Some New $f$-Divergence Measures and Their Basic Properties

Year 2024, Volume: 12 Issue: 2, 43 - 59, 14.04.2024

Sever Dragomır

https://doi.org/10.36753/mathenot.1362706

Abstract

In this paper, we introduce some new $f$-divergence measures that we call $t$-\textit{asymmetric/symmetric divergence measure} and\textit{\ integral divergence measure, }establish their joint convexity and provide some inequalities that connect these $f$-divergences to the classical one introduced by Csiszar in 1963. Applications for the \textit{dichotomy class} of convex functions are provided as well.

Keywords

$f$-divergence measures, Hellinger discrimination, HH $f$-divergence measures, Jeffrey, Kullback-Leibler divergence, $\chi ^{2}$-divergence

References

• [1] Csiszár, I.: Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. Magyar Tud. Akad. Mat. Kutató Int. Közl. 8, 85-108 (1963).
• [2] Cerone, P., Dragomir, S. S., Österreicher, F.: Bounds on extended f-divergences for a variety of classes. Kybernetika (Prague). 40(6), 745-756 (2004) Preprint, RGMIA Res. Rep. Coll. 6(1), 5 (2003). http://rgmia.vu.edu.au/v6n1.html].
• [3] Kafka, P., Österreicher, F., Vincze, I.: On powers of f-divergence defining a distance. Studia Scientiarum Mathematicarum Hungarica. 26, 415-422 (1991).
• [4] Österreicher, F. Vajda, I.: A new class of metric divergences on probability spaces and its applicability in statistics. Annals of the Institute of Statistical Mathematics. 55 (3), 639-653 (2003).
• [5] Liese, F., Vajda, I.: Convex Statistical Distances. Teubuer-Texte zur Mathematik, Band, Leipzig. 95 1987.
• [6] Cerone, P., Dragomir, S. S.: Approximation of the integral mean divergence and f-divergence via mean results. Mathematical and Computer Modelling. 42(1-2), 207-219 (2005).
• [7] Dragomir, S. S.: Some inequalities for (m;M)-convex mappings and applications for the Csiszár -divergence in information theory. Mathematical Journal of Ibaraki University. 33, 35-50 (2001).
• [8] Dragomir, S. S.: Some inequalities for two Csiszár divergences and applications. Matematichki Bilten. 25, 73-90 (2001).
• [9] Dragomir, S. S.: An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. Journal of Inequalities in Pure and Applied Mathematics. 3 (2), 31 (2002).
• [10] Dragomir, S. S.: An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. Journal of Inequalities in Pure and Applied Mathematics. 3(3), 35 (2002).
• [11] Dragomir, S. S.: An upper bound for the Csiszár f-divergence in terms of the variational distance and applications. Panamerican Mathematical Journal. 12(4), 43-54 (2002).
• [12] Dragomir, S. S.: Upper and lower bounds for Csiszár f-divergence in terms of Hellinger discrimination and applications. Nonlinear Analysis Forum. 7(1), 1-13 (2002).
• [13] Dragomir, S. S.: Bounds for f-divergences under likelihood ratio constraints. Applications of Mathematics. 48(3), 205-223 (2003).
• [14] Dragomir, S. S.: New inequalities for Csiszár divergence and applications. Acta Mathematica Vietnamica. 28(2), 123-134 (2003).
• [15] Dragomir, S. S.: A generalized f-divergence for probability vectors and applications. Panamerican Mathematical Journal. 13(4), 61-69 (2003).
• [16] Dragomir, S. S.: Some inequalities for the Csiszár '-divergence when ' is an L-Lipschitzian function and applications. Italian Journal of Pure and Applied Mathematics. 15, 57-76 (2004).
• [17] Dragomir, S. S.: A converse inequality for the Csiszár -divergence. Tamsui Oxford Journal of Mathematical Sciences. 20(1), 35-53 (2004).
• [18] Dragomir, S. S.: Some general divergence measures for probability distributions. Acta Mathematica Hungarica. 109(4), 331-345 (2005).
• [19] Dragomir, S. S.: Bounds for the normalized Jensen functional. Bulletin of the Australian Mathematical Society. 74(3), 471-478 (2006).
• [20] Dragomir, S. S.: A refinement of Jensen’s inequality with applications for f-divergence measures. Taiwanese Journal of Mathematics. 14(1), 153-164 (2010).
• [21] Dragomir, S. S.: A generalization of f-divergence measure to convex functions defined on linear spaces. Communications in Mathematical Analysis. 15(2), 1-14 (2013).