Statistics and Computing

, Volume 7, Issue 4, pp 253–261 | Cite as

The calibration of P-values, posterior Bayes factors and the AIC from the posterior distribution of the likelihood

  • Murray Aitkin
Article

Abstract

The posterior distribution of the likelihood is used to interpret the evidential meaning of P-values, posterior Bayes factors and Akaike's information criterion when comparing point null hypotheses with composite alternatives. Asymptotic arguments lead to simple re-calibrations of these criteria in terms of posterior tail probabilities of the likelihood ratio. (‘Prior’) Bayes factors cannot be calibrated in this way as they are model-specific.

P-value likelihood posterior distribution Bayes factor fractional Bayes factor posterior Bayes factor AIC 

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References

  1. Aitkin, M. (1991) Posterior Bayes factors (with Discussion). Journal of the Royal Statistical Society, B 53, 111–42.Google Scholar
  2. Aitkin, M. (1992) Evidence and the posterior Bayes factor. Mathematical Scientist, 17, 15–25.Google Scholar
  3. Akaike, H. (1973) Information theory and the extension of the maximum likelihood principle. In Proc. 2nd Int. Symp. Information Theory (eds B. N. Petior and F. Csaki), pp. 267–81. Akademiai Kiado, Budapest.Google Scholar
  4. Anscombe, F. J. (1964) Normal likelihood functions. Annals of the Institute of Statistical Mathematics, 26, 1–19.Google Scholar
  5. Bartlett, M. S. (1957) A comment on D. V. Lindley's statistical paradox. Biometrika, 44, 533.Google Scholar
  6. Berger, J. O. (1985) Statistical Decision Theory and Bayesian Analysis, 2nd edn. Springer-Verlag, New York.Google Scholar
  7. Berger, J. O. and Sellke, T. (1987) Testing a point null hypothesis: the irreconcilability of P-values and evidence. Journal of the American Statistical Association, 82, 112–22.Google Scholar
  8. Dempster, A. P. (1974) The direct use of likelihood for significance testing. In Proc. Conf. Foundational Questions in Statistical Inference (eds O. Barndorff-Nielsen, P. Blaesild and G. Sihon), pp. 335–52. University of Aarhus.Google Scholar
  9. Geisser, S. (1992) Some statistical issues in medicine and forensics. Journal of the American Statistical Association, 87, 607–14.Google Scholar
  10. Lindley, D. V. (1957) A statistical paradox. Biometrika, 44, 187–92.Google Scholar
  11. Lindley, D. V. (1993) On the presentation of evidence. Mathematical Scientist, 18, 60–63.Google Scholar
  12. O'Hagan, A. (1995) Fractional Bayes factors for model comparisons (with Discussion). Journal of the Royal Statistical Society, B 57, 99–138.Google Scholar
  13. Schwartz, G. (1978) Estimating the dimension of a model. Annals of Statistics 6, 461–64.Google Scholar

Copyright information

© Chapman and Hall 1997

Authors and Affiliations

  • Murray Aitkin
    • 1
  1. 1.Department of StatisticsUniversity of NewcastleNewcastle-upon-Tyre

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