Abstract
When combining belief functions by conjunctive rules of combination, conflicts often appear, which are assigned to empty set by the non-normalised conjunctive rule or normalised by Dempster’s rule of combination in Dempster-Shafer theory. Combination of conflicting belief functions and interpretation of their conflicts is often questionable in real applications; hence a series of alternative combination rules were suggested and a series of papers on conflicting belief functions have been published and conflicts of belief functions started to be investigated.
This theoretical contribution introduces a new definition of conflict between two belief functions on a general finite frame of discernment. Its idea is based on Hájek-Valdés algebraic analysis of belief functions, on our previous study of conflicts of belief functions, where internal conflicts of belief functions are distinguished from a conflict between belief functions, and on the decomposition of a belief function into its conflicting and non-conflicting parts. Basic properties of this newly defined conflict are presented, analyzed and briefly compared with our previous approaches to conflict as well as with Liu’s degree of conflict.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Almond, R.G.: Graphical Belief Modeling. Chapman & Hall, London (1995)
Ayoun, A., Smets, P.: Data association in multi-target detection using the transferable belief model. International Journal of Intelligent Systems 16(10), 1167–1182 (2001)
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.: Lp consonant approximation of belief functions. IEEE Transactions on Fuzzy Systems 22(2), 420–436 (2014)
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.: Distribution of Contradictive Belief Masses in Combination of Belief Functions. In: Bouchon-Meunier, B., Yager, R.R., Zadeh, L.A. (eds.) Information, Uncertainty and Fusion, pp. 431–446. Kluwer Academic Publishers, Boston (2000)
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: Proceedings of the 7th ISIPTA, pp. 149–158. Studia Universitätsverlag, Innsbruck (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.: 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.: An Interpretation of Conflicting Parts of Belief Functions on Two-Element Frame of Discrement. In: Kratochvíl, V., Vejnarová, J. (eds.) Proceedings of the 16th Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty (CJS 2013), pp. 187–196. University of Economics (2013)
Daniel, M.: Conflicts of Belief Functions: about a New Measure Based on their Non-Conflicting Parts. Technical report V-1205, ICS AS CR, Prague (2014)
Daniel, M., Ma, J.: Conflicts of Belief Functions: Continuity and Frame Resizement. In: Straccia, U., Cali, A. (eds.) SUM 2014. LNCS (LNAI), vol. 8720, pp. 106–119. Springer, Heidelberg (2014)
Destercke, S., Burger, T.: Toward an axiomatic definition of conflict between belief functions. IEEE Transactions on Cybernetics 43(2), 585–596 (2013)
Dubois, D., Prade, H.: Representation an combination of uncertainty with belief functions and possibility measures. Computational Intelligence 4, 244–264 (1988)
Dubois, D., Prade, H.: Consonant Approximations of Belief Functions. International Journal of Approximate Reasoning 4, 419–649 (1990)
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 syst (dempsteroids). Computers and Artificial Intelligence 10(1), 29–42 (1991)
Lefèvre, E., Elouedi, Z.: How to preserve the conflict as an alarm in the combination of belief functions? Decision Support Systems 56(1), 326–333 (2013)
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)
Roquel, A., Le Hégarat-Mascle, S., Bloch, I., Vincke, B.: Decomposition of conflict as a distribution on hypotheses in the framework on belief functions. International Journal of Approximate Reasoning 55(5), 1129–1146 (2014)
Schubert, J.: The Internal Conflict of a Belief Function. In: Denœux, T., Masson, M.-H. (eds.) Belief Functions: Theory & Appl. AISC, vol. 164, pp. 169–177. Springer, Heidelberg (2012)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Smets, P.: Analyzing the combination of conflicting belief functions. Information Fusion 8, 387–412 (2007)
Valdés, J.J.: Algebraic and logical foundations of uncertainty processing in rule-based expert systems of Artificial Intelligence. PhD Thesis, ČSAV, Prague (1987)
Yager, R.R.: On the Demspter-Shafer framework and new combination rules. Information Sciences 41, 93–138 (1987)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Daniel, M. (2014). Conflict between Belief Functions: A New Measure Based on Their Non-conflicting Parts. In: Cuzzolin, F. (eds) Belief Functions: Theory and Applications. BELIEF 2014. Lecture Notes in Computer Science(), vol 8764. Springer, Cham. https://doi.org/10.1007/978-3-319-11191-9_35
Download citation
DOI: https://doi.org/10.1007/978-3-319-11191-9_35
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11190-2
Online ISBN: 978-3-319-11191-9
eBook Packages: Computer ScienceComputer Science (R0)