A Kernel Test for Three-Variable Interactions

TL;DR

This paper presents kernel-based nonparametric tests for detecting three-variable interactions and total independence, demonstrating high sensitivity and effectiveness in complex causal structures.

Contribution

It introduces new kernel-based tests for three-variable interactions and total independence, with a focus on detecting subtle combined effects in directed graphical models.

Findings

Lancaster test detects weak individual influences with strong combined effects

Test outperforms competitors in identifying V-structures in graphical models

Tests are computationally straightforward and consistent against all alternatives

Abstract

We introduce kernel nonparametric tests for Lancaster three-variable interaction and for total independence, using embeddings of signed measures into a reproducing kernel Hilbert space. The resulting test statistics are straightforward to compute, and are used in powerful interaction tests, which are consistent against all alternatives for a large family of reproducing kernels. We show the Lancaster test to be sensitive to cases where two independent causes individually have weak influence on a third dependent variable, but their combined effect has a strong influence. This makes the Lancaster test especially suited to finding structure in directed graphical models, where it outperforms competing nonparametric tests in detecting such V-structures.