Abstract
Compositional models were initially described for discrete probability theory, and later extended for possibility theory, and Dempster-Shafer (D-S) theory of evidence. Valuation-based systems (VBS) can be considered as a generic uncertainty framework that has many uncertainty calculi, such as probability theory, a version of possibility theory where combination is the product t-norm, Spohn’s epistemic belief theory, and D-S belief function theory, as special cases. In this paper, we describe compositional models for the VBS framework using the semantics of no-double counting. We show that the compositional model defined here for belief functions differs from the one studied by Jiroušek, Vejnarová, and Daniel. The latter model can be described in the VBS framework, but with a combination operation that is different from Dempster’s rule.
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References
Ben-Yaghlane, B., Smets, P., Mellouli, K.: Belief function independence: II. The conditional case. International Journal of Approximate Reasoning 31(1-2), 31–75 (2002)
Couso, I., Moral, S., Walley, P.: Examples of independence for imprecise probabilities. In: Proceedings of the First International Symposium on Imprecise Probabilities and Their Applications, ISIPTA 1999 (1999)
Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics 38, 325–339 (1967)
Jiroušek, R.: Composition of probability measures on finite spaces. In: Geiger, D., Shenoy, P.P. (eds.) Proceedings of the 13th Conference on Uncertainty in Artificial Intelligence (UAI 1997), pp. 274–281. Morgan Kaufmann, San Francisco (1997)
Jiroušek, R.: Foundations of compositional model theory. International Journal of General Systems 40(6), 623–678 (2011)
Jiroušek, R., Vejnarová, J., Daniel, M.: Compositional models of belief functions. In: de Cooman, G., Vejnarová, J., Zaffalon, M. (eds.) Proceedings of the 5th Symposium on Imprecise Probabilities and Their Applications (ISIPTA 2007), Prague, Czech Republic, pp. 243–252. Charles University Press (2007)
Jiroušek, R., Shenoy, P.P.: Compositional models in valuation-based systems. Working Paper No. 325, School of Business, University of Kansas, Lawrence, KS (2011)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Shenoy, P.P.: On Spohn’s rule for revision of beliefs. International Journal of Approximate Reasoning 5(2), 149–181 (1991)
Shenoy, P.P.: Conditional independence in valuation-based systems. International Journal of Approximate Reasoning 10(3), 203–234 (1994)
Shenoy, P.P.: Binary join trees for computing marginals in the Shenoy-Shafer architecture. International Journal of Approximate Reasoning 17(2-3), 239–263 (1997)
Shenoy, P.P.: No double counting semantics for conditional independence. In: Cozman, F.G., Nau, R., Seidenfeld, T. (eds.) Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications (ISIPTA 2005), pp. 306–314. Society for Imprecise Probabilities and Their Applications (2005)
Srivastava, R.P., Cogger, K.: Beliefs on Individual Variables from a Single Source to Beliefs on the Joint Space under Dempster-Shafer Theory: An Algorithm. In: Filipe, J., Fred, A., Sharp, B. (eds.) Proceedings of the First International Conference on Agents and Artificial Intelligence, ICAART 2009, pp. 191–197. INSTICC Press, Porto (2009)
Vejnarová, J.: Composition of possibility measures on finite spaces: Preliminary results. In: Bouchon-Meunier, B., Yager, R.R. (eds.) Proceedings of the 7th International Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems (IPMU 1998), pp. 25–30 (1998)
Walley, P., Fine, T.L.: Towards a frequentist theory of upper and lower probability. Annals of Statistics 10, 741–761 (1982)
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Jiroušek, R., Shenoy, P.P. (2012). Compositional Models in Valuation-Based Systems. In: Denoeux, T., Masson, MH. (eds) Belief Functions: Theory and Applications. Advances in Intelligent and Soft Computing, vol 164. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29461-7_26
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DOI: https://doi.org/10.1007/978-3-642-29461-7_26
Publisher Name: Springer, Berlin, Heidelberg
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