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Part of the book series: Handbook of Defeasible Reasoning and Uncertainty Management Systems ((HAND,volume 1))

Abstract

The lack of a unique model to represent quantified uncertainty constitutes a problem for the user: which model should be applied in what situation? We present some models for the quantified representation of uncertainty, focusing on their applicability more than on their mathematical structure. Imprecision often underlies uncertainty, hence we will also study imprecision.

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References

Logical Approach

  1. R. Bradley and N. Swartz. Possible Worlds. Basil Blackwell, Oxford, 1979.

    Google Scholar 

  2. B. F. Chellas. Modal Logic. Cambridge University Press, 1980.

    Google Scholar 

  3. Goodman, Nguyen and Walker, 19911 I. R. Goodman, H. T. Nguyen and E. A. Walker. Conditional Inference and Logic for Intelligent Systems. Elsevier, Amsterdam, 1991.

    Google Scholar 

  4. W. L. Harper, R. Stalnaker and G. Pearce. (1981) Ifs. Reidel, Dordrecht, 1981.

    Google Scholar 

  5. J. Hintikka. Knowledge and Belief. Cornell University Press, Ithaca, NY, 1962. [Hughes and Cresswell, 1968 ] G. E. Hughes and M. J. Cresswell. An Introduction to Modal Logic. Methuem, London, 1968.

    Google Scholar 

  6. N. Rescher. Many-valued Logics. McGraw-Hill, 1969.

    Google Scholar 

  7. F. Voorbraak. As Far as I Know: Epistemic Logic and Uncertainty. Dissertation, Utrecht University, 1993.

    Google Scholar 

Dynamic of Belief

  1. P. Gärdenfors. Knowledge in Flux. Modelling the Dynamics of Epistemic States. MIT Press, Cambridge, MA, 1988.

    Google Scholar 

  2. P. Forrest. The Dynamics of Belief. Basil Blackwell, Oxford, 1986.

    Google Scholar 

  3. G. Harman. Change in View. MIT Press, Cambridge, MA, 1986.

    Google Scholar 

  4. Shoham, 19881 Y. Shoham. Reasoning About Change, Time and Causation from the Standpoint of Artificial Intelligence. The MIT Press, Cambridge, MA, 1988.

    Google Scholar 

Fuzzy Sets

  1. B. Bouchon-Meunier. La Logique Floue. Presses Universitaires de France, 1993.

    Google Scholar 

  2. D. Dubois and H. Prade. Fuzzy Sets and Systems; Theory and Applications. Academic Press, 1980.

    Google Scholar 

  3. D. Dubois, H. Prade and R. R. Yager. Readings in Fuzzy Sets for Intelligent Systems. Morgan Kaufman, San Mateo, CA, 1993.

    Google Scholar 

  4. Z. Wang and G. Klir. Fuzzy Measure Theory. Plenum, New York, 1992.

    Book  Google Scholar 

  5. R. R. Yager, S. Ovchinnikov, R. M. Tong and H. T. Nguyen. Fuzzy Sets and Applications. Selected Papers by L. A. Zadeh. Wiley, New York, 1987.

    Google Scholar 

  6. L. A. Zadeh. Fuzzy sets. Information and Control, 8, 338–353, 1965.

    Article  Google Scholar 

Uncertainty

  1. B. Bouchon-Meunier andH. T. Nguyen. Les Incertitudes dans les Systémes Intelligents. Presses Universitaires de France, Paris, 1996.

    Google Scholar 

  2. M. R. B. Clarke, C. Froidevaux, E. Gregoire and Ph. Smets. Guest Editors of the Special Issue on `Uncertainty, Conditional and Non Monotonicity. Positions and Debates in Non-Standard Logics’. Journal of Applied Non-Classical Logics, 1,103–310, 1991.

    Google Scholar 

  3. D. Dubois, H. Prade and R. R. Yager. Readings in Fuzzy Sets for Intelligent Systems. Morgan Kaufman, San Mateo, CA, 1993.

