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# Geometric Structure of Lower Probabilities

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## Abstract

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## Preview

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## References

- [1]P.K. Black, An
*Examination of Belief Functions and Other Monotone Capacities*Ph.D. Dissertation, Department of Statistics, Carnegie Mellon University, 1996.Google Scholar - [2]P.K. Black
*Methods for `conditioning’ on sets of joint probability distributions induced by upper and lower marginals*Presented at the 149^{th}Meeting of the American Statistical Association, 1988.Google Scholar - [3]P.K. Black
*Is Shafer general Bayes?*Proceedings of the^{3rd}Workshop on Uncertainty in Artificial Intelligence, pp. 2–9, Seattle, Washington, July 1987.Google Scholar - [4]P.K. Black and W.F. Eddy
*The implementation of belief functions in a rule-based system*Technical Report 371, Department of Statistics, Carnegie Mellon University, 1986.Google Scholar - [5]P.K. Black and A.W. Martin
*Shipboard evidential reasoning algorithms*Technical Report 90–11, Decision Science Consortium, Inc., Fairfax, Virginia, 1990.Google Scholar - [6]A. Chateauneuf and J.Y. Jaffray
*Some characterizations of lower probabilities and other monotone capacities*Unpublished Manuscript, Groupe de Mathematique Economiques, Universite Paris I, 12, Place du Pantheon, Paris, France, 1986.Google Scholar - [7]G. Choquet
*Theory of capacities*Annales de l’Institut Fourier, V (1954), pp. 131–295.MathSciNetCrossRefGoogle Scholar - [8]A.P. Dempster
*Upper and lower probabilities induced by a multivalued mapping*The Annals of Mathematical Statistics, 38 (1967), pp. 325–339.MathSciNetzbMATHCrossRefGoogle Scholar - [9]J.Y. Jaffray
*Linear utility theory for belief functions*Unpublished Manuscript, Universite P. et M. Curie (Paris 6), 4 Place Jussieu, 75005 Paris, France, 1989.Google Scholar - [10]H.E. Kyburg
*Bayesian and non-Bayesian evidential updating*Artificial Intelligence, 31 (1987), pp. 279–294.MathSciNetCrossRefGoogle Scholar - [11]I. Levi
*The Enterprise of Knowledge*MIT Press, 1980.Google Scholar - [12]A. Papamarcou and T.L. Fine
*A note on undominated lower probabilities*Annals of Probability, 14 (1986), pp. 710–723.MathSciNetzbMATHCrossRefGoogle Scholar - [13]T. Seidenfeld, M.J. Schervish, and J.B. Kadane
*Decisions without ordering*Technical Report 391, Department of Statistics, Carnegie Mellon University, Pittsburgh, PA 152–13, 1987.Google Scholar - [14]T. Seidenfeld and L.A. Wasserman
*Dilation for sets of probabilities*Annals of Statistics, 21 (1993), pp. 1139–1154.MathSciNetzbMATHCrossRefGoogle Scholar - [15]G. Shafer
*A Mathematical Theory of Evidence*Princeton University Press, 1976.zbMATHGoogle Scholar - [16]P. Smets
*Constructing the pignistic probability function in the context of*uncertainty, 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]C.A.B. Smith
*Consistency in statistical inference and decision*(with Discussion), Journal of the Royal Statistical Society, Series B, 23 (1961), pp. 1–25.zbMATHGoogle Scholar - [18]P. Suppes
*The measurement of belief*Journal of the Royal Statistical Society, Series B, 36 (1974), pp. 160–191.MathSciNetzbMATHGoogle Scholar - [19]H.M. Thoma
*Factorization of Belief Functions*Doctoral Dissertation, Department of Statistics, Harvard University, 1989.Google Scholar - [20]P. Walley
*Coherent lower (and upper) probabilities*Tech. Report No. 22, Department of Statistics, University of Warwick, Coventry, U.K., 1981.Google Scholar - [21]P. Walley
*Statistical Reasoning with Imprecise Probabilities*Chapman-Hall, 1991.zbMATHGoogle Scholar - [22]P. Walley and T.L. Fine
*Towards a frequentist theory of upper and lower probabilities*Annals of Statistics, 10 (1982), pp. 741–761.MathSciNetzbMATHCrossRefGoogle Scholar - [23]L.A. Wasserman and J.B. Kadane
*Bayes’ theorem for Choquet capacities*Annals of Statistics, 18 (1990), pp. 1328–1339.MathSciNetzbMATHCrossRefGoogle Scholar - [24]M. Wolfenson and T.L. Fine
*Bayes-like decision making with*upper*and lower probabilities*Journal of the American Statistical Association, 77 (1982), pp. 80–88.MathSciNetzbMATHCrossRefGoogle Scholar