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Upper Probabilities Attainable by Distributions of Measurable Selections

Upper Probabilities Attainable by Distributions of Measurable Selections,10.1007/978-3-642-02906-6_30,Enrique Miranda,Inés Couso,Pedro Gil

Upper Probabilities Attainable by Distributions of Measurable Selections   (Citations: 5)
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A random set can be regarded as the result of the imprecise observation of a random variable. Following this interpretation, we study to which extent the upper and lower probabilities induced by the random set keep all the information about the values of the probability distribution of the random variable. We link this problem to the existence of selectors of a multi-valued mapping and with the inner approximations of the upper probability, and prove that under fairly general conditions (although not in all cases), the upper and lower probabilities are an adequate tool for modelling the available information. Finally, we study the particular case of consonant random sets and we also derive a relationship between Aumann and Choquet integrals.
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    • ...This is an updated version, with proofs and additional comments, of a paper [42] presented at ECSQARU’09, the Tenth European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty...

    Enrique Mirandaet al. Approximations of upper and lower probabilities by measurable selectio...

    • ...Although working with the upper and lower probabilities leads to a number of mathematical simplications ([20, 21]), the information they provide is in general more imprecise than the one given by the set of distributions of the measurable selections ([16, 18])...
    • ...It was investigated for the case of X nite in [16], and for some particular innite spaces in [3, 6, 10, 15, 18]...
    • ...In order to give conditions for the equality, we must see if the maximum and minimum values of P(Γ)(A) coincide with P (A) and P (A), and also if P(Γ)(A) is convex. This problem was studied in [18]...
    • ...Theorem 1 [18] Consider (Ω; A; P) a probability space, (X; τ) a topological space and Γ : Ω! P(X) a random set...

    Enrique Mirandaet al. Study of the Probabilistic Information of a Random Set

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