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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.

28E10 Fuzzy measure theory
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