Skip to main content

Conditioning of Imprecise Probabilities Based on Generalized Credal Sets

  • Conference paper
  • First Online:
Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11726))

Abstract

Recently, generalized credal sets have been introduced for modeling contradiction (incoherence) in the information. In previous papers, we did not discuss how such information could be updated if some events occur. In this paper, we show that it can be done by the conjunctive rule based on generalized credal sets. We show that the application of generalized credal sets results in several types of conditioning for imprecise probabilities.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    The value \(n.ext({\varPhi ^{(1)}})(f)\) for \(f \in K\) is computed by \(n.ext({\varPhi ^{(1)}})(f) = {\overline{E} _{{\mathbf {P}}({\varPhi ^{(1)}})}}(f)\), where \({\mathbf {P}}({\varPhi ^{(1)}})\) is the usual credal set that corresponds to \({\varPhi ^{(1)}}\).

  2. 2.

    A monotone measure \(\mu \) is called 2-alternating if \(\mu (A) + \mu (B) \geqslant \mu (A \cap B) + \mu (A \cup B)\) for all \(A,B \in {2^X}\).

References

  1. Augustin, T., Coolen, F.P.A., de Cooman, G., Troffaes, M.C.M. (eds.): Introduction to Imprecise Probabilities. Wiley, New York (2014)

    MATH  Google Scholar 

  2. Bronevich, A.G., Rozenberg, I.N.: Incoherence correction and decision making based on generalized credal sets. In: Antonucci, A., Cholvy, L., Papini, O. (eds.) ECSQARU 2017. LNCS (LNAI), vol. 10369, pp. 271–281. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-61581-3_25

    Chapter  MATH  Google Scholar 

  3. Bronevich, A.G., Rozenberg, I.N.: Modelling uncertainty with generalized credal sets: application to conjunction and decision. Int. J. Gen. Syst. 27(1), 67–96 (2018)

    Article  MathSciNet  Google Scholar 

  4. Brozzi, A., Capotorti, A., Vantaggi, B.: Incoherence correction strategies in statistical matching. Int. J. Approx. Reason. 53, 1124–1136 (2012)

    Article  MathSciNet  Google Scholar 

  5. Cattaneo, M.: Likelihood decision functions. Electron. J. Stat. 7, 2924–2946 (2013)

    Article  MathSciNet  Google Scholar 

  6. Denneberg, D.: Non-Additive Measure and Integral. Kluwer, Dordrecht (1997)

    MATH  Google Scholar 

  7. Dempster, A.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 38, 325–339 (1967)

    Article  MathSciNet  Google Scholar 

  8. Destercke, S., Antoine, V.: Combining imprecise probability masses with maximal coherent subsets: application to ensemble classification. In: Kruse, R., Berthold, M.R., Moewes, C., Gil, M.A., Grzegorzewski, P., Hryniewicz, O. (eds.) Synergies of Soft Computing and Statistics for Intelligent Data Analysis Advances in Intelligent Systems and Computing, vol. 190, pp. 27–35. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-33042-1_4

    Chapter  Google Scholar 

  9. Good, I.J.: Good Thinking: The Foundations of Probability and Its Applications. University of Minnesota Press Minneapolis, Minn (1983)

    MATH  Google Scholar 

  10. Moral, S., De Campos, L.M.: Updating uncertain information. In: Bouchon-Meunier, B., Yager, R.R., Zadeh, L.A. (eds.) IPMU 1990. LNCS, vol. 521, pp. 58–67. Springer, Heidelberg (1991). https://doi.org/10.1007/BFb0028149

    Chapter  Google Scholar 

  11. Moral, S.: Calculating uncertainty intervals from conditional convex sets of probabilities. In: Dubois, D., Wellman, M.P., D’Ambrosio, B. Smets, Ph. (eds.) Proceedings of 8th Conference on Uncertainty in Artificial Intelligence, pp. 199–206, Stanford University (1992)

    Google Scholar 

  12. Moral, S., Sagrado, J.: Aggregation of imprecise probabilities. In: Bouchon Meunier, B. (ed.) Aggregation and Fusion of Imperfect Information, pp. 162–188. Physica-Verlag, Heidelberg (1997)

    Google Scholar 

  13. Nau, R.: The aggregation of imprecise probabilities. J. Stat. Plan. Inference 105(1), 265–282 (2002)

    Article  MathSciNet  Google Scholar 

  14. Quaeghebeur, E.: Characterizing coherence, correcting incoherence. Int. J. Approx. Reason. 56, 208–223 (2015). (Part B)

    Article  MathSciNet  Google Scholar 

  15. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)

    MATH  Google Scholar 

  16. Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)

    Book  Google Scholar 

  17. Walter, G., Augustin, T.: Imprecision and prior-data conflict in generalized Bayesian inference. J. Stat. Theory Pract. 3, 255–271 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgment

This work was partially supported by the grant 18-01-00877 of RFBR (Russian Foundation for Basic Research).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andrey G. Bronevich .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bronevich, A.G., Rozenberg, I.N. (2019). Conditioning of Imprecise Probabilities Based on Generalized Credal Sets. In: Kern-Isberner, G., Ognjanović, Z. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2019. Lecture Notes in Computer Science(), vol 11726. Springer, Cham. https://doi.org/10.1007/978-3-030-29765-7_31

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-29765-7_31

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-29764-0

  • Online ISBN: 978-3-030-29765-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics