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Transitive Reasoning with Imprecise Probabilities

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Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2015)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9161))

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Abstract

We study probabilistically informative (weak) versions of transitivity by using suitable definitions of defaults and negated defaults in the setting of coherence and imprecise probabilities. We represent \(\text{ p-consistent }\) sequences of defaults and/or negated defaults by g-coherent imprecise probability assessments on the respective sequences of conditional events. Finally, we present the coherent probability propagation rules for Weak Transitivity and the validity of selected inference patterns by proving p-entailment of the associated knowledge bases.

N. Pfeifer—Supported by the DFG grants PF 740/2-1 and PF 740/2-2 (both within the DFG Priority Programme SPP1516) and the Alexander von Humboldt-Foundation.

G. Sanfilippo—Supported by the INdAM–GNAMPA Project 2015 and by the FFR 2012-ATE-0585 Project of University of Palermo.

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Notes

  1. 1.

    For proving total coherence of \(\mathcal {I}\) on \(\mathcal {F}\) (resp., \(\mathcal {F}'\)) it is sufficient to check that the assessment \(\{0,1\}^3\) on \(\mathcal {F}\) (resp., \(\mathcal {F}'\)) is totally coherent ([19, Theorem 7]), i.e., each of the eight vertices of the unit cube is coherent. Coherence can be checked, for example, by applying Algorithm 1 of [19] or by the CkC-package [2].

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Acknowledgments

We thank two anonymous referees for their very useful comments and suggestions.

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Authors and Affiliations

  1. Department SBAI, University of Rome “La Sapienza”, Rome, Italy

    Angelo Gilio

  2. Munich Center for Mathematical Philosophy, LMU Munich, München, Germany

    Niki Pfeifer

  3. Department of Mathematics and Computer Science, University of Palermo, Palermo, Italy

    Giuseppe Sanfilippo

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  1. Angelo Gilio
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Correspondence to Giuseppe Sanfilippo .

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Editors and Affiliations

  1. Université de Technologie de Compiègne, Compiègne, France

    Sébastien Destercke

  2. Centre de Recherches de Royallieu, Université de Technologie de Compiègne, Compiègne, France

    Thierry Denoeux

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Gilio, A., Pfeifer, N., Sanfilippo, G. (2015). Transitive Reasoning with Imprecise Probabilities. In: Destercke, S., Denoeux, T. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2015. Lecture Notes in Computer Science(), vol 9161. Springer, Cham. https://doi.org/10.1007/978-3-319-20807-7_9

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