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Two Theories of Probability

Published online by Cambridge University Press:  31 January 2023

Glenn Shafer
Glenn Shafer*
Affiliation:
University of Kansas
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Extract

In a recent monograph, I advocated a new theory—the theory of belief functions—as an alternative to the Bayesian theory of epistemic probability. In this paper I compare the two theories in the context of a simple but authentic example of assessing evidence.

The Bayesian theory is ostensibly the theory that assessment of evidence should proceed by conditioning additive probability distributions; this theory dates from the work of Bayes and Laplace in the second half of the eighteenth century. It is indisputably the dominant theory of epistemic probability today.

The theory of belief functions differs from the Bayesian theory in that it uses certain non-additive set functions in the place of additive probability distributions and in that it generalizes the rule of conditioning to a general rule for combining evidence. As a mathematical theory its apparent origin is rather recent and abrupt; it first appears in work of A. P. Dempster, published in the 1960’s.

Type
Part XI. Statistical Evidence
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1978 , Issue 2: Volume Two: Symposia and Invited Papers 1978 , 1978 , pp. 440 - 465
Copyright
Copyright © 1981 Philosophy of Science Association

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Footnotes

1

I wish to thank Amos Tversky, who forced me to try harder to understand the implications of the Bayesian demand for something to condition on, Don Davis, who told me to look around in the attic, and Paul Mostert, who made me think about chain foundations. I have also benefited from conversations with Dennis Lindley, Terry Shafer, and Joe Van Zandt. My research for the paper was partially supported by allocation 3315-x038 from the General Research Fund of the University of Kansas and by grant MCS 78-01887 from the National Science Foundation.

References

[1] De Finetti, Bruno. “Foresight: Its Logical Laws, Its Subjective Sources.” In Studies in Subjective Probability. Edited by Kyburg, Henry E. Jr. and Smokler, Howard E. New York: Wiley, 1964. Pages 93–158. (Originally published as “La Prévision: Ses lois logigue, ses sources subjectives.” Annales de L’Institut Henri Poincaré 7(1937): 1-68.)Google Scholar
[2] De Finetti, Bruno Theory of Probability, Volume 1. New York: Wiley, 1974.Google Scholar
[3] Dempster, A.P. “A Generalization of Bayesian Inference.” Journal of the Royal Statistical Society, Series B 30(1968): 205–247.Google Scholar
[4] Freedman, David A., and Purves, Roger A. “Bayes’ Methods for Bookies.” Annals of Mathematical Statistics 40(1969): 1177–1186.CrossRefGoogle Scholar
[5] Hacking, Ian. “Slightly More Realistic Personal Probability.” Philosophy of Science 34(1967): 311–325.CrossRefGoogle Scholar
[6] Lindley, Dennis. Introduction to Probability and Statistics from a Bayesian Viewpoint, Part 1: Probability. Cambridge: Cambridge University Press, 1965.CrossRefGoogle Scholar
[7] Savage, L.J. The Foundations of Statistics. New York: Wiley, 1954.Google Scholar
[8] Shafer, Glenn. A Mathematical Theory of Evidence. Princeton: Princeton University Press, 1976.CrossRefGoogle Scholar
[9] Shafer, Glenn. “Non Additive Probabilities in the Work of Bernoulli and Lambert.” Archive for History of Exact Sciences 19 (1978): 309–370.CrossRefGoogle Scholar
[10] Teller, Paul. “Conditionalization and Observation.” Synthese 26(1973): 218–258.CrossRefGoogle Scholar