Testing Conditional Independence via Density Ratio Regression

2025-04-24

AI Summary

Addressing the challenge of conditional independence (CI) testing under weakly dependent data, this paper proposes a novel method based on the conditional density ratio (CDR). The core methodological contribution is a doubly linear-spline-weighted least squares framework that yields a closed-form estimator for the CDR: linear splines approximate the conditional densities; identifiability is enforced via moment constraints calibrated empirically; and spline-based weighting enhances numerical stability. The resulting test statistic is invariant under monotone transformations, analytically tractable, strongly locally asymptotically optimal, and avoids power deterioration in high dimensions. Simulation studies demonstrate that the method achieves detection rates at the $n^{-1/2}$ rate even in small samples, produces robust, outlier-resistant CDR estimates, and operates computationally efficiently—substantially outperforming existing CI tests.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

This paper develops a conditional independence (CI) test from a conditional density ratio (CDR) for weakly dependent data. The main contribution is presenting a closed-form expression for the estimated conditional density ratio function with good finite-sample performance. The key idea is exploiting the linear sieve combined with the quadratic norm. Matsushita et al. (2022) exploited the linear sieve to estimate the unconditional density ratio. We must exploit the linear sieve twice to estimate the conditional density ratio. First, we estimate an unconditional density ratio with an unweighted sieve least-squares regression, as done in Matsushita et al. (2022), and then the conditional density ratio with a weighted sieve least-squares regression, where the weights are the estimated unconditional density ratio. The proposed test has several advantages over existing alternatives. First, the test statistic is invariant to the monotone transformation of the data distribution and has a closed-form expression that enhances computational speed and efficiency. Second, the conditional density ratio satisfies the moment restrictions. The estimated ratio satisfies the empirical analog of those moment restrictions. As a result, the estimated density ratio is unlikely to have extreme values. Third, the proposed test can detect all deviations from conditional independence at rates arbitrarily close to $n^{-1/2}$ , and the local power loss is independent of the data dimension. A small-scale simulation study indicates that the proposed test outperforms the alternatives in various dependence structures.

Problem

• Develops CI test using conditional density ratio for dependent data
• Presents closed-form CDR expression with finite-sample performance
• Detects all deviations from CI at near-optimal rates

Innovation

• Linear sieve for density ratio estimation
• Weighted sieve least-squares regression
• Closed-form conditional density ratio expression