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Lower and Upper Probability Bounds for Some Conjunctions of Two Conditional Events

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Scalable Uncertainty Management (SUM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11142))

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Abstract

In this paper we consider, in the framework of coherence, four different definitions of conjunction among conditional events. In each of these definitions the conjunction is still a conditional event. We first recall the different definitions of conjunction; then, given a coherent probability assessment (x, y) on a family of two conditional events \(\{A|H,B|K\}\), for each conjunction \((A|H) \wedge (B|K)\) we determine the (best) lower and upper bounds for the extension \(z=P[(A|H) \wedge (B|K)]\). We show that, in general, these lower and upper bounds differ from the classical Fréchet-Hoeffding bounds. Moreover, we recall a notion of conjunction studied in recent papers, such that the result of conjunction of two conditional events A|H and B|K is (not a conditional event, but) a suitable conditional random quantity, with values in the interval [0, 1]. Then, we remark that for this conjunction, among other properties, the Fréchet-Hoeffding bounds are preserved.

G. Sanfilippo was partially supported by the National Group for Mathematical Analysis, Probability and their Applications (GNAMPA – INdAM).

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Acknowledgments

We thank Angelo Gilio and the three anonymous reviewers for their useful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Giuseppe Sanfilippo

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  1. Giuseppe Sanfilippo
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Correspondence to Giuseppe Sanfilippo .

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Editors and Affiliations

  1. University of Milano-Bicocca, Milan, Italy

    Davide Ciucci

  2. University of Milano-Bicocca, Milan, Italy

    Gabriella Pasi

  3. Sapienza University of Rome, Rome, Italy

    Barbara Vantaggi

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Sanfilippo, G. (2018). Lower and Upper Probability Bounds for Some Conjunctions of Two Conditional Events. In: Ciucci, D., Pasi, G., Vantaggi, B. (eds) Scalable Uncertainty Management. SUM 2018. Lecture Notes in Computer Science(), vol 11142. Springer, Cham. https://doi.org/10.1007/978-3-030-00461-3_18

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  • DOI: https://doi.org/10.1007/978-3-030-00461-3_18

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