Two-sample Bayesian Nonparametric Hypothesis Testing

Abstract

In this article we describe Bayesian nonparametric procedures for two-sample hypothesis testing. Namely, given two sets of samples ${\mathbf{y}}^{(1)}\stackrel{\mathrm{iid}}{~}{F}^{(1)}$ and ${\mathbf{y}}^{(2)}\stackrel{\mathrm{iid}}{~}{F}^{(2)}$, with $F^{(1)},F^{(2)}$ unknown, we wish to evaluate the evidence for the null hypothesis $H_0:F^{(1)}\equiv F^{(2)}$ versus the alternative $H_1:F^{(1)}\ne F^{(2)}$. Our method is based upon a nonparametric Pólya tree prior centered either subjectively or using an empirical procedure. We show that the Pólya 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 $\mathrm{Pr}(H_0|\{{\mathbf{y}}^{(1)},{\mathbf{y}}^{(2)}\}\mathbf{)}$