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On generalized entropies, Bayesian decisions and statistical diversity. (English) Zbl 1143.94006
Let $$X$$ be a random discrete variable with distribution $$p = (p(i):i \in {\mathcal I})$$ with finite $${\mathcal I}$$. The first author [Theory of statistical inference and information. Theory and Decision Library, Series B: Mathematical and Statistical Methods, 11. (Dordrecht) etc.: Kluwer Academic Publishers. (1989; Zbl 0711.62002)] studied $$\psi$$ -entropies $$H_\psi (p) \equiv H_\psi (X) = \sum\limits_x {p(x)\psi (p(x))}$$ , where $$\psi$$ is a decreasing continuous function on $$(0,1]$$ and $$\psi (1) = 0$$. The power entropies $$H_\alpha (X)$$ [J. Havrda, F. Charvat, Kybernetika, Praha 3, 30–35(1967; Zbl 0178.22401)] are obtained using for $$\psi$$ the power function $$\psi _\alpha (\pi ) = (1 - \pi ^{\alpha - 1} )/(\alpha - 1)$$ , $$\alpha \in \mathbb{R}$$ , where $\psi _\alpha (0) = \mathop {\lim }\limits_{\pi \downarrow 0} \psi _\alpha (\pi )$ for $$\alpha \neq 1$$ and $$\psi _1 (\pi ) = - \ln \pi$$ , i.e $$\alpha = 1$$ gives the Shannon entropy; another important subcase is the quadratic entropy $$H_2 (X)$$. It is shown that generalized entropies of information sources as generalized informations in direct observations lead to nonconcave entropies, in particular infinitely many nonconcave power entropies. Relations between the entropies $$H_\alpha$$, $$\alpha \geq 0$$ and the errors of Bayesian decisions about $$X$$ are investigated; it is shown that the quadratic entropy provides estimates which are in average more than 100 and $$H_2 (X)$$ .

##### MSC:
 94A17 Measures of information, entropy 62C10 Bayesian problems; characterization of Bayes procedures
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##### References:
 [1] Cover T., Thomas J.: Elements of Information Theory. Wiley, New York 1991 · Zbl 1140.94001 [2] Cressie N., Read T. R. C.: Multinomial goodness-of-fit tests. J. Roy. Statist. Soc. Ser. B 46 (1984), 440-464 · Zbl 0571.62017 [3] Csiszár I.: Informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. Publ. Math. Inst. Hungar. Acad. Sci. Ser. A 8 (1963), 85-108 · Zbl 0124.08703 [4] Csiszár I.: Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar. 2 (1967), 299-318 · Zbl 0157.25802 [5] Csiszár I.: A class of measures of informativity of observation channels. Period. Math. Hungar. 2 (1972), 191-213 · Zbl 0247.94018 [6] Dalton H.: The Inequality of Incomes. Ruthledge & Keagan Paul, London 1925 [7] Devijver P., Kittler J.: Pattern Recognition: A Statistical Approach. Prentice Hall, Englewood Cliffs, NJ 1982 · Zbl 0542.68071 [8] Devroy L., Györfi, L., Lugosi G.: A Probabilistic Theory of Pattern Recognition. Springer, Berlin 1996 · Zbl 0853.68150 [9] Emlen J. M.: Ecology: An Evolutionary Approach. Adison-Wesley, Reading 1973 [10] Gini C.: Variabilitá e Mutabilitá. Studi Economico-Giuridici della R. Univ. di Cagliari. 3 (1912), Part 2, p. 80 [11] Harremöes P., Topsøe F.: Inequalities between etropy and index of concidence. IEEE Trans. Inform. Theory 47 (2001), 2944-2960 · Zbl 1017.94005 [12] Havrda J., Charvát F.: Concept of structural $$a$$-entropy. Kybernetika 3 (1967), 30-35 · Zbl 0178.22401 [13] Höffding W.: Masstabinvariante Korrelationstheorie. Teubner, Leipzig 1940 · JFM 66.0649.02 [14] Höffding W.: Stochastische Abhängigkeit und funktionaler Zusammenhang. Skand. Aktuar. Tidskr. 25 (1942), 200-207 · Zbl 0027.41401 [15] Kovalevskij V. A.: The problem of character recognition from the point of view of mathematical statistics. Character Readers and Pattern Recognition, 3-30. Spartan Books, New York 1967 [16] Liese F., Vajda I.: Convex Statistical Distances. Teubner, Leipzig 1987 · Zbl 0656.62004 [17] Liese F., Vajda I.: On divergences and informations in statistics and information theory. IEEE Trans. Inform. Theory 52 (2006), 4394-4412 · Zbl 1287.94025 [18] Marshall A. W., Olkin I.: Inequalities: Theory of Majorization and its Applications. Academic Press, New York 1979 · Zbl 1219.26003 [19] Morales D., Pardo, L., Vajda I.: Uncertainty of discrete stochastic systems. IEEE Trans. Systems, Man Cybernet. Part A 26 (1996), 681-697 [20] Pearson K.: On the theory of contingency and its relation to association and normal correlation. Drapers Company Research Memoirs, Biometric Ser. 1, London 1904 · JFM 36.0313.12 [21] Perez A.: Information-theoretic risk estimates in statistical decision. Kybernetika 3 (1967), 1-21 · Zbl 0153.48403 [22] Rényi A.: On measures of dependence. Acta Math. Acad. Sci. Hungar. 10 (1959), 441-451 · Zbl 0091.14403 [23] Rényi A.: On measures of entropy and information. Proc. Fourth Berkeley Symposium on Probab. Statist., Volume 1, Univ. Calif. Press, Berkeley 1961, pp. 547-561 [24] Sen A.: On Economic Inequality. Oxford Univ. Press, London 1973 [25] Simpson E. H.: Measurement of diversity. Nature 163 (1949), 688 · Zbl 0032.03902 [26] Tschuprow A.: Grundbegriffe und Grundprobleme der Korrelationstheorie. Berlin 1925 · JFM 51.0392.03 [27] Vajda I.: Bounds on the minimal error probability and checking a finite or countable number of hypotheses. Information Transmission Problems 4 (1968), 9-17 [28] Vajda I.: Theory of Statistical Inference and Information. Kluwer, Boston 1989 · Zbl 0711.62002 [29] Vajda I., Vašek K.: Majorization concave entropies and comparison of experiments. Problems Control Inform. Theory 14 (1985), 105-115 · Zbl 0601.62006 [30] Vajda I., Zvárová J.: On relations between informations, entropies and Bayesian decisions. Prague Stochastics 2006 (M. Hušková and M. Janžura, Matfyzpress, Prague 2006, pp. 709-718 [31] Zvárová J.: On measures of statistical dependence. Čas. pěst. matemat. 99 (1974), 15-29 · Zbl 0282.62048 [32] Zvárová J.: On medical informatics structure. Internat. J. Medical Informatics 44 (1997), 75-81 [33] Zvárová J.: Information Measures of Stochastic Dependence and Diversity: Theory and Medical Informatics Applications. Doctor of Sciences Dissertation, Academy of Sciences of the Czech Republic, Institute of Informatics, Prague 1998 [34] Zvárová J., Mazura I.: Stochastic Genetics (in Czech). Charles University, Karolinum, Prague 2001 [35] Zvárová J., Vajda I.: On genetic information, diversity and distance. Methods of Inform. in Medicine 2 (2006), 173-179
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