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The Unnormalized Dempster’s Rule of Combination: A New Justification from the Least Commitment Principle and Some Extensions

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Abstract

In the Transferable Belief Model, belief functions are usually combined using the unnormalized Dempster’s rule (also called the TBM conjunctive rule). This rule is used because of its intuitive appeal and because it has received formal justifications as opposed to the many other rules of combination that have been proposed in the literature. This article confirms the singularity of the TBM conjunctive rule by presenting a new formal justification based on (1) the canonical decomposition of belief functions, (2) the least commitment principle and (3) the requirement of having the vacuous belief function as neutral element of the combination. A similar result is also presented for the TBM disjunctive rule. Eventually, the existence of infinite families of rules having similar properties as those two rules is pointed out.

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References

  1. Ayoun, A., Smets, Ph.: Data association in multi-target detection using the transferable belief model. Int. J. Intell. Syst. 10(16), 1167–1182 (2001)

    Article  Google Scholar 

  2. Benavoli, A., Chisci, L., Ristic, B., Farina, A., Graziano, A.: Reasoning under uncertainty: From Bayes to valuation based systems. Application to target classification and threat evaluation. Selex Sistemi Integrati, Università degli Studi di Firenze, Italy (2007)

  3. Cattaneo, M.E.G.V.: Combining belief functions issued from dependent sources. In: Bernard, J.M., Seidenfeld, T., Zaffalon, M. (eds.) Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications (ISIPTA’03), Lugano, Switzerland, 2003, pp. 133–147. Carleton Scientific, Waterloo (2003)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Denœux, T.: A neural network classifier based on Dempster-Shafer theory. IEEE Trans. Syst. Man Cybern. A 30(2), 131–150 (2000)

    Article  Google Scholar 

  6. Denœux, T.: Conjunctive and disjunctive combination of belief functions induced by non distinct bodies of evidence. Artif. Intell. 172, 234–264 (2008)

    Article  MATH  Google Scholar 

  7. Denœux, T., Smets, Ph.: Classification using belief functions: the relationship between the case-based and model-based approaches. IEEE Trans. Syst. Man Cybern. B 36(6), 1395–1406 (2006)

    Article  Google Scholar 

  8. Destercke, S., Dubois, D., Chojnacki, E.: Cautious conjunctive merging of belief functions. In: Mellouli, K. (ed.) Proceedings of the 9th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU’07), Hammamet, Tunisia, 2007. Lecture Notes in Computer Science. Lecture Note in Artificial Intelligence, vol. 4724, pp. 332–343. Springer, New York (2007)

    Chapter  Google Scholar 

  9. Dubois, D., Prade, H.: On the unicity of Dempster rule of combination. Int. J. Intell. Syst. 1, 133–142 (1986)

    Article  MATH  Google Scholar 

  10. Dubois, D., Prade, H.: A set-theoretic view of belief functions: logical operations and approximations by fuzzy sets. Int. J. Gen. Syst. 12(3), 193–226 (1986)

    Article  MathSciNet  Google Scholar 

  11. Dubois, D., Prade, H.: Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum, New York (1988)

    MATH  Google Scholar 

  12. Dubois, D., Prade, H., Smets, Ph.: A definition of subjective possibility. Int. J. Approx. Reason. 48(2), 352–364 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dubois, D., Prade, H., Smets, Ph.: New semantics for quantitative possibility theory. In: Benferhat, S., Besnard, Ph. (eds.) Proceedings of the 6th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU’01), Toulouse, France, 2001. Lecture Notes in Computer Science, vol. 2143, pp. 410–421. Springer–Verlag (2001)

  14. Elouedi, Z., Mellouli, K.: Pooling dependent expert opinions using the theory of evidence. In: Proceedings of the Seventh Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’98), Paris, France, 1998. Congrès Internationaux, vol. 1, pp. 32–39. E.D.K., France (1998)

  15. Haenni, R.: Uncover Dempster’s rule where it is hidden. In: Proceedings of the 9th International Conference on Information Fusion, Florence, Italy (2006)

