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Causal Discovery for Linear DAGs with Dependent Latent Variables via Higher-order Cumulants

Ming Cai, Penggang Gao, Hisayuki Hara

TL;DR

This work tackles causal discovery for LvLiNGAM when latent variables can causally influence both other latents and observed variables. It introduces a cumulant-based three-stage algorithm that (i) identifies over-segmented clusters and latent parents (Stage I), (ii) infers the causal order among latent variables via source-detection and influence-removal using higher-order cumulants (Stage II), and (iii) reconstructs the full latent DAG (Stage III). The approach provides identifiability guarantees for a class of LvLiNGAM under given assumptions and demonstrates superior performance to existing methods on simulations and a real-world political democracy dataset. This cumulant-driven framework enables reliable causal structure learning in linear, non-Gaussian models with latent confounders and latent interactions, with practical implications for SEM-like analyses in domains with unobserved drivers.

Abstract

This paper addresses the problem of estimating causal directed acyclic graphs in linear non-Gaussian acyclic models with latent confounders (LvLiNGAM). Existing methods assume mutually independent latent confounders or cannot properly handle models with causal relationships among observed variables. We propose a novel algorithm that identifies causal DAGs in LvLiNGAM, allowing causal structures among latent variables, among observed variables, and between the two. The proposed method leverages higher-order cumulants of observed data to identify the causal structure. Extensive simulations and experiments with real-world data demonstrate the validity and practical utility of the proposed algorithm.

Causal Discovery for Linear DAGs with Dependent Latent Variables via Higher-order Cumulants

TL;DR

This work tackles causal discovery for LvLiNGAM when latent variables can causally influence both other latents and observed variables. It introduces a cumulant-based three-stage algorithm that (i) identifies over-segmented clusters and latent parents (Stage I), (ii) infers the causal order among latent variables via source-detection and influence-removal using higher-order cumulants (Stage II), and (iii) reconstructs the full latent DAG (Stage III). The approach provides identifiability guarantees for a class of LvLiNGAM under given assumptions and demonstrates superior performance to existing methods on simulations and a real-world political democracy dataset. This cumulant-driven framework enables reliable causal structure learning in linear, non-Gaussian models with latent confounders and latent interactions, with practical implications for SEM-like analyses in domains with unobserved drivers.

Abstract

This paper addresses the problem of estimating causal directed acyclic graphs in linear non-Gaussian acyclic models with latent confounders (LvLiNGAM). Existing methods assume mutually independent latent confounders or cannot properly handle models with causal relationships among observed variables. We propose a novel algorithm that identifies causal DAGs in LvLiNGAM, allowing causal structures among latent variables, among observed variables, and between the two. The proposed method leverages higher-order cumulants of observed data to identify the causal structure. Extensive simulations and experiments with real-world data demonstrate the validity and practical utility of the proposed algorithm.
Paper Structure (34 sections, 26 theorems, 109 equations, 7 figures, 3 tables, 6 algorithms)

This paper contains 34 sections, 26 theorems, 109 equations, 7 figures, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. LvLiNGAM
  5. Cumulants
  6. Proposed Method
  7. Stage I: Estimating Over-segmented Clusters
  8. Stage II: Identifying the Causal Order among Latent Variables
  9. Determining the Source Latent Variable
  10. Determining the Causal Order of Latent Variables
  11. Stage III: Identifying Causal Structure among Latent Variables
  12. Summary
  13. Simulations
  14. Settings
  15. Implementation
  16. ...and 19 more sections

Key Result

Proposition 2.2

For two observed variables $X_{i}$ and $X_{j}$ where $X_{j} \notin Anc(X_{i})$. Let $m := \min(\sum^{k_{2}-k_{1}+1}_{i=1}i, k_{1})$. Then,

Figures (7)

  • Figure 2.1: Examples of LvLiNGAMs
  • Figure 3.1: Two examples of LvLiNGAM with impure children
  • Figure 3.2: An example of merging clusters in Stage II
  • Figure 3.3: Examples of LvLiNGAMs
  • Figure 4.1: Six models for simulations
  • ...and 2 more figures

Theorems & Definitions (56)

  • Definition 1.1: Causal cluster cai2019triadxie2020generalized
  • Definition 2.1: Cumulants Brillinger
  • Proposition 2.2: Theorem 3 in schkoda2024causal
  • Proposition 2.3: Theorem 4 in schkoda2024causal
  • Proposition 2.4: Lemma 5 in schkoda2024causal
  • Definition 3.1: Triad constraint cai2019triad
  • Proposition 3.2: cai2019triad
  • Proposition 3.3: cai2019triad
  • Theorem 3.4
  • Theorem 3.5
  • ...and 46 more