Mutual Dependence: A Novel Method for Computing Dependencies Between Random Variables

TL;DR

This paper introduces a new data-driven method to compute mutual dependence between random variables, outperforming traditional measures in convergence speed, computational efficiency, and ability to detect complex dependencies.

Contribution

The paper develops a direct, non-parametric estimator for mutual dependence based on a maximum likelihood approach, filling a gap in existing dependency measures.

Findings

Requires fewer samples for convergence

Faster computation than standard measures

Captures complex, nonlinear dependencies

Abstract

In data science, it is often required to estimate dependencies between different data sources. These dependencies are typically calculated using Pearson's correlation, distance correlation, and/or mutual information. However, none of these measures satisfy all the Granger's axioms for an "ideal measure". One such ideal measure, proposed by Granger himself, calculates the Bhattacharyya distance between the joint probability density function (pdf) and the product of marginal pdfs. We call this measure the mutual dependence. However, to date this measure has not been directly computable from data. In this paper, we use our recently introduced maximum likelihood non-parametric estimator for band-limited pdfs, to compute the mutual dependence directly from the data. We construct the estimator of mutual dependence and compare its performance to standard measures (Pearson's and distance correlation) for different known pdfs by computing convergence rates, computational complexity, and the ability to capture nonlinear dependencies. Our mutual dependence estimator requires fewer samples to converge to theoretical values, is faster to compute, and captures more complex dependencies than standard measures.