Abstract
Traditional Dempster Shafer belief theory does not provide a simple method for judging the effect of statistical and probabilistic data on belief functions and vice versa. This puts belief theory in isolation from probability theory and hinders fertile cross-disciplinary developments, both from a theoretic and an application point of view. It can be shown that a bijective mapping exists between Dirichlet distributions and Dempster-Shafer belief functions, and the purpose of this paper is to describe this correspondence. This has three main advantages; belief based reasoning can be applied to statistical data, statistical and probabilistic analysis can be applied to belief functions, and it provides a basis for interpreting and visualizing beliefs for the purpose of enhancing human cognition and the usability of belief based reasoning systems.
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References
Dempster, A.P.: Upper and lower probabilities induced by a multiple valued mapping. Annals of Mathematical Statistics 38, 325–339 (1967)
Gelman, A., et al.: Bayesian Data Analysis, 2nd edn. Chapman and Hall/CRC, Florida (2004)
Jøsang, A.: A Logic for Uncertain Probabilities. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 9(3), 279–311 (2001)
Jøsang, A.: The Consensus Operator for Combining Beliefs. Artificial Intelligence Journal 142(1-2), 157–170 (2002)
Jøsang, A.: Probabilistic Logic Under Uncertainty. In: Proceedings of Computing: The Australian Theory Symposium (CATS 2007), CRPIT, Ballarat, Australia, vol. 65 (January 2007)
jøsang, A., Marsh, S.: The Cumulative Rule of Belief Fusion. In: The Proceedings of the International Conference on Information Fusion, Quebec, Canada (July 2007) (to appear)
Pope, S., Jøsang, A.: Analsysis of Competing Hypotheses using Subjective Logic. In: Proceedings of the 10th International Command and Control Research and Technology Symposium (ICCRTS). United States Department of Defense Command and Control Research Program (DoDCCRP) (2005)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Smets, P.: Belief Functions. In: Smets, P., et al. (eds.) Non-Standard Logics for Automated Reasoning, pp. 253–286. Academic Press, London (1988)
Utkin, L.V.: Extensions of belief functions and possibility distributions by using the imprecise Dirichlet model. Fuzzy Sets and Systems 154, 413–431 (2005)
Walley, P.: Inferences from Multinomial Data: Learning about a Bag of Marbles. Journal of the Royal Statistical Society 58(1), 3–57 (1996)
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Jøsang, A., Elouedi, Z. (2007). Interpreting Belief Functions as Dirichlet Distributions. In: Mellouli, K. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2007. Lecture Notes in Computer Science(), vol 4724. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75256-1_36
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DOI: https://doi.org/10.1007/978-3-540-75256-1_36
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