Two new Bayesian approximations of belief functions based on convex geometry

IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics

Volumn 37, Issue 4, 2007, Pages 993-1008

  Cuzzolin, Fabio ^a  

^a INRIA RHÔNE ALPES   (France)

Author keywords

Bayesian belief functions (b.f.); Commutativity; Geometric approach; Intersection probability; Orthogonal projection; Theory of evidence

Indexed keywords

APPROXIMATION THEORY; BAYESIAN NETWORKS; COMPUTATIONAL GEOMETRY; PROBABILITY; PROBABILITY DENSITY FUNCTION;

BAYESIAN APPROXIMATIONS; BAYESIAN BELIEF FUNCTIONS; CONVEX GEOMETRY; INTERSECTION PROBABILITY;

CYBERNETICS;

ALGORITHM; ARTICLE; ARTIFICIAL INTELLIGENCE; AUTOMATED PATTERN RECOGNITION; BAYES THEOREM; COMPUTER SIMULATION; DECISION SUPPORT SYSTEM; METHODOLOGY; STATISTICAL MODEL;

ALGORITHMS; ARTIFICIAL INTELLIGENCE; BAYES THEOREM; COMPUTER SIMULATION; DECISION SUPPORT TECHNIQUES; MODELS, STATISTICAL; PATTERN RECOGNITION, AUTOMATED;

EID: 34547136700     PISSN: 10834419     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSMCB.2007.895991     Document Type: Article

References (39)

