A second-order probability Q(P) may be understood as the probability that the true probability of something has the value P. “True” may be interpreted as the value that would be assigned if certain information were available, including information from reflection, calculation, other people, or ordinary evidence. A rule for combining evidence from two independent sources may be derived, if each source i provides a function Q i (P). Belief functions of the sort proposed by Shafer (1976) also provide a formula for combining independent evidence, Dempster's rule, and a way of representing ignorance of the sort that makes us unsure about the value of P. Dempster's rule is shown to be at best a special case of the rule derived in connection with second-order probabilities. Belief functions thus represent a restriction of a full Bayesian analysis.
KeywordsBayesian Analysis Independent Source Belief Function True Probability Independent Evidence
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- Baron, J.: 1985, Rationality and Intelligence, Cambridge: Cambridge University Press.Google Scholar
- Brown, R. V.: 1986, Assessment Uncertainty and the Firmness of Information: A Decision-Oriented Methodology, Falls Church, VA: Decision Sciences Consortium, Inc.Google Scholar
- Einhorn, H. J. and Hogarth, R. M.: 1985, ‘Ambiguity and Uncertainty in Probabilistic Inference’, Psychological Review 92, 433–461.Google Scholar
- Ellsberg, D.: 1961, ‘Risk, Ambiguity, and the Savage Axioms’, Quaterly Journal of Economics 75, 643–699.Google Scholar
- Gärdenfors, P. and Sahlin, N.-E.: 1983, ‘Decision Making with Unreliable Probabilities’, British Journal of Mathematical and Statistical Psychology 36, 240–251.Google Scholar
- Krantz, D. H., Luce, R. D., Suppes, P., and Tversky, A.: 1971, Foundations of Measurement (Vol. 1), New York: Academic Press.Google Scholar
- Lindley, D. V., Tversky, A., and Brown, R. V.: 1979, ‘On the Reconciliation of Probability Assessments’, Journal of the Royal Statistical Association A. 142, 146–180 (with commentary).Google Scholar
- Raiffa, H.: 1968, Decision Analysis, Reading, Mass.: Addison-Wesley.Google Scholar
- Savage, L. J.: 1954, The Foundations of Statistics, New York: Wiley, 1954.Google Scholar
- Shafer, G.: 1976, A Mathematical Theory of Evidence, Princeton, N. J.: Princeton University Press.Google Scholar
- Shafer, G. and Tversky, A.: 1985, ‘Languages and Designs for Probability Judgment’, Cognitive Science 9, 309–339.Google Scholar