WIKS: A general Bayesian nonparametric index for quantifying differences between two populations

TL;DR

This paper introduces WIKS, a Bayesian nonparametric index for measuring differences between two populations, which is computationally efficient, theoretically justified, and more powerful than existing methods.

Contribution

The paper proposes WIKS, a novel Bayesian nonparametric index based on the Kolmogorov-Smirnov distance, with a flexible prior, fast computation, and proven consistency.

Findings

WIKS outperforms competing methods in simulation studies.

WIKS maintains control over significance levels under the null hypothesis.

WIKS effectively detects differences in real-world Alzheimer patient data.

Abstract

The problem of deciding whether two samples arise from the same distribution is often the question of interest in many research investigations. Numerous statistical methods have been devoted to this issue, but only few of them have considered a Bayesian nonparametric approach. We propose a nonparametric Bayesian index (WIKS) which has the goal of quantifying the difference between two populations $P_1$ and $P_2$ based on samples from them. The WIKS index is defined by a weighted posterior expectation of the Kolmogorov-Smirnov distance between $P_1$ and $P_2$ and, differently from most existing approaches, can be easily computed using any prior distribution over $(P_1,P_2)$. Moreover, WIKS is fast to compute and can be justified under a Bayesian decision-theoretic framework. We present a simulation study that indicates that the WIKS method is more powerful than competing approaches in several settings, even in multivariate settings. We also prove that WIKS is a consistent procedure and controls the level of significance uniformly over the null hypothesis. Finally, we apply WIKS to a data set of scale measurements of three different groups of patients submitted to a questionnaire for Alzheimer diagnostic.