# Trek-Based Parameter Identification for Linear Causal Models With Arbitrarily Structured Latent Variables

arXiv: https://arxiv.org/abs/2507.18170 · HTML: https://mlsanity.com/paper/2507.18170/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 38 references that are in the map.
- First posted on arXiv 24 Jul 2025.

## Builds on (most cited first, 7 of 7)

- [Linear Causal Disentanglement via Interventions](https://mlsanity.com/paper/2211.16467/) (2022)
- [Learning Linear Non-Gaussian Causal Models in the Presence of Latent Variables](https://mlsanity.com/paper/1908.03932/) (2019)
- [Unpaired Multi-Domain Causal Representation Learning](https://mlsanity.com/paper/2302.00993/) (2023)
- [Identifying Representations for Intervention Extrapolation](https://mlsanity.com/paper/2310.04295/) (2023)
- [Causal Effect Identification in LiNGAM Models with Latent Confounders](https://mlsanity.com/paper/2406.02049/) (2024)
- [On the Parameter Identifiability of Partially Observed Linear Causal Models](https://mlsanity.com/paper/2407.16975/) (2024)
- [Combining Graphical and Algebraic Approaches for Parameter Identification in Latent Variable Structural Equation Models](https://mlsanity.com/paper/2302.13220/) (2023)

## Abstract (arXiv)

We develop a criterion to certify whether causal effects are identifiable in linear structural equation models with latent variables. Linear structural equation models correspond to directed graphs whose nodes represent the random variables of interest and whose edges are weighted with linear coefficients that correspond to direct causal effects. In contrast to previous identification methods, we do not restrict ourselves to settings where the latent variables constitute independent latent factors (i.e., to source nodes in the graphical representation of the model). Our novel latent-subgraph criterion is a purely graphical condition that is sufficient for identifiability of causal effects by rational formulas in the covariance matrix. To check the latent-subgraph criterion, we provide a sound and complete algorithm that operates by solving an integer linear program. While it targets effects involving observed variables, our new criterion is also useful for identifying effects between latent variables, as it allows one to transform the given model into a simpler measurement model for which other existing tools become applicable.