Geodesic
Joint Range of Rényi entropies
Kybernetika, Tome 45 (2009) no. 6, pp. 901-911 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The exact range of the joined values of several Rényi entropies is determined. The method is based on topology with special emphasis on the orientation of the objects studied. Like in the case when only two orders of the Rényi entropies are studied, one can parametrize the boundary of the range. An explicit formula for a tight upper or lower bound for one order of entropy in terms of another order of entropy cannot be given.
The exact range of the joined values of several Rényi entropies is determined. The method is based on topology with special emphasis on the orientation of the objects studied. Like in the case when only two orders of the Rényi entropies are studied, one can parametrize the boundary of the range. An explicit formula for a tight upper or lower bound for one order of entropy in terms of another order of entropy cannot be given.
Classification : 62B10, 94A17
Keywords: generalized Vandermonde determinant; orientation; Rényi entropies; Shannon entropy 
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     url = {http://geodesic.mathdoc.fr/item/KYB_2009_45_6_a1/}
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Harremoës, Peter. Joint Range of Rényi entropies. Kybernetika, Tome 45 (2009) no. 6, pp. 901-911. http://geodesic.mathdoc.fr/item/KYB_2009_45_6_a1/

[1] E. Arikan: An inequality on guessing and its application to sequential decoding. IEEE Trans. Inform. Theory 42 (1996), 1, 99–105. | MR | Zbl

[2] C. Arndt: Information Measures. Springer, Berlin 2001. | MR | Zbl

[3] M. Ben-Bassat: $f$-entropies, probability of error, and feature selection. Inform. and Control 39 (1978), 227–242. | MR | Zbl

[4] I. Csiszár: Generalized cutoff rates and Rényi information measures. IEEE Trans. Inform. Theory 41 (1995), 1, 26–34. | MR

[5] M. Feder and N. Merhav: Relations between entropy and error probability. IEEE Trans. Inform. Theory 40 (1994), 259–266.

[6] J. D. Golić: On the relationship between the information measures and the Bayes probability of error. IEEE Trans. Inform. Theory 35 (1987), 5, 681–690. | MR

[7] A. György and T. Linder: Optimal entropy-constrained scalar quantization of a uniform source. IEEE Trans. Inform. Theory 46 (2000), 7, 2704–2711. | MR

[8] P. Harremoës and F. Topsøe: Inequalities between entropy and index of coincidence derived from information diagrams. IEEE Trans. Inform. Theory 47 (2001), 7, 2944–2960. | MR

[9] P. Harremoës and I. Vajda: Efficiency of entropy testing. In: Internat. Symposium on Information Theory, pp. 2639–2643. IEEE 2008.

[10] P. Harremoës and I. Vajda: On the Bahadur-efficient testing of uniformity by means of the entropy. IEEE Trans. Inform. Theory 54 (2008), 1, 321–331. | MR

[11] V. A. Kovalevskij: The Problem of Character Recognition from the Point of View of Mathematical Statistics. Spartan, New York 1967, pp. 3–30.

[12] J. W. Robbin and D. A. Salamon: The exponential Vandermonde matrix. Linear Algebra Appl. 317 (2000), 1–3, 225 – 226. | MR

[13] W. Rudin: Principles of Mathematical Analysis. (Internat. Series in Pure and Applied Mathematics.) Third edition. McGraw-Hill, New York 1976. | MR | Zbl

[14] E. H. Spanier: Algebraic Topology. Springer, Berlin 1982. | MR | Zbl

[15] D. L. Tebbe and S. J. Dwyer: Uncertainty and the probability of error. IEEE Trans. Inform. Theory 14 (1968), 14, 516–518.

[16] K. Zyczkowski: Rényi extrapolation of Shannon entropy. Open Systems and Information Dynamics 10 (2003), 297–310. | MR | Zbl