Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science pp 175-257 | Cite as

# Confidence Intervals vs Bayesian Intervals

## Abstract

For many years, statistics textbooks have followed this ‘canonical’ procedure: (1) the reader is warned not to use the discredited methods of Bayes and Laplace, (2) an orthodox method is extolled as superior and applied to a few simple problems, (3) the corresponding Bayesian solutions are *not* worked out or described in any way. The net result is that no evidence whatsoever is offered to substantiate the claim of superiority of the orthodox method.

To correct this situation we exhibit the Bayesian and orthodox solutions to six common statistical problems involving confidence intervals (including significance tests based on the same reasoning). In every case, we find that the situation is exactly the opposite; i.e., the Bayesian method is easier to apply and yields the same or better results. Indeed, the orthodox results are satisfactory only when they agree closely (or exactly) with the Bayesian results. No contrary example has yet been produced.

By a refinement of the orthodox statistician’s own criterion of performance, the best confidence interval for any location or scale parameter is proved to be the Bayesian posterior probability interval. In the cases of point estimation and hypothesis testing, similar proofs have long been known. We conclude that orthodox claims of superiority are totally unjustified; today, the original statistical methods of Bayes and Laplace stand in a position of proven superiority in actual performance, that places them beyond the reach of mere ideological or philosophical attacks. It is the continued teaching and use of orthodox methods that is in need of justification and defense.

## Keywords

Common Sense Bayesian Method Bayesian Solution Bayesian Test Ancillary Statistic## Preview

Unable to display preview. Download preview PDF.

