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aequationes mathematicae

, Volume 54, Issue 1–2, pp 1–30 | Cite as

Characterizations of sum form information measures on open domains

  • Bruce R. EbanksEmail author
  • Palaniappan Kannappan
  • Prasanna K. Sahoo
  • Wolfgang Sander
Survey Paper

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
$$f(pq) + f(p(1 - q)) + f((1 - p)q) - f((1 - p)(1 - q)) = 0, p,q \in ]0, 1[^k (FE)$$
and
$$f(pq) + f(p(1 - q)) + f((1 - p)q) - f((1 - p)(1 - q)) = g(p)g(q), p,q \in ]0, 1[^k , (LI)$$
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.

AMS (1991) subject classification

39B05 39B52 94A17 

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References

  1. [1]
    Abou-Zaid, S. H. S.,Functional Equations and Related Measurements. M. Phil. Thesis, Dept. of Pure Math. University of Waterloo, Waterloo, Canada, 1984.Google Scholar
  2. [2]
    Aczél, J.,Lectures of Functional Equations and Their Applications. Academic Press, New York, 1966.Google Scholar
  3. [3]
    Aczél, J.,Some recent results on characterizations of measures of information related to coding. IEEE Trans. Inform. Theory,IT-24 (1978), 592–595.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    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.Google Scholar
  5. [5]
    Aczél, J.,Measuring information beyond communication theory—Why some generalized information measures may be useful, others not. Aequationes Math.27 (1984), 1–19.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    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.Google Scholar
  7. [7]
    Aczél, J. andDaróczy, Z.,On measures of information and their characterisations. Academic Press, New York-San Francisco-London, 1975.Google Scholar
  8. [8]
    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.Google Scholar
  9. [9]
    Aczél, J., Ng, C. T. andWagner, C.,Aggregation theorems for allocation problems. SIAM J. Algebraic Discrete Methods5 (1984), 1–8.zbMATHMathSciNetGoogle Scholar
  10. [10]
    Behara, M.,Additive and Nonadditive Measures of Entropy. John Wiley & Sons, New York, 1990.zbMATHGoogle Scholar
  11. [11]
    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.CrossRefGoogle Scholar
  12. [12]
    Capocilli, R. M. andTaneja, I. J.,On some inequalities and generalized entropies: A unified approach. Cybern. Systems16 (1985), 341–375.CrossRefGoogle Scholar
  13. [13]
    Chaundy, T. W. andMcLeod, J. B.,On a functional equation. Proc. Edinburgh Math. Soc. (2), 12, Edinburgh Math. Notes43 (1960), 7–8.MathSciNetGoogle Scholar
  14. [14]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Csiszár, I.,Information measures: A critical survey. InTrans. Seventh Prague Conf. Information Theory, Statistical Decision Functions, Random Processes. Prague: Academia, 1978, pp. 73–86.Google Scholar
  16. [16]
    Daróczy, Z.,Generalized information functions Inform. and Control (Shenyang)16 (1970), 36–51.zbMATHCrossRefGoogle Scholar
  17. [17]
    Daróczy, Z.,On the measurable solutions of a functional equation. Acta. Math. Acad. Sci. Hungar.22 (1971), 11–14.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Ebanks, B. R.,Measurable solutions of functional equations connected with information measures on open domains. Utilitas Math.27 (1985), 217–223.zbMATHMathSciNetGoogle Scholar
  20. [20]
    Ebanks, B. R.,Generalized characteristic equation of branching information measures. Aequationes Math.37 (1989), 162–178.CrossRefMathSciNetzbMATHGoogle Scholar
  21. [21]
    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.MathSciNetGoogle Scholar
  22. [22]
    Ebanks, B. R. andLosonczi, L.,On the linear independence of some functions. Publ. Math. Debrecen41 (1992), 135–146.MathSciNetzbMATHGoogle Scholar
  23. [23]
    Ebanks, B. R., Sahoo, P. K. andSander, W.,General solution of two functional equations concerning measures of information. Results. Math.18 (1989), 10–17.MathSciNetGoogle Scholar
  24. [24]
    Ebanks, B. R., Sahoo, P. K. andSander, W.,Characterization of Information Measures on Open Domains. Manuscript, 1995.Google Scholar
  25. [25]
