Some theoretical properties of interval-valued conditional probability assessments

In this paper we consider interval-valued conditional probability assessments on finite families of conditional events. Based on the coherence principle of de Finetti, we give some preliminary results on precise and imprecise probability assessments, by recalling the properties of avoiding uniform loss (AUL), which coincides with the notion of g-coherence, and of coherence introduced by Walley. Among other results, we generalize to interval-valued assessments a connection property, obtained in a previous paper, for the set of precise coherent assessments on a finite family F of conditional events. More specifically, we prove that, with any pair of AUL interval-valued assessments X' ,X'' on F, we can associate an infinite class of AUL interval-valued imprecise assessments which are convex combination between X' and X'' and connect them. Then, we examine the extension of g-coherent imprecise assessments. We also give a result on totally coherent imprecise assessments, by examining its relationship with a necessary and sufficient condition of total coherence for interval-valued assessments.