IDEAS home Printed from
   My bibliography  Save this article

The Consensus of Subjective Probability Distributions


  • Robert L. Winkler

    (Indiana University)


"`But we can't agree whether A or B is correct,' he concluded, `and so we're collecting expert opinions, weighting them appropriately, and programming WESCAC to arbitrate the whole question.'" (John Barth, Giles Goat-Boy, p. 664.) In the Bayesian framework, quantified judgments about uncertainty are an indispensable input to methods of statistical inference and decision. If a decision maker has little knowledge with regard to the parameters of interest, he may decide to consult a number of experts and obtain their quantified judgments in the form of subjective probability distributions. If this is the case, the decision maker must somehow combine the distributions assessed by the experts and form a single distribution to be used as an input to a formal Bayesian analysis. Several methods for combining the distributions are suggested, some involving mathematical formulae and some involving feedback and/or group discussion. These methods are compared under certain assumptions regarding the form of the distributions and also under experimental conditions.

Suggested Citation

  • Robert L. Winkler, 1968. "The Consensus of Subjective Probability Distributions," Management Science, INFORMS, vol. 15(2), pages 61-75, October.
  • Handle: RePEc:inm:ormnsc:v:15:y:1968:i:2:p:b61-b75

    Download full text from publisher

    File URL:
    Download Restriction: no


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. François Bachoc & Emile Contal & Hassan Maatouk & Didier Rullière, 2017. "Gaussian processes for computer experiments," Post-Print hal-01665936, HAL.
    2. John W. Boudreau, 2004. "50th Anniversary Article: Organizational Behavior, Strategy, Performance, and Design in Management Science," Management Science, INFORMS, vol. 50(11), pages 1463-1476, November.
    3. Hurley, W. J. & Lior, D. U., 2002. "Combining expert judgment: On the performance of trimmed mean vote aggregation procedures in the presence of strategic voting," European Journal of Operational Research, Elsevier, vol. 140(1), pages 142-147, July.
    4. Tommaso Proietti & Martyna Marczak & Gianluigi Mazzi, 2017. "Euromind‐ D : A Density Estimate of Monthly Gross Domestic Product for the Euro Area," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 32(3), pages 683-703, April.
    5. Cho, Sungbin, 2009. "A linear Bayesian stochastic approximation to update project duration estimates," European Journal of Operational Research, Elsevier, vol. 196(2), pages 585-593, July.
    6. repec:eee:reensy:v:170:y:2018:i:c:p:146-158 is not listed on IDEAS
    7. Palfrey, Thomas R. & Wang, Stephanie W., 2009. "On eliciting beliefs in strategic games," Journal of Economic Behavior & Organization, Elsevier, vol. 71(2), pages 98-109, August.
    8. Li, Yongquan & Zhu, Kaijie, 2009. "Information acquisition in new product introduction," European Journal of Operational Research, Elsevier, vol. 198(2), pages 618-625, October.
    9. Greig, I.D., 1981. "Agricultural Research Management and the Ex Ante Evaluation of Research Proposals : A Review," Review of Marketing and Agricultural Economics, Australian Agricultural and Resource Economics Society, vol. 49(02), August.
    10. Utkin, Lev V., 2006. "A method for processing the unreliable expert judgments about parameters of probability distributions," European Journal of Operational Research, Elsevier, vol. 175(1), pages 385-398, November.
    11. Fu, Qi & Zhu, Kaijie, 2010. "Endogenous information acquisition in supply chain management," European Journal of Operational Research, Elsevier, vol. 201(2), pages 454-462, March.
    12. Marcello Basili & Luca Pratelli, 2013. "Aggregation of not necessarily independent opinions," Department of Economics University of Siena 677, Department of Economics, University of Siena.
    13. Patrizio Frederic & Mario Di Bacco & Frank Lad, 2012. "Combining expert probabilities using the product of odds," Theory and Decision, Springer, vol. 73(4), pages 605-619, October.
    14. Wang, W., 1997. "Subjective estimation of the delay time distribution in maintenance modelling," European Journal of Operational Research, Elsevier, vol. 99(3), pages 516-529, June.
    15. repec:eee:reensy:v:95:y:2010:i:7:p:729-741 is not listed on IDEAS
    16. repec:eee:reensy:v:121:y:2014:i:c:p:72-82 is not listed on IDEAS
    17. Jozsef Mezei & Peter Sarlin, 2014. "Aggregation operators for the measurement of systemic risk," Papers 1412.5452,, revised Dec 2014.
    18. Irene Valsecchi, 2008. "Learning from Experts," Working Papers 2008.35, Fondazione Eni Enrico Mattei.
    19. Christian Kascha & Francesco Ravazzolo, 2010. "Combining inflation density forecasts," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 29(1-2), pages 231-250.
    20. Anderson, Jock R. & Hardaker, J. Brian, 1972. "An Appreciation of Decision Analysis in Management," Review of Marketing and Agricultural Economics, Australian Agricultural and Resource Economics Society, vol. 40(04), December.
    21. Joseph Lipscomb & Giovanni Parmigiani & Vic Hasselblad, 1998. "Combining Expert Judgment by Hierarchical Modeling: An Application to Physician Staffing," Management Science, INFORMS, vol. 44(2), pages 149-161, February.
    22. Timo Henckel & Shaun Vahey & Liz Wakerly, 2011. "Probabilistic Interest Rate Setting With A Shadow Board: A Description Of The Pilot Project," CAMA Working Papers 2011-27, Centre for Applied Macroeconomic Analysis, Crawford School of Public Policy, The Australian National University.
    23. Robert Inman, 1981. "On setting the agenda for Pennsylvania school finance reform: An exercise in giving policy advice," Public Choice, Springer, vol. 36(3), pages 449-474, January.
    24. Joseph Kadane & Javier Girón & Daniel Peña & Peter Fishburn & Simon French & D. Lindley & Giovanni Parmigiani & Robert Winkler, 1993. "Several Bayesians: A review," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 2(1), pages 1-32, December.
    25. Stephen C. Hora, 2004. "Probability Judgments for Continuous Quantities: Linear Combinations and Calibration," Management Science, INFORMS, vol. 50(5), pages 597-604, May.

    More about this item


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:ormnsc:v:15:y:1968:i:2:p:b61-b75. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Mirko Janc). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.