Credal Sets Approximation by Lower Probabilities: Application to Credal Networks

  • Alessandro Antonucci
  • Fabio Cuzzolin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6178)


Credal sets are closed convex sets of probability mass functions. The lower probabilities specified by a credal set for each element of the power set can be used as constraints defining a second credal set. This simple procedure produces an outer approximation, with a bounded number of extreme points, for general credal sets. The approximation is optimal in the sense that no other lower probabilities can specify smaller supersets of the original credal set. Notably, in order to be computed, the approximation does not need the extreme points of the credal set, but only its lower probabilities. This makes the approximation particularly suited for credal networks, which are a generalization of Bayesian networks based on credal sets. Although most of the algorithms for credal networks updating only return lower posterior probabilities, the suggested approximation can be used to evaluate (as an outer approximation of) the posterior credal set. This makes it possible to adopt more sophisticated decision making criteria, without having to replace existing algorithms. The quality of the approximation is investigated by numerical tests.


Imprecise probability lower probabilities credal sets credal networks interval dominance maximality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Levi, I.: The Enterprise of Knowledge. MIT Press, London (1980)Google Scholar
  2. 2.
    Walley, P.: Measures of uncertainty in expert systems. Artificial Intelligence 83(1), 1–58 (1996)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Cozman, F.G.: Credal networks. Artificial Intelligence 120, 199–233 (2000)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Antonucci, A., Zaffalon, M.: Decision-theoretic specification of credal networks: A unified language for uncertain modeling with sets of Bayesian networks. International Journal of Approximate Reasoning 49(2), 345–361 (2008)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    de Campos, C.P., Cozman, F.G.: The inferential complexity of Bayesian and credal networks. In: Proceedings of the International Joint Conference on Artificial Intelligence, Edinburgh, pp. 1313–1318 (2005)Google Scholar
  6. 6.
    Cozman, F.G.: Graphical models for imprecise probabilities. International Journal of Approximate Reasoning 39(2-3), 167–184 (2005)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    de Campos, C.P., Cozman, F.G.: Inference in credal networks through integer programming. In: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications, Prague, Action M Agency, pp. 145–154 (2007)Google Scholar
  8. 8.
    Antonucci, A., Sun, Y., de Campos, C.P., Zaffalon, M.: Generalized loopy 2U: a new algorithm for approximate inference in credal networks. International Journal of Approximate Reasoning (to appear, 2010)Google Scholar
  9. 9.
    Wallner, A.: Maximal number of vertices of polytopes defined by f-probabilities. In: Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications, SIPTA, pp. 126–139 (2005)Google Scholar
  10. 10.
    Cuzzolin, F.: On the credal structure of consistent probabilities. In: Hölldobler, S., Lutz, C., Wansing, H. (eds.) JELIA 2008. LNCS (LNAI), vol. 5293, pp. 126–139. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Avis, D., Fukuda, K.: A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete and Computational Geometry 8, 295–313 (1992)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)MATHGoogle Scholar
  13. 13.
    Antonucci, A., Piatti, A.: Modeling unreliable observations in Bayesian networks by credal networks. In: Proceedings of the 3rd International Conference on Scalable Uncertainty Management table of contents, pp. 28–39. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Troffaes, M.C.M.: Decision making under uncertainty using imprecise probabilities. International Journal of Approximate Reasoning 45(1), 17–29 (2007)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Alessandro Antonucci
    • 1
  • Fabio Cuzzolin
    • 2
  1. 1.Istituto Dalle Molle di Studi sull’Intelligenza ArtificialeManno-LuganoSwitzerland
  2. 2.Department of ComputingOxford Brookes UniversityOxfordUnited Kingdom

Personalised recommendations