    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.zbMATHMathSciNetGoogle Scholar
  26. [26]
    Forte, B.,Entropies with and without probabilities. Applications to questionnnaires. Inform. Process. & Management20 (1984), 397–405.zbMATHCrossRefGoogle Scholar
  27. [27]
    Havrda, J. andCharvat, F.,Quantification method of classification process, concept of structural a-entropy. Kybernetika (Prague)3 (1972), 95–100.MathSciNetGoogle Scholar
  28. [28]
    Járai, A.,On measurable solutions of functional equations. Publ. Math. Debrecen26 (1979), 17–35.zbMATHMathSciNetGoogle Scholar
  29. [29]
    Járai, A.,On regular solutions of functional equations. Aequationes Math.30 (1986), 21–54.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    Kannappan, Pl.,On some functional equations from additive and nonadditive measures—I. Proc. Edinburgh Math. Soc.23 (1986), 145–150.MathSciNetCrossRefGoogle Scholar
  31. [31]
    Kannappan, Pl.,On some functional equations from additive and nonadditive measures—IV. Kybernetika (Prague)17 (1981), 394–400.zbMATHMathSciNetGoogle Scholar
  32. [32]
    Kannappan, Pl.,On some functional equations from additive and nonadditive measures—V. Utilitas Math.22 (1982), 141–147.zbMATHMathSciNetGoogle Scholar
  33. [33]
    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.MathSciNetGoogle Scholar
  34. [34]
    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.Google Scholar
  35. [35]
    Kannappan, Pl. andNg, C. T.,On functional equations and measures of information II. J. Appl. Probab.17 (1980), 271–277.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    Kannappan, Pl. andNg, C. T.,On functional equations and measures of information I. Publ. Math. Debrecen32 (1985), 243–249.zbMATHMathSciNetGoogle Scholar
  37. [37]
    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.zbMATHMathSciNetGoogle Scholar
  38. [38]
    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.MathSciNetGoogle Scholar
  39. [39]
    Kannappan, Pl. andSahoo, P. K.,On a functional equation connected to sum form nonadditive information measures on open domains—III. Stochastica9 (1985), 111–124.zbMATHMathSciNetGoogle Scholar
  40. [40]
    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.zbMATHMathSciNetGoogle Scholar
  41. [41]
    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.MathSciNetGoogle Scholar
  42. [42]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    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.zbMATHMathSciNetGoogle Scholar
  44. [44]
    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.MathSciNetGoogle Scholar
  45. [45]
    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.MathSciNetGoogle Scholar
  46. [46]
    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.MathSciNetGoogle Scholar
  47. [47]
    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.MathSciNetGoogle Scholar
  48. [48]
    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.Google Scholar
  49. [49]
    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.zbMATHMathSciNetGoogle Scholar
  50. [50]
    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.Google Scholar
  51. [51]
    Kannappan, Pl. andSahoo, P. K.,Sum form equation of multiplicative type. Acta. Math. Hungar.61 (1993), 205–219.CrossRefMathSciNetGoogle Scholar
  52. [52]
    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.Google Scholar
  53. [53]
    Kannappan, Pl. andSander, W.,On entropies with the sum property on open domain. Analysis9 (1989), 253–267.zbMATHMathSciNetGoogle Scholar
  54. [54]
    Kerridge, D. F.,Inaccuracy and inference. J. Roy. Statist. Soc. Ser. B23 (1961), 184–194.zbMATHMathSciNetGoogle Scholar
  55. [55]
    Kuczma, M.,Note on additive functions of several variables. Uniw. Ślaski w Katowicach Prace Nauk. Prace Mat.2 (1972), 49–51.MathSciNetGoogle Scholar
  56. [56]
    Kuczma, M.,An introduction to the theory of functional equations and inequality. PWN, Uniw Ślaski, Warszawa-Krakow-Katowice, 1985.zbMATHGoogle Scholar
  57. [57]
    Kullback, S.,Information Theory and Statistics. John Wiley and Sons, New York, 1959.zbMATHGoogle Scholar
  58. [58]
