# Generalized Independent Noise Condition for Estimating Causal Structure with Latent Variables

arXiv: https://arxiv.org/abs/2308.06718 · HTML: https://mlsanity.com/paper/2308.06718/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 64 references that are in the map.
- First posted on arXiv 13 Aug 2023.

## Builds on (most cited first, 13 of 13)

- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Causal Inference in the Presence of Latent Variables and Selection Bias](https://mlsanity.com/paper/1302.4983/) (2013)
- [Learning Sparse Causal Models is not NP-hard](https://mlsanity.com/paper/1309.6824/) (2013)
- [Learning Linear Non-Gaussian Causal Models in the Presence of Latent Variables](https://mlsanity.com/paper/1908.03932/) (2019)
- [Large-Scale Kernel Methods for Independence Testing](https://mlsanity.com/paper/1606.07892/) (2016)
- [ParceLiNGAM: A causal ordering method robust against latent confounders](https://mlsanity.com/paper/1303.7410/) (2013)
- [Causal Discovery with Unobserved Confounding and non-Gaussian Data](https://mlsanity.com/paper/2007.11131/) (2020)
- [Robust causal structure learning with some hidden variables](https://mlsanity.com/paper/1708.01151/) (2017)
- [Calculation of Entailed Rank Constraints in Partially Non-Linear and Cyclic Models](https://mlsanity.com/paper/1309.7004/) (2013)
- [Marginal likelihood and model selection for Gaussian latent tree and forest models](https://mlsanity.com/paper/1412.8285/) (2014)
- [Temporal-Spatial Causal Interpretations for Vision-Based Reinforcement Learning](https://mlsanity.com/paper/2112.03020/) (2021)
- [A Survey on Latent Tree Models and Applications](https://mlsanity.com/paper/1402.0577/) (2014)
- [Latent Tree Analysis](https://mlsanity.com/paper/1610.00085/) (2016)

## Built on by (most cited first, 11 of 11)

- [Unveiling and Causalizing CoT: A Causal Pespective](https://mlsanity.com/paper/2502.18239/) (2025)
- [Parameter identification in linear non-Gaussian causal models under general confounding](https://mlsanity.com/paper/2405.20856/) (2024)
- [The Third Pillar of Causal Analysis? A Measurement Perspective on Causal Representations](https://mlsanity.com/paper/2505.17708/) (2025)
- [Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learning](https://mlsanity.com/paper/2603.04780/) (2026)
- [Automating the Selection of Proxy Variables of Unmeasured Confounders](https://mlsanity.com/paper/2405.16130/) (2024)
- [Deep Discrete Encoders: Identifiable Deep Generative Models for Rich Data with Discrete Latent Layers](https://mlsanity.com/paper/2501.01414/) (2025)
- [On Theoretical Identifiability of Discrete Latent Causal Graphical Models](https://mlsanity.com/paper/2505.18410/) (2025)
- [Position: Beyond Sensitive Attributes, ML Fairness Should Quantify Structural Injustice via Social Determinants](https://mlsanity.com/paper/2508.08337/) (2025)
- [Linear Causal Representation Learning by Topological Ordering, Pruning, and Disentanglement](https://mlsanity.com/paper/2509.22553/) (2025)
- [Iterative Identification Closure: Amplifying Causal Identifiability in Linear SEMs](https://mlsanity.com/paper/2604.09309/) (2026)
- [A Recursive Decomposition Framework for Causal Structure Learning in the Presence of Latent Variables](https://mlsanity.com/paper/2605.10651/) (2026)

## Abstract (arXiv)

We investigate the task of learning causal structure in the presence of latent variables, including locating latent variables and determining their quantity, and identifying causal relationships among both latent and observed variables. To this end, we propose a Generalized Independent Noise (GIN) condition for linear non-Gaussian acyclic causal models that incorporate latent variables, which establishes the independence between a linear combination of certain measured variables and some other measured variables. Specifically, for two observed random vectors bfY and bfZ, GIN holds if and only if ωintercalY and Z are independent, where ω is a non-zero parameter vector determined by the cross-covariance between Y and Z. We then give necessary and sufficient graphical criteria of the GIN condition in linear non-Gaussian acyclic models. Roughly speaking, GIN implies the existence of a set S such that S is causally earlier (w.r.t. the causal ordering) than Y, and that every active (collider-free) path between Y and Z must contain a node from S. Interestingly, we find that the independent noise condition (i.e., if there is no confounder, causes are independent of the residual derived from regressing the effect on the causes) can be seen as a special case of GIN. With such a connection between GIN and latent causal structures, we further leverage the proposed GIN condition, together with a well-designed search procedure, to efficiently estimate Linear, Non-Gaussian Latent Hierarchical Models (LiNGLaHs), where latent confounders may also be causally related and may even follow a hierarchical structure. We show that the causal structure of a LiNGLaH is identifiable in light of GIN conditions. Experimental results show the effectiveness of the proposed method.