Summary
The goal of this paper is to give a survey of all important characterizations of sum form information measures that depend uponk discrete complete probability distributions (without zero probabilities) of lengthn and which satisfy a generalized additivity property. It turns out that most of the problems have been solved, but some open problems lead to the very simple looking functional equations
and
wheref, g: ]0, , 1[k → ℝ andk ∈ ℕ. Moreover new entropies analogous to the Shannon entropy, entropies of degree α, entropies of degree (α, β) are introduced for χ, β ∈ ℕ.
Similar content being viewed by others
References
Abou-Zaid, S. H. S.,Functional Equations and Related Measurements. M. Phil. Thesis, Dept. of Pure Math. University of Waterloo, Waterloo, Canada, 1984.
Aczél, J.,Lectures of Functional Equations and Their Applications. Academic Press, New York, 1966.
Aczél, J.,Some recent results on characterizations of measures of information related to coding. IEEE Trans. Inform. Theory,IT-24 (1978), 592–595.
Aczél, J.,Some recent results on information measures, a new generalization and some ‘real life’ interpretations of ‘old’ and new measures. In J. Aczél (ed.),Functional Equations: History, Applications and Theory, D. Reidel Publishing Company, 1984, pp. 175–189.
Aczél, J.,Measuring information beyond communication theory—Why some generalized information measures may be useful, others not. Aequationes Math.27 (1984), 1–19.
Aczél, J.,Characterizing information measures: Approaching the end of an era. In Lecture Notes in Computer Science, Vol. 286 (Uncertainty in Knowledge-Based Systems) Springer, 1986, pp. 359–384.
Aczél, J. andDaróczy, Z.,On measures of information and their characterisations. Academic Press, New York-San Francisco-London, 1975.
Aczél, J., Ng, C. T.,Determination of all semisymmetric recursive information measures of multiplicative type on n positive discrete probability distributions. Linear Algebra Appl.52/53 (1983), 1–30.
Aczél, J., Ng, C. T. andWagner, C.,Aggregation theorems for allocation problems. SIAM J. Algebraic Discrete Methods5 (1984), 1–8.
Behara, M.,Additive and Nonadditive Measures of Entropy. John Wiley & Sons, New York, 1990.
Behara, M. andNath, P.,Additive and non-additive entropies of finite measurable partitions. InProbability and Information Theory, II (Lecture Notes in Mathematics) Vol. 296. Springer, Berlin, 1973, pp. 102–138.
Capocilli, R. M. andTaneja, I. J.,On some inequalities and generalized entropies: A unified approach. Cybern. Systems16 (1985), 341–375.
Chaundy, T. W. andMcLeod, J. B.,On a functional equation. Proc. Edinburgh Math. Soc. (2), 12, Edinburgh Math. Notes43 (1960), 7–8.
Chung, J. K., Kannappan, Pl., Ng, C. T. andSahoo, P. K.,Measures of distance between probability distributions. J. Math. Anal. Appl.138 (1989), 280–292.
Csiszár, I.,Information measures: A critical survey. InTrans. Seventh Prague Conf. Information Theory, Statistical Decision Functions, Random Processes. Prague: Academia, 1978, pp. 73–86.
Daróczy, Z.,Generalized information functions Inform. and Control (Shenyang)16 (1970), 36–51.
Daróczy, Z.,On the measurable solutions of a functional equation. Acta. Math. Acad. Sci. Hungar.22 (1971), 11–14.
Daróczy, Z. andJárai, A.,On the measurable solution of a functional equation arising in information theory. Acta. Math. Acad. Sci. Hungar.34 (1979), 105–116.
Ebanks, B. R.,Measurable solutions of functional equations connected with information measures on open domains. Utilitas Math.27 (1985), 217–223.
Ebanks, B. R.,Generalized characteristic equation of branching information measures. Aequationes Math.37 (1989), 162–178.
Ebanks, B. R.,Determination of all measurable sum form information measures satisfying (2, 2)-additivity of degree (α, β)—II: The whole story. Rad. Mat.8 (1992), 159–169.
Ebanks, B. R. andLosonczi, L.,On the linear independence of some functions. Publ. Math. Debrecen41 (1992), 135–146.
Ebanks, B. R., Sahoo, P. K. andSander, W.,General solution of two functional equations concerning measures of information. Results. Math.18 (1989), 10–17.
Ebanks, B. R., Sahoo, P. K. andSander, W.,Characterization of Information Measures on Open Domains. Manuscript, 1995.
