The multivariate Behrens-Fisher problem consists of testing the equality of the mean vectors of two independent multivariate normal populations with unknown and arbitrary covariance matrices. In this work we provide a Bayesian solution to the multivariate Behrens-Fisher problem based on Bayes factors. For this purpose, the problem is formulated as a homogeneity testing problem and a hierarchical model is considered that allows us to derive a Bayes factor. The use of such a hierarchical model solves the problem that arises in the calculation of the Bayes factor when improper prior distributions are used. The relationship of the proposed Bayes factor with the multivariate Behrens-Fisher distribution is analysed and it is also shown to be consistent. Finally, some examples and comparisons with some frequentist tests are given.
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