Skip to main content
SpringerLink
Log in

Coherent combination of experts' opinions

  • Published:
Test Aims and scope Submit manuscript

Summary

Anexpert (for You) is here defined as someone who shares Your world-view, but knows more than You do, so that were She to reveal Her current opinion to You, You would adopt it as Your own. When You have access to different experts, with differing information, You require acombination formula to aggregate their various opinions. A number of formulae have been suggested, but here we explore the fundamental requirement ofcoherence to relate such a formula to Your joint distribution for the experts' opinions. In particular, in the context of opinions about an uncertain eventA, we investigate coherence properties of the linear, harmonic and logarithmic opinion pools. Some general results on coherence of the joint forecast distribution are also developed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Berger, J. O. and Mortera, J. (1991). Bayesian limited communication.J.Statist. Planning and Inference 28, 1–24.

    Article  MathSciNet  MATH  Google Scholar 

  • Bordley, R. F. (1982). A multiplicative formula for aggregating probability assessments.Manag. Sci. 28, 1137–1148.

    MATH  Google Scholar 

  • Carnap, R. (1950).Logical Foundations of Probability. Chicago: University Press.

    MATH  Google Scholar 

  • Clemen, R. T. (1985). Extraneous expert information.J. Forecasting 4, 329–348.

    Google Scholar 

  • Clemen, R. T. (1986). Calibration and the aggregation of probabilities.Manag. Sci. 32, 312–314.

    Google Scholar 

  • Clemen, R. T. and Murphy, A. H. (1986). Objective and subjective precipitation probability forecasts: statistical analysis of some interrelationships.Weather and Forecasting 1, 56–65.

    Article  Google Scholar 

  • Dawid, A. P. (1979). Conditional independence in statistical theory.J. Roy. Statist. Soc. B 41, 1–31, (with discussion).

    MathSciNet  MATH  Google Scholar 

  • Dawid, A. P. (1982). The well-calibrated Bayesian.J. Amer. Statist. Assoc. 77, 605–610.

    Article  MathSciNet  MATH  Google Scholar 

  • Dawid, A. P. (1986). Probability forecasting.Encyclopedia of the Statistical Sciences (S. Kotz, N. L. Johnson and C. B. Read, eds.), vol.7. New York: Wiley, 210–218.

    Google Scholar 

  • DeGroot, M. H. (1974). Reaching a consensus.J. Amer. Statist. Assoc. 69, 118–121.

    Article  MATH  Google Scholar 

  • DeGroot, M. H. (1988). A Bayesian view of assessing uncertainty and comparing expert opinion.J. Statist. Planning and Inference 20, 295–306.

    Article  MathSciNet  MATH  Google Scholar 

  • DeGroot, M. H. and Eriksson, E. A. (1985). Probability forecasting, stochastic dominance and the Lorenz curve.Bayesian Statistics 2, (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.), Amsterdam: North-Holland, 99–118, (with discussion).

    Google Scholar 

  • DeGroot M. H. and Fienberg, S. E. (1983). The comparison and evaluation of forecasters.The Statistician 32, 12–22.

    Article  Google Scholar 

  • DeGroot, M. H. and Mortera, J. (1991). Optimal linear opinion pools.Manag. Sci. 37, 546–558.

    MATH  Google Scholar 

  • French, S. (1980). Updating belief in the light of someone else's opinion.J. Roy. Statist. Soc. A 143, 43–48.

    MathSciNet  MATH  Google Scholar 

  • French, S. (1981). Consensus of opinion.Europ. J. Oper. Res. 7, 332–340.

    Article  MathSciNet  MATH  Google Scholar 

  • French, S. (1985). Group consensus probability distribution: a critical survey.Bayesian Statistics 2 (J. M. Bernardo M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.), Amsterdam: North-Holland, 183–201, (with discussion).

    Google Scholar 

  • French, S. (1986). Calibration and the expert problem.Manag. Sci. 32, 315–320.

    Google Scholar 

  • Genest, C. (1984). A characterization theorem for externally bayesian groups.Ann. Statist. 12, 1100–1105.

    MathSciNet  MATH  Google Scholar 

  • Genest, C. and McConway K. J. (1990). Allocating the weights in the linear opinion pool.J. Forecasting 9, 53–73.

    Google Scholar 

  • Genest, C., McConway K. J. and Schervish, M. J. (1986). Characterization of externally Bayesian pooling operators.Ann. Statist. 14, 487–501.

    MathSciNet  MATH  Google Scholar 

  • Genest, C. and Schervish, M. J. (1985). Modelling expert judgments for Bayesian updating.Ann. Statist. 13, 1198–1212.

    MathSciNet  MATH  Google Scholar 

  • Genest, C. and Zidek, J. V. (1986). Combining probability distributions: a critique and an annotated bibliography.Statist. Sci. 1, 114–148.

    MathSciNet  Google Scholar 

  • Gutmann, S., Kemperman, J. H. B., Reeds J. A. and Shepp, L. A. (1991). Existence of probability measures with given marginals.Annals of Prob. 19, 1781–1797.

    MathSciNet  MATH  Google Scholar 

  • Jeffreys H. (1939).Theory of Probability. Oxford: Clarendon Press.

    Google Scholar 

  • Kellerer, H. G. (1961). Funktionen und Produkträumen mit vorgegeben Marginal-Funktionen.Math. Ann. 144, 323–344.

    Article  MathSciNet  MATH  Google Scholar 

  • Keynes, J. M. (1921).A Treatise on Probability. London. Macmillan.

    MATH  Google Scholar 

  • Lauritzen, S. L. (1980).Statistical Models as Extremal Families, Aalborg: University Press.

