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A Set-Theoretic View of Belief Functions

Logical Operations and Approximations by Fuzzy Sets

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Classic Works of the Dempster-Shafer Theory of Belief Functions

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 219))

Abstract

A body of evidence in the sense of Shafer can be viewed as an extension of a probability measure, but as a generalized set as well. In this paper we adopt the second point of view and study the algebraic structure of bodies of evidence on a set, based on extended set union, intersection and complementation. Several notions of inclusion are exhibited and compared to each other. Inclusion is used to compare a body of evidence to the product of its projections. Lastly, approximations of a body of evidence under the form of fuzzy sets are derived, in order to squeeze plausibility values between two grades of possibility. Through all the paper, it is pointed out that a body of evidence can account for conjunctive as well as a disjunctive information, i.e. the focal elements can be viewed either as sets of actual values or as restrictions on the (unique) value of a variable.

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Authors
  1. Didier Dubois
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  2. Henri Prade
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Editors and Affiliations

  1. Machine Intelligence Institute, Iona College, 715 N. Avenue, New Rochelle, NY 10801-1890, USA

    Roland R. Yager (Dr.) (Dr.)

  2. Department of Management and Information Systems, University of Akron College of Business Administration, 351 Akron, OH 4325-4801, USA

    Liping Liu

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Dubois, D., Prade, H. (2008). A Set-Theoretic View of Belief Functions. In: Yager, R.R., Liu, L. (eds) Classic Works of the Dempster-Shafer Theory of Belief Functions. Studies in Fuzziness and Soft Computing, vol 219. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44792-4_14

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  • DOI: https://doi.org/10.1007/978-3-540-44792-4_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25381-5

  • Online ISBN: 978-3-540-44792-4

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