TESTING A POINT NULL HYPOTHESIS: THE COMPARISON OF P-VALUES AND BAYESIAN EVIDENCE IN MULTIVARIATE NORMAL DISTRIBUTION

Testing of point null hypothesis on the mean vector of a multivariate normal distribution is studied. Of interest is relationship between p-value (or observed significance level) and Bayesian measure of evidence against the null hypothesis, expressed in terms of the infimum of the posterior probability of the null hypothesis. When the dimension of the parameter space is one, the p-value is much smaller than the Bayesian evidence. For dimension greater than one the p-value is typically larger than the Bayesian evidence. If the symmetric class of priors were used, like the uninvariate distributions, the Bayesian evidence is much larger than the corresponding p-value and remains constant for large dimension of parameter. This means that p-values and Bayesian evidence are irreconcilable.