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Identification of Causal Direction under an Arbitrary Number of Latent Confounders

Wei Chen, Linjun Peng, Zhiyi Huang, Haoyue Dai, Zhifeng Hao, Ruichu Cai, Kun Zhang

TL;DR

This paper tackles the problem of determining causal direction between two observed variables in the presence of an arbitrary number of latent confounders under a linear, non-Gaussian framework (LvLiNGAM). It introduces a cumulant-matrix approach, constructing CM^{(k)} from higher-order joint cumulants and showing that the rank or determinant of these matrices reveals causal direction, even with latent confounding. The authors establish identifiability theorems for both latent-free and latent-affected scenarios, propose a practical, non-iterative learning algorithm that infers the number of latent variables and the causal direction via determinant criteria, and validate the method on simulated data across three cases and real stock-market data. Results demonstrate strong asymptotic performance and robustness to latent confounding, outperforming state-of-the-art methods in many settings. This work offers a scalable, principled solution for causal direction identification under widespread latent confounding, with potential impact in domains where non-Gaussianity and hidden structure are common.

Abstract

Recovering causal structure in the presence of latent variables is an important but challenging task. While many methods have been proposed to handle it, most of them require strict and/or untestable assumptions on the causal structure. In real-world scenarios, observed variables may be affected by multiple latent variables simultaneously, which, generally speaking, cannot be handled by these methods. In this paper, we consider the linear, non-Gaussian case, and make use of the joint higher-order cumulant matrix of the observed variables constructed in a specific way. We show that, surprisingly, causal asymmetry between two observed variables can be directly seen from the rank deficiency properties of such higher-order cumulant matrices, even in the presence of an arbitrary number of latent confounders. Identifiability results are established, and the corresponding identification methods do not even involve iterative procedures. Experimental results demonstrate the effectiveness and asymptotic correctness of our proposed method.

Identification of Causal Direction under an Arbitrary Number of Latent Confounders

TL;DR

This paper tackles the problem of determining causal direction between two observed variables in the presence of an arbitrary number of latent confounders under a linear, non-Gaussian framework (LvLiNGAM). It introduces a cumulant-matrix approach, constructing CM^{(k)} from higher-order joint cumulants and showing that the rank or determinant of these matrices reveals causal direction, even with latent confounding. The authors establish identifiability theorems for both latent-free and latent-affected scenarios, propose a practical, non-iterative learning algorithm that infers the number of latent variables and the causal direction via determinant criteria, and validate the method on simulated data across three cases and real stock-market data. Results demonstrate strong asymptotic performance and robustness to latent confounding, outperforming state-of-the-art methods in many settings. This work offers a scalable, principled solution for causal direction identification under widespread latent confounding, with potential impact in domains where non-Gaussianity and hidden structure are common.

Abstract

Recovering causal structure in the presence of latent variables is an important but challenging task. While many methods have been proposed to handle it, most of them require strict and/or untestable assumptions on the causal structure. In real-world scenarios, observed variables may be affected by multiple latent variables simultaneously, which, generally speaking, cannot be handled by these methods. In this paper, we consider the linear, non-Gaussian case, and make use of the joint higher-order cumulant matrix of the observed variables constructed in a specific way. We show that, surprisingly, causal asymmetry between two observed variables can be directly seen from the rank deficiency properties of such higher-order cumulant matrices, even in the presence of an arbitrary number of latent confounders. Identifiability results are established, and the corresponding identification methods do not even involve iterative procedures. Experimental results demonstrate the effectiveness and asymptotic correctness of our proposed method.
Paper Structure (20 sections, 7 theorems, 9 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 20 sections, 7 theorems, 9 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Latent-Variable Linear Non-Gaussian Acyclic Model
  4. Cumulants and its Role in Capturing Distributional Information
  5. Intuition: Cumulants Contain Information of Causal Asymmetry
  6. Proposed Method
  7. Cumulant Matrix
  8. Causal Discovery without Latent Variables
  9. Causal Discovery with Latent Variables
  10. Learning Algorithm
  11. Experiments
  12. Results on Simulated Data
  13. Experiment Setups.
  14. Evaluation in Case 1.
  15. Evaluation in Case 2.
  16. ...and 5 more sections

Key Result

Theorem 4.2

Assume that $X$ and $Y$ are generated by Eq. eq:path coefficients LvLiNGAM X,Y and there are no shared latent variables influencing them. Define $CM^{(k)}_{(X,Y)}$ and $CM^{(k)}_{(Y,X)}$ as cumulant matrices of $X$ and $Y$ with $k = 2$. Then, 1) $X$ is a cause of $Y$ if and only if $rank(CM^{(k)}_{(

Figures (3)

  • Figure 1: Three causal structures over two observed variables $X$ and $Y$ with causal relationship $X \to Y$, each influenced by a different number of latent variables. (a) No latent variables. (b) One latent variable $L_1$. (c) Multiple latent variables $L_1, \dots, L_m$.
  • Figure 2: Accuracy scores on various sample sizes and different numbers of latent variables.
  • Figure 3: Accuracy scores on various True Number (TN) and Assumed Number (AN) of latent variables.

Theorems & Definitions (9)

  • Definition 2.1: Cumulants brillinger2001time
  • Definition 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 4.5
  • Theorem 4.6
  • Theorem 4.7
  • Corollary 4.8