Decision Analytic and Bayesian Uncertainty Quantification for Decision Support

  • D. Warner NorthEmail author
Reference work entry


This essay introduces probability in support of decision-making, as a state of mind and not of things. Probability provides a framework for reasoning coherently about uncertainty. According to Cox’s theorem, it is the only way to reason coherently about uncertainty. Probability summarizes states of information. A basic desideratum is that states of information judged equivalent should lead to the same probability distributions. Some widely used probabilistic models have both sufficient statistics and maximum entropy characterizations. In practice, outside of certain physics and engineering applications, human judgment is usually needed to quantify uncertainty as probabilities. Reality is highly complex, but one can test judgments against what is known, drawing upon many specialized areas of human knowledge. Sensitivity analysis can evaluate the importance of multiple uncertainties in a decision context. When the choice between alternatives is close, what is the value of further information (VOI)? Important concepts discussed in this chapter are the expected value of perfect information and the expected value of further experimental testing.

Practical application is illustrated in two case studies. The first involves assessing the probability for replication of a terrestrial microbe on Mars, in the decision context of a constraint on planetary exploration. The second involves weather modification of hurricanes/typhoons: whether to deploy this technology to reduce damage from hurricanes impacting US coastal areas and valuing experimental testing offshore. The decision context is that of a natural disaster turned political by deployment of new technology where such deployment might be followed by a reduction, or by an increase, in adverse impacts, compared to not deploying the technology.

Decision support should be viewed as an iterative process of working with those with decision responsibility and with experts who are the best available sources of information. The decision support process includes characterizing uncertainties and values for outcomes in the context of a choice among decision alternatives. Sensitivity analysis and VOI can give insight on whether to act now or to seek more information and more refined analysis. In a public policy decision context, the process can facilitate stakeholders sharing and critiquing information as the basis for characterizing uncertainties and learning which uncertainties and value judgments are most important for decisions.


Bayes’ theorem Decision analysis Value of information 



This chapter draws from presentations on the two case studies at the November 2014 meeting of INFORMS (Institute for Operations Research and the Management Sciences) in San Francisco, on the occasion of the 50th Anniversary of Decision Analysis honoring its founders, Stanford University Professor Ronald Howard and Harvard University Professor Howard Raiffa. The author wishes to acknowledge the collaboration with colleagues in the two case studies, and appreciation to Tony Cox for detailed comments on the initial draft of this chapter. The author expresses gratitude to these and other colleagues over 50 years in decision and risk analysis for the many insights learned about assessing uncertainties and values in a decision support context.


