Probabilistic Opinion Pooling with Imprecise Probabilities

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 1964; Lehrer and Wagner 1981 ; McConway Journal of the American Statistical Association, 76(374), 410-414, 1981; Bordley Management Science, 28(10), 1137-1148, 1982; Genest et al. The Annals of Statistics, 487-501, 1986; Genest and Zidek Statistical Science, 114-135, 1986; Mongin Journal of Economic Theory, 66(2), 313-351, 1995; Clemen and Winkler Risk Analysis, 19(2), 187-203, 1999; Dietrich and List 2014; Herzberg Theory and Decision, 1-19, 2014). 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, 1985), we also argue that IP models admit of a much better philosophical motivation as a model of rational consensus.