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A Bayesian analysis of classical hypothesis testing

  • Hypothesis Testing
  • Invited Papers
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Trabajos de Estadistica Y de Investigacion Operativa

Summary

The procedure of maximizing the missing information is applied to derive reference posterior probabilities for null hypotheses. The results shed further light on Lindley’s paradox and suggest that a Bayesian interpretation of classical hypothesis testing is possible by providing a one-to-one approximate relationship between significance levels and posterior probabilities.

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References

  • Bartlett, M.S. (1957). A comment on D. v. Lindley’s statistical paradox.Biometrika 44, 533–534.

    MATH  MathSciNet  Google Scholar 

  • Bernardo, J.M. (1979a). Expected information as expected utility.Ann. Statist. 7, 686–690.

    Article  MATH  MathSciNet  Google Scholar 

  • — (1979b). Reference posterior distributions for Bayesian inference.J. Roy. Statist. Soc. B 41, 113–147 (with discussion).

    MATH  MathSciNet  Google Scholar 

  • Demster, A.P. (1971). Model searching and estimation in the logic of inference. InFoundations of Statistical Inference (Godambe, V.P. and Sprott, D.A. eds.), 56–81. Toronto: Holt, Rinehart and Winston.

    Google Scholar 

  • Good, I.J. (1966). A derivation of the probabilistic explication of information.J. Roy. Statist. Soc. B 28, 578–581.

    MATH  MathSciNet  Google Scholar 

  • Ibragimov, I.A. andHasminsky, R.Z. (1973). On the information in a sample about a parameter.2nd Internat. Symp. Information Theory. (Petrov and Csaki eds.) 295–309. Budapest: Akedemiai Kiado.

    Google Scholar 

  • Jeffreys, H. (1939/67).Theory of probability. Oxford: Clarendon Press.

    Google Scholar 

  • Lindley, D.V. (1956). On a measure of the information provided by an experiment.Ann. Math. 27, 986–1005.

    MATH  MathSciNet  Google Scholar 

  • — (1957). A statistical paradox.Biometrika 44, 187–192.

    MATH  MathSciNet  Google Scholar 

  • POPPER, K.R. (1958).The logic of scientific discovery. London: Hutchinson and Co.

    Google Scholar 

  • SHANNON, C.E. (1948). A mathematical theory of communication.Bell System Tech. J. 27, 379–423, 623–656.

    MathSciNet  Google Scholar 

  • SOAL, S.G. and BATEMAN, F. (1954).Modern Experiments in Telepathy. London: Faber and Faber.

    Google Scholar 

  • STONE, M. (1958).Studies with a measure of information. Ph. D. Thesis, University of Cambridge.

References in the Discussion

  • AKAIKE, H. (1978). A Bayesian analysis of the minimum AIC procedure.Ann. Inst. Statist. Math. 30, 9–14.

    Article  MATH  MathSciNet  Google Scholar 

  • ATKINSON, A.C. (1978). Posterior probabilities for choosing a regression model.Biometrika 65, 39–48.

    Article  MATH  MathSciNet  Google Scholar 

  • COCCONI, G. and SALPETER, E.E. (1958).Nuovo Cimento 10, 646.

    Article  Google Scholar 

  • COX, R.T. (1946).Amer. J. Phys. 14, 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  • — (1961).The Algebra of Probable Inference. Baltimore: John Hopkins.

    MATH  Google Scholar 

  • — (1978). Of inference and inquiry. InThe Maximum-Entropy Formalism. (Levine, R.D. & Tribus, M. eds.) 119–167. Cambridge, Mass.: M.I.T. Press.

    Google Scholar 

  • DEMPSTER, A.P. (1973). The direct use of likelihood for significance testing.Proceedings of the Conference on Foundational Questions in Statistical Inference. (Barndorff-Nielsen, O., Blaesild, P. and Schou, G., eds.) 335–352. University of Aarhus.

  • DICKEY, J.M. (1971). The weighted, likelihood ratio, linear hypotheses on normal location parameters.Ann. Math. Statist. 42, 204–223.

    Article  MATH  MathSciNet  Google Scholar 

  • — (1978). Approximate coherence for regression model inference with a new analysis of Fisher’s Broadbalk Wheatfield example.Bayesian Analysis in Econometric and Statistics: Essays in Honor of Harold Jeffreys (Zellner, A. ed.) 333–354. Amsterdam: North-Holland.

    Google Scholar 

  • GEISSER, S. (1980). The contributions of Sir Harold Jeffreys to Bayesian Inference. InBayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys. (Zellner, A., ed.) 13–20. Amsterdam: North Holland.

