A Bayesian Solution to the Multivariate Behrens—Fisher Problem

Abstract We consider the problem of finding the posterior probabilities of ellipsoids for the difference between two multivariate normal means. We do not require the two population covariance matrices to be equal, therefore, our results pertain to the multivariate Behrens—Fisher problem. A Hotelling T 2-like quantity is defined, and its cdf is expressed as a linear combination of the cdf's of some F distributions. This representation enables us to compute desired probabilities exactly using only one-dimensional numerical integrations. The calculation can easily be done with natural conjugate priors as well as with diffuse priors. Our main theoretical result, concerning convolutions of multivariate t distributions, is of some general interest. For the univariate case, we suggest a much simpler proof than that of Ruben (1960). We also give some bounds and approximations for the posterior probabilities of ellipsoids. The computation of these approximations requires much less computer time than the computation of the exact probabilities, even though the latter are quite straightforward. A numerical example and comments regarding the computation establish that this is an easily implemented approach to the analysis of the multivariate Behrens—Fisher problem.