Skip to main content
SpringerLink
Log in

Non standard probabilistic and non probabilistic representations of uncertainty

  • Fundamental Issues
  • Conference paper
  • First Online:
Advances in Intelligent Computing — IPMU '94 (IPMU 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 945))

Abstract

Survey of the mathematical models proposed to represent quantified beliefs, and their comparison. The models considered are separated into non standard probability and non probability models, according to the fact they are based on probability theory or not. The first group concerns the upper and lower probability models, the second the possibility theory and the transferable belief model.

Research work has been partly supported by the Action de Recherches Concertées BELON funded by a grant from the Communauté Française de Belgique and the ESPRIT III, Basic Research Action 6156 (DRUMS II) funded by a grant from the Commission of the European Communities.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography.

  • CHEESEMAN P. (1985) In defense of probability. IJCAI-85, Morgan Kaufman, San Mateo, Ca, 1002–1009.

    Google Scholar 

  • CHELLAS B.F. (1980) Modal logic. Cambridge Univ. Press, G.B.

    Google Scholar 

  • CHOQUET G. (1953) Theory of capacities. Annales de l'Institut Fourier, Université de Grenoble, 5 131–296.

    Google Scholar 

  • COHEN L. J., (1993) What has probability to do with strengh of belief. in DUBUCS J.P. (ed.) Philosophy of Probability, Kluwer, Dordrecht, pg. 129–143.

    Google Scholar 

  • COX R.T. (1946) Probability, frequency and reasonable expectation. Amer.J.Phys. 14:1–13.

    Google Scholar 

  • DEGROOT M.H. (1970) Optimal statistical decisions. McGraw-Hill, New York.

    Google Scholar 

  • DEMPSTER A.P. (1967) Upper and lower probabilities induced by a multplevalued mapping. Ann. Math. Statistics 38: 325–339.

    Google Scholar 

  • DEMPSTER A.P. A generalization of Bayesian inference. J. Roy. Statist. Soc. B.30 (1968) 205–247.

    Google Scholar 

  • D'ESPAGNAT B. (1976) Conceptual foundations of quantum mechanics. W.A. Benjamin, Reading, Mass.

    Google Scholar 

  • DUBOIS D., GARBOLINO P., KYBURG H.E., PRADE H. and SMETS P. (1991) Quantified Uncertainty. J. Applied Non-Classical Logics 1:105–197.

    Google Scholar 

  • DUBOIS D. and PRADE H. (1985) Theorie des possibilités. Masson, Paris.

    Google Scholar 

  • EKELOF P.O. (1982) Rättegång IV. Fifth edition, Stockholm.

    Google Scholar 

  • FAGIN R. and HALPERN J. (1990) A new approach to updating beliefs. 6th Conf. on Uncertainty in AI.

    Google Scholar 

  • FINE T. (1973) Theories of probability. Academic Press, New York.

    Google Scholar 

  • GÄRDENFORS P., HANSSON B. and SAHLIN N.E. (1983) Evidentiary value: philosophical, judicial and psychological aspects of a theory. C.W.K. Gleerups, Lund.

    Google Scholar 

  • GÄRDENFORS P. (1988) Knowledge in flux. Modelling the dynamics of epistemic states. MIT Press, Cambridge, Mass.

    Google Scholar 

  • GOOD I.J. (1950) Probability and the weighting of evidence. Hafner.

    Google Scholar 

  • GOOD I.J. (1983) Good thinking: the foundations of probability and its applications. Univ. Minnesota Press, Minneapolis.

    Google Scholar 

  • HACKING I. (1965) Logic of statistical inference. Cambridge University Press, Cambridge, U.K.

    Google Scholar 

  • HACKING I.(1975) The emergence of probability. Cambridge University Press, Cambridge.

    Google Scholar 

  • HINTIKKA J. (1962) Knowledge and belief. Cornell Univ. Press, Ithaca, NY.

    Google Scholar 

  • JAFFRAY J.Y. (1989) Coherent bets under partially resolving uncertainty and belief functions. Theory and Decision 26:99–105.

    Google Scholar 

  • JAFFRAY J.Y. (1992) Bayesian updating and belief functions. IEEE Trans. SMC, 22: 1144–1152.

