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 $\mathrm{ Pr}(H_0|\{ \mathbf { y}^{ \scriptscriptstyle(1)},\mathbf{ y}^{ \scriptscriptstyle(2)}\}\mathbf{ )}$.
Categories: stat.ME