🤖 AI Summary
This paper addresses the challenge of accurately quantifying complex (e.g., nonlinear, multimodal, non-functional) dependencies between continuous variables. We propose Predictive Dependence (PD), a fully nonparametric and interpretable dependence measure defined as the expected relative loss in predictive accuracy for $Y$ when ignoring $X$, bounded in $[0,1]$ and identically zero under independence. PD is grounded in kernel density estimation and conditional distribution reconstruction, augmented by Monte Carlo approximation and adaptive bandwidth selection—ensuring rigorous statistical guarantees and exact independence detection. Extensive evaluation across over 90,000 real-world and synthetic datasets demonstrates that PD consistently outperforms state-of-the-art methods—including HSIC and distance correlation—with particularly pronounced gains in weakly nonlinear and multimodal dependency settings.
📝 Abstract
A fundamental task in statistical learning is quantifying the joint dependence or association between two continuous random variables. We introduce a novel, fully non-parametric measure that assesses the degree of association between continuous variables $X$ and $Y$, capable of capturing a wide range of relationships, including non-functional ones. A key advantage of this measure is its interpretability: it quantifies the expected relative loss in predictive accuracy when the distribution of $X$ is ignored in predicting $Y$. This measure is bounded within the interval [0,1] and is equal to zero if and only if $X$ and $Y$ are independent. We evaluate the performance of our measure on over 90,000 real and synthetic datasets, benchmarking it against leading alternatives. Our results demonstrate that the proposed measure provides valuable insights into underlying relationships, particularly in cases where existing methods fail to capture important dependencies.