Mutual Dependence: A Novel Method for Computing Dependencies Between Random Variables
This work addresses a fundamental issue in data science for researchers and practitioners needing reliable dependency measures, offering a significant improvement over existing methods.
The paper tackled the problem of estimating dependencies between random variables by introducing a novel method to compute mutual dependence, which satisfies Granger's axioms and was previously not directly computable from data. The result shows that their estimator requires fewer samples to converge, is faster to compute, and captures more complex dependencies than standard measures like Pearson's and distance correlation.
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.