Empirical Models for the Dempster-Shafer-Theory

  • Mieczysław Alojzy Kłopotek
  • Sławomir Tadeusz Wierzchoń
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 88)


In spite of many useful properties, the Dempster-Shafer Theory of evidence (DST) experienced sharp criticism from many sides. The basic line of criticism is connected with the relationship between the belief function (the basic concept of DST) and frequencies [65,18]. A number of attempts to interpret belief functions in terms of probabilities have failed so far to produce a fully compatible interpretation with DST — see e.g. [34,18,14] etc. As a way out of those difficulties, in the paper we will explain our three model proposals: (1) “the marginally correct approximation”, (2) “the qualitative model”, (3) “the quantitative model”. All of them fit the framework of DST, especially the Dempster rule of combination of evidence that was the hardest point and the point of failure of previously known attempts.


Conditional Independence Decision Table Belief Function Basic Probability Assignment Conditional Belief 
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  1. 1.
    A. Acid, L.M. deCampos, A. Gonzales, R. Molina, N. Perez de la Blanca: Learning with CASTLE. In: Kruse R., Siegel P., eds, Symbolic and Quantitative Approaches to Uncertainty. Lecture Notes In Computer Science 548, Springer-Verlag (1991), 99–106Google Scholar
  2. 2.
    A. Bendjebbour, W. Pieczynski:Traitement du Signal, Vol. 14, No. 5, 1997, pp. 453–464.Google Scholar
  3. 3.
    C. Beeri, R. Fagin, D. Maier, Y. Yannakakis On the desirability of acyclic database schemes,J. Assoc. Comput. Mach., vol. 30, no 3, 1983, 479–513.CrossRefGoogle Scholar
  4. 4.
    Cano J., Delgado M., Moral S.: An axiomatic framework for propagating uncertainty in directed acyclic networks,Int. J. of Approximate Reasoning. 1993: 8, 253–280.CrossRefGoogle Scholar
  5. 5.
    C.K. Chow, C.N. Liu: Approximating discrete probability distributions with dependence trees. IEEE Transactions on Information Theory IT-14(1968), 462–467Google Scholar
  6. 6.
    G.F. Cooper, E. Herskovits: A Bayesian method for the induction of probabilistic networks from data,Machine Learning 9 (1992), 309–347.Google Scholar
  7. 7.
    deKorvin A., Kleyle R., Lea R.: The object recognition problem when features fail to be homogeneous,IJAR 1993: 8, 141–162.Google Scholar
  8. 8.
    C. Delobel: Normalization and hierarchical dependencies in the relational data model,ACM Transactions on Database Systems, vol. 3, no 3, 1978, 201–222.CrossRefGoogle Scholar
  9. 9.
    A.P. Dempster: Upper and lower probabilities induced by a multi-valued mapping.Ann. Math. Stat. 38 (1967), 325–339.CrossRefGoogle Scholar
  10. 10.
    A DromignyBadin, S Rossato, Y.M. Zhu: Radioscopic and ultrasonic data fusion via the evidence theory,Traitement du Signal, Vol. 14, No. 5, 1997, pp. 499–510.Google Scholar
  11. 11.
    Dubois D., Prade H.: Evidence, knowledge and belief functions,International Journal of Approximate Reasoning, 1992: 6, 295–319.CrossRefGoogle Scholar
  12. 12.
    Durham S.D., Smolka J.S., Valtorta M.: Statistical consistency with Dempster’s rule on diagnostic trees having uncertain performance parameters,International Journal of Approximate Reasoning 1992: 6, 67–81.CrossRefGoogle Scholar
  13. 13.
    Dutta, S. K., K. Harrison, and R. P. Srivastava, “The Audit Risk Model Under the Risk of Fraud,” in Applications of Fuzzy Sets and The Theory of Evidence to Accounting II, Vol. 7, edited by P. Siegel, K. Omer, A. Korvin, and A. Zebda, published by Jai Press Inc., 1998, pp. 221–244.Google Scholar
  14. 14.
    R. Fagin, J.Y. Halpern: Uncertainty, belief, and probability.Comput. Intell. 7 (1991), 160–173.CrossRefGoogle Scholar
  15. 15.
    Geiger D., Verma T., Pearl J.: d-Separation: From theorems to algorithms. In:Henrion M., Shachter R.D., Kamal L.N. and Lemmer J.F., eds,Uncertainty in Artificial Intelligence 5. Elsevier Science Publishers B.V. ( North-Holland ), 1990, 139–148.Google Scholar
  16. 16.
