Precision-imprecision equivalence in a broad class of imprecise hierarchical uncertainty models (Q1611810)

The two level hierarchical models are considered. There is a first-order uncertainty model about a phenomenon of an interest. The second-order uncertainty represents the modeler's uncertainty. A subject represents his information using a precise model or an imprecise model. The author shows, that it does not matter, whether we assume, that the underlying ideal first-order model is precise or imprecise (precision-imprecision equivalence). An overview of basic notions of the behavioral theory of imprecise probabilities is given. The author uses the term ``gamble'' and explains the relation to the associated term ``event''. The hierarchical model in terms of lower desirability functions is introduced. The second-order model is studied under assumptions, that the first-order model is precise or imprecise. The behavioral implications of lower desirability functions do not depend on whether the underlying first-order model is assumed to be precise or not. The present model is related to the theory of first-order imprecise probabilities, Bayesian first- and second-order models and the theory of fuzzy probabilities and buying functions.