Probability boxes (pairs of cumulative distribution functions) are among the most popular models used in imprecise probability theory. 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 lower previsions. We allow p-boxes to be defined on arbitrary totally preordered spaces, hence thereby also admitting multivariate p-boxes. We discuss the construction of multivariate p-boxes under various independence assumptions. An example demonstrates the practical feasibility of our results.
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