Ebanks, B. R., Sahoo, P. K. andSander, W.,Determination of measurable sum form information measures satisfying (2,2)-additivity of degree (α, β). Rad. Mat.6 (1990), 77–96.
Forte, B.,Entropies with and without probabilities. Applications to questionnnaires. Inform. Process. & Management20 (1984), 397–405.
Havrda, J. andCharvat, F.,Quantification method of classification process, concept of structural a-entropy. Kybernetika (Prague)3 (1972), 95–100.
Járai, A.,On measurable solutions of functional equations. Publ. Math. Debrecen26 (1979), 17–35.
Járai, A.,On regular solutions of functional equations. Aequationes Math.30 (1986), 21–54.
Kannappan, Pl.,On some functional equations from additive and nonadditive measures—I. Proc. Edinburgh Math. Soc.23 (1986), 145–150.
Kannappan, Pl.,On some functional equations from additive and nonadditive measures—IV. Kybernetika (Prague)17 (1981), 394–400.
Kannappan, Pl.,On some functional equations from additive and nonadditive measures—V. Utilitas Math.22 (1982), 141–147.
Kannappan, Pl.,Characterization of some measures of information theory and the sum form functional equations—Generalized directed divergence—I. Lecture Notes in Math.286 (1987), 285–394.
Kannappan, Pl. andNg, C. T.,Representations of measures of information. Trans. of the Eighth Prague Conference, Academic Publ. House of the Czec. Acad. Sci. Prague, C (1979), 203–207.
Kannappan, Pl. andNg, C. T.,On functional equations and measures of information II. J. Appl. Probab.17 (1980), 271–277.
Kannappan, Pl. andNg, C. T.,On functional equations and measures of information I. Publ. Math. Debrecen32 (1985), 243–249.
Kannappan, Pl. andSahoo, P. K.,On a functional equation connected to sum form nonadditive information measures on an open domain. C.R. Math. Rep. Acad. Sci. Canada7 (1985), 45–50.
Kannappan, Pl. andSahoo, P. K.,On a functional equation in two variables connected to sum form information measures on an open domain. Indian J. Math.27 (1985), 33–40.
Kannappan, Pl. andSahoo, P. K.,On a functional equation connected to sum form nonadditive information measures on open domains—III. Stochastica9 (1985), 111–124.
Kannappan, Pl. andSahoo, P. K.,On a functional equation connected to sum form nonaditive information measures on an open domain—I. Kybernetika (Prague)22 (1986), 268–275.
Kannappan, Pl. andSahoo, P. K.,On the general solution of a functional equation connected to the sum form information measure—I. Publ. De L’Institut Math. (Beograd)40 (54) (1986), 57–62.
Kannappan, Pl. andSahoo, P. K.,On the general solution of a functional equation connected to the sum form information measure—I. Internat. J. Math. Math. Sci.9 (1986), 545–550.
Kannappan, Pl. andSahoo, P. K.,On the general solution of a functional equation connected to the sum form information measure—IV. Utilitas Math.30 (1986), 191–197.
Kannappan, Pl. andSahoo, P. K.,On the general solution of a functional equation connected to the sum form information measure—II. Mathematica (Cluj)29 (52) (1987), 131–137.
Kannappan, Pl. andSahoo, P. K.,On a functional equation connected to sum form nonadditive information measures on an open domain—II. Glas. Mat.22 (42) (1987), 343–351.
Kannappan, Pl. andSahoo, P. K.,On the general solution of a functional equation connected to the sum form information measure—V. Acta Math. Univ. Commenian. (N.S.),54/55 (1988), 89–102.
Kannappan, Pl. andSahoo, P. K.,Weighted entropy of degree β on open domain. Proc. of the Ramanujan Centennial Inter. Conference, RMS Publication No 1 (1988), 119–125.
Kannappan, Pl. andSahoo, P. K.,Representation of sum form information measures with additivity of type (α, β) on open domain. In R. Janicki and W. W. Kockodaj (eds.),Computing and Information, Elsevier Science Publishers B.V., North-Holland, 1989, 243–253.
Kannappan, Pl. andSahoo, P. K.,Representation of sum form information measures with weighted additivity of type (α, β) on open domain. J. Math. Phy. Sci.24 (1990), 89–99.
Kannappan, Pl. andSahoo, P. K.,Parametrically additive sum form information measures. In T. M. Rassias (ed.),Constatine Caratheodory: An International Tribute, World Scientific Publishers, Vol I, 1991, 574–580.
