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On possibilistic marginal problem. (English) Zbl 1152.28020
Analogously to the probabilistic marginal problem, a possibilistic marginal problem is formulated in a form generalizing the first attempts from [L. M. de Campos and J. F. Huete, Fuzzy Sets Syst. 103, No. 1, 127–152 (1999; Zbl 0951.68150) and ibid., No. 3, 487–505 (1999; Zbl 0971.68150)]. For a given system of marginal possibility distributions, the existence of their common extension is discussed. If such an extension exists, then also $$\pi_{\min}$$ given by $$\pi_{\min}(x)=\min+K\pi_K(x_K)$$ is shown to be a solution of this possibilistic marginal problem. The author has also shown a necessary condition for the positive solution of possibilistic marginal problem. Note that the set of all solutions is not convex, in general (in constrast to the analogous probabilistic marginal problem), however, it is an upper semilattice, i.e., it is closed under maximization. Triangular norm-based extensions are also discussed, based on a conditional independence requirement. The paper opens several new problems. For example, if there is no solution of the extension problem, in the probabilistic framework there are several approximation techniques, which are still missing in the possibilistic framework.

##### MSC:
 2.8e+11 Fuzzy measure theory
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##### References:
 [1] Campos L. M. de, Huete J. F.: Independence concepts in possibility theory: Part 1. Fuzzy Sets and Systems 103 (1999), 127-152 · Zbl 0951.68150 · doi:10.1016/S0165-0114(97)00160-7 [2] Campos L. M. de, Huete J. F.: Independence concepts in possibility theory: Part 2. Fuzzy Sets and Systems 103 (1999), 487-505 · Zbl 0971.68150 · doi:10.1016/S0165-0114(97)00161-9 [3] Cooman G. de: Possibility theory I - III. Internat. J. Gen. Systems 25 (1997), 291-371 · Zbl 0955.28014 [4] Fonck P.: Conditional independence in possibility theory. Proc. 10th Conference UAI (R. L. de Mantaras and P. Poole, Morgan Kaufman, San Francisco 1994, pp. 221-226 [5] Janssen H., Cooman, G. de, Kerre E. E.: First results for a mathematical theory of possibilistic Markov processes. Proc. IPMU’96, volume III (Information Processing and Management of Uncertainty in Knowledge-Based Systems), Granada 1996, pp. 1425-1431 [6] Jiroušek R.: Composition of probability measures on finite spaces. Proc. 13th Conference UAI (D. Geiger and P. P. Shennoy, Morgan Kaufman, San Francisco 1997, pp. 274-281 [7] Malvestuto F. M.: Existence of extensions and product extensions for discrete probability distributions. Discrete Math. 69 (1988), 61-77 · Zbl 0637.60021 · doi:10.1016/0012-365X(88)90178-1 [8] Perez A.: $$\varepsilon$$-admissible simplification of the dependence structure of a set of random variables. Kybernetika 13 (1977), 439-450 · Zbl 0382.62003 · eudml:28225 [9] Perez A.: A probabilistic approach to the integration of partial knowledge for medical decisionmaking (in Czech). Proc. 1st Czechoslovak Congress of Biomedical Engineering (BMI’83), Mariánské Lázně 1983, pp. 221-226 [10] Vejnarová J.: Composition of possibility measures on finite spaces: Preliminary results. Proc. 7th Internat. Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU’98, Paris 1998, pp. 25-30 [11] Vejnarová J.: Possibilistic independence and operators of composition of possibility measures. Prague Stochastics’98 (M. Hušková, J. Á. Víšek, and P. Lachout, Union of the Czech Mathematicians and Physicists, Prague 1998, pp. 575-580 [12] Vejnarová J.: Conditional independence relations in possibility theory. Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 8 (2000), 253-269 · Zbl 1113.68536 · doi:10.1142/S0218488500000186 [13] Vejnarová J.: Markov properties and factorization of possibility distributions. Ann. Math. Artif. Intell. 35 (2002), 357-377 · Zbl 1014.68155 · doi:10.1023/A:1014568208202 [14] Walley P., Cooman G. de: Coherence rules for defining conditional possibility. Internat. J. Approx. Reason. 21 (1999), 63-104 · Zbl 0957.68115 · doi:10.1016/S0888-613X(99)00007-9
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