    Google Scholar 

  • GEISSER, S. and EDDY, W.F. (1979). A predictive approach to model selection.J. Amer. Statist. Assoc. 74, 153–160.

    Article  MATH  MathSciNet  Google Scholar 

  • GOOD, I.J. (1950).Probability and the Weighing of Evidence. New York: Haffners.

    MATH  Google Scholar 

  • — (1957). Saddle-point methods for the multinomial distribution.Ann. Math. Statist. 28, 861–880.

    Article  MathSciNet  Google Scholar 

  • — (1969). What is the use of a distribution? InMultivariate Analysis II. (Krishaiah, P.R., ed.) 183–203. New York: Academic Press.

    Google Scholar 

  • GOOD, I.J. and CROOK, J.F. (1974). The Bayes/Non Bayes compromise and the multinomial distribution.J. Amer. Statist. Assoc. 69. 711–720.

    Article  MATH  Google Scholar 

  • JAYNES, E.T. (1976). Confidence intervals vs. Bayesian intervals (with discussion). InFoundations of Probability Theory, Statistical Inference, and Statistical Theories of Science. (Harper, W.L. & Hooker, C.A., eds.) 175–257. Dordrecht, Holland: D. Reidel.

    Google Scholar 

  • — (1979). Marginalization and prior probabilities (with discussion). InBayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys. (Zellner, A., ed.) 43–87. Amsterdam: North-Holland.

    Google Scholar 

  • JEFFREYS, H. (1963). Review of L.J. Savage, et. al.The Foundations of Statistical Inference, (1962).Technometrics 5, 407–410.

    Article  Google Scholar 

  • — (1967).Theory of Probability. (3rdrev.ed.). Oxford: University Press.

    Google Scholar 

  • JEFFREYS, H. (1979). Personal communication.

  • KLEIN, F. (1939).The Monist 39, 350–364.

    Google Scholar 

  • LINDLEY, D.V. (1957). A statistical paradox.Biometrika 44, 187–192.

    MATH  MathSciNet  Google Scholar 

  • — (1961). The use of prior probability distributions in statistical inference and decision.Proc. 4th. Berkeley Symposium 1, 453–468.

    MathSciNet  Google Scholar 

  • — (1965).Introduction to Probability and Statistics from a Bayesian Viewpoint: Part 2, Inference. Cambridge: University Press.

    MATH  Google Scholar 

  • MOLINA, E.C. (1963). Some comments on Bayes’ Essay. InTwo Papers by Bayes (Deming, W.E., ed.) 7–12. New York: Hafner Pub. Co.

    Google Scholar 

  • PEARSON, E.S. (1962). Some thoughts on statistical inference.Ann. Math. Statist. 33, 394–403.

    Article  MATH  MathSciNet  Google Scholar 

  • SAVAGE, L.J. (1962).The Foundations of Statistical Inference: A Discussion. London: Methuen.

    Google Scholar 

  • SCHWARZ, G. (1978). Estimating the dimension of a model.Ann. Statist. 6, 461–464.

    Article  MATH  MathSciNet  Google Scholar 

  • SHERWIN, C.W.,et al. (1960).Phys. Rev. Lett. 4, 399–400.

    Article  Google Scholar 

  • SMITH, A.F.M. and SPIEGELHALTER, D.J. (1980). Bayes factors and choice criteria for linear models.J. Roy. Statist. Soc. B. 42, 213–220.

    MATH  MathSciNet  Google Scholar 

  • WEISSKOPF, V.F. (1961). Selected topics in theoretical physics. InLectures in Theoretical Physics,3, 54–105. (Brittin, W.,et al. eds.) New York: Interscience Publishers, Inc.

    Google Scholar 

  • ZELLNER, A. (1971).An Introduction to Bayesian Inference in Econometrics. New York: Wiley.

    MATH  Google Scholar 

  • — (1977). Maximal data information prior distributions. InNew Developments in the Applications of Bayesian Methods. (Aykac, A. and Brumat, C., eds.) Ch. 12, 211–232. Amsterdam: North-Holland.

    Google Scholar 

  • ZELLNER, A. and SIOW, A. (1979). On posterior odds ratios for sharp null hypotheses and one-sided alternatives.Tech. Rep. University of Chicago.

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Authors and Affiliations

  1. Universidad de Valencia, Valencia, España

    José M. Bernardo

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  1. José M. Bernardo
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Bernardo, J.M. A Bayesian analysis of classical hypothesis testing. Trabajos de Estadistica Y de Investigacion Operativa 31, 605–647 (1980). https://doi.org/10.1007/BF02888370

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