Abstract
A research program whose objective is to study uncertainty and uncertainty-based information in all their manifestations was introduced in the early 1990’s under the name “generalized information theory” (GIT). This research program, motivated primarily by some fundamental methodological issues emerging from the study of complex systems, is based on a two-dimensional expansion of classical, probability-based information theory. In one dimension, additive probability measures, which are inherent in classical information theory, are expanded to various types of nonadditive measures. In the other dimension, the formalized language of classical set theory, within which probability measures are formalized, is expanded to more expressive formalized languages that are based on fuzzy sets of various types. As in classical information theory, uncertainty is the primary concept in GIT and information is defined in terms of uncertainty reduction. This restricted interpretation of the concept of information is described in GIT by the qualified term “uncertainty-based information”. Each uncertainty theory that is recognizable within the expanded framework is characterized by: (i) a particular formalized language (a theory of fuzzy sets of some particular type); and (ii) a generalized measure of some particular type (additive or nonadditive). The number of possible uncertainty theories is thus equal to the product of the number of recognized types of fuzzy sets and the number of recognized types of generalized measures. This number has been growing quite rapidly with the recent developments in both fuzzy set theory and the theory of generalized measures. In order to fully develop any of these theories of uncertainty requires that issues at each of the following four levels be adequately addressed: (i) the theory must be formalized in terms of appropriate axioms; (ii) a calculus of the theory must be developed by which the formalized uncertainty is manipulated within the theory; (iii) a justifiable way of measuring the amount of relevant uncertainty (predictive, diagnostic, etc.) in any situation formalizable in the theory must be found; and (iv) various methodological aspects of the theory must be developed. Among the many uncertainty theories that are possible within the expanded conceptual framework, only a few theories have been sufficiently developed so far. By and large, these are theories based on various types of generalized measures, which are formalized in the language of classical set theory. Fuzzification of these theories, which can be done in different ways, has been explored only to some degree and only for standard fuzzy sets. One important result of research in the area of GIT is that the tremendous diversity of uncertainty theories made possible by the expanded framework is made tractable due to some key properties of these theories that are invariant across the whole spectrum or, at least, within broad classes of uncertainty theories. One important class of uncertainty theories consists of theories that are viewed as theories of imprecise probabilities. Some of these theories are based on Choquet capacities of various orders, especially capacities of order infinity (the well known theory of evidence), interval-valued probability distributions, and Sugeno λ-measures. While these theories are distinct in many respects, they share several common representations, such as representation by lower and upper probabilities, convex sets of probability distributions, and so-called Möbius representation. These representations are uniquely convertible to one another, and each may be used as needed. Another unifying feature of the various theories of imprecise probabilities is that two types of uncertainty coexist in each of them. These are usually referred to as nonspecificity and conflict. It is significant that well-justified measures of these two types of uncertainty are expressed by functionals of the same form in all the investigated theories of imprecise probabilities, even though these functionals are subject to different calculi in different theories. Moreover, equations that express relationship between marginal, joint, and conditional measures of uncertainty are invariant across the whole spectrum of theories of imprecise probabilities. The tremendous diversity of possible uncertainty theories is thus compensated by their many commonalities.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abellan, J., and Moral, S., 2002, A non-specificity measure for convex sets of probability distributions, Intern. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 8(3):357–367.
Atanassov, K. T., 2000, Intuitionistic Fuzzy Sets, Springer-Verlag, New York.
Choquet, G., 1953–54, Theory of capacities, Annales de L’Institut Fourier 5:131–295.
De Cooman, G., 1997, Possibility theory-I, II, III, Intern. J. of General Systems 25,291–371.
Denneberg, D., 1994, Non-additive Measure and Integral, Kluwer, Boston.
Dubois, D., and Prade, H., 1982, A class of fuzzy measures based on triangular norms, Intern. J. of General Systems 8(1):43–61.
Dubois, D., and Prade, H., 1990, Rough fuzzy sets and fuzzy rough sets, Intern. J. of General Systems 17(2–3):191–209.
Gil, M. A., ed., 2001, Special Issue on Fuzzy Random Variables, Information Sciences 133(1–2):1–100.
Goguen, J. A., 1967, L-fuzzy sets, J. of Math. Analysis and Applications 18(1):145–174.
Gottwald, S., 1979, Set theory for fuzzy sets of higher level, Fuzzy Sets and Systems 2(2):125–151.
Grabisch, M., 1997, k-order additive discrete fuzzy measures and their representation, Fuzzy Sets and Systems 92(2):167–189.
Halmos, P. R., 1950, Measure Theory, D. Van Nostrand, Princeton, NJ.
Hartley, R.V.L., 1928, Transmission of information, The Bell System Technical J., 7:535–563.
Klir, G. J., 1991, Generalized information theory, Fuzzy Sets and Systems 40(1):127–142.
Klir, G. J., 1997, Fuzzy arithmetic with requisite constraints, Fuzzy Sets and Systems 91(2):165–175.
