Two-sample Bayesian Nonparametric Hypothesis Testing

摘要

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