Validation of imprecise probability models

Validation is the assessment of the match between a model’s predictions and any empirical observations relevant to those predictions. This comparison is straightforward when the data and predictions are deterministic, but is complicated when either or both are expressed in terms of uncertain numbers (i.e., intervals, probability distributions, p-boxes, or more general imprecise probability structures). There are two obvious ways such comparisons might be conceptualized. Validation could measure the discrepancy between the shapes of the uncertain numbers representing prediction and data, or it could characterize the differences between realizations drawn from the respective uncertain numbers. When both prediction and data are represented with probability distributions, comparing shapes would seem to be the most intuitive choice because it sidesteps the issue of stochastic dependence between the prediction and the data values which would accompany a comparison between realizations. However, when prediction and observation are represented as intervals, comparing their shapes seems overly strict as a measure for validation. Intuition demands that the measure of mismatch between two intervals be zero whenever the intervals overlap at all. Thus, intervals are in perfect agreement even though they may have very different shapes. The unification between these two concepts relies on defining the validation measure between prediction and data as the shortest possible distance given the imprecision about the distributions and their dependencies.