  This paper deals with a new Bayesian approach to the two-sample problem. More
specifically, let $x=(x_1,\ldots,x_{n_1})$ and $y=(y_1,\ldots,y_{n_2})$ be two
independent samples coming from unknown distributions $F$ and $G$,
respectively. The goal is to test the null hypothesis $\mathcal{H}_0:~F=G$
against all possible alternatives. First, a Dirichlet process prior for $F$ and
$G$ is considered. Then the change of their Cram\'{e}r-von Mises distance from
a priori to a posteriori is compared through the relative belief ratio. Many
theoretical properties of the procedure have been developed and several
examples have been discussed, in which the proposed approach shows excellent
performance.
