 3(2) (2002), Art. 31.
• [11] S. S. Dragomir, An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure and Appl. Math., 3(3) (2002), Art. 35.
• [12] S. S. Dragomir, Upper and lower bounds for Csiszar f -divergence in terms of Hellinger discrimination and applications, Nonlinear Anal. Forum, 7(1) (2002), 1–13.
• [13] S. S. Dragomir, Bounds for f -divergences under likelihood ratio constraints, Appl. Math., 48(3) (2003), 205–223.
• [14] S. S. Dragomir, New inequalities for Csiszar divergence and applications, Acta Math. Vietnam., 28(2) (2003), 123–134.
• [15] S. S. Dragomir, A generalized f -divergence for probability vectors and applications, Panamer. Math. J., 13(4) (2003), 61–69.
• [16] S. S. Dragomir, Some inequalities for the Csiszar j-divergence when j is an L-Lipschitzian function and applications, Ital. J. Pure Appl. Math., 15 (2004), 57–76.
• [17] S. S. Dragomir, A converse inequality for the Csiszar F-divergence, Tamsui Oxf. J. Math. Sci., 20(1) (2004), 35–53.
• [18] S. S. Dragomir, Some general divergence measures for probability distributions, Acta Math. Hungar., 109(4) (2005), 331–345.
• [19] S. S. Dragomir, Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc., 74(3)(2006), 471–476.
• [20] S. S. Dragomir, A refinement of Jensen’s inequality with applications for f -divergence measures, Taiwanese J. Math., 14(1) (2010), 153–164.
• [21] J. Burbea, C. R. Rao, On the convexity of some divergence measures based on entropy functions, IEEE Tran. Inf. Theor., Vol. IT-28(3) (1982), 489–495.
• [22] S. S. Dragomir, C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, (2000), https://rgmia.org/papers/monographs/Master.pdf.