# Article

 Title: Conditional problem for objective probability (English) Author: Kříž, Otakar Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 34 Issue: 1 Year: 1998 Pages: [27]-40 Summary lang: English . Category: math . Summary: Marginal problem (see [Kel]) consists in finding a joint distribution whose marginals are equal to the given less-dimensional distributions. Let’s generalize the problem so that there are given not only less-dimensional distributions but also conditional probabilities. It is necessary to distinguish between objective (Kolmogorov) probability and subjective (de Finetti) approach ([Col,Sco]). In the latter, the coherence problem incorporates both probabilities and conditional probabilities in a unified framework. Different algorithms available for its solution are described e. g. in ([Gil,Col,Vic]). In the context of the former approach, it will be shown that it is possible to split the task into solving the marginal problem independently and to subsequent solving pure “conditional" problem as certain type of optimization. First, an algorithm (Conditional problem) that generates a distribution whose conditional probabilities are equal to the given ones is presented. Due to the multimodality of the criterion function, the algorithm is only heuristical. Due to the computational complexity, it is efficient for small size problems e. g. 5 dichotomical variables. Second, a method is mentioned how to unite marginal and conditional problem to a more general consistency problem for objective probability. Due to computational complexity, both algorithms are effective only for limited number of variables and conditionals. The described approach makes possible to integrate in the solution of the consistency problem additional knowledge contained e. g. in an empirical distribution. (English) Keyword: marginal problem Keyword: algorithm MSC: 60A99 MSC: 65C50 MSC: 68T30 MSC: 68T35 idZBL: Zbl 1274.60013 idMR: MR1619053 . Date available: 2009-09-24T19:13:25Z Last updated: 2015-03-27 Stable URL: http://hdl.handle.net/10338.dmlcz/135183 . Reference: [1] Cheeseman P.: A method of computing generalized Bayesian probability values of expert systems.In: Proceedings of the 6-th Joint Conference on Artificial Intelligence (IJCAI-83), Karlsruhe, pp. 198–202 Reference: [2] Deming W. E., Stephan F. F.: On a least square adjustment of sampled frequency table when the expected marginal totals are known.Ann. Math. Statist. 11 (1940), 427–444 MR 0003527, 10.1214/aoms/1177731829 Reference: [3] Gilio A., Ingrassia S.: Geometrical aspects in checking coherence of probability assessments.In: IPMU’96: Proceedings of the 6th International IPMU Conference (B. Bouchon–Meunier, M. Delgado, J. L. Verdegay, M. A. Vila, R. Yager, eds.), Granada 1996, pp. 55–59 Reference: [4] Coletti G., Scozzafava R.: Characterization of coherent conditional probabilities as a tool for their assessment and extension.Internat. J. Uncertainty, Fuzziness and Knowledge–Based Systems, 4 (1996), 2, 103–127 Zbl 1232.03010, MR 1390898, 10.1142/S021848859600007X Reference: [5] Kellerer H. G.: Verteilungsfunktionen mit gegebenen Marginalverteilungen.Z. Wahrsch. verw. Gebiete 3 (1964), 247–270 Zbl 0126.34003, MR 0175158, 10.1007/BF00534912 Reference: [6] Kříž O.: Invariant moves for constructing extensions of marginals.In: IPMU’94: Proceedings of the 5th International IPMU Conference (B. Bouchon–Meunier, R. Yager, eds.), Paris 1994, pp. 984–989 Reference: [7] Kříž O.: Optimizations on finite–dimensional distributions with fixed marginals.In: WUPES 94: Proceedings of the 3-rd Workshop on Uncertainty Processing (R. Jiroušek, ed.), Třešť 1994, pp. 143–156 Reference: [8] Kříž O.: Marginal problem on finite sets.In: IPMU’96: Proceedings of the 6-th International IPMU Conference (B. Bouchon–Meunier, M. Delgado, J. L. Verdegay, M. A. Vila, R. Yager, eds.), Granada 1996, Vol. II, pp. 763–768 Reference: [9] Kříž O.: Inconsistent marginal problem on finite sets.In: Distributions with Given Marginals and Moment Problems (J. Štěpán, V. Beneš, eds.), Kluwer Academic Publishers, Dordrecht – Boston – London 1997, pp. 235–242 Zbl 0907.60003 Reference: [10] Scozzafava R.: A probabilistic background for the management of uncertainty in Artificial Intelligence.European J. Engineering Education 20 (1995), 3, 353–363 10.1080/03043799508923366 Reference: [11] Vicig P.: An algorithm for imprecise conditional probability assesment in expert systems.In: IPMU’96: Proceedings of the 6-th International IPMU Conference (B. Bouchon–Meunier, M. Delgado, J. L. Verdegay, M. A. Vila, R. Yager, eds.), Granada, 1996, Vol. I, pp. 61–66 .

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