# Probabilistic Opinion Pooling with Imprecise Probabilities

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## 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.

## Keywords

Aggregation Consensus Imprecise probabilities Pooling Social epistemology## Notes

### Acknowledgments

Several people gave us very helpful feedback on the ideas in this paper. Thanks to Robby Finley and Yang Liu for their comments on a presentation given to the Formal Philosophy reading group at Columbia University, and to members of the audiences at the Columbia Graduate Student Workshop, the Probability and Belief Workshop organized by Hans Rott at the University of Regensburg, and a presentation at CUNY organized by Rohit Parikh. We are grateful to Arthur Heller, Michael Nielsen, Teddy Seidenfeld, Reuben Stern, and Mark Swails for their excellent comments on drafts of the paper. We would like to especially thank Isaac Levi for extensive discussion of the content of this paper and comments on drafts and a presentation. Finally, thanks to both the editor and the anonymous referee. The referee provided engaged and thorough feedback that has undoubtedly improved the essay in a number of ways.

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