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Probability Judgments for Continuous Quantities: Linear Combinations and Calibration


  • Stephen C. Hora

    () (Department of Business Administration, University of Hawaii-Hilo, Hilo, Hawaii 96720.)


Expert judgment elicitation is often required in probabilistic decision making and the evaluation of risk. One measure of the quality of probability distributions given by experts is calibration--the faithfulness of the probabilities in an empirically verifiable sense. A method of measuring calibration for continuous probability distributions is presented here. A discussion of the impact of using linear rules for combining such judgments is given and an empirical demonstration is given using data collected from experts participating in a large-scale risk study. It is shown by theoretical argument that combining well-calibrated distributions of individual experts using linear rules can only result in reducing calibration. In contrast, it is demonstrated, both by example and empirically, that an equally weighted linear combination of experts who tend to be "overconfident" can produce distributions that are better calibrated than the experts' individual distributions. Using data from training exercises, it is shown that the improvement in calibration is rapid as the number of experts is increased from one to five or six, but there is only modest improvement from increasing the number of experts beyond that point.

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  • Stephen C. Hora, 2004. "Probability Judgments for Continuous Quantities: Linear Combinations and Calibration," Management Science, INFORMS, vol. 50(5), pages 597-604, May.
  • Handle: RePEc:inm:ormnsc:v:50:y:2004:i:5:p:597-604

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    References listed on IDEAS

    1. James E. Matheson & Robert L. Winkler, 1976. "Scoring Rules for Continuous Probability Distributions," Management Science, INFORMS, vol. 22(10), pages 1087-1096, June.
    2. Peter A. Morris, 1974. "Decision Analysis Expert Use," Management Science, INFORMS, vol. 20(9), pages 1233-1241, May.
    3. Peter A. Morris, 1977. "Combining Expert Judgments: A Bayesian Approach," Management Science, INFORMS, vol. 23(7), pages 679-693, March.
    4. Robert L. Winkler, 1968. "The Consensus of Subjective Probability Distributions," Management Science, INFORMS, vol. 15(2), pages 61-75, October.
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    Cited by:

    1. Hammitt, James K. & Zhang, Yifan, 2012. "Combining Experts’ Judgments: Comparison of Algorithmic Methods using Synthetic Data," TSE Working Papers 12-293, Toulouse School of Economics (TSE).
    2. Hammitt, James & Roman, Henry & Stieb, David & Walsh, Tyra, 2012. "Expert Elicitation of the Value per Statistical Life in an Air Pollution Context," LERNA Working Papers 12.10.367, LERNA, University of Toulouse.
    3. Roopesh Ranjan & Tilmann Gneiting, 2010. "Combining probability forecasts," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(1), pages 71-91.
    4. Baker, Erin & Bosetti, Valentina & Anadon, Laura Diaz & Henrion, Max & Aleluia Reis, Lara, 2015. "Future costs of key low-carbon energy technologies: Harmonization and aggregation of energy technology expert elicitation data," Energy Policy, Elsevier, vol. 80(C), pages 219-232.
    5. repec:eee:reensy:v:121:y:2014:i:c:p:72-82 is not listed on IDEAS
    6. Erin Baker & Olaitan Olaleye & Lara Aleluia Reis, 2015. "Decision Frameworks and the Investment in R&D," Working Papers 2015.42, Fondazione Eni Enrico Mattei.
    7. James E. Smith & Detlof von Winterfeldt, 2004. "Anniversary Article: Decision Analysis in Management Science," Management Science, INFORMS, vol. 50(5), pages 561-574, May.


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