Abstract
In this paper we introduce indeed two alternative formulations of the theory of evidence by proving that both plausibility and commonality functions share the same combinatorial structure of sum function of belief functions, and computing their Moebius inverses called basic plausibility and commonality assignments. The equivalence of the associated formulations of the ToE is mirrored by the geometric congruence of the related simplices. Applications to the probabilistic approximation problem are briefly presented.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Dempster, A.: Upper and lower probabilities generated by a random closed interval. Annals of Mathematical Statistics 39, 957–966 (1968)
Xiong, W., Luo, X., Ju, S.: An analysis of a defect in Dempster-Shafer theory. In: Proceedings of the 10th IEEE International Conference on Fuzzy Systems, pp. 793–796 (2001)
Kohlas, J.: Mathematical foundations of evidence theory. In: Coletti, G., Dubois, D., Scozzafava, R. (eds.) Mathematical Models for Handling Partial Knowledge in Artificial Intelligence, pp. 31–64. Plenum Press (1995)
Shafer, G.: Allocations of probability. Annals of Probability 7(5), 827–839 (1979)
Aigner, M.: Combinatorial Theory. In: Classics in Mathematics. Springer, Heidelberg (1979)
Cuzzolin, F.: A geometric approach to the theory of evidence. IEEE Transactions on Systems, Man and Cybernetics - Part C (2008)
Smets, P.: Decision making in the TBM: the necessity of the pignistic transformation. International Journal of Approximate Reasoning 38(2), 133–147 (2005)
Voorbraak, F.: A computationally efficient approximation of Dempster-Shafer theory. International Journal on Man-Machine Studies 30, 525–536 (1989)
Bauer, M.: Approximation algorithms and decision making in the Dempster-Shafer theory of evidence–an empirical study. International Journal of Approximate Reasoning 17, 217–237 (1997)
Nguyen, H.T.: On random sets and belief functions. J. Mathematical Analysis and Applications 65, 531–542 (1978)
Hestir, H., Nguyen, H., Rogers, G.: A random set formalism for evidential reasoning. In: Conditional Logic in Expert Systems, pp. 309–344. North-Holland, Amsterdam (1991)
Fagin, R., Halpern, J.Y.: Uncertainty, belief and probability. In: Proc. of IJCAI 1988, pp. 1161–1167 (1988)
Cuzzolin, F.: Geometry of upper probabilities. In: Proceedings of ISIPTA 2003 (2003)
Smets, Ph.: The nature of the unnormalized beliefs encountered in the transferable belief model. In: Proceedings of UAI 1992, p. 292. Morgan Kaufmann, San Mateo (1992)
Cobb, B., Shenoy, P.: On the plausibility transformation method for translating belief function models to probability models. International Journal of Approximate Reasoning 41(3), 314–330 (2006)
Cuzzolin, F.: Two new Bayesian approximations of belief functions based on convex geometry. IEEE Transactions on Systems, Man and Cybernetics part B 37(4), 993–1008 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cuzzolin, F. (2008). Alternative Formulations of the Theory of Evidence Based on Basic Plausibility and Commonality Assignments. In: Ho, TB., Zhou, ZH. (eds) PRICAI 2008: Trends in Artificial Intelligence. PRICAI 2008. Lecture Notes in Computer Science(), vol 5351. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89197-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-540-89197-0_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-89196-3
Online ISBN: 978-3-540-89197-0
eBook Packages: Computer ScienceComputer Science (R0)