    Google Scholar 

  4. R. Fagin, J. Y. Halpern, Y. Moses and M. Y. Vardi. Reasoning about Knowledge. MIT Press Cambridge, MA, 1995.

    Google Scholar 

  5. A. Hunter. Uncertainty in Information Systems. McGraw-Hill, London, 1996.

    Google Scholar 

  6. R Krause and D. Clark. Representing Uncertain Knowledge. An AI approach. Intellect, Oxford, 1993.

    Google Scholar 

  7. R. Kruse, E. Schwecke and J. Heinsohn. Uncertainty and Vagueness in Knowledge Based Systems, Numerical Methods. Springer Verlag, Berlin, 1991.

    Book  Google Scholar 

  8. Léa Sombé. Raisonnements sur des informations incompltes in intelligence arti-ficielle. Teknea, Toulouse, 1989. English translation: Reasoning under Incomplete Information in Artificial Intelligence. Wiley, New York, 1990.

    Google Scholar 

  9. Léa Sombé. A glance at revision and updating in knowledge bases. International Journal of Intelligent Systems, 9, 1–28, 1994.

    Google Scholar 

  10. R. Lopez de Mantaras. Approximate Reasoning Models. Ellis Horwood, Chichester, 1990.

    Google Scholar 

  11. A. Motro and Ph. Smets. Uncertainty Management in Information Systems: From Needs to Solutions. Kluwer, Boston, 1997.

    Book  Google Scholar 

  12. J. Paris. The Uncertainty Reasoner’s Companion. Cambridge University Press, Cambridge, 1994.

    Google Scholar 

  13. G. Shafer and J. Pearl. Readings in Uncertainty Reasoning. Morgan Kaufmann, San Mateo, CA, 1990.

    Google Scholar 

  14. Ph. Smets, E. H. Mamdani, D. Dubois and H. Prade, eds. Non-Standard Logics for Automated Reasoning. Academic Press, London, 1988, 334 pages.

    Google Scholar 

  15. M. Smithson. Ignorance and Uncertainty: Emerging Paradigms. Springer-Verlag, New York, 1989.

    Book  Google Scholar 

Probability Theory

  1. T. Bayes. An essay toward solving a problem in the doctrine of chances. Philos. Trans. Roy. Soc., London, 53, 370–418, 1793.

