Uncertainty and the Residual Hypothesis

  • Sergio Parrinello


A well-known criticism raised against the use of probability as a measure of uncertainty can be summarised as follows: it is very doubtful whether a single number (probability) has any meaning when it is attributed to events which are not repeatable or, in general, when the probability assessment is vague for whatever reason. In such cases the reliability of the evidence should also be taken into account in making decisions and assessed as distinct from the determination of probability. Keynes suggested the notion of ’weight of the argument’ as distinct from the notion of probability: ’New evidence will sometimes decrease the probability of an argument, but it will always increase its weight.’1 According to other proposals the reliability of a probability assessment can be measured by the probability of a statement of probability. This idea appears in the work of Reichenbach2 and a similar concept has subseyuently been proposed by J. Marschak3 and criticized by de Finetti.4


Decision Maker Prime Consequence Bayesian Approach Residual Event Attribute Level 
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  1. 1.
    J. M. Keynes (1921) A Treatise on Probability (London: Macmillan) p.12.Google Scholar
  2. 2.
    H. Reichenbach (1935) Experience and Prediction (University of Chicago Press) p. 314.Google Scholar
  3. 3.
    J. Marschak (1975) ’Personal Probabilities of Probabilities’, Theory of Decision, 6, n.2.Google Scholar
  4. 4.
    B. de Finetti, Probabilities of Probabilities: a real problem or a misunderstanding?, mimeo.Google Scholar
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    D. Lindley (1973) Making Decisions (London: Wiley-Interscience) pp. 116–117.Google Scholar
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    See Glen Shafer (1976) ’A Mathematical Theory of Evidence’ (Prince-ton University Press) and S. Geisser (1985) ’On the Prediction of Observables: a selective update’, in J. M. Bernardo, M. H. De Groote, D. V. Lindley and A. T. M. Smith (eds) Bayesian Statistics II (Amsterdam: North Holland).Google Scholar
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    We quote: ’Terminal actions represent the best of one’s existing combination of information and ignorance. ... Informational actions are non-terminal in that a final decision is deferred while awaiting or actively seeking new evidence which will, it is anticipated, reduce uncertainty’, from J. Hirshleifer and J. G. Riley (1979) ’The Analytics of Uncertainty and Information’, Journal of Economic Literatnre, 17, December, P. 1377–8.Google Scholar
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    G. L. S. Shackle, ibid, pp. 50–51.Google Scholar
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    Let us denote: p(Ei/H) the probability of the event Ei prior to the message, given the original information H; p(Ei/M and H) the probability of the same event Ei, given both the original information and the new knowledge that the message M is true; p(M/Ei and H) the probability of the message, given the original information and the knowledge that the event Ei is true. The Bayes’s theorem states:Google Scholar
  13. 13.
    Quoted from S. French (1983) ’A Survey and Interpretation of Multi-Attribute Utility Theory’, in Mudtiobjective Decision Making, S. French, R. Hartley, L. C. Thomas and D. J. White (eds) (London: Academic Press) pp. 266–7.Google Scholar

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© J. A. Kregel 1989

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  • Sergio Parrinello

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