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GaussDetect-LiNGAM:Causal Direction Identification without Gaussianity test

Ziyi Ding, Xiao-Ping Zhang

TL;DR

GaussDetect-LiNGAM addresses causal direction identification in the bivariate LiNGAM setting without requiring a priori Gaussianity checks. It establishes a theoretical equivalence between forward-model Gaussianity and reverse-regression residual independence, enabling robust kernel independence testing to drive direction decisions. The method abstains when residuals appear Gaussian and relies on independence evidence otherwise, reducing the reliance on Gaussianity tests and improving efficiency. Experiments show high consistency across noise types and sample sizes, with notably lower tests-per-decision than classic LiNGAM variants, enhancing practical applicability.

Abstract

We propose GaussDetect-LiNGAM, a novel approach for bivariate causal discovery that eliminates the need for explicit Gaussianity tests by leveraging a fundamental equivalence between noise Gaussianity and residual independence in the reverse regression. Under the standard LiNGAM assumptions of linearity, acyclicity, and exogeneity, we prove that the Gaussianity of the forward-model noise is equivalent to the independence between the regressor and residual in the reverse model. This theoretical insight allows us to replace fragile and sample-sensitive Gaussianity tests with robust kernel-based independence tests. Experimental results validate the equivalence and demonstrate that GaussDetect-LiNGAM maintains high consistency across diverse noise types and sample sizes, while reducing the number of tests per decision (TPD). Our method enhances both the efficiency and practical applicability of causal inference, making LiNGAM more accessible and reliable in real-world scenarios.

GaussDetect-LiNGAM:Causal Direction Identification without Gaussianity test

TL;DR

GaussDetect-LiNGAM addresses causal direction identification in the bivariate LiNGAM setting without requiring a priori Gaussianity checks. It establishes a theoretical equivalence between forward-model Gaussianity and reverse-regression residual independence, enabling robust kernel independence testing to drive direction decisions. The method abstains when residuals appear Gaussian and relies on independence evidence otherwise, reducing the reliance on Gaussianity tests and improving efficiency. Experiments show high consistency across noise types and sample sizes, with notably lower tests-per-decision than classic LiNGAM variants, enhancing practical applicability.

Abstract

We propose GaussDetect-LiNGAM, a novel approach for bivariate causal discovery that eliminates the need for explicit Gaussianity tests by leveraging a fundamental equivalence between noise Gaussianity and residual independence in the reverse regression. Under the standard LiNGAM assumptions of linearity, acyclicity, and exogeneity, we prove that the Gaussianity of the forward-model noise is equivalent to the independence between the regressor and residual in the reverse model. This theoretical insight allows us to replace fragile and sample-sensitive Gaussianity tests with robust kernel-based independence tests. Experimental results validate the equivalence and demonstrate that GaussDetect-LiNGAM maintains high consistency across diverse noise types and sample sizes, while reducing the number of tests per decision (TPD). Our method enhances both the efficiency and practical applicability of causal inference, making LiNGAM more accessible and reliable in real-world scenarios.

Paper Structure

This paper contains 16 sections, 3 theorems, 18 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Formulation
  4. Consistent Equivalence of Gaussianity Test and Independence Test in the LiNGAM
  5. Algorithms
  6. Experiments
  7. Experimental Setup
  8. Sample Generation
  9. Consistency Rate Experiment
  10. Experiment Procedure
  11. Results
  12. Tests per Decision (TPD) Experiment
  13. Experiment Procedure
  14. Results
  15. Experiment Summary
  16. ...and 1 more sections

Key Result

Lemma 4.1

Let $X_1, X_2, \ldots, X_n$ be independent random variables. If there exist non-zero constants $c_1, c_2, \dots, c_n$, $d_1, d_2, \dots, d_n$ such that the linear combinations $Y_1 = \sum_{i=1}^{n} c_i X_i$ and $Y_2 = \sum_{j=1}^{n} d_j X_j$ are independent, then $X_1, X_2, \ldots, X_n$ are normally

Figures (3)

  • Figure 1: This figure compares the Traditional Pairwise LiNGAM and GaussDetect-LiNGAM algorithms. Pairwise LiNGAM requires completed assumptions (linearity, acyclicity, exogeneity, and non-Gaussianity) and involves two steps: a Gaussianity test followed by an independence test. In contrast, GaussDetect-LiNGAM only requires the released assumptions (linearity, acyclicity, and exogeneity) and can infer causality with just one independence test, without needing the Gaussianity test.
  • Figure 2: The figure illustrates the Consistency Rate across different noise distributions (Exponential, Laplace, and Poisson) and sample sizes (n=400, n=800, n=1600). The bars in different colors represent the performance of the consistency rate for each sample size.
  • Figure 3: This figure illustrates the Tests per Decision (TPD) required by GaussDetect-LiNGAM and Pairwise-LiNGAM across different noise distributions (Exponential, Laplace, and Poisson) and sample sizes (n=400, n=800, n=1600). The bars in different colors represent the performance of GaussDetect-LiNGAM (in orange) and Pairwise-LiNGAM (in green).

Theorems & Definitions (7)

  • Definition 2.1: Bivariate LiNGAM
  • Definition 2.2: Consistent Equivalence of Hypothesis Tests
  • Lemma 4.1: Skitovich--Darmois Theorem
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof