A Note on Undominated Lower Probabilities

Abstract

Interest in lower probability has largely focussed on lower envelopes and, more particularly, on belief functions. We consider those lower probabilities that do not admit of a dominating probability measure and hence are not lower envelopes. A simple and useful family of such undominated lower probabilities is constructed. We briefly explore the geometry of several important classes of lower probabilities and note that the class of undominated lower probabilities has the dimension of the set of all lower probabilities when these are modeled as vectors. While joint experiments can always be formed from given individual experiments characterized by probability measures, the existence of joint experiments is an open question as regards characterizations by lower probabilities. We constructively show the existence of joint experiments for a wide, but not exhaustive, range of characterizations of the marginal experiments. We also consider extensions of lower probabilities and show that a lower probability (including a measure) on a finite algebra can always be extended to an undominated lower probability on a larger, but still finite algebra. Finally we construct continuous undominated extensions of lower probabilities given on finite algebras.