Coherent restrictions of vague conditional lower-upper probability extensions

In this paper we propose a way to restrict extension bounds induced by coherent conditional lower-upper probability assessments. Such shrinkage turns out to be helpful whenever the natural bounds are too vague to be used. Since coherence of a conditional lower-upper probability assessment can be characterized through a class of conditional probability distributions, the idea is to take the intersection of the extension bounds induced by each single element of the class instead of the convex combination, as it is usually done. Coherence of such method is proved for extensions performed on both conditional events logical dependent and not-dependent on the initial domain.