MixCIT: A Kernel Based Local-Polynomial Debiased Test for Conditional Independence on Mixed-Type Data

2026-07-14

AI Summary

Existing nonparametric conditional independence tests lack a unified, efficient, and theoretically guaranteed approach for mixed-type data involving both discrete and continuous variables. This work proposes a graph-based kernel method that constructs composite neighborhoods by exact matching on discrete variables and k-nearest neighbors on continuous variables, augmented with local polynomial debiasing to correct smoothing bias. The method establishes, for the first time, asymptotic null distribution theory across all combinations of variable types, achieving a dimension-agnostic detection rate of \(n^{-1/4}\). It circumvents the phase transition issues inherent in high-dimensional geometric estimators and reduces computational complexity from cubic to nearly quadratic. The proposed test thus offers statistical consistency, computational scalability, and rigorous theoretical guarantees.

Abstract

Conditional independence testing (CIT) is fundamental to modern statistical inference in areas related to causal discovery and variable selection. While marginal independence is relatively well-understood, despite multiple advances, no existing non-parametric CIT provides a unified, efficient, and statistically guaranteed solution across heterogeneous data. We introduce a graph-based test statistic comparing kernel similarities of the response within composite neighborhoods that use exact matching on discrete components and $k_n$-nearest-neighbor matching on continuous ones. The raw statistic, related to prior constructions, suffices under fully discrete conditioning. However, when at least one conditioning variable is continuous, we instead use a local-polynomial debiased variant that cancels the local smoothing bias. We rigorously establish its asymptotic null distribution across all data-type combinations. We further prove a dimension-free $n^{-1/4}$ detection threshold under local alternatives, eliminating the phase transition that affects geometric estimators in high dimensions. Finally, we develop efficient algorithms with near-quadratic complexity and analytic graph-based calibration, bypassing the cubic bottlenecks of global kernel methods.

Problem

• conditional independence testing
• mixed-type data
• non-parametric methods
• heterogeneous data
• statistical inference

Innovation

• conditional independence testing
• mixed-type data
• local-polynomial debiasing
• kernel similarity
• graph-based statistic