Abstract
Belief functions usually contain some internal conflict. Based on Hájek-Valdés algebraic analysis of belief functions, a unique decomposition of a belief function into its conflicting and non-conflicting part was introduced at ISIPTA’11 symposium for belief functions defined on a two-element frame of discernment.
This contribution studies the conditions under which such a decomposition exists for belief functions defined on a three-element frame. A generalisation of important Hájek-Valdés homomorphism f of semigroup of belief functions onto its subsemigroup of indecisive belief functions is found and presented. A class of quasi-Bayesian belief functions, for which the decomposition into conflicting and non-conflicting parts exists is specified. A series of other steps towards a conflicting part of a belief function are presented. Several open problems from algebra of belief functions which are related to the investigated topic and are necessary for general solution of the issue of decomposition are formulated.
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Cobb, B.R., Shenoy, P.P.: A Comparison of Methods for Transforming Belief Function Models to Probability Models. In: Nielsen, T.D., Zhang, N.L. (eds.) ECSQARU 2003. LNCS (LNAI), vol. 2711, pp. 255–266. Springer, Heidelberg (2003)
Cuzzolin, F.: The geometry of consonant belief functions: Simplicial complexes of necessity measures. Fuzzy Sets and Systems 161(10), 1459–1479 (2010)
Daniel, M.: Algebraic structures related to Dempster-Shafer theory. In: Bouchon-Meunier, B., Yager, R.R., Zadeh, L.A. (eds.) IPMU 1994. LNCS, vol. 945, pp. 51–61. Springer, Heidelberg (1995)
Daniel, M.: Probabilistic Transformations of Belief Functions. In: Godo, L. (ed.) ECSQARU 2005. LNCS (LNAI), vol. 3571, pp. 539–551. Springer, Heidelberg (2005)
Daniel, M.: Conflicts within and between Belief Functions. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. LNCS (LNAI), vol. 6178, pp. 696–705. Springer, Heidelberg (2010)
Daniel, M.: Non-conflicting and Conflicting Parts of Belief Functions. In: Coolen, F., de Cooman, G., Fetz, T., Oberguggenberger, M. (eds.) ISIPTA 2011, pp. 149–158, Studia Universitätsverlag, Innsbruck (2011)
Daniel, M.: Morphisms of Dempster’s Semigroup: A Rev. and Interpret. In: Barták, R. (ed.) Proceedings of 14th Czech-Japan Seminar on Data Analysis and Decision Making Under Uncertainty, CJS 2011, pp. 26–34. Matfyzpress, Prague (2011)
Daniel, M.: Introduction to an Algebra of Belief Functions on Three-Element Frame of Discernment — A Quasi Bayesian Case. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012, Part III. CCIS, vol. 299, pp. 532–542. Springer, Heidelberg (2012)
Daniel, M.: Introduction to Algebra of Belief Functions on Three-element Frame of Discernment - A General Case. In: Kroupa, T., Vejnarová, J. (eds.) WUPES 2012. Proceedings of the 9th Workshop on Uncertainty Processing, pp. 46–57. University of Economics, Prague (2012)
Daniel, M.: Properties of Plausibility Conflict of Belief Functions. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2013, Part I. LNCS (LNAI), vol. 7894, pp. 235–246. Springer, Heidelberg (2013)
Daniel, M.: Belief Functions: A Revision of Plausibility Conflict and Pignistic Conflict. In: Liu, W., Subrahmanian, V.S., Wijsen, J. (eds.) SUM 2013. LNCS (LNAI), vol. 8078, pp. 190–203. Springer, Heidelberg (2013)
Daniel, M.: Steps Towards a Conflicting Part of a Belief Function. Technical report V-1179, ICS AS CR, Prague (2013)
Destercke, S., Burger, T.: Revisiting the Notion of Conflicting Belief Functions. In: Denœux, T., Masson, M.-H. (eds.) Belief Functions: Theory & Appl. AISC, vol. 164, pp. 153–160. Springer, Heidelberg (2012)
Hájek, P., Havránek, T., Jiroušek, R.: Uncertain Information Processing in Expert Systems. CRC Press, Boca Raton (1992)
Hájek, P., Valdés, J.J.: Generalized algebraic foundations of uncertainty processing in rule-based expert systems (dempsteroids). Computers and Artificial Intelligence 10(1), 29–42 (1991)
Lefèvre, É., Elouedi, Z., Mercier, D.: Towards an Alarm for Opposition Conflict in a Conjunctive Combination of Belief Functions. In: Liu, W. (ed.) ECSQARU 2011. LNCS (LNAI), vol. 6717, pp. 314–325. Springer, Heidelberg (2011)
Liu, W.: Analysing the degree of conflict among belief functions. Artificial Intelligence 170, 909–924 (2006)
Martin, A.: About Conflict in the Theory of Belief Functions. In: Denœux, T., Masson, M.-H. (eds.) Belief Functions: Theory & Appl. AISC, vol. 164, pp. 161–168. Springer, Heidelberg (2012)
Quaeghebeur, E., de Cooman, G.: Extreme lower probabilities. Fuzzy Sets and Systems 159(16), 2163–2175 (1990)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Smets, P.: The combination of evidence in the transferable belief model. IEEE-Pattern Analysis and Machine Intelligence 12, 447–458 (1990)
Smets, P.: Analyzing the combination of conflicting belief functions. Information Fusion 8, 387–412 (2007)
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Daniel, M. (2014). Towards a Conflicting Part of a Belief Function. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2014. Communications in Computer and Information Science, vol 444. Springer, Cham. https://doi.org/10.1007/978-3-319-08852-5_22
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DOI: https://doi.org/10.1007/978-3-319-08852-5_22
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