  1. 1.
    Arrow, K.J.: A difficulty in the concept of social welfare. J. Political Econ. 58, 328–346 (1950)CrossRefGoogle Scholar
  2. 2.
    Arrow, K.J.: Alternative approaches to the theory of choice in risk-taking situations. Econometrica 19(4), 404–437 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arrow, K.J.: Essays in the Theory of Risk-Bearing. North Holland Pub. Co., Amsterdam (1970)zbMATHGoogle Scholar
  4. 4.
    Boyd, D.W., Howard, R.A., Matheson, J.E., North, D.W.: Decision Analysis of Hurricane Modification, Final Report, SRI Project 8503, Stanford Research Institute, Menlo Park (1971)Google Scholar
  5. 5.
    Budnitz, R.J., Apostolakis, G., Boore, D.M., Cluff, L.S., Coppersmith, K.J., Cornell, C.A., Morris, P.A.: Recommendations for Probabilistic Seismic Hazard Analysis: Guidance on Uncertainty and Use of Experts, Report prepared for the Nuclear Regulatory Commission, NUREG/CR-6372, vol. 1, main report, vol. 2, appendices. (1997)
  6. 6.
    Budnitz, R.J., Apostolakis, G., Boore, D.M., Cluff, L.S., Coppersmith, K.J., Cornell, C.A., Morris, P.A.: Use of technical expert panels: applications to probabilistic seismic hazard analysis. Risk Anal. 18(4), 463–469 (1998)CrossRefGoogle Scholar
  7. 7.
    Cooke, R.M.: Experts in Uncertainty: Opinion and Subjective Probability in Science. Oxford University Press, Oxford (1991)Google Scholar
  8. 8.
    Cox, R.T.: Probability, Frequency, and Reasonable Expectation. Am. J. Phys. 14, 1–10 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cox, R.T.: The Algebra of Probable Inference. Johns Hopkins University Press, Baltimore (1961); See also;
  10. 10.
    De Finetti, B.: La Prevision: ses lois logiques, ses sources subjectives. Annales de L’Institut Henri Poincaré (1937). An English translation is in Studies in Subjective Probability, Kyburg and Smokler (eds), 1964Google Scholar
  11. 11.
    Fischhoff, B.: The realities of risk-cost-benefit analysis. Science 350(6260) (2015). doi:10.1126/science.aaa6516Google Scholar
  12. 12.
    Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York (1970)zbMATHGoogle Scholar
  13. 13.
    Fishburn, P.C.: Personal communication (1970)Google Scholar
  14. 14.
    Haidt, J.: The Righteous Mind: Why Good People Are Divided by Politics and Religion. Vintage Books, New York (2013)Google Scholar
  15. 15.
    Howard, R.A.: Decision analysis: applied decision theory. In: Hertz, D.B., Melese, J. (eds.) Proceedings of the Fourth International Conference on Operation Research, pp. 55–71. Wiley-Interscience, New York (1966)Google Scholar
  16. 16.
    Howard, R.A.: The foundations of decision analysis. IEEE Trans. Syst. Sci. Cybern. SSC-4(3), 1–9 (1968)Google Scholar
  17. 17.
    Howard, R.A., Abbas, A.E.: Foundations of Decision Analysis. Pearson, New York (2015)Google Scholar
  18. 18.
    Howard, R.A., Matheson, J.E.: Influence diagrams. Decis. Anal. 2(3), 127–143 (2005)CrossRefGoogle Scholar
  19. 19.
    Howard, R.A., Matheson, J.E., North, D.W.: The decision to seed hurricanes. Science 176, 1191–1202 (1972)CrossRefGoogle Scholar
  20. 20.
    Jaynes, E.T.: Prior probabilities. IEEE Trans. Syst. Sci. Cybern. 4(3), 227–241 (1968)CrossRefzbMATHGoogle Scholar
  21. 21.
    Jaynes, E.T.: Probability Theory: The Logic of Science. Cambridge University Press, Cambridge (2003). Earlier versions of this book were circulated as colloquium lectures, Socony Mobil Oil Company, 1958, then in a version available online, prior to Jaynes’ death in 1998. The Cambridge University Press version was edited by G. Larry Bretthorst and published five years after Jaynes’ deathGoogle Scholar
  22. 22.
    Jeffreys, H.: Theory of Probability, 3rd edn. Clarendon Press, Oxford (1939, 1961)Google Scholar
  23. 23.
  24. 24.
    Jet Prolusion Laboratory News: (2015)
  25. 25.
    Judd, B.R., Warner North, D., Pezier, J.P.: Assessment of the Probability of Contaminating Mars., Report No. MSU-2788, prepared for the Planetary Programs Division, National Aeronautics and Space Administration by SRI International, Menlo Park, 156 pp. (1974). Available at:
  26. 26.
    Kahneman, D.: Thinking, Fast and Slow. Farrar, Strauss, and Giroux, New York (2011)Google Scholar
  27. 27.
    Kahneman, D., Tversky, A., Slovic, P.: Judgment Under Uncertainty: Heuristics and Biases. Cambridge University Press, New York (1982)CrossRefGoogle Scholar
  28. 28.
    Kok, J.F., Parteli, E.J.R., Michaels, T.I., Karam, D.B.: The physics of wind-blown sand and dust. Rep. Prog. Phys. 75, 106901 (2012). Available at CrossRefGoogle Scholar
  29. 29.
    Kyburg, H.E., Jr., Smokler, H.E. (eds.): Studies in Subjective Probability. Wiley, New York (1964)zbMATHGoogle Scholar
  30. 30.
    Laplace, P.-S., marquis de, Essai philosophique sur les probabilities (1814). English translation, Dover, New York (1951)Google Scholar
  31. 31.
    Loève, M.: Probability Theory, 4th edn. 1963, Springer, New York (1977)Google Scholar
  32. 32.
    Luce, R.D., Howard R.: Games and Decisions: Introduction and Critical Survey. Wiley, New York (1957) Reprinted by Dover (1989)Google Scholar
  33. 33.
    Morgan, M.G.: The use (and abuse) of expert elicitation in support of decision making for public policy. Proc. Natl. Acad. Sci. 111, 7176–7184 (2014)CrossRefGoogle Scholar
  34. 34.
