Abstract
The basic definitions and propositions of the well-known Campos–Bolanos probability representations (CBR) and Murofushi–Sugeno probability representations (MSR) of several monotone measures are presented. These include Choquet capacities of order two (in general), Dempster–Shafer belief structure, Sugeno λ-measures, and possibility measure. Connections between the CBR and MSR are considered. Analogously to CBR, theorems of existence of Choquet capacities of order two are proved for MSR. A definition of a distance between monotone measures in MSR is introduced, which is similar to the definition of a distance for CBR. Some properties of monotone expectation in relation to MSR are also obtained. On the basis of CBR of a monotone measure, its restoring problem is considered using insufficient expert data. Most typical weighted expected values on insufficient expert data are defined based on the CBR.
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Sirbiladze, G. (2013). Monotone Measure Probability Representations and Weighted Fuzzy Statistics. In: Extremal Fuzzy Dynamic Systems. IFSR International Series on Systems Science and Engineering, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4250-9_2
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