Nonparametric Independence Testing for Small Sample Sizes

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 operator in Reproducing Kernel Hilbert Spaces (RKHSs). Our main contribution is strong empirical evidence that by employing 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 estimating the true operator; however, the non-linearly shrunk FCOSE usually achieves greater improvements in test power. This work is important for more powerful nonparametric detection of subtle nonlinear dependencies for small samples.