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Random Sets pp 361-383 | Cite as

Geometric Structure of Lower Probabilities

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

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 

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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

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

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