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Causal Discovery of Linear Non-Gaussian Causal Models with Unobserved Confounding

Daniela Schkoda, Elina Robeva, Mathias Drton

TL;DR

This work tackles causal discovery in linear non-Gaussian acyclic models with latent confounding, where the full causal effects are not uniquely identifiable but the latent structure is. It introduces ReLVLiNGAM, a recursive, cumulant-based algorithm that identifies a source and its latent parents, estimates their effects on descendants via polynomial equations, and iteratively removes their influence to recover the entire graph. The approach relies on rank conditions on higher-order cumulant-based matrices to infer the number of latents and to recover a compatible path matrix $B$, with asymptotic correctness when locally $\ell\le p$. In simulations, ReLVLiNGAM achieves competitive performance with overcomplete ICA techniques without requiring the latent count in advance, and it demonstrates robustness to misspecification of the latent bound, offering a practical tool for causal effect identification in the presence of latent confounding.

Abstract

We consider linear non-Gaussian structural equation models that involve latent confounding. In this setting, the causal structure is identifiable, but, in general, it is not possible to identify the specific causal effects. Instead, a finite number of different causal effects result in the same observational distribution. Most existing algorithms for identifying these causal effects use overcomplete independent component analysis (ICA), which often suffers from convergence to local optima. Furthermore, the number of latent variables must be known a priori. To address these issues, we propose an algorithm that operates recursively rather than using overcomplete ICA. The algorithm first infers a source, estimates the effect of the source and its latent parents on their descendants, and then eliminates their influence from the data. For both source identification and effect size estimation, we use rank conditions on matrices formed from higher-order cumulants. We prove asymptotic correctness under the mild assumption that locally, the number of latent variables never exceeds the number of observed variables. Simulation studies demonstrate that our method achieves comparable performance to overcomplete ICA even though it does not know the number of latents in advance.

Causal Discovery of Linear Non-Gaussian Causal Models with Unobserved Confounding

TL;DR

This work tackles causal discovery in linear non-Gaussian acyclic models with latent confounding, where the full causal effects are not uniquely identifiable but the latent structure is. It introduces ReLVLiNGAM, a recursive, cumulant-based algorithm that identifies a source and its latent parents, estimates their effects on descendants via polynomial equations, and iteratively removes their influence to recover the entire graph. The approach relies on rank conditions on higher-order cumulant-based matrices to infer the number of latents and to recover a compatible path matrix , with asymptotic correctness when locally . In simulations, ReLVLiNGAM achieves competitive performance with overcomplete ICA techniques without requiring the latent count in advance, and it demonstrates robustness to misspecification of the latent bound, offering a practical tool for causal effect identification in the presence of latent confounding.

Abstract

We consider linear non-Gaussian structural equation models that involve latent confounding. In this setting, the causal structure is identifiable, but, in general, it is not possible to identify the specific causal effects. Instead, a finite number of different causal effects result in the same observational distribution. Most existing algorithms for identifying these causal effects use overcomplete independent component analysis (ICA), which often suffers from convergence to local optima. Furthermore, the number of latent variables must be known a priori. To address these issues, we propose an algorithm that operates recursively rather than using overcomplete ICA. The algorithm first infers a source, estimates the effect of the source and its latent parents on their descendants, and then eliminates their influence from the data. For both source identification and effect size estimation, we use rank conditions on matrices formed from higher-order cumulants. We prove asymptotic correctness under the mild assumption that locally, the number of latent variables never exceeds the number of observed variables. Simulation studies demonstrate that our method achieves comparable performance to overcomplete ICA even though it does not know the number of latents in advance.
Paper Structure (17 sections, 11 theorems, 73 equations, 8 figures, 2 algorithms)

This paper contains 17 sections, 11 theorems, 73 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contribution
  4. Notation
  5. Preliminaries
  6. Linear Structural Equation Models
  7. Tensors and Cumulants
  8. Two Observed Variables
  9. Distinguish Cause and Effect
  10. Estimating Effect Sizes
  11. Cumulants of the Latents
  12. Arbitrary Number of Variables
  13. Inferring a Source and its Effects
  14. Next Iteration
  15. Practical Implementation
  16. ...and 2 more sections

Key Result

Lemma 1

If $P^X \in \mathcal{M}_\mathcal{G}$ is defined via generic coefficients, then $\mathcal{G}$ is the unique minimal graph such that $P^X \in \mathcal{M}_\mathcal{G}$ and the number of choices for $(\Lambda, \Gamma)$ compatible with $P^X$ and $\mathcal{G}$ is

Figures (8)

  • Figure 1: Two parameter sets yielding the same observed distribution.
  • Figure 2: New edges introduced by swapping exogenous sources.
  • Figure 3: Graph $\mathcal{G}_{2,\ell}$.
  • Figure 4: Two graphs with the same number of confounders between the source and its descendants.
  • Figure 5: Cumulant estimation might fail
  • ...and 3 more figures

Theorems & Definitions (14)

  • Example 1
  • Lemma 1
  • Lemma 2
  • Example 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • ...and 4 more