## References

- Bayes, Rev. Thomas, ‘An Essay Toward Solving a Problem in the Doctrine of Chances’, Phil. Trans. Roy. Soc. 330–418 (1763). Reprint, with biographical note by G. A. Barnard, in Biometrika 45, 293–315 (1958) and in Studies in the History of Statistics and Probability, E. S. Pearson and M. G. Kendall, (eds), C. Griffin and Co. Ltd., London, (1970). Also reprinted in Two Papers by Bayes with Commentaries, ( W. E. Deming, ed.), Hafner Publishing Co., New York, (1963).Google Scholar
- Birnbaum, Allen, ‘On the Foundations of Statistical Inference’, J. Am. Stat. Ass’n 57 269 (1962).Google Scholar
- Crow, E. L., Davis, F. A., and Maxfield, M. W.,
*Statistics Manual*, Dover Publications, Inc., New York (1960).Google Scholar - Fisher, R. A.,
*Statistical Methods and Scientific Inference*, Hafner Publishing Co., New York (1956).Google Scholar - Friedman, K. and Shimony, A., ‘Jaynes’ Maximum Entropy Prescription and Probability Theory’,.
*Stat. Phys*3, 381–384 (1971).CrossRefGoogle Scholar - Good, I. J.,
*Probability and The Weighing of Evidence*, C. Griffin and Co. Ltd., London (1950).Google Scholar - Good, I. J.,
*The Estimation of Probabilities*, Research Monograph #30, The MIT Press, Cambridge, Mass. (1965); paperback edition, 1968.Google Scholar *Jaynes, E. T*Probability Theory in Science and Engineering*, No. 4 of*Colloquium Lectures on Pure and Applied Science*, Socony-Mobil Oil Co., Dallas, Texas*Google Scholar- Jaynes, E. T., ‘New Engineering Applications of Information Theory’, in
*Engineering Uses of Random Function Theory and Probability*, J. L. Bogdanoff and F. Kozin, (eds.), J. Wiley & Sons, Inc., N.Y. (1963); pp. 163–203.Google Scholar - Jaynes, E. T., ‘Information Theory and Statistical Mechanics’, in
*Statistical Physics*, K. W. Ford, (ed.), W. A. Benjamin, Inc., (1963); pp. 181–218.Google Scholar - Jaynes, E. T., ‘Gibbs vs. Boltzmann Entropies’,
*Am. J. Phys*33, 391 (1965).CrossRefGoogle Scholar - Jaynes, E. T. T., ‘Foundations of Probability Theory and Statistical Mechanics’, Chap. 6 in
*Delaware Seminar in Foundations of Physics*, M. Bunge, (ed.), Springer-Verlag, Berlin (1967); Spanish translation in*Modern Physics*, David Webber, (ed.), Alianza Editorial s/a, Madrid 33 (1973).Google Scholar - Jaynes, E. T., ‘The Well-Posed Problem’, in
*Foundations of Statistical Inference*, V. P. Godambe and D. A. Sprott, (eds.), Holt, Rinehart and Winston of Canada, Toronto (1971).Google Scholar - Jaynes, E. T., ‘Survey of the Present Status of Neoclassical Radiation Theory’, in
*Coherence and Quantum Optics*, L. Mandel and E. Wolf, (eds.), Plenum Publishing Corp., New York (1973), pp. 35–81.Google Scholar - Jeffreys, H.,
*Theory of Probability*, Oxford University Press (1939).Google Scholar - Jeffreys, H.,
*Scientific Inference*, Cambridge University Press (1957).Google Scholar - Kempthorne, O., ‘Probability, Statistics, and the Knowledge Business’, in
*Foundations of Statistical Inference*, V. P. Godambe and D. A. Sprott, (eds.), Holt, Rinehart and Winston of Canada, Toronto (1971).Google Scholar - Kendall, M. G. and Stuart, A.,
*The Advanced Theory of Statistics*Volume 2, C. Griffin and Co., Ltd., London C1961).Google Scholar - Lehmann, E. L.,
*Testing Statistical Hypotheses*, J. Wiley & Sons, Inc., New York, (1959) p. 62.Google Scholar - Pearson, E. S., Discussion in Savage (1962); p. 57.Google Scholar
- Roberts, Norman A.,
*Mathematical Methods in Reliability Engineering*, McGraw-Hill Book Co., Inc., New York (1964) pp. 86–88.Google Scholar - Rowlinson, J. S., ‘Probability, Information and Entropy’,
*Nature**225*, 1196–1198 (1970).CrossRefGoogle Scholar - Savage, L. J.,
*The Foundations of Statistics*, John Wiley, & Sons, Inc., New York (1954).Google Scholar - Savage, L. J.,
*The Foundations of Statistical Inference*, John Wiley & Sons, Inc., New York (1962).Google Scholar *Schlaifer, R*Probability and Statistics for Business Decisions*, McGraw-Hill Book Co. Inc*., New York (1959).Google Scholar- Sobel, M. and Tischendorf, J. A., Proc. Fifth Nat’l Symposium on Reliability andGoogle Scholar
- Quality Control, I.R.E., pp. 108–118 (1959).Google Scholar
- Smith, C. A. B., Discussion in Savage (1962); p. 60.Google Scholar
- Cornfield, J., 1969, ‘The Bayesian Outlook and its Applications’, Biometrics 25, 617–642, for a good listing of background material.Google Scholar
