On the fiber bundle structure of the space of belief functions. (English) Zbl 1311.60010
Author’s abstract: The study of finite non-additive measures or “belief functions” has been recently posed in connection with combinatorics and convex geometry. As a matter of fact, as belief functions are completely specified by the associated belief values on the events of the frame on which they are defined, they can be represented as points of a Cartesian space. The space of all belief functions \(\mathcal{B}\) or “belief space” is a simplex whose vertices are BF focused on single events. In this paper, we present an alternative description of the space of belief functions in terms of differential geometric notions. The belief space possesses indeed a recursive bundle structure inherently related to the mass assignment mechanism, in which basic probability is recursively assigned to events of increasing size. A formal proof of the decomposition of \(\mathcal{B}\) together with a characterization of its bases and fibers as simplices are provided.
Reviewer: Hiroaki Shimomura (Tokyo)
MSC:
| 60A99 | Foundations of probability theory |
| 52B99 | Polytopes and polyhedra |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 03E72 | Theory of fuzzy sets, etc. |
| 55R10 | Fiber bundles in algebraic topology |
References:
| [1] | Aigner, M.: Combinatorial Theory. Springer-Verlag, Berlin-New York (1979) · Zbl 0415.05001 |
| [2] | Black, P.K.: Geometric structure of lower probabilities. In: Goutsias, J., Mahler, R.P.S., Nguyen, H.T. (eds.), Random Sets, pp. 361-383. Springer, New York (1997) · Zbl 0898.60029 |
| [3] | Cuzzolin F.: Two new Bayesian approximations of belief functions based on convex geometry. IEEE Trans. Syst., Man Cybern. B, Cybern. 37(4), 993-1008 (2007) · doi:10.1109/TSMCB.2007.895991 |
| [4] | Cuzzolin F.: A geometric approach to the theory of evidence. IEEE Trans. Syst., Man, Cybern. C, Appl. Rev. 38(4), 522-534 (2008) · doi:10.1109/TSMCC.2008.919174 |
| [5] | Cuzzolin F.: On the relative belief transform. Internat. J. Approx. Reason. 53(5), 786-804 (2012) · Zbl 1246.68229 · doi:10.1016/j.ijar.2011.12.009 |
| [6] | Cuzzolin F.: The geometry of consonant belief functions: Simplicial complexes of necessity measures. Fuzzy Sets and Systems 161(10), 1459-1479 (2010) · Zbl 1207.68385 · doi:10.1016/j.fss.2009.09.024 |
| [7] | Cuzzolin F.: Geometry of relative plausibility and relative belief of singletons. Ann. Math. Artif. Intell. 59(1), 47-79 (2010) · Zbl 1205.62001 · doi:10.1007/s10472-010-9186-x |
| [8] | Cuzzolin F.: Geometry of Dempster’s rule of combination. IEEE Trans. Syst., Man Cybern. B, Cybern. 34(2), 961-977 (2004) · doi:10.1109/TSMCB.2003.818431 |
| [9] | Dempster A.P.: Upper and lower probabilities induced by a multivariate mapping. Ann. Math. Stat. 38, 325-339 (1967) · Zbl 0168.17501 · doi:10.1214/aoms/1177698950 |
| [10] | Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Sovremennaja Geometrija: Metody i Prilozenija. Nauka, Moscow (1986) · Zbl 1205.62001 |
| [11] | Fagin, R., Halpern, J.Y.: Uncertainty, belief, and probability. In: Proceeding of the 9th International Joint Conference in AI (IJCAI-89), pp. 1161-1167. Morgan Kaufmann Publishers Inc., San Francisco (1989) · Zbl 0718.68066 |
| [12] | Gould, H.W.: Combinatorial Identities. Morgantown, W.Va. (1972) · Zbl 0263.05013 |
| [13] | Ha, V., Haddawy, P.: Theoretical foundations for abstraction-based probabilistic planning. In: Horvitz, E., Jensen, F. (eds.) Uncertainty in Artificial Intelligence, pp. 291-298. Morgan Kaufmann, San Francisco, CA (1996) · Zbl 0807.68087 |
| [14] | Hestir, H.T., Nguyen, H.T., Rogers, G.S.: A random set formalism for evidential reasoning. In: Goodman, I.R. et al. (eds.) Conditional Logic in Expert Systems, pp. 309-344. North Holland, Amsterdam (1991) |
| [15] | Jøsang, A., Pope, S.: Normalising the consensus operator for belief fusion. In: Proceedings of the International Conference on Information Processing and Management of Uncertainty (IPMU 2006). Paris (2006) |
| [16] | Miranda, P., Grabisch, M., Gil, P.: On some results of the set of dominating k-additive belief. In: Proceedings of the 10th International Conference on Information Processing and Management of Uncertainty (IPMU’04), pp. 625-632. Perugia (2004) |
| [17] | Nguyen H.T.: On random sets and belief functions. J. Math. Anal. Appl. 65(3), 531-542 (1978) · Zbl 0409.60016 · doi:10.1016/0022-247X(78)90161-0 |
| [18] | Ruspini, E.H.: Epistemic logics, probability and the calculus of evidence. In: Proceedings of the Tenth International Conference on Artificial Intelligence (IJCAI-87), pp. 924-931. Morgan Kaufmann, San Mateo (1987) |
| [19] | Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976) · Zbl 0359.62002 |
| [20] | Smets P., Kennes R.: The transferable belief model. Artificial Intelligence 66(2), 191-234 (1994) · Zbl 0807.68087 · doi:10.1016/0004-3702(94)90026-4 |
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