Journal of the Italian Statistical Society

, Volume 1, Issue 1, pp 17–32 | Cite as

The application of robust Bayesian analysis to hypothesis testing and Occam's Razor

  • James O. Berger
  • William H. Jefferys


Robust Bayesian analysis deals simultaneously with a class of possible prior distributions, instead of a single distribution. This paper concentrates on the surprising results that can be obtained when applying the theory to problems of testing precise hypotheses when the “objective” class of prior distributions is assumed. First, an example is given demonstrating the serious inadequacy of P-values for this problem. Next, it is shown how the approach can provide statistical quantification of Occam's Razor, the famous principle of science that advocates choice of the simpler of two hypothetical explanations of data. Finally, the theory is applied to multinomial testing.


Bayes factor P-value Occam's Razor Monte-Carlo integration Mercury's perihilion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Berger J. (1985),Statistical Decision Theory and Bayesian Analysis. Springer-Verlag, New York.zbMATHGoogle Scholar
  2. Berger J., (1990), Robust Bayesian analysis: sensitivity to the prior.J. Statist. Planning and Inference, 25, 303–328.zbMATHCrossRefGoogle Scholar
  3. Berger J. andDelampady, M., (1987), Testing precise hypotheses (with Discussion).Statistical Science, 2, 317–352.zbMATHMathSciNetGoogle Scholar
  4. Berger J. andSellke T. (1987), Testing of a point null hypothesis: the irreconciliability of significance levels and evidence (with Discussion).J. Amer. Statist. Assoc., 82, 112–139.zbMATHCrossRefMathSciNetGoogle Scholar
  5. Brush S., (1989), Prediction and theory evaluation: The case of light bending.Science, 246, 1124–1129. See also the responses to this article inScience, 248, 422–423.CrossRefGoogle Scholar
  6. Casella G. andBerger R. L. (1987), Reconciling Bayesian and frequentist evidence in the one-sided testing problem.J. Amer. Statist. Assoc., 82, 106–111.zbMATHCrossRefMathSciNetGoogle Scholar
  7. Degroot M. H., (1973), Doing what comes naturally: Interpreting a tail area as a posterior probability or as a like lihood ratio.J. Amer. Statist. Assoc., 68, 966–969.zbMATHCrossRefMathSciNetGoogle Scholar
  8. Delampady M., (1989a), Lower bounds on Bayes factors for interval hypotheses.J. Amer. Statist. Assoc., 84, 120–124.zbMATHCrossRefMathSciNetGoogle Scholar
  9. Delampady M. (1989b), Lower bounds on Bayes factors for invariant testing situations.J. Multivariate Anal. 28, 227–246.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Delampady M. andBerger J., (1990). Lower bounds on posterior probabilities for multinomial and chi-squared tests.Ann. Statist., 18, 1295–1316.zbMATHMathSciNetGoogle Scholar
  11. Dempster A. P., (1973), The direct use of likelihood for significance testing. InProc. of the Conference on Foundational Questions in Statistical Inference (O. Barndorff-Nielsen et al., eds.) 335–352. Dept. Theoretical Statistics, Univ. Aarhus.Google Scholar
  12. Dickey J. M., (1977), Is the tail area useful as an approximate Bayes factor?J. Amer. Statist. Assoc. 72, 138–142.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Edwards W., Lindman H. andSavage L. J. (1963), Bayesian statistical inference for psychological research.Psychological Review, 70, 193–242.CrossRefGoogle Scholar
  14. Good I. J., (1967), A Bayesian significance test for the multinomial distribution.J. Roy. Statist. Soc. B, 29, 399–431.zbMATHMathSciNetGoogle Scholar
  15. Good I. J., (1983),Good Thinking: The Foundations of Probability and Its Applications. Univ. Minnesota Press, Minneapolis.zbMATHGoogle Scholar
  16. Good I. J., (1984), Notes C140, C144, C199, C200 and C201.J. Statist. Comput. Simulation, 19.Google Scholar
  17. Gull S., (1988), Bayesian inductive inference and maximum entropy, In G. J. Erickson and C. R. Smith (eds.),Maximum Entropy and Bayesian Methods in Science and Engineering (Vol. 1) 53–74. Dordrecht: Kluwer Academic Publishers.Google Scholar
  18. Jaynes E. T., (1979), Inference, method, and decision: Towards a Bayesian philosophy of science.J. Amer. Statist. Assoc., 74, 740–741.CrossRefGoogle Scholar
  19. Jefferys W. H., (1990), Bayesian analysis of random event generator data.Journal for Scientific Exploration, 4, 153–169.Google Scholar
  20. Jeffreys H. (1921), Secular perturbations of the inner planets.Science, 54, 248.CrossRefGoogle Scholar
  21. Jeffreys H., (1939),Theory of Probability, Third Edition, (1961), Oxford: Clarendon Press.Google Scholar
  22. Lindley D. V., (1957), A statistical paradox.Biometrika, 44, 187–192.zbMATHMathSciNetGoogle Scholar
  23. Loredo T. J., (1990), From Laplace to Supernova 1987 A: Bayesian inference in astrophysics. In P. Fougere (ed.),Maximum Entropy and Bayesian Methods, 81–142. Dordrecht: Kluwer Academic Publishers.Google Scholar
  24. Nobili A. M. andWill C. M., (1986), The real value of Mercury's perihelion advance.Nature, 320, 39–41.CrossRefGoogle Scholar
  25. Poor C. L., (1921), The motions of the planets and the relativity theory.Science, 54, 30–34.CrossRefGoogle Scholar
  26. Roseveare N. T. (1982),Mercury's Perihelion from Le Verrier to Einstein, Oxford: Clarendon Press.zbMATHGoogle Scholar
  27. Shafer G. (1982), Lindley's paradox (with discussion).J. Amer. Statist. Assoc. 77, 325–351.zbMATHCrossRefMathSciNetGoogle Scholar
  28. Smith, A. F. M. andSpiegelhalter D. J. (1980). Bayes factors and choice criteria for linear models.J. Roy. Statist. Soc. B, 42, 213–220.zbMATHMathSciNetGoogle Scholar
  29. Thorburn W. M., (1918). The myth of Occam's razor.Mind 27, 345–353.CrossRefGoogle Scholar
  30. Zellner A., (1984), Posterior odds ratios for regression hypotheses: General considerations and some specific results. InBasic Issues in Econometrics (A. Zellner, ed.), 275–305.Google Scholar

Copyright information

© Societa Italiana di Statistica 1992

Authors and Affiliations

  • James O. Berger
    • 2
  • William H. Jefferys
    • 1
  1. 1.University of TexasUSA
  2. 2.Department of Statistics, Math. Sciences BdgPurdue UniversityWest LafayetteUSA

Personalised recommendations