1. G. Shafer, A Mathematical Theory of Evidence. Princeton, NJ: Princeton Univ. Press, 1976.
2. D. Dubois and H. Prade, "Consonant approximations of belief functions," Int. J. Approx. Reason., vol. 4, no. 5/6, pp. 419-449, Sep./Nov. 1990.
3. A. B. Yaghlane, T. Denoeux, and K. Mellouli, "Coarsening approximations of belief functions," in Proc. ECSQARU, S. Benferhat and P. Besnard, Eds., 2001, pp. 362-373.
4. T. Denoeux, "Inner and outer approximation of belief structures using a hierarchical clustering approach," Int. J. Uncertain. Fuzziness Knowl.-Based Syst., vol. 9, no. 4, pp. 437-460, Aug. 2001.
5. T. Denoeux and A. B. Yaghlane, "Approximating the combination of belief functions using the fast Moebius transform in a coarsened frame," Int. J. Approx. Reason., vol. 31, no. 1/2, pp. 77-101, Oct. 2002.
6. R. Haenni and N. Lehmann, "Resource bounded and anytime approximation of belief function computations," Int. J. Approx. Reason., vol. 31, no. 1/2, pp. 103-154, Oct. 2002.
7. M. Bauer, "Approximation algorithms and decision making in the Dempster-Shafer theory of evidence - An empirical study," Int. J. Approx. Reason., vol. 17, no. 2/3, pp. 217-237, Aug.-Oct. 1997.
8. M. Bauer, "Approximations for decision making in the Dempster-Shafer theory of evidence," in Proc. 12th Conf. Uncertainty Artif. Intell. F. Horvitz and E. Jensen, Eds., Portland, OR, Aug. 1-4, 1996, pp. 73-80.
9. B. Tessem, "Approximations for efficient computation in the theory of evidence," Artif. Intell., vol. 61, no. 2, pp. 315-329, Jun. 1993.
10. J. D. Lowrance, T. D. Garvey, and T. M. Strat, "A framework for evidential-reasoning systems," in Proc. Nat. Conf. Artif. Intell. 1986, pp. 896-903.
11. P. Smets, "Belief functions versus probability functions," in Uncertainty and Intelligent Systems, L. Saitta, B. Bouchon, and R. Yager, Eds. Berlin, Germany: Springer-Verlag, 1988, pp. 17-24.
12. P. Smets, "Decision making in the TBM: The necessity of the pignistic transformation," Int. J. Approx. Reason., vol. 38, no. 2, pp. 133-147, Feb. 2005.
13. F. Voorbraak, "A computationally efficient approximation of Dempster-Shafer theory," Int. J. Man-Mach. Stud., vol. 30, no. 5, pp. 525-536, May 1989.
14. B. Cobb and P. Shenoy, "On transforming belief function models to probability models," Univ. Kansas, Sch. Bus., Lawrence, KS, Feb. 2003. Working Paper 293, Tech. Rep.
15. B. R. Cobb and P. P. Shenoy, "A comparison of Bayesian and belief function reasoning," Inf. Syst. Frontiers, vol. 5, no. 4, pp. 345-358, Dec. 2003.
16. B. R. Cobb and P. P. Shenoy, "A comparison of methods for transforming belief function models to probability models," in Proc. ECSQARU, Aalborg, Denmark, Jul. 2003, pp. 255-266.
17. B. Cobb and P. Shenoy, "On the plausibility transformation method for translating belief function models to probability models," Int. J. Approx. Reason., vol. 41, no. 3, pp. 314-330, Apr. 2006.
18. V. Ha and P. Haddawy, "Theoretical foundations for abstraction-based probabilistic planning," in Proc. 12th Conf. Uncertainty Artif. Intell., Aug. 1996, pp. 291-298.
19. P. Black, "Geometric structure of lower probabilities," in Random Sets: Theory and Applications, J. Goutsias, R. P. S. Mahler, and H. T. Nguyen, Eds. New York: Springer-Verlag, 1997, pp. 361-383.
20. P. Black, "An examination of belief functions and other monotone capacities," Ph.D. dissertation, Dept. Statist., Carnegie Mellon Univ., Pittsburg, PA, 1996.
21. V. Ha and P. Haddawy, "Geometric foundations for interval-based probabilities," in KR'98: Principles of Knowledge Representation and Reasoning, A. G. Cohn, L. Schubert, and S. C. Shapiro, Eds. San Francisco, CA: Morgan Kaufmann, 1998, pp. 582-593. [Online]. Available: citeseer.ist.psu.edu/ha98geometric.html
22. F. Cuzzolin and R. Frezza, "Geometric analysis of belief space and conditional subspaces," in Proc. 2nd ISIPTA, Ithaca, NY, Jun. 26-29, 2001, pp. 122-132.
23. F. Cuzzolin, "Geometry of upper probabilities," in Proc. 3rd ISIPTA Jul. 2003, pp. 188-203.
24. F. Cuzzolin, "A geometric approach to the theory of evidence," IEEE Trans. Syst., Man, Cybern. C, Appl. Rev., 2007. to be published.
25. A. P. Dempster, "Upper and lower probability inferences based on a sample from a finite univariate population," Biometrika, vol. 54, no. 3/4, pp. 515-528, Dec. 1967.
26. A. Dempster, "Upper and lower probabilities generated by a random closed interval," Ann. Math. Stat., vol. 39, no. 3, pp. 957-966, Jun. 1968.
27. A. Dempster, "Upper and lower probabilities inferences for families of hypothesis with monotone density ratios," Ann. Math. Stat., vol. 40, no. 3, pp. 953-969, Jun. 1969.
28. F. Cuzzolin, "Visions of a generalized probability theory," Ph.D. dissertation, Università di Padova, Dipartimento di Elettronica e Informatica, Padova, Italy, Feb. 2001.
29. F. Cuzzolin, "Geometry of Dempster's rule of combination," IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34, no. 2, pp. 961-977, Apr. 2004.
30. B. Dubrovin, S. Novikov, and A. Fomenko, Sovremennaja Geometrija. Metody i Prilozenija. Moscow, Russia: Nauka, 1986.
31. M. Aigner, Combinatorial Theory. New York: Springer-Verlag, 1979.
32. A. Chateauneuf and J. Y. Jaffray, "Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion," Math. Soc. Sci., vol. 17, no. 3, pp. 263-283, Jun. 1989.
33. D. Dubois, H. Prade, and P. Smets, "New semantics for quantitative possibility theory," in Proc. ISIPTA, 2001, pp. 152-161.
34. P. Smets, "Constructing the pignistic probability function in a context of uncertainty," in Proc. Uncertainty Artif. Intell., 5, M. Henrion, R. Shachter, L. Kanal, and J. Lemmer, Eds., 1990, pp. 29-39.
35. P. Smets and R. Kennes, "The transferable belief model," Artif. Intell., vol. 66, no. 2, pp. 191-234, 1994.
36. F. Cuzzolin, "The geometry of relative plausibility and belief of singletons," Ann. Math. Artif. Intell., May 2007. submitted for publication.
37. F. Cuzzolin, "The geometry of relative plausibilities," in 11th Int. Conf. IPMU, Special Session Fuzzy Measures and Integrals, Capacities and Games, 2006.
38. P. Smets, "The nature of the unnormalized beliefs encountered in the transferable belief model," in Proc. 8th Annu. Conf. UAI, 1992, pp. 292-297.
39. P. Smets, "The application of the matrix calculus to belief functions," Int. J. Approx. Reason., vol. 31, no. 1/2, pp. 1-30, Oct. 2002.