Epistemic independence in numerical possibility theory

Numerical possibility measures can be interpreted as systems of upper betting rates for events. As such, they have a special part in the unifying behavioural theory of imprecise probabilities, proposed by Walley. On this interpretation, they should arguably satisfy certain rationality, or consistency, requirements, such as avoiding sure loss and coherence. Using a version of Walley’s notion of epistemic independence suitable for possibility measures, we study in detail what these rationality requirements tell us about the construction of independent product possibility measures from given marginals, and we obtain necessary and sufficient conditions for a product to satisfy these criteria. In particular, we show that the well-known minimum and product rules for forming independent joint distributions from marginal ones, are only coherent when at least one of these distributions assume just the values zero and one.