Klir, G. J., 1999, On fuzzy-set interpretation of possibility theory, Fuzzy Sets and Systems 108(3):263–273.
Klir, G. J., 2001a, Facets of Systems Science, Plenum Press, New York.
Klir, G. J., 2001b, Foundations of fuzzy set theory and fuzzy logic: A Historical Overview, Intern. J. of General Systems 30(2):91–132.
Klir, G. J., 2002, Uncertainty in Economics: The heritage of G.L.S. Shackle, Fuzzy Economic Review VII(2):3–21.
Klir, G. J., 2005, Uncertainty and Information: Foundations and Applications of Generalized Information Theory (in production).
Klir, G. J., and Wierman, M. J., 1999, Uncertainty-Based Information: Elements of Generalized Information Theory, Physica-Verlag/Springer-Verlag, Heidelberg and New York.
Klir, G. J., and Yuan, B., 1995a, Fuzzy Sets and Fuzzy Logic: Theory and Applications Prentice Hall, PTR, Upper Saddle River, NJ.
Klir, G. J., and Yuan, B., 1995b, On nonspecificity of fuzzy sets with continuous membership functions, in: Proc. Intern. Conf. on Systems, Man, and Cybernetics, Vancouver, pp.25–29.
Klir, G. J., and Yuan, B. (eds.), 1996, Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers by Lotfi A. Zadeh, World Scientific, Singapore.
Kolmogorov, A. N., 1965, Three approaches to the quantitative definition of information, Problems of Information Transmission 1:1–7.
Kramosil, I., 2001, Probabilistic Analysis of Belief Functions, Kluwer Academic/Plenum Publishers, New York.
Kyburg, H. E., 1987, Bayesian and non-Bayesian evidential updating, Artificial Intelligence 31:271–293.
Mendel, J. M., 2001, Uncertain Rule-Based Fuzzy Logic Systems, Prentice Hall PTR, Upper Saddle River, NJ.
Pan, Y., and Klir, G. J., 1997, Bayesian inference based on interval probabilities, J. of Intelligent and Fuzzy Systems 5(3):193–203.
Pan, Y., and Yuan, B., 1997, Bayesian inference of fuzzy probabilities, Intern. J. of General Systems 26(1–2):73–90.
Pap, E., 1995, Null-Additive Set Functions, Kluwer, Boston.
Ramer, A., and Padet, C., 2001, Nonspecificity in Pn, Intern. J. of General Systems 30(6):661–680.
Rènyi, A., 1970, Probability Theory, North-Holland, Amsterdam (Chapter IX, Introduction to Information Theory, pp.540–616).
Rodabaugh, S. E., Klement, E. P., and Höhle, Y., 1992, Applications of Category Theory to Fuzzy Subsets, Kluwer, Boston.
Shafer, G., 1976, A Mathematical Theory of Evidence, Princeton Univ. Press, Princeton, NJ.
Shannon, C. E., 1948, The mathematical theory of communication, The Bell System Technical J. 27:379–423, 623–656.
Smets, P., 1988, Belief functions, in: Non-standard Logics for Automated Reasoning, P. Smets et al., ed., Academic Press, San Diego, pp 253–286.
Stoll, R. R., 1961, Set Theory and Logic, W. H. Freeman, San Francisco and London.
Walley, P., 1991, Statistical Reasoning With Imprecise Probabilities, Chapman and Hall, London.
Wang, Z., and Klir, G. J., 1992, Fuzzy Measure Theory, Plenum Press, New York.
Weichselberger, K., and Pöhlmann, S., 1990, A Methodology for Uncertainty in Knowledge-Based Systems, Springer-Verlag, New York.
Yager, R. R., et al., eds., 1987, Fuzzy Sets and Applications-Selected Papers by L. A. Zadeh, John Wiley, New York.
Yang, M., Chen, T., and Wu, K., 2003, Generalized belief function, plausibility function, and Dempster’s combination rule to fuzzy sets, Intern. J. of Intelligent Systems 18(8):925–937.
Yen, J., 1990, Generalizing the Dempster-Shafer theory to fuzzy sets, IEEE Transactions on Systems, Man, and Cybernetics 20(3):559–570.
Zadeh, L. A., 1968a, Probability measures and fuzzy events, J. of Math. Analysis and Applications 23(2):421–427.
Zadeh, L. A., 1978, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1(1):3–28.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer Science+Business Media, Inc.
About this paper
Cite this paper
Klir, G.J. (2006). Uncertainty and Information: Emergence of Vast New Territories. In: Minati, G., Pessa, E., Abram, M. (eds) Systemics of Emergence: Research and Development. Springer, Boston, MA. https://doi.org/10.1007/0-387-28898-8_1
Download citation
DOI: https://doi.org/10.1007/0-387-28898-8_1
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-28899-4
Online ISBN: 978-0-387-28898-7
eBook Packages: Business and EconomicsBusiness and Management (R0)