Abstract
In this contribution we generalize belief functions to many-valued events represented by elements of the finite product of standard MV-algebras. Our definition is based on the mass assignment approach from Dempster-Shafer theory of evidence. The generalized belief function is totally monotone and it has Choquet integral representation w.r.t. a classical belief function.
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References
Butnariu, D., Klement, E.P.: Triangular Norm Based Measures and Games with Fuzzy Coalitions. Kluwer Academic Publishers, Dordrecht (1993)
Choquet, G.: Theory of capacities. Ann. Inst. Fourier, Grenoble 5, 131–295 (1953–1954) (1955)
Cignoli, R.L.O., D’Ottaviano, I.M.L., Mundici, D.: Algebraic foundations of many-valued reasoning. Trends in Logic—Studia Logica Library, vol. 7. Kluwer Academic Publishers, Dordrecht (2000)
Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Statist. 38, 325–339 (1967)
Denneberg, D.: Non-additive measure and integral, Theory and Decision Library. Series B: Mathematical and Statistical Methods, vol. 27. Kluwer Academic Publishers, Dordrecht (1994)
Fedel, M., Keimel, K., Montagna, F., Roth, W.: Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic (submitted for publication, 2010)
Flaminio, T., Godo, L., Marchioni, E.: On the logical formalization of possibilistic counterparts of states over n-valued Łukasiewicz events. J. Logic Comput. (2010), doi:10.1093/logcom/exp012
Kroupa, T.: Every state on semisimple MV-algebra is integral. Fuzzy Sets Syst. 157(20), 2771–2782 (2006)
Kroupa, T.: Belief functions on formulas in Łukasiewicz logic. In: Kroupa, T., Vejnarova, J. (eds.) Proceedings of the 8th Workshop on Uncertainty Processing, WUPES 2009, Liblice, Czech Republic (2009)
Mundici, D.: Averaging the truth-value in Łukasiewicz logic. Studia Logica 55(1), 113–127 (1995)
Mundici, D.: Bookmaking over infinite-valued events. Internat. J. Approx. Reason. 43(3), 223–240 (2006)
Navara, M.: Triangular norms and measures of fuzzy sets. In: Klement, E., Mesiar, R. (eds.) Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, pp. 345–390. Elsevier, Amsterdam (2005)
Panti, G.: Invariant measures in free MV-algebras. Comm. Algebra 36(8), 2849–2861 (2008)
Riečan, B., Mundici, D.: Probability on MV-algebras. In: Handbook of Measure Theory, vol. I, II, pp. 869–909. North-Holland, Amsterdam (2002)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Shafer, G.: Allocations of probability. Ann. Probab. 7(5), 827–839 (1979)
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Kroupa, T. (2010). From Probabilities to Belief Functions on MV-Algebras. In: Borgelt, C., et al. Combining Soft Computing and Statistical Methods in Data Analysis. Advances in Intelligent and Soft Computing, vol 77. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14746-3_48
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DOI: https://doi.org/10.1007/978-3-642-14746-3_48
Publisher Name: Springer, Berlin, Heidelberg
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