Two-sample Bayesian Nonparametric Hypothesis Testing

TL;DR

This paper introduces Bayesian nonparametric methods using Pólya tree priors for two-sample hypothesis testing, providing explicit formulas for evaluating the probability of the null hypothesis that two unknown distributions are equal.

Contribution

It develops a novel Bayesian nonparametric approach for two-sample testing using Pólya tree priors, enabling explicit computation of the null hypothesis probability.

Findings

Provides an analytic expression for the marginal likelihood under both hypotheses.

Enables explicit calculation of the probability of the null hypothesis.

Applicable with subjective or empirical Pólya tree priors.

Abstract

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