Kannappan, Pl. andSahoo, P. K.,Sum form equation of multiplicative type. Acta. Math. Hungar.61 (1993), 205–219.
Kannappan, Pl. andSahoo, P. K.,Parametrically additive sum form weighted information measures. In S. G. Akl, F. Fiala and W. Koczkodaj (eds.),Advances in Computing and Information, Canadian Scholars’ Press Inc., Toronto, 1990, pp. 26–31.
Kannappan, Pl. andSander, W.,On entropies with the sum property on open domain. Analysis9 (1989), 253–267.
Kerridge, D. F.,Inaccuracy and inference. J. Roy. Statist. Soc. Ser. B23 (1961), 184–194.
Kuczma, M.,Note on additive functions of several variables. Uniw. Ślaski w Katowicach Prace Nauk. Prace Mat.2 (1972), 49–51.
Kuczma, M.,An introduction to the theory of functional equations and inequality. PWN, Uniw Ślaski, Warszawa-Krakow-Katowice, 1985.
Kullback, S.,Information Theory and Statistics. John Wiley and Sons, New York, 1959.
Losonczi, L.,A characterization of entropies of degree α. Metrika28 (1981), 237–244.
Losonczi, L.,Functional equations of sum form. Publ. Math. Debrecen32 (1985), 57–71.
Losonczi, L.,Sum form equations on an open domain I. C.R. Math. Rep. Acad. Sci. Canada7 (1985), 85–90.
Losonczi, L.,Sum form equations on an open domain II. Utilitas Math.29 (1986), 125–132.
Losonczi, L.,Measurable solutions of a functional equation related to (2,2)-additive entropies of degree α. Publ. Math. Debrecen42 (1993), 109–137.
Losonczi, L. andMaksa, Gy.,On some functional equations of the information theory. Acta Math. Acad. Sci. Hungar.39 (1982), 73–82.
Maska, Gy.,On the bounded solutions of a functional equation. Acta Math. Acad. Sci. Hungar.37 (1981), 445–450.
Maksa, Gy.,The general solution of a functional equation arising in information theory. Acta Math. Acad. Sci. Hungar.49 (1987), 213–217.
Mathai, A. M. andRathie, P. N.,Recent contributions to axiomatic definitions of information and statistical measures through functional equations. Essays in Probability and Statistics, 1976, pp. 607–633. Shinko Tsusho, Tokyo.
Nath, P.,On some functional equations and their applications. Publ. De L’Institut Math. (Beograd)20 (34) (1976), 191–201.
Ng, C. T.,Representation of measures of information with the branching property. Inform. and Control25 (1974), 45–56.
Roy, D.,Axiomatic characterization of second order information improvement. J. Combin. Inform. Syst. Sci.5 (1980), 107–111.
Sahoo, P. K.,On some functional equations connected to sum form information measures on open domains. Utilitas Math.23 (1983), 161–175.
Sahoo, P. K.,Theory and Applications of Some Measures of Uncertainty. Ph.D. Thesis, Dept. of Applied Math., University of Waterloo, Waterloo, Canada, 1986.
Sahoo, P. K.,Determination of all additive sum form information measures of k positive discrete probability distributions. J. Math. Anal. Appl.194 (1995), 235–249.
Sahoo, P. K.,Three open problems in functional equations. Amer. Math. Monthly102 (1995), 741–742.
Sahoo, P. K. andSander, W. (1989),Sum form information measures on open domain. Rad. Mat.5 (2) (1989), 261–270.
Sander, W.,The fundamental equation of information and its generalizations. Aequationes Math.33 (1987), 150–182.
Sander, W.,Information measures on the open domain. Analysis8 (1988), 207–224.
Shannon, C. E.,A mathematical theory of communication. Bell Systems Technical Journal27 (1948), 379–423, 623–656.
Sharma, B. D. andTaneja, I. J.,Entropy of type (α, β) and other generalized measures in information theory. Metrika22 (1975), 205–215.
Taneja, I. J.,Some contributions to information theory—I (a Survey: On measures of information). J. Combin. Inform. System Sci.4 (1979), 253–274.
Theil, H.,Economics and Information Theory. North-Holland, Amsterdam Rand McNally, Chicago, 1967.
Vajda, I.,Axioms for a-entropy of a generalized probability scheme (Czech). Kybernetika 4 (Prague), 1968, 105–112.