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Graphoid properties of epistemic irrelevance and independence

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This paper investigates Walley's concepts of epistemic irrelevance and epistemic independence for imprecise probability models. We study the mathematical properties of irrelevance and independence, and their relation to the graphoid axioms. Examples are given to show that epistemic irrelevance can violate the symmetry, contraction and intersection axioms, that epistemic independence can violate contraction and intersection, and that this accords with informal notions of irrelevance and independence.

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References

  1. J.-M. Bernard, T. Seidenfeld and M. Zaffalon, eds., in: Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications (Carleton Scientific, Lugano, Switzerland, 2003).

  2. I. Couso, S. Moral and P. Walley, A survey of concepts of independence for imprecise probabilities, Risk, Decision and Policy 5 (2000) 165–181.

    Article  Google Scholar 

  3. F.G. Cozman, Irrelevance and independence relations in quasi-Bayesian networks, in: XIV Conference on Uncertainty in Artificial Intelligence, eds. G. Cooper and S. Moral (Morgan Kaufmann, San Francisco, 1998) pp. 89–96.

    Google Scholar 

  4. F.G. Cozman, Irrelevance and independence axioms in quasi-Bayesian theory, in: European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, eds. A. Hunter and S. Parsons (Springer, Berlin Heidelberg New York, 1999) pp. 128–136.

    Chapter  Google Scholar 

  5. F.G. Cozman, Credal networks, Artificial Intelligence 120 (2000) 199–233.

    Article  MATH  MathSciNet  Google Scholar 

  6. F.G. Cozman, Algorithms for conditioning on events of zero lower probability, in: Fifteenth International Florida Artificial Intelligence Research Society Conference, eds. S. Haller and G. Simmons (AAAI, Menlo Park, California, 2002) pp. 248–252.

    Google Scholar 

  7. A.P. Dawid, Conditional independence, in: Encyclopedia of Statistical Sciences, Update Volume 2, eds. S. Kotz, C.B. Read and D.L. Banks (Wiley, New York, 1999) pp. 146–153.

    Google Scholar 

  8. A.P. Dawid, Separoids: A mathematical framework for conditional independence and irrelevance, Annals of Mathematics and Artificial Intelligence 32 (2001) 335–372.

    Article  MathSciNet  Google Scholar 

  9. C.P. de Campos and F.G. Cozman, Belief updating and learning in semi-qualitative probabilistic networks, in: Conference on Uncertainty in Artificial Intelligence, eds. F. Bacchus and T. Jaakkola (AUAI, Edinburgh, 2005).

  10. L. de Campos and S. Moral, Independence concepts for convex sets of probabilities, in: XI Conference on Uncertainty in Artificial Intelligence, eds. P. Besnard and S. Hanks (Morgan Kaufmann, San Francisco, 1995) pp. 108–115.

    Google Scholar 

  11. G. de Cooman, F.G. Cozman, S. Moral and P. Walley, eds., in: Proceedings of the First International Symposium on Imprecise Probabilities and Their Applications (Imprecise Probabilities Project, Ghent, Belgium, 1999).

  12. G. de Cooman, T.L. Fine and T. Seidenfeld, eds., in: Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications (Shaker, Maastricht, The Netherlands, 2001).

  13. D. Galles and J. Pearl, Axioms of causal relevance, Artificial Intelligence 97 (1997) 9–43.

    Article  MATH  MathSciNet  Google Scholar 

  14. D. Geiger, T. Verma and J. Pearl, Identifying independence in Bayesian networks, Networks 20 (1990) 507–534.

    Article  MATH  MathSciNet  Google Scholar 

  15. J.Y. Halpern, Reasoning About Uncertainty (MIT, Cambridge, Massachusetts, 2003).

    MATH  Google Scholar 

  16. P. Hansen and B. Jaumard, Probabilistic satisfiability. Technical Report, Les Cahiers du GERAD, École Polytechique de Montréal, G-96-31, 1996.

  17. I. Levi, The Enterprise of Knowledge (MIT, Cambridge, Massachusetts, 1980).

    Google Scholar 

  18. F. Matús and M. Studený, eds., in: Workshop on Conditional Independence Structures and Graphical Models (Fields Institute for Research in Mathematical Sciences, Toronto, 1999).

