Bayes Theorem Bounds for Convex Lower Previsions

In this paper we consider some bounds for lower previsions that are either coherent or, more generally, centered convex. We focus on bounds concerning the classical product and Bayes’ rules, discussing first weak product rules and some of their implications for coherent lower previsions. We then generalise a well-known lower bound, which is a (weak) version for events and coherent lower probabilities of Bayes’ theorem, to the case of random variables and (centered) convex previsions. We obtain a family of bounds and show that one of them is undominated in all cases. Some applications are outlined, and it is shown that 2-monotonicity, which ensures that the bound is sharp in the case of events, plays a much more limited role in this general framework.