A Bayes test for simple versus one-sided hypothesis on the mean vector of a multivariate normal distribution

A Bayes test for simple versus one-sided hypothesis on the mean vector of a multivariate normal distribution is developed. Expressions for Bayes factors are derived in various cases and Monte Carlo approximation methods are suggested. For comparison of the Bayes test with a classical test, lower bounds of the posterior probability of the null hypothesis over some reasonable classes of prior distributions are derived and compared with the p-value of the classical likelihood ratio test.