Coherence graphs

We consider the task of proving Walley’s (joint or strong) coherence of a number of probabilistic assessments, when these assessments are represented as a collection of conditional lower previsions. In order to maintain generality in the analysis, we assume to be given nearly no information about the numbers that make up the lower previsions in the collection. Under this condition, we investigate the extent to which the above global task can be decomposed into simpler and more local ones. This is done by introducing a graphical representation of the conditional lower previsions, that we call the coherence graph: we show that the coherence graph allows one to isolate some subsets of the collection whose coherence is sufficient for the coherence of all the assessments. The situation is shown to be completely analogous in the case of Walley’s notion of weak coherence, for which we prove in addition that the subsets found are optimal, in the sense that they embody the maximal degree to which the task of checking weak coherence can be decomposed. In doing all of this, we obtain a number of related results: we give a new characterisation of weak coherence; we characterise, by means of a special kind of coherence graph, when the local notion of separate coherence is sufficient for coherence; and we provide an envelope theorem for collections of lower previsions whose graph is of the latter type.