Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T06:33:00.279Z Has data issue: false hasContentIssue false
  • English
  • Français

Can Coherent Predictions be Contradictory?

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  17 March 2021

Krzysztof Burdzy  and
Soumik Pal
Krzysztof Burdzy*
Affiliation:
University of Washington
Soumik Pal*
Affiliation:
University of Washington
*
*Postal address: Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195.
*Postal address: Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195.
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the sharp bound for the probability that two experts who have access to different information, represented by different $\sigma$-fields, will give radically different estimates of the probability of an event. This is relevant when one combines predictions from various experts in a common probability space to obtain an aggregated forecast. The optimizer for the bound is explicitly described. This paper was originally titled ‘Contradictory predictions’.

Keywords

Conditional probabilityopinionjoint distribution of conditional expectations

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 1 , March 2021 , pp. 133 - 161
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Burdzy, K. (2009). The Search for Certainty. On the Clash of Science and Philosophy of Probability. World Scientific, Hackensack, NJ.CrossRefGoogle Scholar
Burdzy, K. (2016). Resonance—from Probability to Epistemology and Back. Imperial College Press, London.CrossRefGoogle Scholar
Burdzy, K. and Pitman, J. (2020). Bounds on the probability of radically different opinions. Electron. Commun. Prob. 25, 14.CrossRefGoogle Scholar
Casarin, R., Mantoan, G. and Ravazzolo, F. (2016). Bayesian calibration of generalized pools of predictive distributions. Econometrics 4, 17.CrossRefGoogle Scholar
Cichomski, S. (2020). Maximal spread of coherent distributions: a geometric and combinatorial perspective. Preprint. Available at https://arxiv.org/abs/2007.08022.Google Scholar
Dawid, A. P., DeGroot, M. H. and Mortera, J. (1995). Coherent combination of experts’ opinions. Test 4, 263–313.CrossRefGoogle Scholar
Dawid, A. P. and Mortera, J. (2018). A note on prediction markets. Preprint. Available at .Google Scholar
DeGroot, M. H. (1988). A Bayesian view of assessing uncertainty and comparing expert opinion. J. Statist. Planning Infer. 20, 295–306.CrossRefGoogle Scholar
DeGroot, M. H. and Mortera, J. (1991). Optimal linear opinion pools. Manag. Sci. 37, 546–558.CrossRefGoogle Scholar
Dubins, L. E. and Pitman, J. (1980). A divergent, two-parameter, bounded martingale. Proc. Amer. Math. Soc. 78, 414–416.CrossRefGoogle Scholar
Easwaran, K., Fenton-Glynn, L., Hitchcock, C. and Velasco, J. D. (2016). Updating on the credences of others: disagreement, agreement, and synergy. Philosophers’ Imprint 16, 1–39.Google Scholar
French, S. (2011). Aggregating expert judgement. RACSAM Rev. R. Acad. A 105, 181–206.Google Scholar
Genest, C. (1984). A characterization theorem for externally Bayesian groups. Ann. Statist. 12, 1100–1105.CrossRefGoogle Scholar
Gneiting, T. and Ranjan, R. (2013). Combining predictive distributions. Electron. J. Statist. 7, 1747–1782.CrossRefGoogle Scholar
Kapetanios, G., Mitchell, J., Price, S. and Fawcett, N. (2015). Generalised density forecast combinations. J. Econometrics 188, 150–165.CrossRefGoogle Scholar
Kchia, Y. and Protter, P. (2015). Progressive filtration expansions via a process, with applications to insider trading. Internat. J. Theoret. Appl. Finance 18, 1550027.CrossRefGoogle Scholar
Krüger, F. and Nolte, I. (2016). Disagreement versus uncertainty: evidence from distribution forecasts. J. Banking Finance 72, S172–S186.CrossRefGoogle Scholar
Lorenz, J., Rauhut, H., Schweitzer, F. and Helbing, D. (2011). How social influence can undermine the wisdom of crowd effect. Proc. Nat. Acad. Sci. USA 108, 9020–9025.CrossRefGoogle Scholar
Möller, A. and Gross, J. (2016). Probabilistic temperature forecasting based on an ensemble autoregressive modification. Quart. J. R. Meteor. Soc. 142, 1385–1394.CrossRefGoogle Scholar
Moral-Benito, E. (2015). Model averaging in economics: an overview. J. Econom. Surveys 29, 46–75.CrossRefGoogle Scholar
Ranjan, R. and Gneiting, T. (2010). Combining probability forecasts. J. R. Statist. Soc. B [Statist. Methodology] 72, 71–91.CrossRefGoogle Scholar
Satopää, V. A., Pemantle, R. and Ungar, L. H. (2016). Modeling probability forecasts via information diversity. J. Amer. Statist. Assoc. 111, 1623–1633.CrossRefGoogle Scholar
Taylor, J. W. and Jeon, J. (2018). Probabilistic forecasting of wave height for offshore wind turbine maintenance. Europ. J. Operat. Res. 267, 877–890.CrossRefGoogle Scholar