    Losonczi, L.,A characterization of entropies of degree α. Metrika28 (1981), 237–244.zbMATHCrossRefMathSciNetGoogle Scholar
  59. [59]
    Losonczi, L.,Functional equations of sum form. Publ. Math. Debrecen32 (1985), 57–71.zbMATHMathSciNetGoogle Scholar
  60. [60]
    Losonczi, L.,Sum form equations on an open domain I. C.R. Math. Rep. Acad. Sci. Canada7 (1985), 85–90.zbMATHMathSciNetGoogle Scholar
  61. [61]
    Losonczi, L.,Sum form equations on an open domain II. Utilitas Math.29 (1986), 125–132.zbMATHMathSciNetGoogle Scholar
  62. [62]
    Losonczi, L.,Measurable solutions of a functional equation related to (2,2)-additive entropies of degree α. Publ. Math. Debrecen42 (1993), 109–137.zbMATHMathSciNetGoogle Scholar
  63. [63]
    Losonczi, L. andMaksa, Gy.,On some functional equations of the information theory. Acta Math. Acad. Sci. Hungar.39 (1982), 73–82.zbMATHCrossRefMathSciNetGoogle Scholar
  64. [64]
    Maska, Gy.,On the bounded solutions of a functional equation. Acta Math. Acad. Sci. Hungar.37 (1981), 445–450.CrossRefMathSciNetGoogle Scholar
  65. [65]
    Maksa, Gy.,The general solution of a functional equation arising in information theory. Acta Math. Acad. Sci. Hungar.49 (1987), 213–217.zbMATHMathSciNetGoogle Scholar
  66. [66]
    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.Google Scholar
  67. [67]
    Nath, P.,On some functional equations and their applications. Publ. De L’Institut Math. (Beograd)20 (34) (1976), 191–201.MathSciNetGoogle Scholar
  68. [68]
    Ng, C. T.,Representation of measures of information with the branching property. Inform. and Control25 (1974), 45–56.CrossRefzbMATHGoogle Scholar
  69. [69]
    Roy, D.,Axiomatic characterization of second order information improvement. J. Combin. Inform. Syst. Sci.5 (1980), 107–111.zbMATHGoogle Scholar
  70. [70]
    Sahoo, P. K.,On some functional equations connected to sum form information measures on open domains. Utilitas Math.23 (1983), 161–175.zbMATHMathSciNetGoogle Scholar
  71. [71]
    Sahoo, P. K.,Theory and Applications of Some Measures of Uncertainty. Ph.D. Thesis, Dept. of Applied Math., University of Waterloo, Waterloo, Canada, 1986.Google Scholar
  72. [72]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  73. [73]
    Sahoo, P. K.,Three open problems in functional equations. Amer. Math. Monthly102 (1995), 741–742.zbMATHCrossRefMathSciNetGoogle Scholar
  74. [74]
    Sahoo, P. K. andSander, W. (1989),Sum form information measures on open domain. Rad. Mat.5 (2) (1989), 261–270.zbMATHMathSciNetGoogle Scholar
  75. [75]
    Sander, W.,The fundamental equation of information and its generalizations. Aequationes Math.33 (1987), 150–182.zbMATHMathSciNetGoogle Scholar
  76. [76]
    Sander, W.,Information measures on the open domain. Analysis8 (1988), 207–224.MathSciNetzbMATHGoogle Scholar
  77. [77]
    Shannon, C. E.,A mathematical theory of communication. Bell Systems Technical Journal27 (1948), 379–423, 623–656.MathSciNetzbMATHGoogle Scholar
  78. [78]
    Sharma, B. D. andTaneja, I. J.,Entropy of type (α, β) and other generalized measures in information theory. Metrika22 (1975), 205–215.zbMATHCrossRefMathSciNetGoogle Scholar
  79. [79]
    Taneja, I. J.,Some contributions to information theory—I (a Survey: On measures of information). J. Combin. Inform. System Sci.4 (1979), 253–274.zbMATHMathSciNetGoogle Scholar
  80. [80]
    Theil, H.,Economics and Information Theory. North-Holland, Amsterdam Rand McNally, Chicago, 1967.Google Scholar
  81. [81]
    Vajda, I.,Axioms for a-entropy of a generalized probability scheme (Czech). Kybernetika 4 (Prague), 1968, 105–112.MathSciNetzbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag 1997

Authors and Affiliations

  • Bruce R. Ebanks
    • 1
    Email author
  • Palaniappan Kannappan
    • 2
  • Prasanna K. Sahoo
    • 3
  • Wolfgang Sander
    • 4
  1. 1.Department of MathematicsMarshall UniversityHuntingtonUSA
  2. 2.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  3. 3.Department of MathematicsUniversity of LouisvilleLouisvilleUSA
  4. 4.Institute for AnalysisTechnische Universität BraunschweigBraunschweigGermany

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