Information Fusion and Revision in Qualitative and Quantitative Settings

Steps Towards a Unified Framework
  • Didier Dubois
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6717)

Abstract

Fusion and revision are two key topics in knowledge representation and uncertainty theories. However, various formal axiomatisations of these notions were proposed inside specific settings, like logic, probability theory, possibility theory, kappa functions, belief functions and imprecise probability. For instance, the revision rule in probability theory is Jeffrey’s rule, and is characterized by two axioms. The AGM axioms for revision are stated in the propositional logic setting. But there is no bridge between these axiomatizations. Likewise, Dempster rule of combination was axiomatized by Smets among others, and a logical syntax-independent axiomatization for merging was independently proposed by Koniezny and Pino-Perez, while a belief function can be viewed as a weighted belief set. Moreover the distinction between fusion and revision is not always so clear and comparing sets of postulates for each of them can be enlightening. This paper presents a tentative set of basic principles for revision and another set of principles for fusion that could be valid regardless of whether information is represented qualitatively or quantitatively. In short, while revision obeys a success postulate and a minimal change principle, fusion is essentially symmetric, and obeys a principle of optimism, that tries to take advantage of all sources of information. Moreover, when two pieces of information are consistent, revising one by the other comes down to merging them symmetrically. Finally, there is a principle of minimal commitment at work in all settings, and common to the two operations.

