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On Aggregating Probabilistic Evidence

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Book cover Logical Foundations of Computer Science (LFCS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9537))

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Abstract

Imagine a database – a set of propositions \(\varGamma =\{F_1,\ldots ,F_n\}\) with some kind of probability estimates, and let a proposition X logically follow from \(\varGamma \). What is the best justified lower bound of the probability of X? The traditional approach, e.g., within Adams’ Probability Logic, computes the numeric lower bound for X corresponding to the worst-case scenario. We suggest a more flexible parameterized approach by assuming probability events \(u_1,u_2,\ldots ,u_n\) which support \(\varGamma \), and calculating aggregated evidence \(e(u_1,u_2,\ldots ,u_n)\) for X. The probability of e provides a tight lower bound for any, not only a worst-case, situation. The problem is formalized in a version of justification logic and the conclusions are supported by corresponding completeness theorems. This approach can handle conflicting and inconsistent data and allows the gathering both positive and negative evidence for the same proposition.

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Notes

  1. 1.

    Since \(P(B\mid A)= P(A\cap B)/P(A)\).

  2. 2.

    This axiom can be replaced by an explicit list of its instances corresponding to a standard algorithm for deciding \(s\preceq t\) (cf. [14]).

  3. 3.

    \(\varGamma \) is not necessarily compatible with set \(u_1^*,u_2^*,\ldots ,u_n^*\) but we ignore this question for now by assuming that the given evidence is consistent with \(\varGamma \).

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Acknowledgements

The author is very grateful to Melvin Fitting, Vladimir Krupski, Elena Nogina, Tudor Protopopescu, Çağıl Taşdemir, and participants of the Trends in Logic XV conference in Delft for inspiring discussions and helpful suggestions. Special thanks to Karen Kletter for editing and proofreading this text.

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Authors and Affiliations

  1. The City University of New York, The Graduate Center, 365 Fifth Avenue, New York City, NY, 10016, USA

    Sergei Artemov

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  1. Sergei Artemov
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Correspondence to Sergei Artemov .

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Editors and Affiliations

  1. City University of New York , New York, New York, USA

    Sergei Artemov

  2. Cornell University , Ithaca, New York, USA

    Anil Nerode

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Artemov, S. (2016). On Aggregating Probabilistic Evidence. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2016. Lecture Notes in Computer Science(), vol 9537. Springer, Cham. https://doi.org/10.1007/978-3-319-27683-0_3

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  • DOI: https://doi.org/10.1007/978-3-319-27683-0_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-27682-3

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