Probabilistic Opinion Pooling with Imprecise Probabilities

Abstract

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, 1964;Lehrer and Wagner, 1981;McConway, 1981;Bordley, 1982;Genest and Zidek, 1986;Mongin, 1995;Clemen and Winkler, 1999;Dietrich and List, 2014;Herzberg, 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 (1985), we also argue that IP models admit of a much better philosophical motivation as a model of rational consensus.

Learning and Pooling, Pooling and Learning

“…In the precise setting, we are out of luck. Again, not so in the IP setting, as our results in conjunction with those of (Stewart and Ojea Quintana, 2017) attest. Philosophical positions that argue for or assume that pooling should be of a particular format are answerable for the limitations of those methods.…”

“…Such results are not the intended contribution of this paper (though Proposition 2 generalizes a characterization result due to Wagner to the imprecise probabilities setting). Instead, we continue an exploration of the potential of imprecise probabilities in the context of learning and pooling (Stewart and Ojea Quintana, 2017). There, we argue that collective opinion is more properly represented by imprecise probabilities (IP).…”

“…We explore which types of probabilistic updating commute with convex IP pooling (Stewart and Ojea Quintana, 2017). Positive results are stated for Bayesian conditionalization (and a mild generalization of it), imaging, and a certain parameterization of Jeffrey conditioning.…”

“…Similar points can be made regarding other properties of particular pooling methods. For example, neither linear nor geometric pooling preserves probabilistic independence in general (Genest and Wagner, 1987), though convex IP pooling preserves Levi's confirmational irrelevance, a generalization of probabilistic independence (Stewart and Ojea Quintana, 2017, Proposition 4). Seidenfeld, Schervish, and Kadane offer a decision-theoretic counterexample to the reasonableness of linear pooling on the basis of its failure to preserve independence (2010).…”

Resolving Peer Disagreements Through Imprecise Probabilities

“…See Stewart and Ojea Quintana () for an excellent review of Bayesian pooling methods and their properties, and Wheeler () for an objection to Williamson's Objective Bayesian approach.…”