Nonparametric Independence Testing for Small Sample Sizes
This work addresses the challenge of detecting subtle nonlinear dependencies in small datasets, which is important for fields like causality and neuroscience, though it appears incremental as it builds on existing RKHS-based methods.
The paper tackles the problem of nonparametric independence testing for small sample sizes by applying shrinkage operators to the HSIC test statistic, demonstrating improved test power at low false positive rates through empirical evidence.
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.