Precision–imprecision equivalence in a broad class of imprecise hierarchical uncertainty models

Abstract

Hierarchical models are rather common in uncertainty theory. They arise when there is a ‘correct’ or ‘ideal’ (the so-called first-order) uncertainty model about a phenomenon of interest, but the modeler is uncertain about what it is. The modeler's uncertainty is then called second-order uncertainty. For most of the hierarchical models in the literature, both the first- and the second-order models are precise, i.e., they are based on classical probabilities. In the present paper, I propose a specific hierarchical model that is imprecise at the second level, which means that at this level, lower probabilities are used. No restrictions are imposed on the underlying first-order model: that is allowed to be either precise or imprecise. I argue that this type of hierarchical model generalizes and includes a number of existing uncertainty models, such as imprecise probabilities, Bayesian models, and fuzzy probabilities. The main result of the paper is what I call precision–imprecision equivalence: the implications of the model for decision making and statistical reasoning are the same, whether the underlying first-order model is assumed to be precise or imprecise.