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On Measuring Uncertainty and Uncertainty-Based Information: Recent Developments

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Abstract

It is shown in this paper how the emergence of fuzzy set theory and the theory of monotone measures considerably expanded the framework for formalizing uncertainty and suggested many new types of uncertainty theories. The paper focuses on issues regarding the measurement of the amount of relevant uncertainty (predictive, prescriptive, diagnostic, etc.) in nondeterministic systems formalized in terms of the various uncertainty theories. It is explained how information produced by an action can be measured by the reduction of uncertainty produced by the action. Results regarding measures of uncertainty (and uncertainty-based information) in possibility theory, Dempster–Shafer theory, and the various theories of imprecise probabilities are surveyed. The significance of these results in developing sound methodological principles of uncertainty and uncertainty-based information is discussed.

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References

  1. J. Abellan and S. Moral, A non-specificity measure for convex sets of probability distributions, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 8(3) (2000) 357-367.

    Google Scholar 

  2. A. Chateauneuf and J.Y. Jaffray, Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion, Mathematical Social Sciences 17 (1989) 263-283.

    Google Scholar 

  3. G. Choquet, Theory of capacities, Annales de L'Institut Fourier 5 (1953-1954) 131-295.

    Google Scholar 

  4. R. Christensen, Entropy Minimax Sourcebook (Entropy Limited, Lincoln, MA, 1980-1981).

    Google Scholar 

  5. R. Christensen, Entropy minimax multivariate statistical modeling-I: Theory, International Journal of General Systems 11(3) (1985) 231-277.

    Google Scholar 

  6. R. Christensen, Entropy minimax multivariate statistical modeling-II: Applications, International Journal of General Systems 12(3) (1986) 227-305.

    Google Scholar 

  7. G. De Cooman, Possibility theory, International Journal of General Systems 25(4) (1997) 291-371.

    Google Scholar 

  8. D. Dubois and H. Prade, A note on measures of specificity for fuzzy sets, International Journal of General Systems 10(4) (1985) 279-283.

    Google Scholar 

  9. D. Dubois and H. Prade, Possibility Theory (Plenum, New York, 1988).

    Google Scholar 

  10. D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems 17(2-3) (1990) 191-209.

    Google Scholar 

  11. D. Dubois, J. Lang and H. Prade, Possibilistic logic, in: eds. D.M. Gabbay, et al., Handbook of Logic in Artificial Intelligence and Logic Programming (Clarendon, Oxford, UK, 1994), pp. 439-513.

    Google Scholar 

  12. J.F. Geer and G.J. Klir, A mathematical analysis of information preserving transformations between probabilistic and possibilistic formulations of uncertainty, International Journal of General Systems 20(2) (1992) 143-176.

    Google Scholar 

  13. P.R. Halmos, Measure Theory (Van Nostrand, Princeton, NJ, 1950).

    Google Scholar 

  14. R.V.L. Hartley, Transmission of information, The Bell SystemTechnical Journal 7(3) (1928) 535-563.

    Google Scholar 

  15. M. Higashi and G.J. Klir, Measures of uncertainty and information based on possibility distributions, International Journal of General Systems 9(1) (1983) 43-58.

    Google Scholar 

  16. G.E. Hughes and M.J. Cresswell, A New Introduction to Modal Logic (Routledge, London and New York, 1996).

    Google Scholar 

  17. E.T. Jaynes, in: Papers on Probability, Statistics and Statistical Physics, ed. R.D. Rosenkrantz (Reidel, Dordrecht, 1983).

    Google Scholar 

  18. J.N. Kapur, Maximum Entropy Models in Science and Engineering (Wiley, New York, 1989).

    Google Scholar 

  19. G.J. Klir, A principle of uncertainty and information invariance, International Journal of General Systems 17(2-3) (1990) 249-275.

    Google Scholar 

  20. G.J. Klir, Principles of uncertainty: What are they? Why do we need them?, Fuzzy Sets and Systems 74(1) (1995) 15-31.

    Google Scholar 

  21. G.J. Klir, On fuzzy-set interpretation of possibility theory, Fuzzy Sets and Systems 108(3) (1999) 263-273.

    Google Scholar 

  22. G.J. Klir and M. Mariano, On the uniqueness of possibilistic measure of uncertainty and information, Fuzzy Sets and Systems 24(2) (1987) 197-219.

    Google Scholar 

  23. G.J. Klir and B. Parviz, Probability-possibility transformations: A comparison, International Journal of General Systems 21(3) (1992) 291-310.

    Google Scholar 

  24. G.J. Klir and M.J. Wierman, Uncertainty-Based Information: Elements of Generalized Information Theory (Physica-Verlag/Springer, Heidelberg and New York, 1999).