    Gordon J., Shortliffe E.H.: The Dempster-Shafer theory of evidence, in: Shafer G., Pearl J.,eds,Readings in Uncertain Reasoning. Morgan Kaufmann Pub.lnc., San Mateo CA, (1990), 529–539.Google Scholar
  17. 17.
    J.W. Grzymala-Busse:Managing Uncertainty in Expert Systems, Kluwer, 1991.Google Scholar
  18. 18.
    J.Y. Halpern, R. Fagin: Two views of belief: belief as generalized probability and belief as evidence.ArtificialIntelligence 54 (1992), 275–317Google Scholar
  19. 19.
    R. Hummel, M. Landy: A statistical viewpoint on the theory of evidence.IEEE Trans.PAMI (1988), 235–247.Google Scholar
  20. 20.
    M. Itoh, T. Inagaki: Evidential theoretic revision of beliefs under a mixed strategy for safety-control of a large-complex plant.Transactions of the Society of Instrumental and Control Engineering, Vol. 34, no. 5, 1998, pp. 415–421Google Scholar
  21. 21.
    Jafray, J.Y. (1989) Coherent bets under partially resolving uncertainty and belief functions. Theory and Decisions, 26, 99–105CrossRefGoogle Scholar
  22. 22.
    M.A. Klopotek: Partial dependency separation–a new concept for expressing dependence/independence relations in causal networks.Demonstratio Mathem, atica. Vol XXXII No 1, 1999, pp. 207–226.Google Scholar
  23. 23.
    M.A. Klopotek: Beliefs in Markov trees - From local computations to local valuation. In: Trappl R., ed., Cybernetics and Systems Research. Vol.1., World Scientific Publishers (1994), pp. 351–358Google Scholar
  24. 24.
    M.A. Klopotek: Learning belief network structure from data under causal insufficiency. In: Bergadano F., DeRaed L., eds, Machine Learning ECML-94. Lecture Notes in Artificial Intelligence 784, Springer-Verlag (1994), 379–382.CrossRefGoogle Scholar
  25. 25.
    M.A. Klopotek: Handling Causal Insufficiency in Dempster-Shafer Belief Networks.Posters of Information Processing and Management of Uncertainty, Paris, 4–8 July 199.4.41–42.Google Scholar
  26. 26.
    M.A. Klopotek: Restricted Causal Inference Algorithm, In:Pehrson B., Simon I., eds,Proc. World Computer Congress of IFIP, Hamburg 28 August–2 September 1994, Vol.1. Elsevier Scientific Publishers ( North-Holland ), Amsterdam (1994), 342–347Google Scholar
  27. 27.
    M.A. Klopotek: On a Deficiency of the FCI Algorithm Learning Bayesian Networks from Data. to appear inDemonstratio Mathernatica. Vol. XXX III, 2000.Google Scholar
  28. 28.
    M.A. Klopotek: Reasoning from Data in the Mathematical Theory of Evidence, In:Proc. Eighth International Symposium On Methodologies For Intelligent Systems (ISMIS’94), Charlotte, North Carolina, USA, October 16–19, 1994.Oak National Laboratory Publishers (1994), 71–84.Google Scholar
  29. 29.
    M.A. Klopotek: Testumgebung für Entwicklung eines Beratungssystems auf der Basis der Mathematischen Theorie der Evidenz,Oster. Zeitschrift für Statistik and Informatik, 199.4 Google Scholar
  30. 30.
    M.A. Klopotek: Interpretation of belief function in Dempster-Shafer Theory.Foundations of Computing 6 Decision Sciences20(1995)4, pp. 287–306.Google Scholar
  31. 31.
    M.A. Klopotek: Identification of belief structure in Dempster-Shafer Theory.Foundations of Computing 6 Decision Sciences21(1996)1, pp. 35–54.Google Scholar
  32. 32.
    M.A. Klopotek:Methods of Identification and Interpretations of Belief Distributions in the Dempster–Shafer Theory(in Polish). Publisher: Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland, 1998, ISBN 83–900820–8–x.Google Scholar
  33. 33.
    M.A. Klopotek, S.T. Wierzchon: An Interpretation for the Conditional Belief Function in The Theory of Evidence. In:: Z.R.as, A. Skowron eds:Foundations of Intelligent Systems. Lecture Notes in Artificial Intelligence 1609, Springer-Verlag, Berlin, 1999, 494–502.Google Scholar
  34. 34.
    H.E. Kyburg Jr: Bayesian and non-Bayesian evidential updating.Artificial Intelligence 31 (1987), 271–293.CrossRefGoogle Scholar
  35. 35.
