B0877
Title: Kernel partial correlation coefficient: A measure of conditional dependence
Authors: Zhen Huang - Columbia University (United States) [presenting]
Nabarun Deb - University of Chicago (United States)
Bodhisattva Sen - Columbia University (United States)
Abstract: A class of simple, nonparametric, yet interpretable measures of conditional dependence is proposed and studied, which is called kernel partial correlation (KPC) coefficient, between two random variables $Y$ and $Z$ given a third variable $X$, all taking values in general topological spaces. The population KPC captures the strength of conditional dependence, and it is 0 if and only if $Y$ is conditionally independent of $Z$ given $X$, and 1 if and only if $Y$ is a measurable function of $Z$ and $X$. Two consistent methods of estimating KPC are described. The first method is based on the general framework of geometric graphs, including K-nearest neighbor graphs and minimum-spanning trees. A sub-class of these estimators can be computed in near-linear time and converges at a rate that adapts automatically to the intrinsic dimensionality of the underlying distributions. The second strategy involves direct estimation of conditional mean embeddings in the RKHS framework. Using these empirical measures, a fully model-free variable selection algorithm is developed, and the consistency of the procedure is formally proven under suitable sparsity assumptions. Extensive simulation and real-data examples illustrate the superior performance of the methods compared to existing procedures.
