Kernel-based Tests for Joint Independence

TL;DR

This paper introduces a kernel-based method called dHSIC for testing joint independence among multiple variables, extending HSIC to higher dimensions and providing practical non-parametric tests with theoretical guarantees.

Contribution

The paper develops the dHSIC measure for testing joint independence of multiple variables using reproducing kernel Hilbert spaces, along with three non-parametric tests and their theoretical properties.

Findings

Permutation test controls significance level

Bootstrap test is asymptotically consistent

Gamma approximation is fast and effective for small dimensions

Abstract

We investigate the problem of testing whether $d$ random variables, which may or may not be continuous, are jointly (or mutually) independent. Our method builds on ideas of the two variable Hilbert-Schmidt independence criterion (HSIC) but allows for an arbitrary number of variables. We embed the $d$-dimensional joint distribution and the product of the marginals into a reproducing kernel Hilbert space and define the $d$-variable Hilbert-Schmidt independence criterion (dHSIC) as the squared distance between the embeddings. In the population case, the value of dHSIC is zero if and only if the $d$ variables are jointly independent, as long as the kernel is characteristic. Based on an empirical estimate of dHSIC, we define three different non-parametric hypothesis tests: a permutation test, a bootstrap test and a test based on a Gamma approximation. We prove that the permutation test achieves the significance level and that the bootstrap test achieves pointwise asymptotic significance level as well as pointwise asymptotic consistency (i.e., it is able to detect any type of fixed dependence in the large sample limit). The Gamma approximation does not come with these guarantees; however, it is computationally very fast and for small $d$, it performs well in practice. Finally, we apply the test to a problem in causal discovery.