A Bayesian Approach to the Multivariate Behrens-Fisher Problem Under the Assumption of Proportional Covariance Matrices

Two independent random samples of sizesN 1 andN 2 from multivariate normal populationsN p (θ1,∑1) andN p (θ2,∑2) are considered. Under the null hypothesisH 0: θ1=θ2, a single θ is generated from aN p(μ, Σ) prior distribution, while underH 1: θ1≠θ2 two means are generated from the exchangeable priorN p(μ,σ). In both cases Σ will be assumed to have a vague prior distribution. For a simple covariance structure, the Bayes factorB and minimum Bayes factor in favour of the null hypotheses is derived. The Bayes risk for each hypothesis is derived and a strategy is discussed for using the Bayes factor and Bayes risks to test the hypothesis.