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References

  1. 1.
    Domotor, Z.: Probability kinematics and representation of belief change. Philosophy of Science 47, 284–403 (1980)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: Partial meet functions for contraction and revision. Symbolic Logic 50, 510–530 (1985)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Konieczny, S., Pino-Pérez, R.: On the logic of merging. In: Procs. of KR 1998, pp. 488–498 (1998)Google Scholar
  4. 4.
    Jeffrey, R.: The logic of decision, 2nd edn. Chicago University Press, Chicago (1983)Google Scholar
  5. 5.
    Darwiche, A., Pearl, J.: On the logic of iterated belief revision. Artificial Intelligence 89, 1–29 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Friedman, N., Halpern, J.: Belief revision: A critique. In: Proceedings of KR 1996, pp. 421–631 (1996)Google Scholar
  7. 7.
    Dubois, D.: Three scenarios for the revision of epistemic states. Journal of Logic and Computation 18(5), 721–738 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cooke, R.M.: Experts in Uncertainty. Oxford University Press, Oxford (1991)Google Scholar
  9. 9.
    Genest, C., Zidek, J.: Combining probability distributions: A critique and an annoted bibliography. Statistical Science 1(1), 114–135 (1986)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Benferhat, S., Dubois, D., Kaci, S., Prade, H.: Possibilistic merging and distance-based fusion of propositional information. Annals of Mathematics and Artificial Intelligence 34(1-3), 217–252 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)MATHGoogle Scholar
  12. 12.
    Smets, P.: Analyzing the combination of conflicting belief functions. Information Fusion 8(4), 387–412 (2007)CrossRefGoogle Scholar
  13. 13.
    Maynard-Reid II, P., Lehmann, D.: Representing and aggregating conflicting beliefs. In: Proceedings of KR 2000, pp. 153–164 (2000)Google Scholar
  14. 14.
    Spohn, W.: Ordinal conditional functions: A dynamic theory of epistemic states. Causation in Decision, Belief Change, and Statistics 2, 105–134 (1988)CrossRefGoogle Scholar
  15. 15.
    Halpern, J.Y.: Defining relative likelihood in partially-ordered preferential structures. Journal of A.I. Research 7, 1–24 (1997)MathSciNetMATHGoogle Scholar
  16. 16.
    Lewis, D.: Counterfactuals. Basil Blackwell, U.K (1973)MATHGoogle Scholar
  17. 17.
    Dubois, D., Prade, H.: Possibility theory: qualitative and quantitative aspects. In: Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 1, pp. 169–226. Kluwer, Dordrecht (1998)Google Scholar
  18. 18.
    Walley, P.: Statistical reasoning with imprecise Probabilities. Chapman and Hall, New York (1991)CrossRefMATHGoogle Scholar
  19. 19.
    Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. The Annals of Statistics 28, 325–339 (1967)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Katsuno, H., Mendelzon, A.O.: Propositional knowledge base revision and minimal change. Artificial Intelligence 52, 263–294 (1991)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Gärdenfors, P., Makinson, D.: Nonmonotonic inference based on expectations. Artificial Intelligence 65, 197–245 (1994)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Dubois, D., Prade, H.: Belief change and possibility theory. In: Gärdenfors, P. (ed.) Belief Revision, pp. 142–182. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  23. 23.
    Dubois, D., Prade, H.: Epistemic entrenchment and possibilistic logic. Artificial Intelligence 50, 223–239 (1991)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rescher, N., Manor, R.: On inference from inconsistent premises. Theory and Decision 1, 179–219 (1970)CrossRefMATHGoogle Scholar
  25. 25.
    Benferhat, S., Konieczny, S., Papini, O., Pérez, R.P.: Iterated revision by epistemic states: Axioms, semantics and syntax. In: Proc. of ECAI 2000, pp. 13–17 (2000)Google Scholar
  26. 26.
    Benferhat, S., Dubois, D., Prade, H.: Possibilistic and standard probabilistic semantics of conditional knowledge bases. Journal of Logic and Computation 9, 873–895 (1999)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Papini, O.: Iterated revision operations stemming from the history of an agentÕs observations. In: Rott, H., Williams, M.A. (eds.) Frontiers of Belief Revision, pp. 281–293. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  28. 28.
    Maynard-Reid II, P., Shoham, Y.: Belief fusion: Aggregating pedigreed belief states. Journal of Logic, Language and Information 10(2), 183–209 (2001)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Konieczny, S., Pino Pérez, R.: Merging information under constraints: a qualitative framework. Journal of Logic and Computation 12(5), 773–808 (2002)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Delgrande, J., Dubois, D., Lang, J.: Iterated revision as prioritized merging. In: Proc. of KR 2006, pp. 210–220 (2006)Google Scholar
  31. 31.
    Chopra, S., Ghose, A.K., Meyer, T.A.: Social choice theory, belief merging, and strategy-proofness. Information Fusion 7(1), 61–79 (2006)CrossRefGoogle Scholar
  32. 32.
    Chan, H., Darwiche, A.: On the revision of probabilistic beliefs using uncertain evidence. Artif. Intell. 163(1), 67–90 (2005)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Dubois, D., Lang, J., Prade, H.: Possibilistic logic. In: Gabbay, D., et al. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3, pp. 439–513. Oxford University Press, Oxford (1994)Google Scholar
  34. 34.
    Shackle, G.: Decision Order and Time In Human Affairs. Cambridge University Press, U.K (1961)Google Scholar
  35. 35.
    Benferhat, S., Dubois, D., Prade, H., Williams, M.: A framework for revising belief bases using possibilistic counterparts of Jeffrey’s rule. Fundamenta Informaticae 11, 1–18 (2009)Google Scholar
  36. 36.
    Dubois, D., Prade, H.: A synthetic view of belief revision with uncertain inputs in the framework of possibility theory. Int. J. Approx. Reasoning 17(2-3), 295–324 (1997)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Destercke, S., Dubois, D.: Can the minimum rule of possibility theory be extended to belief functions? In: Sossai, C., Chemello, G. (eds.) ECSQARU 2009. LNCS, vol. 5590, pp. 299–310. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  38. 38.
    Dubois, D., Prade, H.: Representation and combination of uncertainty with belief functions and possibility measures. Computational Intelligence 4, 244–264 (1988)CrossRefGoogle Scholar
  39. 39.
    Ichihashi, H., Tanaka, H.: Jeffrey-like rules of conditioning for the Dempster-Shafer theory of evidence. Int. J. of Approximate Reasoning 3, 143–156 (1989)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Smets, P.: Jeffrey’s rule of conditioning generalized to belief functions. In: Proc. of UAI, pp. 500–505 (1993)Google Scholar
  41. 41.
    Ma, J., Liu, W., Dubois, D., Prade, H.: Revision rules in the theory of evidence. In: Procs. of ICTAI 2010, pp. 295–302. IEEE Press, Los Alamitos (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Didier Dubois
    • 1
  1. 1.IRIT, CNRS and Université de ToulouseFrance

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