Conditional independence testing based on a nearest-neighbor estimator of conditional mutual information

TL;DR

This paper introduces a non-parametric, nearest-neighbor based conditional independence test for continuous data that effectively handles nonlinear and high-dimensional dependencies, outperforming kernel-based methods in calibration and power.

Contribution

It presents a novel conditional independence test combining nearest-neighbor estimation with a local permutation scheme, improving calibration and power over existing kernel-based methods.

Findings

Reliable null distribution simulation for small samples and high dimensions

Better calibration than kernel-based tests, especially for non-smooth densities

Comparable or higher power levels than existing methods

Abstract

Conditional independence testing is a fundamental problem underlying causal discovery and a particularly challenging task in the presence of nonlinear and high-dimensional dependencies. Here a fully non-parametric test for continuous data based on conditional mutual information combined with a local permutation scheme is presented. Through a nearest neighbor approach, the test efficiently adapts also to non-smooth distributions due to strongly nonlinear dependencies. Numerical experiments demonstrate that the test reliably simulates the null distribution even for small sample sizes and with high-dimensional conditioning sets. The test is better calibrated than kernel-based tests utilizing an analytical approximation of the null distribution, especially for non-smooth densities, and reaches the same or higher power levels. Combining the local permutation scheme with the kernel tests leads to better calibration, but suffers in power. For smaller sample sizes and lower dimensions, the test is faster than random fourier feature-based kernel tests if the permutation scheme is (embarrassingly) parallelized, but the runtime increases more sharply with sample size and dimensionality. Thus, more theoretical research to analytically approximate the null distribution and speed up the estimation for larger sample sizes is desirable.