    Google Scholar 

  • Lindley, D. V. (1982). The improvement of probability judgementsJ. Roy. Statist. Soc. B 145, 117–126.

    MathSciNet  MATH  Google Scholar 

  • Lindley, D. V. (1985). Reconciliation, of discrete probability distributions.Bayesian Statistics 2 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.), Amsterdam: North-Holland, 375–390, (with discussion).

    Google Scholar 

  • Lindley, D. V. (1986). Another look at an axiomatic approach to expert resolution.Manag. Sci. 32, 303–306.

    Google Scholar 

  • McConway, K. J. (1981). Marginalization and linear opinion pools.J. Amer. Statist. Assoc. 76, 410–414.

    Article  MathSciNet  MATH  Google Scholar 

  • Madansky, A. (1964). Externally Bayesian groups. RAND Memo RM-4141-PR. Santa Monica, CA: The Rand Corporation.

    Google Scholar 

  • Morris, P. A. (1983). An axiomatic approach to expert resolution.Manag. Sci. 29, 24–32.

    Google Scholar 

  • Murphy, A. H. and Winkler R. L. (1992). Diagnostic verification of probability forecasts.Int. J. Forecasting 7, 435–455.

    Article  Google Scholar 

  • Schervish, M. J. (1986). Comments on some axioms for combining expert judgments.Manag. Sci. 32, 306–312.

    Google Scholar 

  • Singpurwalla, N. D. (1988). Discussion of “Gaining weight: A Bayesian Approach” by Bayarri, M. J. and DeGroot, M. H.Bayesian Statistics 2 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.), Amsterdam: North-Holland, 39–42.

    Google Scholar 

  • Stone, M. (1961). The opinion pool.Ann. Math. Statist. 32, 1339–1342.

    MathSciNet  MATH  Google Scholar 

  • Strassen, V. (1965). The existence of probability measures with given marginals.Ann. Math. Statist. 36, 423–439.

    MathSciNet  MATH  Google Scholar 

  • Winkler, R. L. (1981). Combining probability distributions from dependent information sources.Manag. Sci. 27, 479–488.

    MathSciNet  MATH  Google Scholar 

  • Winkler, R. L. (1986). Expert resolution.Manag. Sci. 32, 298–303.

    Google Scholar 

Additional references in the discussion

  • Arrow, K. J. (1951).Social Choice and Individual Values. New York: Wiley.

    MATH  Google Scholar 

  • Clemen, R. T. (1987). Combining overlapping information.Manag. Sci. 33, 373–380.

    Google Scholar 

  • Clemen, R. T. and Winkler, R. L. (1990). Unanimity and compromise among probability forecasters.Manag. Sci. 36, 767–779.

    MATH  Google Scholar 

  • Clemen, R. T. and Jouini, M. N. (1996). Copula models for aggregating expert opinions.Operations Research 44, (to appear).

  • Cooke, R. M. (1993). The ill advised BayesianISBA Newsletter 2, 2–3.

    Google Scholar 

  • Dalkey N. C. (1972). An impossibility theorem for group probability functionsTech. Rep. P-4862. Santa Monica, CA: RAND Corporation.

    Google Scholar 

  • Draper, D. (1995). Assessment and propagation of model uncertainty.J. Roy. Statist. Soc. B 57, 45–97, (with discussion).

    MathSciNet  MATH  Google Scholar 

  • French, S. (1994). Utility: probability's younger twin.Aspects of Uncertainty: a Tribute to D. V. Lindley (P. R. Freeman, and A. F. M. Smith, eds.), Chichester: Wiley, 171–180.

    Google Scholar 

  • Harper, F. T.; Goossens, L. H. J.; Cooke, R. M.; Helton, J. C.; Hora, S. C.; Jones, J. A.; Kraan, B. C. P.; Lui, C.; McKay, M. D.; Miller, L. A.; Päsler Sauer, J. and Young, M. L. (1994).Joint USNRC/CEC Consequence Uncertainty Study: Summary of Objectives, Approach, Application and Results for the Dispersion and Deposition Uncertainty Assessments. NUREG/CR-6244, EUR 15855, SAND94-1453. Sandia National Laboratories and Delft University of Technology.

  • Kelly, F. S. (1978).Arrow Impossibility Theorems. New York: Academic Press.

    MATH  Google Scholar 

  • Makridakis, S. and Winkler, R. L. (1983). Average of forecasts: some empirical results.Manag. Sci. 29, 987–996.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistical Science, University College London, Gower Street, WC1E 6BT, London, UK

    A. P. Dawid

  2. Dipartimento di Economia, Università di Roma III, Via Ostiense, 00154, Rome, Italy

    J. Mortera

  3. Technishe Universiteit Delft, The Netherlands

    R. Cooke

  4. University of Leeds, UK

    S. French

  5. Université Laval, Canada

    C. Genest

  6. Carnegie Mellon University, USA

    M. J. Schervish

  7. Somerset, UK

    D. V. Lindley

  8. The Open University, UK

    K. J. McConway

  9. Duke University, USA

    R. L. Winkler

Authors
  1. A. P. Dawid
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. H. DeGroot
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Mortera
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Cooke
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. S. French
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. C. Genest
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. M. J. Schervish
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. D. V. Lindley
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. K. J. McConway
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. R. L. Winkler
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Morrie DeGroot passed away on 2 November, 1989. This work was initiated by Dawid and DeGroot and was completed by Dawid and Mortera.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dawid, A.P., DeGroot, M.H., Mortera, J. et al. Coherent combination of experts' opinions. Test 4, 263–313 (1995). https://doi.org/10.1007/BF02562628

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02562628

Keywords

Search

Navigation