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Binary Action Based Estimation of Propensities

Published online by Cambridge University Press:  01 January 2025

George T. Duncan
George T. Duncan*
Affiliation:
Carnegie-Mellon University
*
Requests for reprints should be sent to George T. Duncan, Department of Statistics, Carnegie-Mellon University, Pittsburgh, PA 15213.
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Abstract

An observer is to make inference statements about a quantity p, called a propensity and bounded between 0 and 1, based on the observation that p does or does not exceed a constant c. The propensity p may have an interpretation as a proportion, as a long-run relative frequency, or as a personal probability held by some subject. Applications in medicine, engineering, political science, and, most especially, human decision making are indicated. Bayes solutions for the observer are obtained based on prior distributions in the mixture of beta distribution family; these are then specialized to power-function prior distributions. Inference about log p and log odds is considered. Multiple-action problems are considered in which the focus of inference shifts to the process generating the propensities p, both in the case of a process parameter π known to the subject and unknown. Empirical Bayes techniques are developed for observer inference about c when π is known to the subject. A Bayes rule, a minimax rule and a beta-minimax rule are constructed for the subject when he is uncertain about π.

Keywords

estimationsubjective probabilityBayesian inferenceminimaxbeta distributiondecision-makinglog odds
Type
Original Paper
Information
Psychometrika , Volume 43 , Issue 1 , March 1978 , pp. 93 - 107
Copyright
Copyright © 1978 The Psychometric Society

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Footnotes

This research was partially supported by the Defense Advanced Research Projects Agency of the Department of Defense and was monitored by ONR under Contract No. N00014-77-C-0095. Any opinions, findings, conclusions or recommendations expressed herein are those of the author and do not necessarily reflect the views of the Defense Advanced Research Projects Agency, the Office of Naval Research, or Carnegie-Mellon University.

References

Reference Notes

Duncan, G. T. Information and questionnaires in statistical inference, 1970, Minneapolis, Minnesota: School of Statistics, University of Minnesota.Google Scholar
Good, I. J. The estimation of probabilities: An essay on modern Bayesian methods, 1965, Cambridge, Massachusetts: MIT Press.Google Scholar
Marschak, J. Remarks on the economics of information. Contributions to Scientific Research in Management. Los Angeles: Western Data Processing Center, UCLA. 1959, 79–100.Google Scholar

References

Beach, L. R., Rose, R. M., Sayeki, Y., Wise, J. A., & Carter, W. B. Response proportions and verbal estimates. Journal of Experimental Psychology, 1970, 86, 165–170.CrossRefGoogle Scholar
Cox, D. R. Analysis of Binary Data, 1970, London: Methuen & Co. Ltd..Google Scholar
de Finetti, B. Methods for discriminating levels of partial knowledge concerning a test item. The British Journal of Mathematical and Statistical Psychology, 1965, 18, 87–123.CrossRefGoogle Scholar
de Finetti, B. Probability, induction and statistics: The art of guessing, 1972, New York: John Wiley and Sons, Inc..Google Scholar
Ebel, R. L. Measuring educational achievement, 1965, Englewood Cliffs, New Jersey: Prentice-Hall, Inc..Google Scholar
Ferguson, T. S. Mathematical statistics, a decision theoretic approach, 1967, New York: Academic Press.Google Scholar
Good, I. J. Rational decisions. Journal of the Royal Statistical Society, 1952, 14, 107–114.CrossRefGoogle Scholar
Hampton, J. M., Moore, P. G., & Thomas, H. Subjective probability and its measurement. Journal of the Royal Statistical Society, 1973, 136, 21–42.CrossRefGoogle Scholar
Hendrickson, A. D. & Buehler, R. J. Proper scores for probability forecasters. Annals of Mathematical Statistics, 1971, 42, 1916–1921.CrossRefGoogle Scholar
Johnson, N. L. & Kotz, S. Distributions in statistics: Continuous univariate distributions—2, 1970, Boston: Houghton Mifflin Company.Google Scholar
Kroll, N. E. A. Learning of several simultaneous probability learning problems as a function of overall event probability and prior knowledge. Journal of Experimental Psychology, 1970, 83, 200–215.CrossRefGoogle Scholar
Laming, D. R. J. Information theory of choice-reaction times, 1968, New York: Academic Press.Google Scholar
McCarthy, J. Measures of the value of information. Proceedings of the National Academy of Science, 1956, 42, 654–655.CrossRefGoogle ScholarPubMed
Nicks, D. C. Prediction of two-choice decisions. Journal of Experimental Psychology, 1959, 57, 105–114.CrossRefGoogle ScholarPubMed
Rapoport, A. Mathematical methods in theories of international relations: expectations, caveats, and opportunities. In Zinnes, D., Gillespie, J. (Eds.), Mathematical models in international relations, 1976, New York: Praeger.Google Scholar
Roberts, H. V. Probabilistic prediction. Journal of the American Statistical Association, 1965, 60, 50–62.CrossRefGoogle Scholar
Savage, L. J. Elicitation of personal probabilities and expectations. Journal of the American Statistical Association, 1971, 66, 783–801.CrossRefGoogle Scholar
Shuford, E. H., Albert, A., & Massengill, H. E. Admissible probability measurement procedures. Psychometrika, 1966, 31, 125–145.CrossRefGoogle ScholarPubMed
Siegel, S., & Goldstein, D. Decision making behavior in a two-choice uncertain outcome situation. Journal of Experimental Psychology, 1959, 57, 37–42.CrossRefGoogle Scholar
Spragins, J. Learning without a teacher. IEEE Transactions on Information Theory, 1966, IT-12, 223–230.CrossRefGoogle Scholar
Tversky, A. Assessing uncertainty. Journal of the Royal Statistical Society, 1974, 36, 148–159.CrossRefGoogle Scholar
Winkler, R. L. The quantification of judgment: Some methodological suggestions. Journal of the American Statistical Association, 1967, 62, 1105–1120.CrossRefGoogle Scholar
Winkler, R. L. & Murphy, A. H. Evaluation of subjective precipitation forecasts. Proceedings of the First National Conference on Statistical Meteorology, American Meteorological Society, 1968, 148–157.Google Scholar
Winkler, R. L. & Murphy, A. H. “Good” probability assessors. Journal of Applied Meteorology, 1968, 7, 751–758.2.0.CO;2>CrossRefGoogle Scholar