  In this article we describe Bayesian nonparametric procedures for two-sample
hypothesis testing. Namely, given two sets of samples
$\mathbf{y}^{\scriptscriptstyle(1)}\;$\stackrel{\scriptscriptstyle{iid}}{\s
im}$\;F^{\scriptscriptstyle(1)}$ and $\mathbf{y}^{\scriptscriptstyle(2
)}\;$\stackrel{\scriptscriptstyle{iid}}{\sim}$\;F^{\scriptscriptstyle( 2)}$,
with $F^{\scriptscriptstyle(1)},F^{\scriptscriptstyle(2)}$ unknown, we wish to
evaluate the evidence for the null hypothesis
$H_0:F^{\scriptscriptstyle(1)}\equiv F^{\scriptscriptstyle(2)}$ versus the
alternative $H_1:F^{\scriptscriptstyle(1)}\neq F^{\scriptscriptstyle(2)}$. Our
method is based upon a nonparametric P\'{o}lya tree prior centered either
subjectively or using an empirical procedure. We show that the P\'{o}lya tree
prior leads to an analytic expression for the marginal likelihood under the two
hypotheses and hence an explicit measure of the probability of the null
$\mathrm{Pr}(H_0|\{\mathbf
{y}^{\scriptscriptstyle(1)},\mathbf{y}^{\scriptscriptstyle(2)}\}\mathbf{)}$.
