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Quasi-Bayesian analysis using imprecise probability assessments and the generalized Bayes’ rule. (English) Zbl 1087.62040

Summary: The generalized Bayes’ rule (GBR) can be used to conduct ‘quasi-Bayesian’ analyses when prior beliefs are represented by imprecise probability models. We describe a procedure for deriving coherent imprecise probability models when the event space consists of a finite set of mutually exclusive and exhaustive events. The procedure is based on P. Walley’s theory of upper and lower previsions [see ‘Statistical reasoning with imprecise probabilities. (1991; Zbl 0732.62004)] and employs simple linear programming models. We then describe how these models can be updated using F. G. Cozman’s [see Int. J. Approx. Reasoning 24, 191–205 (2000; Zbl 0995.60005)] linear programming formulation of the GBR. Examples are provided to demonstrate how the GBR can be applied in practice. These examples also illustrate the effects of prior imprecision and prior-data conflict on the precision of the posterior probability distribution.

MSC:

62F15 Bayesian inference
90C05 Linear programming
68T37 Reasoning under uncertainty in the context of artificial intelligence
90C90 Applications of mathematical programming
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[7] Breese, J.S. and Fertig, K.W. (1991), Decision making with interval influence diagrams. In Bonissone, P.P., Henrion, M., Kanal, L.N., and Lemmer, J.F. (eds.), Uncertainty in Artificial Intelligence 6, Elsevier Science Publishers, pp. 467–478.
[9] Coletti, G. and Scozzafava, R. (1999), Coherent upper and lower bayesian updating, in G. DeCooman, F.G. Cozman, S. Moral and Walley, P. (eds.), Proceedings of the First International Symposium on Imprecise Probabilities and Their Applications, pp. 101–110, The Imprecise Probabilities Project.
[14] Cozman, F.G. (1999), Computing posterior upper expectations, in G. DeCooman, F.G. Cozman, S. Moral and Walley, P. (eds.), Proceedings of the First International Symposium on Imprecise Probabilities and Their Applications, The Imprecise Probabilities Project, pp. 131–140.
[18] Dickey, J. (2003), Convenient interactive computing for coherent imprecise prevision assessments, in J.M. Bernard, T. Seidenfeld and M.affalon (eds.), Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications,pp. 218–230, Carleton Scientific.
[26] Fortin, V., Parent, E. and Bobée, B. (2001), Posterior previsions for the parameter of a binomial model via natural extension of a finite number of judgments, in G. DeCooman, T. Fine and Seidenfeld, T. (eds.),Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, Shaker Publishing.
[32] Kozine, I. and Krymsky, V. (2003), Reducing uncertainty by imprecise judgments on probability distribution: application to system reliability. in J.M. Bernard, T. Seidenfeld and Zaffalon, M. (eds.), Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications, pp. 335–344, Carleton Scientific.
[34] Lins, G.C.N. and Campello de Souza, F.M. (2001), A protocol for the elicitation of prior distributions, in G. DeCooman, T. Fine and Seidenfeld, T. (eds.), Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, Shaker Publishing.
[39] Schervish, M.J., Seidenfeld, T., Kadane, J.B. and Levi, I. (2003), Extensions of expected utility and some limitations of pairwise comparisons, in J.M. Bernard, T. Seidenfeld and Zaffalon, M. (eds.), Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications, pp. 496–510, Carleton Scientific.
[45] Utkin, L.V. and Kozine I.O. (2001), Computing the reliability of complex systems. in G. DeCooman, T. Fine and Seidenfeld, T. (eds.), Proceedings of the Second Internation Symposium on Imprecise Probabilities and Their Applications, Shaker Publishing.
[50] Walley, P. Pelessoni, R. and Vicig, P. (1999), Direct algorithms for checking coherence and making inferences from confiditional probability assessments. Quad. n. 6/99 del Dipartimento di Matematica Applicata ”Bruno de Finetti”, University of Trieste. · Zbl 1075.62002
[51] White III, C.C. (1986), A posteriori representations based on linear inequality descriptions of a priori and conditional distributions, IEEE Transactions on Systems, Man and Cybernetics, SMC-16, 570–573.
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