Uncertainty modelling and conditioning with convex imprecise previsions

Two classes of imprecise previsions, which we termed convex and centered convex previsions, are studied in this paper in a framework close to Walley's and Williams' theory of imprecise previsions. We show that convex previsions are related with a concept of convex natural extension, which is useful in correcting a large class of inconsistent imprecise probability assessments, characterised by a condition of avoiding unbounded sure loss. Convexity further provides a conceptual framework for some uncertainty models and devices, like unnormalised supremum preserving functions. Centered convex previsions are intermediate between coherent previsions and previsions avoiding sure loss, and their not requiring positive homogeneity is a relevant feature for potential applications. We discuss in particular their usage in (financial) risk measurement. In a final part we introduce convex imprecise previsions in a conditional environment and investigate their basic properties, showing how several of the preceding notions may be extended and the way the generalised Bayes rule applies.