Probabilistic Opinion Pooling with Imprecise Probabilities

Abstract

The question of how the probabilistic opinions of different individuals should be aggregated to form a group opinion is controversial. But one assumption seems to be pretty much common ground: for a group of Bayesians, the representation of group opinion should itself be a unique probability distribution (Madansky [44]; Lehrer and Wagner [34]; McConway Journal of the American Statistical Association, 76(374), 410–414, [45]; Bordley Management Science, 28(10), 1137–1148, [5]; Genest et al. The Annals of Statistics, 487–501, [21]; Genest and Zidek Statistical Science, 114–135, [23]; Mongin Journal of Economic Theory, 66(2), 313–351, [46]; Clemen and Winkler Risk Analysis, 19(2), 187–203, [7]; Dietrich and List [14]; Herzberg Theory and Decision, 1–19, [28]). We argue that this assumption is not always in order. We show how to extend the canonical mathematical framework for pooling to cover pooling with imprecise probabilities (IP) by employing set-valued pooling functions and generalizing common pooling axioms accordingly. As a proof of concept, we then show that one IP construction satisfies a number of central pooling axioms that are not jointly satisfied by any of the standard pooling recipes on pain of triviality. Following Levi (Synthese, 62(1), 3–11, [39]), we also argue that IP models admit of a much better philosophical motivation as a model of rational consensus.