    Morgan, M.G.: Commentary: our knowledge of the world is not simple: policy makers should deal with it. Risk Anal. 35(1), 19–20 (2015)MathSciNetCrossRefGoogle Scholar
  35. 35.
    National Research Council: Understanding Risk: Informing Decisions in a Democratic Society. National Academy Press, Washington, D.C. (1996)Google Scholar
  36. 36.
    National Research Council: Public Participation in Environmental Assessment and Decision Making. National Academy Press, Washington, D.C. (2008)Google Scholar
  37. 37.
    Navarro, D., Perfors, A.: An introduction to the Beta-Binomial Model, University of Adelaide. (undated). Last accessed 2 Feb 2016
  38. 38.
    North, D.W.: The invariance approach to the probabilistic encoding of information. Ph.D. dissertation, Stanford University (1970). Available at: Google Scholar
  39. 39.
    North, D.W.: Limitations, definitions, principles, and methods of risk analysis. Rev. sci. Tech. Off. Int. Epiz. 14(4), 913–923 (1995)CrossRefGoogle Scholar
  40. 40.
    North, D.W.: Review of five books on climate change. Risk Anal. 35(12), 2221–2227 (2015)CrossRefGoogle Scholar
  41. 41.
    North, D.W., Judd, B.R., Pezier, J.P.: New methodology for assessing the probability of contaminating Mars. Life Sci. Space Res. XIII, 103–109. Academie-Verlag, Berlin (1975)Google Scholar
  42. 42.
    Phillips, L., von Winterfeldt, D.: Reflections on the Contributions of Ward Edwards to Decision Analysis and Behavioral Research, Working paper, London School of Economics. (2006). Chapter 5 in Advances in Decision Analysis. Cambridge University Press, Cambridge (2007)
  43. 43.
    Pratt, J.: Risk Aversion in the Small and In the Large. Econometrica 32(1/2), 122–136 (1964)CrossRefzbMATHGoogle Scholar
  44. 44.
    Pratt, J., Raiffa, H., Schlaifer, R.: Introduction to Statistical Decision Theory. MIT, Cambridge (1995)zbMATHGoogle Scholar
  45. 45.
    Raiffa, H.: Decision Analysis: Introductory Lectures on Choices under Uncertainty. Addison-Wesley, Reading (1968)zbMATHGoogle Scholar
  46. 46.
    Raiffa, H., Schlaifer, R.: Applied Statistical Decision Theory. Cambridge, MA: Graduate School of Business, Harvard University, 1961. Reprinted, Wiley Classics Library, New York (2000)Google Scholar
  47. 47.
    Ramsey, F.P.: Truth and Probability (1926). Reprinted in: Studies in Subjective Probability, Kyburg and Smokler (eds.) 1964. Also, the text of this paper appears in the book by Ramsey, The Foundations of Mathematics and Other Essays. Harcourt Brace and Company, New York (1931) and is available at:
  48. 48.
    Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954). 2nd revised edition, Dover, New York (1972)Google Scholar
  49. 49.
    Space Science Board: National Research Council, Biological Contamination of Mars. National Academy Press, Washington, D.C. (1992)
  50. 50.
    Spetzler, C., Stael von Holstein, C.-A.S.: Probability encoding in decision analysis. Manag. Sci. 22, 340–358 (1975)Google Scholar
  51. 51.
    Stanford Encyclopedia of Philosophy, Modal Logic entry: Last accessed 2 Feb 2016
  52. 52.
    Taleb, N.N.: Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets, 2001, updated second edition. Random House, New York (2005)Google Scholar
  53. 53.
    Taleb, N.N.: The Black Swan. Random House, New York (2007)Google Scholar
  54. 54.
    Taleb, N.N.: Antifragile: Things That Gain from Disorder. Random House, New York (2012)Google Scholar
  55. 55.
    Tetlock, P.E., Gardner, D.: Superforecasting: The Art and Science of Prediction. Crown Publishers, New York (2015)Google Scholar
  56. 56.
    Tribus, M.: Rational Descriptions, Decisions, and Designs, Pergamon Press, Elmsford (1969)Google Scholar
  57. 57.
    Vaihinger, H.: The Philosophy of ‘As-If’: A System of the Theoretical, Practical, and Religious Fictions of Mankind. Published in German (1911); in England (1924). Available in a translation by C.K. Ogden, New York, Barnes and Noble. (1968). Last accessed 2 Mar 2015
  58. 58.
    Van Horn, K.S.: Constructing a logic of probable inference: a guide to Cox’s theorem. Int. J. Approx. Reason. 34(1), 3–24 (2003)CrossRefGoogle Scholar
  59. 59.
    Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior, 2nd edn. 1947, paperback edition, 2007, Princeton University Press, PrincetonGoogle Scholar
  60. 60.
    Von Winterfeldt, D., Edwards, W.: Decision Analysis and Behavioral Research. Cambridge University Press, New York (1986)Google Scholar
  61. 61.
    Wald, A.: Statistical Decision Functions. Wiley, New York (1950)zbMATHGoogle Scholar
  62. 62.
    Wiki, Arrow’s impossibility theorem: Last accessed 30 Jan 2016
  63. 63.
    Wiki, Bayesian network: Last accessed 30 Jan 2016
  64. 64.
    Wiki, Cox’s theorem: Last accessed 30 Jan 2016
  65. 65.
    Wiki, De Finetti’s theorem: Last accessed 7 Feb 2016
  66. 66.
    Wiki, Influence diagram: Last accessed on 30 Jan 2016
  67. 67.
  68. 68.
    Wiki, Monty Hall problem: Last accessed 14 Nov 2015
  69. 69.
    Wiki, Probability interpretations: Last accessed 17 Nov 2015
  70. 70.
    Wiki, Project Stormfury: Last accessed 15 Nov 2015
  71. 71.
    Wiki, Voting paradox: Last accessed 30 Jan 2016
  72. 72.
    Winkler, R.: Equal versus differential weighting in combining forecasts. Risk Anal. 31(1), 16–18 (2015)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.NorthWorksSan FranciscoUSA

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