Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baron, J.: 1985, Rationality and Intelligence, Cambridge: Cambridge University Press.
Brown, R. V.: 1986, Assessment Uncertainty and the Firmness of Information: A Decision-Oriented Methodology, Falls Church, VA: Decision Sciences Consortium, Inc.
Einhorn, H. J. and Hogarth, R. M.: 1985, ‘Ambiguity and Uncertainty in Probabilistic Inference’, Psychological Review 92, 433–461.
Ellsberg, D.: 1961, ‘Risk, Ambiguity, and the Savage Axioms’, Quaterly Journal of Economics 75, 643–699.
Gärdenfors, P. and Sahlin, N.-E.: 1983, ‘Decision Making with Unreliable Probabilities’, British Journal of Mathematical and Statistical Psychology 36, 240–251.
Krantz, D. H., Luce, R. D., Suppes, P., and Tversky, A.: 1971, Foundations of Measurement (Vol. 1), New York: Academic Press.
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).
Raiffa, H.: 1968, Decision Analysis, Reading, Mass.: Addison-Wesley.
Savage, L. J.: 1954, The Foundations of Statistics, New York: Wiley, 1954.
Shafer, G.: 1976, A Mathematical Theory of Evidence, Princeton, N. J.: Princeton University Press.
Shafer, G. and Tversky, A.: 1985, ‘Languages and Designs for Probability Judgment’, Cognitive Science 9, 309–339.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Baron, J. Second-order probabilities and belief functions. Theor Decis 23, 25–36 (1987). https://doi.org/10.1007/BF00127335
Issue Date:
DOI: https://doi.org/10.1007/BF00127335