Stochastic Independence for Upper and Lower Probabilities in a Coherent Setting

Abstract

In this paper we extend to upper and lower probabilities our approach to independence, based solely on conditional probability (in a coherent setting). Most difficulties arise (when this notion is put forward in the classical framework) either from the introduction of marginals for upper and lower probabilities (often improperly called, in the relevant literature, “imprecise” probabilities) when trying to extend to them the “product rule”, or from the different ways of introducing conditioning for upper and lower probabilities. Our approach to conditioning in the context of “imprecise” probabilities is instead the most natural: in fact its starting point refers to a direct definition (through coherence) of the “enveloping” conditional “precise” probabilities. The discussion of some critical examples seems to suggest that the intuitive aspects of independence are better captured by referring to just one (precise) probability than to a family (such as that one singling-out a lower or upper probability).