## Abstract

How should a group with different opinions (but the same values) make decisions? In a Bayesian setting, the natural question is how to *aggregate credences*: how to use a single credence function to naturally represent a collection of different credence functions. An extension of the standard Dutch-book arguments that apply to individual decision-makers recommends that group credences should be updated by conditionalization. This imposes a constraint on what aggregation rules can be like. Taking conditionalization as a basic constraint, we gather lessons from the established work on credence aggregation, and extend this work with two new impossibility results. We then explore contrasting features of two kinds of rules that satisfy the constraints we articulate: one kind uses fixed prior credences, and the other uses geometric averaging, as opposed to arithmetic averaging. We also prove a new characterisation result for geometric averaging. Finally we consider applications to neighboring philosophical issues, including the epistemology of disagreement.

## Keywords

Credence aggregation Formal epistemology Social epistemology Conditionalization Disagreement## References

- Arrow, K. J. (1970).
*Social choice and individual values*(2nd ed.). New Haven, CT: Yale University Press.Google Scholar - Christensen, D. (2007). Epistemology of disagreement: The good news.
*Philosophical Review*,*116*(2), 187–217. http://www.jstor.org/stable/20446955. - Elga, A. (2007). Reflection and disagreement.
*Noûs*,*41*(3), 478–502. http://www.jstor.org/stable/4494542. - Elga, A. (2010). Subjective probabilities should be sharp.
*Philosophers’ Imprint*,*10*(05). http://hdl.handle.net/2027/spo.3521354.0010.005. - Field, H. (1978). A note on Jeffrey conditionalization.
*Philosophy of Science*,*45*(3), 361–367. http://www.jstor.org/stable/187023. - Fitelson, B., & Jehle, D. (2009). What is the ‘equal weight view’?
*Episteme*,*6*(3), 280–293. doi: 10.3366/E1742360009000719.CrossRefGoogle Scholar - Genest, C. (1984). A characterization theorem for externally Bayesian groups.
*The Annals of Statistics 12*(3), 1100–1105. http://www.jstor.org/stable/2240984. - Genest, C., & Wagner, C. G. (1987). Further evidence against independence preservation in expert judgement synthesis.
*Aequationes Mathematicae 32*(1), 74–86. doi: 10.1007/BF02311302. - Genest, C., & Zidek, J. V. (1986). Combining probability distributions: A critique and an annotated bibliography.
*Statistical Science 1*(1), 114–135. http://www.jstor.org/stable/2245510. - Genest, C., McConway, K. J., & Schervish, M. J. (1986). Characterization of externally Bayesian pooling operators.
*The Annals of Statistics 14*(2), 487–501. http://www.jstor.org/stable/2241231. - Gilboa, I., Samet, D., & Schmeidler, D. (2004). Utilitarian aggregation of beliefs and tastes.
*Journal of Political Economy 112*(4), 932–938. doi: 10.1086/42117310.1086/421173. - Halmos, P. R. (1950).
*Measure theory*. Princeton: D. Van Nostrand.CrossRefGoogle Scholar - Jeffrey, R. C. (1983a).
*The logic of decision*. Chicago: University of Chicago Press.Google Scholar - Jeffrey, R. C. (1983b). Bayesianism with a human face. In
*Testing scientific theories*. Minneapolis, MN: University of Minnesota Press.Google Scholar - Kelly, T. (2010). Peer disagreement and higher order evidence. In
*Social epistemology: Essential readings*. Oxford: Oxford University Press.Google Scholar - Lackey, J. (2008). What should we do when we disagree? In
*Oxford studies in epistemology*. Oxford: Oxford University Press.Google Scholar - Lehrer, K., & Wagner, C. G. (1983). Probability amalgamation and the independence issue: A reply to Laddaga.
*Synthese 55*(3), 339–346. doi: 10.1007/BF00485827. - Levi, I. (1980).
*The enterprise of knowledge: An essay on knowledge, credal probability, and chance*. Cambridge, MA: The MIT Press.Google Scholar - List, C., & Pettit, P. (2002). Aggregating sets of judgments: An impossibility result.
*Economics and Philosophy*,*18*(1), 89–110. http://eprints.lse.ac.uk/704/. - Loewer, B., & Laddaga, R. (1985). Destroying the consensus.
*Synthese*,*62*(1), 79–95. http://www.jstor.org/stable/20116085.CrossRefGoogle Scholar - McConway, K. J. (1981). Marginalization and linear opinion pools.
*Journal of the American Statistical Association 76*(374), 410–414. doi: 10.2307/2287843. - Mongin, P. (1995). Consistent Bayesian aggregation.
*Journal of Economic Theory 66*(2), 313–351. doi: 10.1006/jeth.1995.1044. - Moss, S. (2011). Scoring rules and epistemic compromise.
*Mind*,*120*(480), 1053–1069. http://www.jstor.org/stable/41494776 CrossRefGoogle Scholar - Pruss, A. (2012). Aggregating data from agents with the same evidence.
*Alexander Pruss’s Blog*. http://alexanderpruss.blogspot.co.uk/2012/03/aggregating-data-from-agents-with-same.html. - Wagner, C. G. (1982). Allocation, Lehrer models, and the consensus of probabilities.
*Theory and Decision 14*(2), 207–220. doi: 10.1007/BF00133978. - Wagner, C. G. (2002). Probability kinematics and commutativity.
*Philosophy of Science 69*(2), 266–278. http://www.jstor.org/stable/10.1086/341053. - Wagner, C. G. (2010a). Jeffrey conditioning and external Bayesianity.
*Logic Journal of IGPL 18*(2), 336–345. http://jigpal.oxfordjournals.org/content/18/2/336.short. - Wagner, C. G. (2010b). Peer disagreement and independence preservation.
*Erkenntnis 74*(2), 277–288. doi: 10.1007/s10670-010-9256-9. - Wilson, A. (2010). Disagreement, equal weight and commutativity.
*Philosophical Studies*,*149*(3), 321–326. http://www.jstor.org/stable/40783268.