Hilbert space embeddings of independence tests of several variables with radial basis functions

TL;DR

This paper introduces a new class of radial basis functions for independence testing of multiple variables, extending existing methods to more complex interactions and providing a unified framework for various types of independence measures.

Contribution

It characterizes new radial basis functions for independence testing, including multivariate extensions and a generalized measure of interaction, filling a gap in current statistical methods.

Findings

New classes of radial basis functions for independence tests

Extension of Lancaster and Streitberg interactions to multivariate cases

Examples derived from high-order completely monotone functions

Abstract

In this paper, we characterize several classes of continuous radial basis functions that can be employed to determine whether a interaction of a probability is zero or not. These functions encompass standard independence tests but also the Lancaster/Streitberg interactions, and are multivariate extensions of Bernstein functions. Addressing a gap in these two probability contexts of interactions, we introduce an indexed measure of independence that generalizes the Lancaster interaction. We present several examples of these functions derived from high-order completely monotone functions.