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Local computation with valuations from a commutative semigroup

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Abstract

This paper studies a variant of axioms originally developed by Shafer and Shenoy (Shafer and Shenoy, 1988). It is investigated which extra assumptions are needed to perform the local computations in a HUGIN-like architecture (Jensen et al., 1990) or in the architecture of Lauritzen and Spiegelhalter (Lauritzen and Spiegelhalter, 1988). In particular it is shown that propagation of belief functions can be performed in these architectures.

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References

  1. R. Almond, Graphical Belief Modelling (Chapman and Hall, London, 1995).

    Google Scholar 

  2. S.K. Andersen, K.G. Olesen, F.V. Jensen and F. Jensen, HUGIN — A shell for building Bayesian belief universes for expert systems, in: Proceedings of the 11th International Joint Conference on Artificial Intelligence (Morgan-Kaufmann, San Mateo, 1990) pp. 1080–1085. Also reprinted in [19].

    Google Scholar 

  3. A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups (American Mathematical Society, Providence, RI, 1961).

    MATH  Google Scholar 

  4. R.G. Cowell and A.P. Dawid, Fast retraction of evidence in a probabilistic expert system, Statistics and Computing 2 (1992) 37–40.

    Article  Google Scholar 

  5. A.P. Dawid, Applications of a general propagation algorithm for probabilistic expert systems, Statistics and Computing 2 (1992) 25–36.

    Article  Google Scholar 

  6. P. Hájek, T. Havránek and R. Jiroušek, Uncertain Information Processing in Expert Systems (CRC Press, Boca Raton, 1992).

    Google Scholar 

  7. E. Hewitt and H.S. Zuckermann, The l 1-algebra of a commutative semigroup, Transactions of the American Mathematical Society 83 (1956) 70–97.

    Article  MATH  MathSciNet  Google Scholar 

  8. F. Jensen, F.V. Jensen and S.L. Dittmer, From influence diagrams to junction trees, in: Proceedings of the 10th Conference on Uncertainty in Artificial Intelligence, eds. R.L. de Mantaras and D. Poole, (Morgan-Kaufmann, San Mateo, 1994) pp. 367–373.

    Google Scholar 

  9. F.V. Jensen, An Introduction to Bayesian Networks (University College London Press, London, 1996).

    Google Scholar 

  10. F.V. Jensen, S.L. Lauritzen and K.G. Olesen, Bayesian updating in causal probabilistic networks by local computation, Computational Statistics Quarterly 4 (1990) 269–282.

    MATH  MathSciNet  Google Scholar 

  11. A. Kong, Multivarate belief functions and graphical models, PhD Thesis, Harvard University, Department of Statistics (1986).

  12. S.L. Lauritzen, Propagation of probabilities, means and variances in mixed graphical association models, Journal of the American Statistical Association 86 (1992) 1098–1108.

    Article  MathSciNet  Google Scholar 

  13. S.L. Lauritzen and D.J. Spiegelhalter, Local computations with probabilities on graphical structures and their application to expert systems (with discussion), Journal of the Royal Statistical Society, Series B 50 (1988) 157–224.

    MATH  MathSciNet  Google Scholar 

  14. E. Neapolitan, Probabilistic Reasoning in Expert Systems (John Wiley and Sons, New York, 1990).

    Google Scholar 

  15. R.M. Oliver and J.Q. Smith, Influence Diagrams, Belief Nets and Decision Analysis (John Wiley and Sons, Chichester, 1990).

    Google Scholar 

  16. G. Shafer, A Mathematical Theory of Evidence (Princeton University Press, Princeton, NJ, 1976).

    MATH  Google Scholar 

  17. G. Shafer, An axiomatic study of computation in hypertrees, Technical Report WP-232, School of Business, University of Kansas (1991).

  18. G. Shafer, Probabilistic Expert Systems (Society for Industrial and Applied Mathematics, Philadelphia, 1996).

    MATH  Google Scholar 

  19. G. Shafer and J. Pearl, eds., Readings in Uncertain Reasoning (Morgan-Kaufmann, San Mateo, 1990).

    MATH  Google Scholar 

  20. G. Shafer and P.P. Shenoy, Local computation in hypertrees, Technical Report WP-201, School of Business, University of Kansas (1988).

  21. P.P. Shenoy, Valuation-based systems for Bayesian decision analysis, Operations Research 40(3) (1992) 463–484.

    Article  MATH  MathSciNet  Google Scholar 

  22. P.P. Shenoy and G. Shafer, Axioms for probability and belief-function propagation, in: Uncertainty in Artificial Intelligence IV, eds. R.D. Shachter, T.S. Levitt, L.N. Kanal and J.F. Lemmer (North-Holland, Amsterdam, 1990) pp. 169–198.

    Google Scholar 

  23. D.J. Spiegelhalter, A.P. Dawid, S.L. Lauritzen and R.G. Cowell, Bayesian analysis in expert systems (with discussion), Statistical Science 8 (1993) 219–283.

    MATH  MathSciNet  Google Scholar 

  24. H.M. Thoma, Factorization of belief functions, PhD Thesis, Harvard University, Department of Statistics (1989).

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Lauritzen, S., Jensen, F. Local computation with valuations from a commutative semigroup. Annals of Mathematics and Artificial Intelligence 21, 51–69 (1997). https://doi.org/10.1023/A:1018953016172

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