# Article

 Title: On possibilistic marginal problem (English) Author: Vejnarová, Jiřina Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 43 Issue: 5 Year: 2007 Pages: 657-674 Summary lang: English . Category: math . Summary: A possibilistic marginal problem is introduced in a way analogous to probabilistic framework, to address the question of whether or not a common extension exists for a given set of marginal distributions. Similarities and differences between possibilistic and probabilistic marginal problems will be demonstrated, concerning necessary condition and sets of all solutions. The operators of composition will be recalled and we will show how to use them for finding a $T$-product extension. Finally, a necessary and sufficient condition for the existence of a solution will be presented. (English) Keyword: marginal problem Keyword: possibility distributions Keyword: triangular norm Keyword: conditioning Keyword: conditional independence Keyword: extension MSC: 28E10 idZBL: Zbl 1152.28020 idMR: MR2376330 . Date available: 2009-09-24T20:28:05Z Last updated: 2012-06-06 Stable URL: http://hdl.handle.net/10338.dmlcz/135805 . Reference: [1] Campos L. M. de, Huete J. F.: Independence concepts in possibility theory: Part 1.Fuzzy Sets and Systems 103 (1999), 127–152 MR 1674018 Reference: [2] Campos L. M. de, Huete J. F.: Independence concepts in possibility theory: Part 2.Fuzzy Sets and Systems 103 (1999), 487–505 MR 1669261 Reference: [3] Cooman G. de: Possibility theory I – III.Internat. J. Gen. Systems 25 (1997), 291–371 MR 1449007 Reference: [4] Fonck P.: Conditional independence in possibility theory.In: Proc. 10th Conference UAI (R. L. de Mantaras and P. Poole, eds.), Morgan Kaufman, San Francisco 1994, pp. 221–226 Reference: [5] Janssen H., Cooman, G. de, Kerre E. E.: First results for a mathematical theory of possibilistic Markov processes.In: Proc. IPMU’96, volume III (Information Processing and Management of Uncertainty in Knowledge-Based Systems), Granada 1996, pp. 1425–1431 Reference: [6] Jiroušek R.: Composition of probability measures on finite spaces.In: Proc. 13th Conference UAI (D. Geiger and P. P. Shennoy, eds.), Morgan Kaufman, San Francisco 1997, pp. 274–281 Reference: [7] Malvestuto F. M.: Existence of extensions and product extensions for discrete probability distributions.Discrete Math. 69 (1988), 61–77 Zbl 0637.60021, MR 0935028 Reference: [8] Perez A.: $\varepsilon$-admissible simplification of the dependence structure of a set of random variables.Kybernetika 13 (1977), 439–450 MR 0472224 Reference: [9] Perez A.: A probabilistic approach to the integration of partial knowledge for medical decisionmaking (in Czech).In: Proc. 1st Czechoslovak Congress of Biomedical Engineering (BMI’83), Mariánské Lázně 1983, pp. 221–226 Reference: [10] Vejnarová J.: Composition of possibility measures on finite spaces: Preliminary results.In: Proc. 7th Internat. Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU’98, Paris 1998, pp. 25–30 Reference: [11] Vejnarová J.: Possibilistic independence and operators of composition of possibility measures.In: Prague Stochastics’98 (M. Hušková, J. Á. Víšek, and P. Lachout, eds.), Union of the Czech Mathematicians and Physicists, Prague 1998, pp. 575–580 Reference: [12] Vejnarová J.: Conditional independence relations in possibility theory.Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 8 (2000), 253–269 Zbl 1113.68536, MR 1770487 Reference: [13] Vejnarová J.: Markov properties and factorization of possibility distributions.Ann. Math. Artif. Intell. 35 (2002), 357–377 Zbl 1014.68155, MR 1899959 Reference: [14] Walley P., Cooman G. de: Coherence rules for defining conditional possibility.Internat. J. Approx. Reason. 21 (1999), 63–104 MR 1693207 .

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