Conditional independence testing based on a nearest-neighbor estimator of conditional mutual information (1709.01447v1)

Abstract: Conditional independence testing is a fundamental problem underlying causal discovery and a particularly challenging task in the presence of nonlinear and high-dimensional dependencies. Here a fully non-parametric test for continuous data based on conditional mutual information combined with a local permutation scheme is presented. Through a nearest neighbor approach, the test efficiently adapts also to non-smooth distributions due to strongly nonlinear dependencies. Numerical experiments demonstrate that the test reliably simulates the null distribution even for small sample sizes and with high-dimensional conditioning sets. The test is better calibrated than kernel-based tests utilizing an analytical approximation of the null distribution, especially for non-smooth densities, and reaches the same or higher power levels. Combining the local permutation scheme with the kernel tests leads to better calibration, but suffers in power. For smaller sample sizes and lower dimensions, the test is faster than random fourier feature-based kernel tests if the permutation scheme is (embarrassingly) parallelized, but the runtime increases more sharply with sample size and dimensionality. Thus, more theoretical research to analytically approximate the null distribution and speed up the estimation for larger sample sizes is desirable.

• The paper introduces a new non-parametric test for conditional independence using a nearest-neighbor estimator of conditional mutual information and a local permutation scheme.
• Experimental results show the proposed test is well-calibrated and powerful, outperforming kernel-based methods in controlling false positives with small samples and complex dependencies.
• The research offers a robust tool for causal inference, adaptable to nonlinear and high-dimensional data, though future work is needed to improve computational efficiency for large datasets.

An Analysis of Conditional Independence Testing via Nearest-Neighbor Estimation of Conditional Mutual Information

This paper introduces a method for conducting conditional independence (CI) tests focused on continuous data, utilizing a non-parametric approach based on conditional mutual information (CMI). The paper addresses a critical challenge in causal discovery, where identifying CI is fundamental for inferring causal relationships among variables. The proposed CI test is designed to handle situations where nonlinear dependencies and high-dimensional datasets complicate traditional analysis. The research distinguishes itself by leveraging a nearest-neighbor estimator of CMI, coupled with a local permutation scheme to enhance adaptability and robustness, particularly under non-smooth density distributions.

Overview of Methodology

The authors present a methodology that estimates CMI using the Kozachenko-Leonenko $k$-nearest neighbor estimator. This estimator is noted for its adaptability to data, thanks to locally variable hypercubes that adjust based on sample density. Despite its practicality, the lack of theoretical underpinnings regarding convergence rates and finite-sample variance in mutual information estimation is a noted limitation. To simulate the null distribution needed for hypothesis testing, the paper introduces a novel nearest-neighbor permutation scheme. This scheme preserves local dependencies between variables, enhancing the alignment of the permuted distribution with the true null distribution of CMI, even with small datasets.

Experimental Results

The paper presents extensive experimental validation, demonstrating that the proposed CMI test is well-calibrated and maintains power, even when dealing with strongly nonlinear dependencies and high-dimensional conditioning sets. In comparisons to kernel-based approaches such as Kernel Conditional Independence Test (KCIT), the Randomized Conditional Independence Test (RCIT), and the Randomized Conditional Correlation Test (RCoT), the nearest-neighbor-based CMI test exhibits superior calibration in terms of false positive control, particularly with small sample sizes or complex dependency structures.

However, computational efficiency is flagged as a challenge, with the runtime of the nearest-neighbor test increasing significantly with larger datasets and higher dimensions. The paper suggests that analytical approximations of the null distribution, alongside potential improvements in NN search algorithms, could mitigate these limitations.

Implications and Future Directions

The research carries important implications for causal inference, providing a tool that can reliably and efficiently test for conditional independence without heavy computational costs associated with large kernel matrices in traditional methods. By eschewing fixed bandwidths, the nearest-neighbor CMI estimator is particularly advantageous in adaptive scenarios, which is critical when dealing with heterogeneous data distributions typical of complex systems.

Future work can focus on addressing computational bottle-necks and exploring the theoretical properties of the nearest-neighbor CMI estimator to provide guidance on choice parameters in practice. Analytical theory to support permutation strategies and null distribution modeling could also extend the practical applicability of this method, especially in big data contexts where computational resources are a constraint.

Conclusion

This paper offers a significant contribution to the field of causal discovery by introducing a CMI-based CI test that effectively accommodates the challenges of high-dimensional and nonlinear dependencies. Its well-calibrated nature assures reliable false positive rates across a broad spectrum of sample sizes, marking it as a robust choice for researchers in statistical causality and related domains. As the method continues to develop, extensions addressing scalability will further enhance its utility across diverse research settings.