- Dawid, A. P., M. Stone and J. V. Sidek, 1973, ‘Marginalization Paradoxes in Bayesian and Structural Inference’,
*J. Roy. Stat. Soc. B*in press.Google Scholar - Dawid, A. P. and M. Stone, 1972, ‘Expectation Consistency of Inverse Probability Distributions’,
*Biometrika*59, 486–489.CrossRefGoogle Scholar - Easterling, R. G., 1972, ‘A Personal View of the Bayesian Controversy in Reliability and Statistics’,
*IEEE Trans*. R-21, 186–194.Google Scholar - Fisher, R. A., 1959,
*Statistical Methods and Scientific Inference*, 2nd ed., Hafner, New York.Google Scholar - Kalbfleisch, J. G., 1971,
*Probability and Statistical Methods*, Dept. of Statistics, Waterloo.Google Scholar - Kalbfleisch, J. G., and D. A. Sprott, 1973, ‘On Tests of Significance’, this volume, p. 259.Google Scholar
- Kempthorne, O., 1969,
*Biometrics*25, 647–654. Discussion of paper by J. Cornfield.Google Scholar - Kempthorne, O., 1971, ‘Probability, Statistics, and the Knowledge Business’, in
*Foundations of Statistical Inference*, V. P. Godambe, and D. A. Sprott, (j, Holt, Rinehart and Winston, New York; 1971.Google Scholar - Kempthorne, O., 1972, ‘Theories of Inference and Data Analysis’, in
*Statistical Papers in Honor of George W. Snedecor*, Iowa State Univ. Press, Ames.Google Scholar - Kempthorne, O. and J. L. Folks, 1971,
*Probability, Statistics, and Data Analysis*, Iowa State Univ. Press, Ames.Google Scholar - Luce, R. D. and H. Raiffa, 1957,
*Games and Decisions*, Wiley, New York.Google Scholar - Ramsey, F. P., 1926, ‘Truth and Probability’, in
*The Foundations of Mathematics and Other Logical Essays*, Kegan, London.Google Scholar - Savage, L. J., 1954,
*The Foundations of Statistics*, Wiley, New York.Google Scholar - Snedecor, G. W. and W. G. Cochran, 1967,
*Statistical Methods*, 6th ed., Iowa State Univ. Press, Ames.Google Scholar - Stone, M. and A. P. Dawid, 1972, ‘Un-Bayesian Implications of Improper Bayes Inference in Routine Statistical Problems’, Biometrika 59, 369–375.Google Scholar
- Barnard, G. A., ‘Comments on Stein’s “ Remark on the Likelihood Principle”Google Scholar
- Cox, R. T.,
*Am. J. Phys*17, 1 (1946).CrossRefGoogle Scholar - Cox, R. T.,
*The Algebra of Probable Inference*, Johns Hopkins University Press, 1961; Reviewed by E. T. Jaynes,*Am. J. Phys*31, 66 (1963).CrossRefGoogle Scholar - Deming, W. E.,
*Statistical Adjustment of Data*, J. Wiley, New York (1943).Google Scholar - Fisher, R. A.,
*Contributions to Mathematical Statistics*, W. A. Shewhart, (ed.), J. Wiley and Sons, Inc. New York (1950); Referred to above as ‘Collected Works’.Google Scholar - Fisher, R. A.,
*Statistical Methods and Scientific Inference*, Hafner Publishing Co., New York (1956).Google Scholar - Fisher, R. A.,
*Statistical Methods for Research Workers*, Hafner Publishing Co., New York: Thirteenth Edition (1958).Google Scholar - Hoel, P. G.,
*Introduction to Mathematical Statistics*, Fourth Edition, J. Wiley and Sons, Inc., New York (1971).Google Scholar - Jaynes, E. T., ‘Review of
*Noise and Fluctuations*’, by D. K. C. MacDonald,*Am. J. Phys*31, 946 (1963).Google Scholar - Kendall, M. G., ‘Ronald Aylmer Fisher, 1890–1962’,
*Biometrika*50, 1–15 (1963); reprinted in*Studies in the History of Statistics and Probability*, E. S. Pearson and M. G. Kendall, (eds)., Hafner Publishing Co., Darien, Conn. (1970).Google Scholar - Mandel, J.,
*The Statistical Analysis of Experimental Data*, Interscience Publishers, New York (1964); p. 290.Google Scholar - McColl, H., ‘The Calculus of Equivalent Statements’,
*Proc. Lond. Math. Soc*28, p. 556 (1897).Google Scholar - Pearson, Karl, ‘Method of Moments and Method of Maximum Likelihood’,
*Biometrika*28, 34 (1936).Google Scholar - Pratt, John W., ‘Review of
*Testing Statistical Hypothesis*’ (Lehmann, 1959); /.*Am. Stat. Assoc*. Vol. 56, pp. 163–166 (1961).CrossRefGoogle Scholar - Roberts, Harry V., ‘Statistical Dogma: One Response to a Challenge’, Multilithed, University of Chicago (1965).Google Scholar
- Thornber, Hodson, ‘An Autoregressive Model: Bayesian Versus Sampling Theory Analysis’, Multilithed, Dept. of Economics, University of Chicago, Chicago, Illinois (1965).Google Scholar
- Wilbraham, H.,
*Phil. Mag. Series*, 4, Vol. vii, (1854).Google Scholar - Zellner, Arnold, ‘Bayesian Inference and Simultaneous Equation Models’, Multilithed, University of Chicago, Chicago, Illinois (1965).Google Scholar