    Y. Ma, D.C. Wilkins. Induction of uncertain rules and the sociopathicity property in DST. In:Kruse R., Siegel P.,eds, Symbolic and Quantitative Approaches to Uncertainty.Lecture Notes In Computer Science 548, Springer-Verlag (1991), 238–245.Google Scholar
  36. 36.
    Michalewicz M., Wierzchon S.T., Klopotek M.A.: Knowledge Acquisition, RepresentationandManipulation in Decision Support Systems. In:, M. Dabrowski Michalewicz M., Rai Z., eds, Intelligent Information Systems.Proceedings of a Workshop held in August6w, Poland, 7–11 June, ICS PAS (1993), 210–238,.Google Scholar
  37. 37.
    Nguyen H.T.: On random sets and belief functions,J. Math. Anal. Appl. 65 (1978), 539–542.CrossRefGoogle Scholar
  38. 38.
    D Pagac, E.M. Nebot, H Durrant-Whyte: An evidential approach to map-building for autonomous vehicles. IEEE Transactions on Robotics and. Automation, vol. 14, no. 4, 1998, pp. 623–629.CrossRefGoogle Scholar
  39. 39.
    J. Pearl:Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Influence. Morgan and Kaufmann, 1988Google Scholar
  40. 40.
    J. Pearl: Bayesian and Belief-Function formalisms for evidential reasoning:A conceptual analysis. In: Shafer G., Pearl J.,eds,Readings in Uncertain Reasoning. Morgan Kaufmann Pub.Inc., San Mateo CA, (1990), 540–569.Google Scholar
  41. 41.
    J. Pearl, T. Verma: A theory of inferred causation. In:Allen J., Fikes R., Sandewell E., eds,Principles of Knowledge Representation and Reasoning. Proc. of the Second IOnternational Conference, Cambridge, Massachusetts, April 22–25, 1991, San Mateo CA: Morgen Kaufmann (1991), 441–452.Google Scholar
  42. 42.
    G.M. Provan: A logic-based analysis of Dempster-Shafer Theory,International Journal of Approximate Reasoning 4 (1990), 451–495.CrossRefGoogle Scholar
  43. 43.
    Provan G.M.: The validity of Dempster-Shafer belief functions,International Journal of Approximate Reasoning, 1992: 6, 389–399.CrossRefGoogle Scholar
  44. 44.
    Z.W. Ras: Query processing in distributed information systems,Fondamenta Informaticae Journal Special Issue on Logics for Artificial Intelligence, IOS Press, Vol. XV, No. 3 /4, 1991, 381–397Google Scholar
  45. 45.
    Ruspini E.II., Lowrance D.J., Strat T.M.: Understanding evidential reasoning,International Journal of Approximate Reasoning, 1992: 6, 401–424.CrossRefGoogle Scholar
  46. 46.
    E.H. Ruspini: The logical foundation of evidential reasoning, Tech. Note 408, SRI International, Menlo Park, Calif. USA, 1986.Google Scholar
  47. 47.
    G. Rebane, J. Pearl: The recovery of causal poly-trees from statistical data. In:Kanal L.N., Levit T.S., Lemmer J.F., eds,Uncertainty in Artificial Intelligence 3. Elsevier Science Publishers B.V., ( North Holland ) (1989), 175–182Google Scholar
  48. 48.
    M. Singh, M. Valtorta: Construction of bayesian network structures from data: A brief survey and an efficient algorithm.International Journal of Approximate Reasoning, 12 (1995), 111–131CrossRefGoogle Scholar
  49. 49.
    Spirtes P., Glymour C., Scheines R.:Causation,Prediction and Search. Lecture Notes in Statistics 81, Springer-Verlag, 1993.Google Scholar
  50. 50.
    G. Shafer: AMathematical Theory of Evidence. Princeton University Press, Princeton, 1976.Google Scholar
  51. 51.
    G. Shafer: Belief functions: An introduction. In: Shafer G., Pearl J.,eds,Readings in Uncertain Reasoning. Morgan Kaufmann Pub.Inc., San Mateo CA, (1990), 473–482.Google Scholar
  52. 52.
    G. Shafer, R. Srivastava: The Bayesian and Belief-Function Formalisms. A General Prospective for Auditing. In: Shafer G., Pearl J.,eds,Readings in Uncertain Reasoning. Morgan Kaufmann Pub.Inc., San Mateo CA, (1990), 482–521.Google Scholar
  53. 53.
    G. Shafer: Perspectivés on the theory and practice of belief functions.International Journal of Approximate Reasoning 4 (1990), 323–362.CrossRefGoogle Scholar
  54. 54.
    P.P. Shenoy: Conditional independence in valuation based systems.International Journal of Approximate Reasoning109(1994)Google Scholar
  55. 55.
