Imprecise probabilities based on generalized intervals for system reliability assessment

Di erent representations of imprecise probabilities have been proposed, where interval-valued probabilities are used such that uncertainty is distinguished from variability. In this paper, we present a new form of imprecise probabilities for reliability assessment based on generalized intervals. Generalized intervals have group properties under the Kaucher arithmetic, which provides a concise representation and calculus structure as an extension of precise probabilities. With the separation between proper and improper interval probabilities, focal and non-focal events are di erentiated based on the associated modalities and logical semantics. Focal events have the semantics of critical, uncontrollable, and speci ed in probabilistic analysis, whereas the corresponding non-focal events are complementary, controllable, and derived. A logic coherence constraint is proposed in the new form. Because of the algebraic properties of generalized intervals, conditional interval probability can be directly de ned based on marginal interval probabilities. A Bayes' rule with generalized intervals allows us to interpret the logic relationship between interval prior and posterior probabilities. The imprecise Dirichlet model is also extended with the logic coherence constraint.