Nonparametric Independence Testing for Small Sample Sizes

TL;DR

This paper explores nonparametric independence testing for small samples, demonstrating that employing shrunk operators improves test power, especially for detecting subtle nonlinear dependencies, with theoretical and empirical analysis of shrinkage methods.

Contribution

It provides empirical evidence that shrunk operators enhance independence test power in small samples and analyzes Stein shrinkage effects on HSIC, comparing SCOSE and FCOSE estimators.

Findings

Shrunk operators improve test power at low false positive rates.

FCOSE often outperforms SCOSE in test power.

SCOSE is optimal for estimating the true operator.

Abstract

This paper deals with the problem of nonparametric independence testing, a fundamental decision-theoretic problem that asks if two arbitrary (possibly multivariate) random variables $X,Y$ are independent or not, a question that comes up in many fields like causality and neuroscience. While quantities like correlation of $X,Y$ only test for (univariate) linear independence, natural alternatives like mutual information of $X,Y$ are hard to estimate due to a serious curse of dimensionality. A recent approach, avoiding both issues, estimates norms of an \textit{operator} in Reproducing Kernel Hilbert Spaces (RKHSs). Our main contribution is strong empirical evidence that by employing \textit{shrunk} operators when the sample size is small, one can attain an improvement in power at low false positive rates. We analyze the effects of Stein shrinkage on a popular test statistic called HSIC (Hilbert-Schmidt Independence Criterion). Our observations provide insights into two recently proposed shrinkage estimators, SCOSE and FCOSE - we prove that SCOSE is (essentially) the optimal linear shrinkage method for \textit{estimating} the true operator; however, the non-linearly shrunk FCOSE usually achieves greater improvements in \textit{test power}. This work is important for more powerful nonparametric detection of subtle nonlinear dependencies for small samples.