    Shenoy P., Shafer G.: Axioms for probability and belief-function propagation, in: Shachter RD., Levitt T.S., Kanal L.N., Lemmer J.F. (eds):Uncertainty in Artificial Intelligence4, Elsevier Science Publishers B.V. (North Holland), 1990Google Scholar
  56. 56.
    A. Skowron: Boolean reasoning for decision rules generation, J. Komorowski, Z.W.R.as (Eds):Methodologies for Intelligent Systems,7th International Symposium, ISMIS’93, Torndheim, Norway, June 1993, Proceedings, Lecture Notes in Artif.Inte1.689, Springer-Verlag, 1993, 295–305Google Scholar
  57. 57.
    A. Skowron: A synthesis of decision rules: applications of discernibility matrix properties, In:M. Dgbrowski, M. Michalewicz, Z. Rai eds: Intelligent Information Systems, Proceedings of a Workshop held in Augusthw, Poland, 7–11 June, 1993, ISBN 83–900820–2–0, 30–46Google Scholar
  58. 58.
    A. Skowron, J.W. Grzymala-Busse: From rough set theory to evidence theory. In:Yager R.R., Kasprzyk J. and Fedrizzi M., eds,Advances in the Dempster-Shafer Theory of Evidence. J. Wiley, New York (1994), 193–236.Google Scholar
  59. 59.
    Ph. Smets: Resolving misunderstandings about belief functions.International Journal of Approximate Reasoning 6 (1992), 321–344.CrossRefGoogle Scholar
  60. 60.
    Ph. Smets, R. Keines: The tranferable belief model.Artif.Intel. 66 (1994), 191–234.CrossRefGoogle Scholar
  61. 61.
    Smets Ph.: Numerical represenation of uncertainty. In: Gabbay DF.M., Smets Ph. Eds.: Handbook of defeasible reasoning and uncertainty management systems. Vol. 3. Kluwer, Doordrecht, 1998, 265–309.Google Scholar
  62. 62.
    Srivastava, R. P., “Decision Making Under Ambiguity: A Belief-Function Perspective,” Archives of Control Sciences, Vol. 6 (XLII), 1997, No. 1–2, pp. 5–27.Google Scholar
  63. 63.
    M. Studeny: Formal properties of conditional independence in different calculi of AI, Sonderforschungsbericht 343, Diskrete Strukturen in der Mathematik, Universität BielefeldGoogle Scholar
  64. 64.
    A. Vang:SQL and Relational Databases. Microtrend Books, Slawson Communications Inc., 1991Google Scholar
  65. 65.
    L. Wasserman: Comments on Shafer’s “Perspectives on the theory and practice of belief functions”,International Journal of Approximate Reasoning 1992: 6: 367–375.CrossRefGoogle Scholar
  66. 66.
    S.T. Wierzchon: On plausible reasoning, in: M.M.Gupta, T.Yamakawa eds.:Fuzzy Logic in Knowledge Based Systems,Decision and Control, North-Holland Amsterdam, 1988, 133–152.Google Scholar
  67. 67.
    K. Weichselberger, S. Poehlmann, AMethodology for Uncertainty in Knowledge-Based Systems. Lecture Notes in AI 419, Springer Verlag, 1990.Google Scholar
  68. 68.
    S.T. Wierzchon, M.A. Klopotek, Modified Component Valuations in Valuation Based Systems As A Way to Optimize Query Processing,Journal of Intelligent Information Systems 9, 157–180 (1997).CrossRefGoogle Scholar
  69. 69.
    S.T. Wierzchon:Melody repreyentacji i pryetwaryania informacji niepewnej w romach teorii Dempstera-Shafera, monography, Instytut Podstaw Informatyki PAN, 1996.Google Scholar
  70. 70.
    Zarley D.K., Hsia Y, Shafer G.: Evidential reasoning using DELIEF. In:Proc. of the Seventh National conference on Artificial Intelligence (AAAI-88), 1. Minneapolis MN (1988), 205–209.Google Scholar
  71. 71.
    Q. Zhu, E.S. Lee: Dempster-Shafer Approach in Propositional Logic.International Journal of Intelligent Systems 8 (1993), 341–349.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Mieczysław Alojzy Kłopotek
    • 1
    • 3
  • Sławomir Tadeusz Wierzchoń
    • 1
    • 2
  1. 1.Institute of Computer SciencePolish Academy of SciencesWarszawaPoland
  2. 2.Institute of Computer ScienceBiałystok University of TechnologyPoland
  3. 3.Institute of Computer ScienceUniversity of PodlasiePoland

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