    Google Scholar 

  2. E Bacchus. Representing and Reasoning with Probabilistic Knowledge. MIT Press, Cambridge, MA, 1990.

    Google Scholar 

  3. R. Carnap. Logical Foundations of Probability. University of Chicago Press, 1962.

    Google Scholar 

  4. R. Chuaqui. Truth, Possibility and Probability. North Holland, Amsterdam, 1991.

    Google Scholar 

  5. R. M. Cooke. Experts in Uncertainty. Oxford University Press, New York, 1991.

    Google Scholar 

  6. B. de Finetti. La prévision: ses lois logiques, ses sources subjectivs. Ann. Inst. H. Poincarré, 7, 1–68, 1937.

    Google Scholar 

  7. B. de Finetti. Theory of Probability, Vol. I and Vol. 2. Wiley, London, 1974.

    Google Scholar 

  8. J. Earman. Bayes or Bust? A Critical Examination of Bayesian Confirmation Theory. MIT Press, Cambridge, MA, 1992.

    Google Scholar 

  9. A. W. E. Edwards. Likelihood. Cambridge University Press, Cambridge, 1972.

    Google Scholar 

  10. T. Fine. Theories of Probability. Academic Press, New York, 1973.

    Google Scholar 

  11. P. Gärdenfors and N. E. Sahlin. Decision, Probability and Utility. Cambridge University Press, Cambridge, 1988.

    Book  Google Scholar 

  12. I. Hacking. Logic of Statistical Inference. Cambridge University Press, Cambridge, 1965.

    Google Scholar 

  13. I. Hacking. The Emergence of Probability. Cambridge University Press, Cambridge, 1975.

    Google Scholar 

  14. H. E. Kyburg, Jr. Probability and the Logic of Rational Belief Wesleyan Univ. Press, 1961.

    Google Scholar 

  15. H. E. Kyburg, Jr., and H. E. Smolker, eds. Studies in Subjective Probability. Wiley, New York, 1964.

    Google Scholar 

  16. J.Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA, 1988.

    Google Scholar 

  17. J. L. Pollock. Nomic Probability and the Foundations of Induction. Oxford University Press, New York, 1990.

    Google Scholar 

  18. H. Reichenbach. The Theory of Probability. University of California Press, Berkeley, 1949.

    Google Scholar 

  19. D. A. Schum. Evidential Foundations of Probabilistic Reasoning. Wiley, New York, 1994.

    Google Scholar 

Decision

  1. M. H. Degroot. Optimal Statistical Decisions. McGraw-Hill, New York, 1970.

    Google Scholar 

  2. R. C. Jeffrey. The Logic of Decision. 2nd Ed. University of Chicago Press, Chicago, 1983.

    Google Scholar 

  3. D. V. Lindley. Making Decisions. 2nd Ed. Wiley, New York, 1985.

    Google Scholar 

  4. I. Levi. Gambling with Truth. An Essay in Induction and the Aims of Science. MIT Press, Cambridge, MA, 1967.

    Google Scholar 

  5. H. Raiffa. Decision Analysis: Introductory Lectures on Choices under Uncertainty. Addison-Wesley, Reading, MA, 1970.

    Google Scholar 

  6. L. J. Savage. Foundations of Statistics. Wiley, New York, 1954.

    Google Scholar 

Set of Probability Functions

  1. I. Levi. The Enterprise of Knowledge. MIT Press, Cambridge, MA, 1980. Upper and Lower Probability

    Google Scholar 

  2. I. J. Good. Probability and the Weighting of Evidence. Hafner, 1950.

    Google Scholar 

  3. I. J. Good. Good Thinking: The Foundations of Probability and its Applications. Univ. Minnesota Press, Minneapolis, 1983.

    Google Scholar 

  4. P. Walley. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, 1991.

    Book  Google Scholar 

Dempster’s Approach

  1. A. P. Dempster. Upper and lower probabilities induced by a multplevalued mapping. Ann. Math. Statistics, 38, 325–339, 1967.

    Article  Google Scholar 

  2. J. Kohlas and P. A. Monney. A Mathematical Theory of Hints. An Approach to Dempster-Shafer Theory of Evidence. Vol. 425 of Lecture Notes in Economics and Mathematical Systems,. Springer-Verlag, Berlin, 1995.

    Google Scholar 

Possibility Theory

  1. D. Dubois and H. Prade. Theorie des Possibilités. Masson, Paris, 1985.

    Google Scholar 

  2. D. Dubois and H. Prade. Possibility Theory. Plenum, London, 1988.

    Book  Google Scholar 

  3. L. Zadeh. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1, 3–28, 1978.

    Article  Google Scholar 

Belief Functions

  1. G. Shafer. A Mathematical Theory of Evidence. Princeton Univ. Press. Princeton, NJ, 1976.

    Google Scholar 

  2. Ph. Smets. Belief functions. In Non Standard Logics for Automated Reasoning, Ph. Smets, A. Mamdani, D. Dubois and H. Prade, eds. pp. 253–286. Academic Press, London, 1988.

    Google Scholar 

  3. Ph. Smets and R. Kennes. The transferable belief model. Artificial Intelligence, 66, 191–234, 1994.

    Article  Google Scholar 

  4. R. R. Yager, J. Kacprzyk and M. Fedrizzi. Advances in the Dempster—Shafer Theory of Evidence. Wiley, New York, 1994.

    Google Scholar 

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Smets, P. (1998). Probability, Possibility, Belief: Which and Where?. In: Smets, P. (eds) Quantified Representation of Uncertainty and Imprecision. Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1735-9_1

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  • DOI: https://doi.org/10.1007/978-94-017-1735-9_1

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-5038-0

  • Online ISBN: 978-94-017-1735-9

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