On the Symbiosis of Two Concepts of Conditional Interval Probability

This paper argues in favor of the thesis that two different concepts of conditional interval probability are needed, in order to serve the huge variety of tasks conditional probability has in the classical setting of precise probabilities. We compare the commonly used intuitive concept of conditional interval probability with the canonical concept, and see, in particular, that the canonical concept is the appropriate one to generalize the idea of transition kernels to interval probability: only the canonical concept allows reconstruction of the original interval probability from the marginals and conditionals, as well as the powerful formulation of Bayes Theorem.