Assume that \(X_ i\sim N(\theta^ 2_ i,\sigma^ 2_ i)\) independently for \(i=1,...,k\). The variances \(\sigma^ 2_ i\) are assumed to be known. The two common questions that are asked are: (i) What is the strength of evidence against the hypothesis \(H_ 0\) of equality of the means \(\theta_ i?\) and (ii) If \(H_ 0\) is false, which mean \(\theta_ i\) is the largest? A Bayesian approach is given in this paper for calculating the posterior probability of \(H_ 0\) and the posterior probability that each \(\theta_ i\) is the largest one conditional on \(H_ 0\) being false.
Let \(\nu\) and 1-\(\nu\) be the prior probabilities of \(H_ 0\) and not \(H_ 0\), respectively. Under not \(H_ 0\), the prescription for the prior distribution on \(\theta =(\theta_ 1,...,\theta_ k)\) follows the hierarchical approach consisting of 2 stages: a distribution \(\pi_ 1(\theta | \beta,\sigma^ 2_{\pi})\) of \(\theta\) given hyperparameters \((\beta,\sigma^ 2_{\pi})\) and a distribution \(\pi_ 2(\beta,\sigma^ 2_{\pi})\).