  16. Hájek, P.: Deriving Dempster’s rule. In: Valverde, L., Yager, R.R. (eds.) Uncertainty in Intelligent Systems, pp. 75–83. North-Holland, Amsterdam (1993)

    Google Scholar 

  17. Kallel, A., Le Hégarat-Mascle, S.: Combination of partially non-distinct beliefs: the cautious-adaptive rule. Int. J. Approx. Reason. 50(7), 1000–1021 (2009)

    Article  MATH  Google Scholar 

  18. Klawonn, F., Schwecke, E.: On the axiomatic justification of Dempster’s rule of combination. Int. J. Intell. Syst. 7(5), 469–478 (1992)

    Article  MATH  Google Scholar 

  19. Klawonn, F., Smets, Ph.: The dynamic of belief in the transferable belief model and specialization-generalization matrices. In: Dubois, D., Wellman, M.P., D’Ambrosio, B., Smets, Ph. (eds.) Proceedings of the 8th Conference on Uncertainty in Artificial Intelligence (UAI’92), pp. 130–137. Morgan Kaufmann, San Meteo (1992)

    Google Scholar 

  20. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000)

    MATH  Google Scholar 

  21. Kohlas, J., Shenoy, P.P.: Computation in valuation algebras. In: Gabbay, D.M., Smets, Ph. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management Systems: Algorithms for Uncertainty and Defeasible Reasoning, vol. 5, pp. 5–39. Kluwer, Dordrecht (2000)

    Google Scholar 

  22. Ling, X.N., Rudd, W.G.: Combining opinions from several experts. Appl. Artif. Intell. 3, 439–452 (1989)

    Article  Google Scholar 

  23. Liu, L.: A theory of Gaussian belief functions. Int. J. Approx. Reason. 14(2–3), 95–126 (1996)

    Article  MATH  Google Scholar 

  24. Mercier, D., Cron, G., Denœux, T., Masson, M.-H.: Decision fusion for postal address recognition using belief functions. Expert Syst. Appl. 36(3), 5643–5653 (2009)

    Article  Google Scholar 

  25. Pichon, F., Denœux, T.: A new justification of the unnormalized Dempster’s rule of combination from the least commitment principle. In: Proceedings of the 21th Florida Artificial Intelligence Research Society Int. Conference (FLAIRS’08), Special Track on Uncertain Reasoning, Coconut Grove, Florida, USA, pp. 666–671, 15–17 May 2008

  26. Pichon, F., Denœux, T.: T-norm and uninorm-based combination of belief functions. In: Proceedings of the North American Fuzzy Information Processing Society Int. Conference (NAFIPS’08), New York, USA, 19–22 May 2008

  27. Pichon, F., Denœux, T.: A new singular property of the unnormalized Dempster’s rule among uninorm-based combination rules. In: Magdalena, L., Ojeda-Aciego, M., Verdegay, J.L. (eds.) Proceedings of the 12th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’08), pp. 314–321. Torremolinos (Malaga), Spain, 22–27 June 2008

  28. Quost, B., Denœux, T., Masson, M.-H.: Pairwise classifier combination using belief functions. Pattern Recogn. Lett. 28(5), 644–653 (2007)

    Article  Google Scholar 

  29. Quost, B., Denœux, T., Masson, M.-H.: Adapting a combination rule to non-independent information sources. In: Magdalena, L., Ojeda-Aciego, M., Verdegay, J.L. (eds.) Proceedings of the 12th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’08), pp. 448–455. Torremolinos (Malaga), Spain, 22–27 June 2008

  30. Quost, B., Denœux, T., Masson, M.-H.: Refined classifier combination using belief functions. In: Proceedings of the 11th Int. Conf. on Information Fusion (FUSION’08), pp. 776–782. Cologne, Germany, 30 June–03 July 2008

  31. Ristic, B., Smets, Ph.: Target identification using belief functions and implication rules. IEEE Trans. Aerosp. Electron. Syst. 41(3), 1097–1103 (2005)

