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A Relationship of Conflicting Belief Masses to Open World Assumption

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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

When combining belief functions by conjunctive rules of combination, conflicting belief masses often appear, which are assigned to empty set by the non-normalized conjunctive rule or normalized by Dempster’s rule of combination in Dempster-Shafer theory.

This theoretical study analyses processing of conflicting belief masses under open world assumption. It is observed that sum of conflicting masses covers not only a possibility of a non-expected hypothesis out of considered frame of discernment. It also covers, analogously to the case of close world assumption, internal conflicts of individual belief functions and conflict between/among two or several combined belief functions.

Thus, for correct and complete interpretation of open world assumption it is recommended to include extra element(s) into used frame of discernment.

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Notes

  1. 1.

    Do not forget that the equality of BFs is not equivalent to their dependence: dependent BFs, BFs from dependent believers should be same or somehow similar, dependence implies similarity, but same (or very similar) BFs do not imply their dependence.

  2. 2.

    Combining two vacuous BFs gives \(m(\varOmega ) =1\), thus \(m(\emptyset ) = 0\), but vacuous BF does not express the same positive arguments for all hypotheses, it expresses the full ignorance.

  3. 3.

    Note, that the pignistic probability gives numerically same results under close and open world assumptions, as normalization is part of pignistic transformation; and that TBM with non-normalized under OWA gives same decisional results as classic Shafer’s approach with \(\oplus \) and pignistic transformation does. The only difference is that TBM explicitly keeps in \(m(\emptyset )\) value of conflict (internal and external conflict together with masses of unexpected hypotheses) until the moment of decision.

  4. 4.

    Note, that normalized plausibility is consistent with conjunctive combination (they mutually commute), while pignistic transformation is not. Pignistic transformation commutes instead of conjunctive combination with linear combination of BFs.

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Acknowledgments

This study is a continuation of author’s research previously conducted at the Institute of Computer Science, The Czech Academy of Sciences.

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Authors and Affiliations

  1. Prague, Czech Republic

    Milan Daniel

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  1. Milan Daniel
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Correspondence to Milan Daniel .

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  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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Daniel, M. (2016). A Relationship of Conflicting Belief Masses to Open World Assumption. 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_15

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  • DOI: https://doi.org/10.1007/978-3-319-45559-4_15

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