Distance between the Distributions of the P-value and the Lower Bound of the Posterior Probability

  • Published : 1999.04.01

Abstract

It has been issued that the irreconcilability of the classical test for a point null and standard Bayesian formulation for testing such a point null. The infimum of the posterior probability of the null hypothesis is used as measure of evidence against the null hypothesis in Bayesian approach; here the infimum is over the family of priors on the alternative hypotheses which includes all density that are a priori reasonable. For iid observations from a multivariate normal distribution in $\textit{p}$ dimensions with an unknown mean and a covariance matrix propotional to the Identity we consider the difference and the Wolfowitz distance of the distributions of the P-value and the lower bound of the posterior probability over the family of all normal priors. The Wolfowitz distance is interpreted as the average difference of the quantiles of the two distrbutions.

Keywords

References

  1. Journal of American Statistical Association v.82 Testing a Point Null Hypothesis: The irreconcilability of Significance levels and Evidence Berger, J.O.;Sellke, T.
  2. Statistical. Science. v.2 Testing precise hypotheses (with discussion) Berger, J.O.;Delampady, M.
  3. Journal of American Statistical Association v.82 Reconciling Bayesian and Frequentist Evidence in the One-sided Testing Problem (with discussion) Casella, G.;Berger, R.L.
  4. Journal of Multivariate Analysis v.28 Lower bounds on Bayes factors for invariant testing situations Delampady, M.
  5. Journal of American Statistical Association v.84 Lower bounds on Bayes factors for interval null hypotheses Delampady, M.
  6. Technical Report #90-47 Bayesian hypothesis testing with symmetric and unimodal priors Delampady, M.
  7. Real Analysis and Probability Dudley, R.M.
  8. Journal of Statistical Planning and Inferences v.76 Comparison of the P-value and posterior probability of a sharp null hypothesis Oh, H.S.;DasGupta, A.