    Article  Google Scholar 

  32. Sentz, K., Ferson, S.: Combination of evidence in Dempster-Shafer theory. Technical report, SANDIA Tech. Report, SAND2002-0835 (2002)

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

    MATH  Google Scholar 

  34. Shenoy, P.P.: Conditional independence in valuation-based systems. Int. J. Approx. Reason. 10, 203–234 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  35. Smets, Ph.: Un modèle mathématico-statistique simulant le processus du diagnostic médical. Ph.D. thesis (in French). Université Libre de Bruxelles, Brussels, Belgium (1978)

  36. Smets, Ph.: The combination of evidence in the transferable belief model. IEEE Trans. Pattern Anal. Mach. Intell. 12(5), 447–458 (1990)

    Article  Google Scholar 

  37. Smets, Ph.: Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem. Int. J. Approx. Reason. 9, 1–35 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  38. Smets, Ph.: What is Dempster-Shafer’s model? In: Yager, R.R., Kacprzyk, J., Fedrizzi, M. (eds.) Advances in the Dempster-Shafer Theory of Evidence, pp. 5–34. Wiley, New York (1994)

    Google Scholar 

  39. Smets, Ph.: The canonical decomposition of a weighted belief. In: Proceedings of the 14th Int. Joint Conf. on Artificial Intelligence (IJCAI’95), San Mateo, California, USA, 1995, pp. 1896–1901. Morgan Kaufmann, San Francisco (1995)

    Google Scholar 

  40. Smets, Ph.: The α-junctions: combination operators applicable to belief functions. In: Gabbay, D.M., Kruse, R., Nonnengart, A., Ohlbach, H.J. (eds.) Proceedings of the First International Joint Conference on Qualitative and Quantitative Practical Reasoning (ECSQARU-FAPR’97), Bad Honnef, Germany, 1997. Lecture Notes in Computer Science, vol. 1244, pp. 131–153. Springer, New York (1997)

    Google Scholar 

  41. Smets, Ph.: Analyzing the combination of conflicting belief functions. Inf. Fusion 8(4), 387–412 (2007)

    Article  MathSciNet  Google Scholar 

  42. Smets, Ph.: The transferable belief model for quantified belief representation. In: Gabbay, D.M., Smets, Ph. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 1, pp. 267–301. Kluwer, Dordrecht (1998)

    Google Scholar 

  43. Smets, Ph., Kennes, R.: The transferable belief model. Artif. Intell. 66, 191–243 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  44. Smets, Ph., Kruse, R.: The transferable belief model for belief representation. In: Motro, A., Smets, Ph. (eds.) Uncertainty Management in Information Systems: From Needs to Solution, pp. 343–368. Kluwer, Boston (1996)

    Google Scholar 

  45. Yager, R.R.: An introduction to applications of possibility theory. Hum. Syst. Manag. 3, 246–269 (1983)

    Google Scholar 

  46. Yager, R.R.: The entailment principle for Dempster-Shafer granules. Int. J. Intell. Syst. 1, 247–262 (1986)

    Article  MATH  Google Scholar 

  47. Yager, R.R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets Syst. 80, 111–120 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Thales Research and Technology, Campus Polytechnique, 1 avenue Augustin Fresnel, 91767, Palaiseau Cedex, France

    Frédéric Pichon

  2. UMR CNRS 6599 Heudiasyc, Université de Technologie de Compiègne, BP20529, 60205, Compiègne Cedex, France

    Thierry Denœux

Authors
  1. Frédéric Pichon
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  2. Thierry Denœux
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Correspondence to Frédéric Pichon.

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This paper is an extended and revised version of [25].

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Pichon, F., Denœux, T. The Unnormalized Dempster’s Rule of Combination: A New Justification from the Least Commitment Principle and Some Extensions. J Autom Reasoning 45, 61–87 (2010). https://doi.org/10.1007/s10817-009-9152-7

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  • DOI: https://doi.org/10.1007/s10817-009-9152-7

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