Practical Kernel Tests of Conditional Independence (2402.13196v1)

Abstract: We describe a data-efficient, kernel-based approach to statistical testing of conditional independence. A major challenge of conditional independence testing, absent in tests of unconditional independence, is to obtain the correct test level (the specified upper bound on the rate of false positives), while still attaining competitive test power. Excess false positives arise due to bias in the test statistic, which is obtained using nonparametric kernel ridge regression. We propose three methods for bias control to correct the test level, based on data splitting, auxiliary data, and (where possible) simpler function classes. We show these combined strategies are effective both for synthetic and real-world data.

• The paper presents SplitKCI, a novel bias reduction method that mitigates inflated false positive rates in kernel-based conditional independence tests.
• It employs data splitting, auxiliary data, and simpler regression functions to reduce bias, leading to superior Type I error control and enhanced test power.
• Empirical results demonstrate SplitKCI's robustness in high-dimensional and unbalanced scenarios, making it a promising tool for causal discovery and complex data analysis.

Strategies for Enhancing Kernel-Based Conditional Independence Tests

Introduction

In the landscape of statistical testing, Conditional Independence (CI) testing is pivotal for deciphering the interdependencies of random variables in the presence of potential confounders. This work scrutinizes the kernel-based approach to CI testing, highlighting its data efficiency and utility across various applications, from basic scientific inquiry to the evaluation of machine learning methods and causal discovery.

Kernel-Based Conditional Independence Tests

Kernel-based CI tests, particularly Kernel-based Conditional Independence (KCI) and Conditional Independence Regression CovariancE (CIRCE), leverage kernel analogues in place of linear regression residuals and encompass a kernel-based alternative to covariance. These methods possess the theoretical capability to detect any form of conditional dependence. However, their practical application is hampered in low-data regimes due to bias in nonparametric regression used for conditional feature means estimation, threatening test validity through inflated false positive rates.

Bias Reduction Strategies

This paper introduces three bias correction techniques - data splitting, the use of auxiliary data, and employment of simpler function classes for regression - culminating in the proposal of a modified KCI/CIRCE method named SplitKCI. By deploying data splitting in the computation of conditional mean embeddings (CME) and differing feature space dimensions for the two regressions involved, SplitKCI markedly reduces test statistic bias without compromising test consistency. Illustratively, improvements in Type I error control and test power under several synthetic data settings underscore the efficacy of these bias mitigation strategies.

Empirical Evaluation

Empirical investigations reveal SplitKCI’s superior Type I error control compared to conventional KCI in both balanced and unbalanced data scenarios. Its robustness is further demonstrated across tasks simulating varying degrees of dependence complexity and confounding, with notable performance in high-dimensional contexts often encountered in causal discovery applications. Moreover, the flexibility in kernel choice for SplitKCI — enabling the use of non-universal kernels without sacrificing asymptotic guarantees — permits adaptation to structured data peculiarities, enhancing the practical applicability of kernel-based CI tests.

Theoretical Implications and Applications

The research posits SplitKCI as a promising solution to the bias challenge inherent in kernel-based CI testing, particularly beneficial when auxiliary information is accessible or when prior knowledge necessitates specific kernel functions. The methodological advancements presented here bear implications for developing more reliable tools for CI testing in complex datasets, which is cornerstone in elucidating causal relationships and ensuring fairness in algorithmic predictions.

Future Directions in Kernel-Based CI Testing

Further exploratory avenues include the integration of kernel-based methods with strategies to counteract potential data sparsity and high dimensionality, and the expansion of kernel choice paradigms to accommodate diverse data characteristics. In addition, addressing interpretability concerns related to conditional (in)dependence assertions — especially critical in domains with significant societal impact — remains an essential frontier in refining kernel-based statistical tests for widespread adoption.

Conclusion

This work not only fortifies the theoretical underpinnings of kernel-based CI testing but also enhances its practical viability through innovative bias correction techniques, exemplified by SplitKCI. By offering a more accurate and flexible testing framework, the contributions made pave the way for nuanced inference of conditional independence across varied scientific and technological fields.