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Uncertainty about Probability: A Decision Analysis Perspective

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  • Ronald A. Howard

Abstract

The issue of how to think about “uncertainty about probability” is framed and analyzed from the viewpoint of a decision analyst. The failure of nuclear power plants is used as an example. The key idea is to think of probability as describing a state of information on an uncertain event, and to pose the issue of uncertainty in this quantity as uncertainty about a number that would be definitive: it has the property that you would assign it as the probability if you knew it. Logical consistency requires that the probability to assign to a single occurrence in the absence of further information be the mean of the distribution of this definitive number, not the median as is sometimes suggested. Any decision that must be made without the benefit of further information must also be made using the mean of the definitive number's distribution. With this formulation, we find further that the probability of r occurrences in n exchangeable trials will depend on the first n moments of the definitive number's distribution. In making decisions, the expected value of clairvoyance on the occurrence of the event must be at least as great as that on the definitive number. If one of the events in question occurs, then the increase in probability of another such event is readily computed. This means, in terms of coin tossing, that unless one is absolutely sure of the fairness of a coin, seeing a head must increase the probability of heads, in distinction to usual thought. A numerical example for nuclear power shows that the failure of one plant of a group with a low probability of failure can significantly increase the probability that must be assigned to failure of a second plant in the group.

Suggested Citation

  • Ronald A. Howard, 1988. "Uncertainty about Probability: A Decision Analysis Perspective," Risk Analysis, John Wiley & Sons, vol. 8(1), pages 91-98, March.
  • Handle: RePEc:wly:riskan:v:8:y:1988:i:1:p:91-98
    DOI: 10.1111/j.1539-6924.1988.tb01156.x
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    Cited by:

    1. Emanuele Borgonovo & Alessandra Cillo, 2017. "Deciding with Thresholds: Importance Measures and Value of Information," Risk Analysis, John Wiley & Sons, vol. 37(10), pages 1828-1848, October.
    2. Gordon Hazen & Emanuele Borgonovo & Xuefei Lu, 2023. "Information Density in Decision Analysis," Decision Analysis, INFORMS, vol. 20(2), pages 89-108, June.
    3. Sarat Sivaprasad & Cameron A. MacKenzie, 2018. "The Hurwicz Decision Rule’s Relationship to Decision Making with the Triangle and Beta Distributions and Exponential Utility," Decision Analysis, INFORMS, vol. 15(3), pages 139-153, September.
    4. Paul K. J. Han & William M. P. Klein & Thomas C. Lehman & Holly Massett & Simon C. Lee & Andrew N. Freedman, 2009. "Laypersons' Responses to the Communication of Uncertainty Regarding Cancer Risk Estimates," Medical Decision Making, , vol. 29(3), pages 391-403, May.
    5. M. Elisabeth Paté-Cornell & Robin L. Dillon, 2006. "The Respective Roles of Risk and Decision Analyses in Decision Support," Decision Analysis, INFORMS, vol. 3(4), pages 220-232, December.
    6. Ronald A. Howard, 2004. "Speaking of Decisions: Precise Decision Language," Decision Analysis, INFORMS, vol. 1(2), pages 71-78, June.

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