    Google Scholar 

  • KOHLAS J. and MONNEY P.A (1994a) Representation of Evidence by Hints. in Advances in the Dempster-Shafer Theory of Evidence. Yager R.R., Kacprzyk J. and Fedrizzi M., eds, Wiley, New York, pg. 473–492.

    Google Scholar 

  • KOHLAS J. and MONNEY P.A. (1994b) Theory of Evidence. A Survey of its Mathematical Foundations, Applications and Computations. ZOR-Mathematical Methods of Operatioanl Research 39:35–68.

    Google Scholar 

  • KOOPMAN B.O. (1940) The bases of probability. Bull. Amer. Math. Soc. 46:763–774.

    Google Scholar 

  • KYBURG H. (1987) Objective probabilities. IJCAI-87, 902–904.

    Google Scholar 

  • LEVI I. (1984) Decisions and revisions: philosophical essays on knowledge and value. Cambridge University Press, Cambridge.

    Google Scholar 

  • PEARL J. (1988) Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann Pub. San Mateo, Ca, USA.

    Google Scholar 

  • RAMSEY F.P. (1931) Truth and probability. in Studies in subjective probability, eds. KYBURG H.E. and SMOKLER H.E., p. 61–92. Wiley, New York.

    Google Scholar 

  • RUSPINI E.H. (1986) The logical foundations of evidential reasoning. Technical note 408, SRI International, Menlo Park, Ca.

    Google Scholar 

  • SAHLIN N. E. (1993) On higher order beliefs. in DUBUCS J.P. (ed.) Philosophy of Probability, Kluwer, Dordrecht, pg. 13–34.

    Google Scholar 

  • SAVAGE L.J. (1954) Foundations of Statistics. Wiley, New York.

    Google Scholar 

  • SHAFER G. (1976a) A mathematical theory of evidence. Princeton Univ. Press. Princeton, NJ.

    Google Scholar 

  • SHAFER G. (1976b) A theory of statistical evidence. in Foundations of probability theory, statistical inference, and statistical theories of science. Harper and Hooker ed. Reidel, Doordrecht-Holland.

    Google Scholar 

  • SHAFER G. (1978) Nonadditive probabilities in the work of Bernoulli and Lambert. Arch. History Exact Sci. 19:309–370.

    Google Scholar 

  • SHAPLEY L.S. (1953) A value for n-person games. In Contributions to the Theory of Games, vol. 2, eds. H. Kuhn and A.W. Tucker. Princeton University Press, pp. 307–317.

    Google Scholar 

  • SHORTLIFFE E.H. and BUCHANAN B.G. (1975) A model of inexact reasoning in medicine. Math. Biosci. 23:351–379.

    Google Scholar 

  • SMETS P. (1978) Un modèle mathématico-statistique simulant le processus du diagnostic médical. Doctoral dissertation, Université Libre de Bruxelles, Bruxelles, (Available through University Microfilm International, 30–32 Mortimer Street, London WIN 7RA, thesis 80-70,003)

    Google Scholar 

  • SMETS P. (1987) Upper and lower probability functions versus belief functions. Proc. International Symposium on Fuzzy Systems and Knowledge Engineering, Guangzhou, China, July 10–16, pg 17–21.

    Google Scholar 

  • SMETS P. (1988) Belief functions. in SMETS Ph, MAMDANI A., DUBOIS D. and PRADE H. ed. Non standard logics for automated reasoning. Academic Press, London pg 253–286.

    Google Scholar 

  • SMETS P. (1990a) The combination of evidence in the transferable belief model. IEEE-Pattern analysis and Machine Intelligence, 12:447–458.

    Google Scholar 

  • SMETS P. (1990b) Constructing the pignistic probability function in a context of uncertainty. Uncertainty in Artificial Intelligence 5, Henrion M., Shachter R.D., Kanal L.N. and Lemmer J.F. eds, North Holland, Amsterdam, 29–40.

    Google Scholar 

  • SMETS P. (1991a) Varieties of ignorance. Information Sciences. 57–58:135–144.

    Google Scholar 

  • SMETS P. (1991b) About updating. in D'Ambrosio B., Smets P., and Bonissone P.P. eds, Uncertainty in AI 91, Morgan Kaufmann, San Mateo, Ca, USA, 1991, 378–385.

