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Entropy of Belief Functions in the Dempster-Shafer Theory: A New Perspective

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Belief Functions: Theory and Applications (BELIEF 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9861))

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Abstract

We propose a new definition of entropy of basic probability assignments (BPA) in the Dempster-Shafer (D-S) theory of belief functions, which is interpreted as a measure of total uncertainty in the BPA. We state a list of five desired properties of entropy for D-S belief functions theory that are motivated by Shannon’s definition of entropy of probability functions, together with the implicit requirement that any definition should be consistent with semantics of D-S belief functions theory. Three of our five desired properties are different from the five properties described by Klir and Wierman. We demonstrate that our definition satisfies all five properties in our list, and is consistent with semantics of D-S theory, whereas none of the existing definitions do. Our definition does not satisfy the sub-additivity property. Whether there exists a definition that satisfies our five properties plus sub-additivity, and that is consistent with semantics for the D-S theory, remains an open question.

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Acknowledgements

This article is a short version of [14], which has been supported in part by funds from grant GAČR 15-00215S to the first author, and from the Ronald G. Harper Distinguished Professorship at the University of Kansas to the second author. We are extremely grateful to Thierry Denoeux, Marc Pouly, Anne-Laure Jousselme, Joaquín Abellán, and Mark Wierman for their comments on earlier drafts of [14]. We are grateful to two anonymous reviewers of Belief-2016 conference for their comments. We are also grateful to Suzanna Emelio for a careful proof-reading of the text.

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Authors and Affiliations

  1. Faculty of Management, University of Economics, Jindřichův Hradec, Czech Republic

    Radim Jiroušek

  2. School of Business, University of Kansas, Lawrence, KS, USA

    Prakash P. Shenoy

  3. Institute of Information Theory and Automation, Academy of Sciences, Prague, Czech Republic

    Radim Jiroušek

Authors
  1. Radim Jiroušek
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  2. Prakash P. Shenoy
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Correspondence to Prakash P. Shenoy .

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Editors and Affiliations

  1. Czech Academy of Sciences, nstitute of Information Theory and Automation, Prague , Czech Republic

    Jiřina Vejnarová

  2. Czech Academy of Sciences, Institute of Information Theory and Automation, Prague, Czech Republic

    Václav Kratochvíl

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Jiroušek, R., Shenoy, P.P. (2016). Entropy of Belief Functions in the Dempster-Shafer Theory: A New Perspective. In: Vejnarová, J., Kratochvíl, V. (eds) Belief Functions: Theory and Applications. BELIEF 2016. Lecture Notes in Computer Science(), vol 9861. Springer, Cham. https://doi.org/10.1007/978-3-319-45559-4_1

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  • DOI: https://doi.org/10.1007/978-3-319-45559-4_1

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