    Google Scholar 

  25. G.J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications (Prentice Hall, Upper Saddle River, NJ, 1995).

    Google Scholar 

  26. G.J. Klir and B. Yuan, On nonspecificity of fuzzy sets with continuous membership functions, in: Proc. 1995 International Conf. on Systems, Man, and Cybernetics, Vancouver (1995).

  27. G.J. Klir and B. Yuan, eds., Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers by Lotfi A. Zadeh (World Scientific, Singapore, 1996).

    Google Scholar 

  28. A.N. Kolmogorov, Foundations of the Theory of Probability (Chelsea, New York, 1950), first published in German in 1933.

    Google Scholar 

  29. H.E. Kyburg, Bayesian and non-Bayesian evidential updating, Artificial Intelligence 31 (1987) 271-293.

    Google Scholar 

  30. H.E. Kyburg and M. Pittarelli, Set-based Bayesianism, IEEE Transactions on Systems, Man, and Cybernetics A 26(3) (1996) 324-339.

    Google Scholar 

  31. Y. Pan and G.J. Klir, Bayesian inference based on interval probabilities, Journal of Intelligent and Fuzzy Systems 5(3) (1997) 193-203.

    Google Scholar 

  32. Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data (Kluwer, Boston, 1991).

    Google Scholar 

  33. H. Prade, Modal semantics and fuzzy set theory, in: Fuzzy Set and Possibility Theory: Recent Developments, ed. R.R. Yager (Pergamon Press, Oxford, 1982) pp. 232-246.

    Google Scholar 

  34. A. Ramer, Uniqueness of information measure in the theory of evidence, Fuzzy Sets and Systems 24(2) (1987) 183-196.

    Google Scholar 

  35. A. Ramer, Euclidean specificity: two solutions and few problems, in: Proc. World Congress of the International Fuzzy Systems Assoc., Vol. 4, Prague (1997) 268-271.

    Google Scholar 

  36. A. Ramer, Nonspecificity in ℝn, International Journal of General Systems (2001) in press.

  37. A. Rènyi, Probability Theory (North-Holland, Amsterdam, 1970) chapter IX, pp. 540-616.

    Google Scholar 

  38. G. Shafer, A Mathematical Theory of Evidence (Princeton Univ. Press, Princeton, NJ, 1976).

    Google Scholar 

  39. C.E. Shannon, The mathematical theory of communication, The Bell System Technical Journal 27 (1948) 379-423, 623-656.

    Google Scholar 

  40. R.M. Smith, Generalized information theory: resolving some old questions and opening some new ones, Ph.D. dissertation, Binghamton University-SUNY, Binghamton (2000).

    Google Scholar 

  41. P. Walley, Statistical Reasoning With Imprecise Probabilities (Chapman and Hall, London, 1991).

    Google Scholar 

  42. P. Walley, Towards a unified theory of imprecise probability, International Journal of Approximate Reasoning 24(2-3) (2000) 125-148.

    Google Scholar 

  43. Z. Wang and G.J. Klir, Fuzzy Measure Theory (Plenum, New York, 1992).

    Google Scholar 

  44. K. Weichselberger and S.K. Pöhlmann, A Methodology for Uncertainty in Knowledge-Based Systems (Springer, New York, 1990).

    Google Scholar 

  45. R.R. Yager, A foundation for a theory of possibility, Journal of Cybernetics 10(1-3) (1980) 77-204.

    Google Scholar 

  46. R.R. Yager, On the Dempster-Shafer framework and new combination rules, Information Sciences 41 (1987) 93-137.

    Google Scholar 

  47. R.R. Yager et al., Fuzzy Sets and Applications-Selected Papers by L.A. Zadeh (Wiley, New York, 1987).

    Google Scholar 

  48. J. Yen, Generalizing the Dempster-Shafer theory to fuzzy sets, IEEE Transactions on Systems, Man, and Cybernetics 20(3) (1990) 559-570.

    Google Scholar 

  49. L.A. Zadeh, Fuzzy sets, Information and Control 8(3) (1965) 338-353.

    Google Scholar 

  50. L.A. Zadeh, Probability measures of fuzzy events, Journal of Mathematical Analysis and Applications 23 (1968) 421-427.

    Google Scholar 

  51. L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1(1) (1978) 3-28.

    Google Scholar 

  52. L.A. Zadeh, Soft computing and fuzzy logic, IEEE Software 11(6) (1994) 48-56.

    Google Scholar 

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Authors and Affiliations

  1. Center for Intelligent Systems and Department of Systems Science and Industrial Engineering, Binghamton University – SUNY, Binghamton, NY, 13902, USA

    George J. Klir & Richard M. Smith

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  1. George J. Klir
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  2. Richard M. Smith
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Klir, G.J., Smith, R.M. On Measuring Uncertainty and Uncertainty-Based Information: Recent Developments. Annals of Mathematics and Artificial Intelligence 32, 5–33 (2001). https://doi.org/10.1023/A:1016784627561

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