    Google Scholar 

  • SMETS P. (1991c) Probability of provability and belief functions. Logique et Analyse, 133–134:177–195.

    Google Scholar 

  • SMETS P. (1992a) The nature of the unnormalized beliefs encountered in the transferable belief model. in Dubois D., Wellman M.P., d'Ambrosio B. and Smets P. Uncertainty in AI 92. Morgan Kaufmann, San Mateo, Ca, USA, 1992, pg.292–297.

    Google Scholar 

  • SMETS P. (1992b) The transferable belief model and random sets. Int. J. Intell. Systems 7:37–46.

    Google Scholar 

  • SMETS P. (1993a) Imperfect information: imprecision — uncertainty. in Smets P. and Motro A. eds. Uncertainty Management in information systems: from needs to solutions. UMIS Workshop, Puerto Andraix.Volume 2. pg. 1–28.

    Google Scholar 

  • SMETS P. (1993b) Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem. Int. J. Approximate Reasoning 9:1–35.

    Google Scholar 

  • SMETS P. (1993c) An axiomatic justifiaction for the use of belief function to quantify beliefs. IJCAI'93 (Inter. Joint Conf. on AI), San Mateo, Ca, pg. 598–603.

    Google Scholar 

  • SMETS P. (1994a) The representation of quantified belief by belief functions: an axiomatic justificiation. TR/IRIDIA/94.3. Submitted for publication.

    Google Scholar 

  • SMETS P. (1994b) What is Dempster-Shafer's model? in Advances in the Dempster-Shafer Theory of Evidence. Yager R.R., Kacprzyk J. and Fedrizzi M., eds, Wiley, New York, pg. 5–34.

    Google Scholar 

  • SMETS P. and KENNES R. (1994) The transferable belief model. Artificial Intelligence 66:191–234.

    Google Scholar 

  • SMITH C.A.B. (1961) Consistency in statistical inference and decision. J. Roy. Statist. Soc. B23:1–37.

    Google Scholar 

  • SMITH C.A.B. (1965) Personal probability and statistical analysis. J. Roy. Statist. Soc. A128, 469–499.

    Google Scholar 

  • SMITH P. and JONES O.R. (1986) The philosophy of mind, an introduction. Cambridge University Press, Cambridge.

    Google Scholar 

  • SMITHSON M. (1989) Ignorance and Uncertainty: Emerging Paradigms. Springer-Verlag, New York.

    Google Scholar 

  • VOORBRAAK F. (1993) As Far as I Know: Epistemic Logic and Uncerttaiunty. Dissertation, Utrecht University.

    Google Scholar 

  • WALLEY P. (1991) Statistical reasoning with imprecise probabilities. Chapman and Hall, London.

    Google Scholar 

  • WILLIAMS P.M. (1976) Indetermiinate probabilities. in Formal methods and the methodology of empirical sciences, Przlecki M., Szaniawski K. and Wojciki R. (eds.) Reidel, Dordrecht.

    Google Scholar 

  • WONG S.K.M., YAO Y.Y., BOLLMANN P. and BÜRGER H.C. (1990) Axiomatization of qualitative belief structure. IEEE Trans. SMC, 21:726–734.

    Google Scholar 

  • ZADEH L.A. (1965) Fuzzy sets. Inform.Control. 8:338–353.

    Google Scholar 

  • ZADEH L. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3–28.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. I.R.I.D.I.A. Université Libre de Bruxelles, 50 av. Roosevelt, CP 194-6, B-1050, Brussels, Belgium

    Philippe Smets

Authors
  1. Philippe Smets
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Bernadette Bouchon-Meunier Ronald R. Yager Lotfi A. Zadeh

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Smets, P. (1995). Non standard probabilistic and non probabilistic representations of uncertainty. In: Bouchon-Meunier, B., Yager, R.R., Zadeh, L.A. (eds) Advances in Intelligent Computing — IPMU '94. IPMU 1994. Lecture Notes in Computer Science, vol 945. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0035934

Download citation

  • DOI: https://doi.org/10.1007/BFb0035934

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60116-6

  • Online ISBN: 978-3-540-49443-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation