# Causal Effect Identification in Heterogeneous Environments from Higher-Order Moments

arXiv: https://arxiv.org/abs/2506.11756 · HTML: https://mlsanity.com/paper/2506.11756/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 35 references that are in the map.
- First posted on arXiv 13 Jun 2025.

## Builds on (most cited first, 23 of 23)

- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [Causal Discovery from Heterogeneous/Nonstationary Data with Independent Changes](https://mlsanity.com/paper/1903.01672/) (2019)
- [Invariant Causal Prediction for Nonlinear Models](https://mlsanity.com/paper/1706.08576/) (2017)
- [Invariance, Causality and Robustness](https://mlsanity.com/paper/1812.08233/) (2018)
- [Anchor regression: heterogeneous data meets causality](https://mlsanity.com/paper/1801.06229/) (2018)
- [Invariant Causal Prediction for Block MDPs](https://mlsanity.com/paper/2003.06016/) (2020)
- [Permutation-Based Causal Structure Learning with Unknown Intervention Targets](https://mlsanity.com/paper/1910.09007/) (2019)
- [Invariant Causal Prediction for Sequential Data](https://mlsanity.com/paper/1706.08058/) (2017)
- [Causal Discovery in Heterogeneous Environments Under the Sparse Mechanism Shift Hypothesis](https://mlsanity.com/paper/2206.02013/) (2022)
- [Nonlinear Invariant Risk Minimization: A Causal Approach](https://mlsanity.com/paper/2102.12353/) (2021)
- [Learning Linear Non-Gaussian Causal Models in the Presence of Latent Variables](https://mlsanity.com/paper/1908.03932/) (2019)
- [Learning Causal Structures Using Regression Invariance](https://mlsanity.com/paper/1705.09644/) (2017)
- [BaCaDI: Bayesian Causal Discovery with Unknown Interventions](https://mlsanity.com/paper/2206.01665/) (2022)
- [Invariant Representation Learning for Treatment Effect Estimation](https://mlsanity.com/paper/2011.12379/) (2020)
- [Optimization-based Causal Estimation from Heterogenous Environments](https://mlsanity.com/paper/2109.11990/) (2021)
- [Causal Discovery with Heterogeneous Observational Data](https://mlsanity.com/paper/2201.12392/) (2022)
- [A Cross-Moment Approach for Causal Effect Estimation](https://mlsanity.com/paper/2306.06263/) (2023)
- [Meta-Learned Invariant Risk Minimization](https://mlsanity.com/paper/2103.12947/) (2021)
- [Causal Discovery of Linear Non-Gaussian Causal Models with Unobserved Confounding](https://mlsanity.com/paper/2408.04907/) (2024)
- [Identifying Invariant Factors Across Multiple Environments with KL Regression](https://mlsanity.com/paper/2002.08341/) (2020)
- [Causal Invariance Learning via Efficient Nonconvex Optimization](https://mlsanity.com/paper/2412.11850/) (2024)

## Built on by (most cited first, 2 of 2)

- [Estimating Treatment Effects with Independent Component Analysis](https://mlsanity.com/paper/2507.16467/) (2025)
- [Sample Complexity of Causal Identification with Temporal Heterogeneity](https://mlsanity.com/paper/2602.06899/) (2026)

## Abstract (arXiv)

We investigate the estimation of the causal effect of a treatment variable on an outcome in the presence of a latent confounder. We first show that the causal effect is identifiable under certain conditions when data is available from multiple environments, provided that the target causal effect remains invariant across these environments. Secondly, we propose a moment-based algorithm for estimating the causal effect as long as only a single parameter of the data-generating mechanism varies across environments -- whether it be the exogenous noise distribution or the causal relationship between two variables. Conversely, we prove that identifiability is lost if both exogenous noise distributions of both the latent and treatment variables vary across environments. Finally, we propose a procedure to identify which parameter of the data-generating mechanism has varied across the environments and evaluate the performance of our proposed methods through experiments on synthetic data.