  19. S. Moral, Epistemic irrelevance on sets of desirable gambles, in: Second International Symposium on Imprecise Probabilities and Their Applications, eds. G. de Cooman, T.L. Fine and T. Seidenfeld (Shaker, Maastricht, The Netherlands, 2001) pp. 247–254.

    Google Scholar 

  20. N.J. Nilsson, Probabilistic logic, Artificial Intelligence 28 (1986) 71–87.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Parsons. Qualitative Approaches to Reasoning Under Uncertainty (MIT, Cambridge, Massachusetts, 2001).

    Google Scholar 

  22. J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference (Morgan Kaufmann, San Mateo, California, 1988).

    Google Scholar 

  23. J. Pearl, Causality: Models, Reasoning, and Inference (Cambridge University Press, Cambridge, UK, 2000).

    MATH  Google Scholar 

  24. J. Pearl, On two pseudo-paradoxes in Bayesian analysis, Annals of Mathematics and Artificial Intelligence 32 (2001) 171–177.

    Article  MathSciNet  Google Scholar 

  25. T. Seidenfeld and L. Wasserman, Dilation for sets of probabilities, Annals of Statistics 21 (1993) 1139–1154.

    Article  MATH  MathSciNet  Google Scholar 

  26. P. Spirtes, C. Glymour and R. Scheines, Causation, Prediction, and Search, 2nd ed. (MIT, Cambridge, Massachusetts, 2003).

    Google Scholar 

  27. M. Studený, Conditional independence relations have no finite complete characterizations, in: Transactions of the Eleventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, eds. S. Kubik and J.A. Visek (Kluwer, Dordrecht, 1992) pp. 377–396.

    Google Scholar 

  28. M. Studený, Semigraphoids and structures of probabilistic conditional independence, Annals of Mathematics and Artificial Intelligence 21 (1997) 71–98.

    Article  MATH  MathSciNet  Google Scholar 

  29. B. Vantaggi, Graphical models for conditional independence structures, in: Second International Symposium on Imprecise Probabilities and Their Applications, eds. G. de Cooman, T.L. Fine and T. Seidenfeld (Shaker, Maastricht, The Netherlands, 2001) pp. 332–341.

    Google Scholar 

  30. J. Vejnarová. Conditional independence relations in possibility theory, in: First International Symposium on Imprecise Probabilities and Their Applications, eds. G. de Cooman, F.G. Cozman, S. Moral and P. Walley (Imprecise Probabilities Project, Ghent, Belgium, 1999) pp. 343–351.

    Google Scholar 

  31. P. Vicig, Epistemic independence for imprecise probabilities, International Journal of Approximate Reasoning 24 (2000) 235–250.

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  33. P. Walley, Measures of uncertainty in expert systems, Artificial Intelligence 83 (1996) 1–58.

    Article  MathSciNet  Google Scholar 

  34. P. Walley and T.L. Fine, Towards a frequentist theory of upper and lower probability, Annals of Statistics 10 (1982) 741–761.

    Article  MATH  MathSciNet  Google Scholar 

  35. P. Walley, R. Pelessoni and P. Vicig, Direct algorithms for checking consistency and making inferences from conditional probability assessments, Journal of Statistical Planning and Inference 126 (2004) 119–151.

    Article  MATH  MathSciNet  Google Scholar 

  36. M.P. Wellman, Fundamental concepts of qualitative probabilistic networks, Artificial Intelligence 44 (1990) 257–303.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Cidade Universitária, Av. Prof. Mello Moraes, 2231, São Paulo, SP, Brazil

    Fabio G. Cozman

  2. Departamento de Ciencias de la Computación e Inteligencia Artificial, Universidad de Granada, Granada, Spain

    Peter Walley

Authors
  1. Fabio G. Cozman
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  2. Peter Walley
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Correspondence to Fabio G. Cozman.

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Cozman, F.G., Walley, P. Graphoid properties of epistemic irrelevance and independence. Ann Math Artif Intell 45, 173–195 (2005). https://doi.org/10.1007/s10472-005-9004-z

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  • DOI: https://doi.org/10.1007/s10472-005-9004-z

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