Random Sets pp 361-383 | Cite as

Geometric Structure of Lower Probabilities

  • Paul K. Black
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 97)


Lower probability theory has gained interest in recent years as a basis for representing uncertainty. Much of the work in this area has focused on lower probabilities termed belief functions. Belief functions correspond precisely to random set representations of uncertainty, although more general classes of lower probabilities are available, such as lower envelopes and 2—monotone capacities. Characterization through definitions of monotone capacities provides a mechanism by which different classes of lower probabilities can be distinguished. Such characterizations have previously been performed through mathematical representations that offer little intuition about differences between lower probability classes. An alternative characterization is offered that uses geometric structures to provide a direct visual representation of the distinctions between different classes of lower probabilities, or monotone capacities. Results are presented in terms of the shape of lower probabilities that provide a characterization of monotone capacities that, for example, distinguishes belief functions from other classes of lower probabilities. Results are also presented in terms of the size, relative size, and location-scale transformation of lower probabilities that provide further insights into their general structure.

Key words

Barycentric Coordinate System Belief Functions Coherent Lower Probabilities Lower Envelopes Möbius Transformations Monotone Capacities 2—Monotone Lower Probabilities Undominated Lower Probabilities 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P.K. Black, AnExamination of Belief Functions and Other Monotone CapacitiesPh.D. Dissertation, Department of Statistics, Carnegie Mellon University, 1996.Google Scholar
  2. [2]
    P.K. BlackMethods for `conditioning’ on sets of joint probability distributions induced by upper and lower marginalsPresented at the 149thMeeting of the American Statistical Association, 1988.Google Scholar
  3. [3]
    P.K. BlackIs Shafer general Bayes?Proceedings of the3rdWorkshop on Uncertainty in Artificial Intelligence, pp. 2–9, Seattle, Washington, July 1987.Google Scholar
  4. [4]
    P.K. Black and W.F. EddyThe implementation of belief functions in a rule-based systemTechnical Report 371, Department of Statistics, Carnegie Mellon University, 1986.Google Scholar
  5. [5]
    P.K. Black and A.W. MartinShipboard evidential reasoning algorithmsTechnical Report 90–11, Decision Science Consortium, Inc., Fairfax, Virginia, 1990.Google Scholar
  6. [6]
    A. Chateauneuf and J.Y. JaffraySome characterizations of lower probabilities and other monotone capacitiesUnpublished Manuscript, Groupe de Mathematique Economiques, Universite Paris I, 12, Place du Pantheon, Paris, France, 1986.Google Scholar
  7. [7]
    G. ChoquetTheory of capacitiesAnnales de l’Institut Fourier, V (1954), pp. 131–295.MathSciNetCrossRefGoogle Scholar
  8. [8]
    A.P. DempsterUpper and lower probabilities induced by a multivalued mappingThe Annals of Mathematical Statistics, 38 (1967), pp. 325–339.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    J.Y. JaffrayLinear utility theory for belief functionsUnpublished Manuscript, Universite P. et M. Curie (Paris 6), 4 Place Jussieu, 75005 Paris, France, 1989.Google Scholar
  10. [10]
    H.E. KyburgBayesian and non-Bayesian evidential updatingArtificial Intelligence, 31 (1987), pp. 279–294.MathSciNetCrossRefGoogle Scholar
  11. [11]
    I. LeviThe Enterprise of KnowledgeMIT Press, 1980.Google Scholar
  12. [12]
    A. Papamarcou and T.L. FineA note on undominated lower probabilitiesAnnals of Probability, 14 (1986), pp. 710–723.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    T. Seidenfeld, M.J. Schervish, and J.B. KadaneDecisions without orderingTechnical Report 391, Department of Statistics, Carnegie Mellon University, Pittsburgh, PA 152–13, 1987.Google Scholar
  14. [14]
    T. Seidenfeld and L.A. WassermanDilation for sets of probabilitiesAnnals of Statistics, 21 (1993), pp. 1139–1154.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    G. ShaferA Mathematical Theory of EvidencePrinceton University Press, 1976.zbMATHGoogle Scholar
  16. [16]
    P. SmetsConstructing the pignistic probability function in the context ofuncertainty, Uncertainty in Artificial Intelligence V, M. Henrion, R.D. Shachter, L.N. Kanal, and J.F. Lemmer (Eds.), North-Holland, Amsterdam (1990), pp. 29–40.Google Scholar
  17. [17]
    C.A.B. SmithConsistency in statistical inference and decision(with Discussion), Journal of the Royal Statistical Society, Series B, 23 (1961), pp. 1–25.zbMATHGoogle Scholar
  18. [18]
    P. SuppesThe measurement of beliefJournal of the Royal Statistical Society, Series B, 36 (1974), pp. 160–191.MathSciNetzbMATHGoogle Scholar
  19. [19]
    H.M. ThomaFactorization of Belief FunctionsDoctoral Dissertation, Department of Statistics, Harvard University, 1989.Google Scholar
  20. [20]
    P. WalleyCoherent lower (and upper) probabilitiesTech. Report No. 22, Department of Statistics, University of Warwick, Coventry, U.K., 1981.Google Scholar
  21. [21]
    P. WalleyStatistical Reasoning with Imprecise ProbabilitiesChapman-Hall, 1991.zbMATHGoogle Scholar
  22. [22]
    P. Walley and T.L. FineTowards a frequentist theory of upper and lower probabilitiesAnnals of Statistics, 10 (1982), pp. 741–761.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    L.A. Wasserman and J.B. KadaneBayes’ theorem for Choquet capacitiesAnnals of Statistics, 18 (1990), pp. 1328–1339.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    M. Wolfenson and T.L. FineBayes-like decision making withupperand lower probabilitiesJournal of the American Statistical Association, 77 (1982), pp. 80–88.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Paul K. Black
    • 1
  1. 1.Neptune and Company, Inc.Los AlamosUSA

Personalised recommendations