Consider the following optimization problem: maximize a bounded real-valued function f-defined on a set X-over all x in X that satisfy the constraint xRY , where Y is a random variable taking values in a set Y and R is a relation on X × Y. The aim of our active research is to reduce this problem to a (constrained) optimization problem from which the uncertainties present in the description of the constraint are eliminated. We investigate what results can be obtained for different types of uncertainty models for the random variable Y-linear previsions, vacuous previsions, possibility distributions, p-boxes, etc. [see, e.g., 1, 2, 4]-and for two different optimality criteria-maximinity and maximality [see, e.g., 3]. We work with general X , Y , and R for the combinations of uncertainty model and optimality criterion that allow it and restrict our attention to more concrete situations otherwise. In our poster, we will present the problem statement, give illustrated solutions for the most interesting cases we have investigated, as well as discuss the strengths and weaknesses of our approach. In the remainder of this abstract, we sketch this approach and give the solutions of two of the cases we have already fully investigated. The first thing we do is reformulate the optimization problem as a decision problem. To wit, with every x in X we associate a utility function G x on Y that gives the constant value f (x) for y in Y such that xRy and that gives a penalty-value L otherwise (i.e., when x Ry). As the name suggest, it penalizes the fact that x such that x Ry are considered; we therefore choose L < inf f. The uncertainty model associated to Y is formulated as a coherent lower prevision P (and its conjugate upper prevision P) for a sufficiently rich set of gambles on Y [see, e.g., 4, for terminology]. Using the maximinity criterion, the optimal values for X are those that maximize P(G x). Using the more conservative maximality criterion, the optimal values for X are those for which min z∈X P(G x − G z) ≥ 0. Both finding the prevision on the sufficiently rich set of gambles {G x : x ∈ X } or {G x − G z : x, z ∈ X } and checking the optimality criteria are, in general, nontrivial steps. For the general case, when Y is described by a linear prevision P, we find that under both maximinity and maximality the optimal x are those that maximize the function on X that takes the value (f (x) − L)P({y ∈ Y : xRy}) in x. For the concrete case X = Y := R and R :=≤, and when Y is described by a triangular possibility distribution with basis [a, b] ⊂ R and top c ∈ (a, b), we find that under maximinity the optimal x are those that maximize the function on (−∞, c] that takes the value f (x) for x ≤ a and x−a c−a L + c−x c−a f (x) for x in (a, c].
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A pair of lower and upper cumulative distribution functions, also called probability box or p-box, is among the most popular models used in imprecise probability theory. They arise naturally in expert elicitation, for instance in cases where bounds are specified on the quantiles of a random variable, or when quantiles are specified only at a finite number of points. Many practical and formal results concerning p-boxes already exist in the literature. In this paper, we provide new efficient tools to construct multivariate p-boxes and develop algorithms to draw inferences from them. For this purpose, we formalise and extend the theory of p-boxes using Walley's behavioural theory of imprecise probabilities, and heavily rely on its notion of natural extension and existing results about independence modeling. In particular, we allow p-boxes to be defined on arbitrary totally preordered spaces, hence thereby also admitting multivariate p-boxes via probability bounds over any collection of nested sets. We focus on the cases of independence (using the factorization property), and of unknown dependence (using the Fréchet bounds), and we show that our approach extends the probabilistic arithmetic of Williamson and Downs. Two design problems-a damped oscillator, and a river dikedemonstrate the practical feasibility of our results.
We study n-monotone functionals, which constitute a generalisation of n-monotone set functions. We investigate their relation to the concepts of exactness and natural extension, which generalise the notions of coherence and natural extension in the behavioural theory of imprecise probabilities. We improve upon a number of results in the literature, and prove among other things a representation result for exact n-monotone functionals in terms of Choquet integrals
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