# A Skewness-Based Criterion for Addressing Heteroscedastic Noise in Causal Discovery

arXiv: https://arxiv.org/abs/2410.06407 · HTML: https://mlsanity.com/paper/2410.06407/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 61 references that are in the map.
- First posted on arXiv 8 Oct 2024.

## Builds on (most cited first, 27 of 27)

- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [CAM: Causal additive models, high-dimensional order search and penalized regression](https://mlsanity.com/paper/1310.1533/) (2013)
- [Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game](https://mlsanity.com/paper/2102.13647/) (2021)
- [DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization](https://mlsanity.com/paper/2209.08037/) (2022)
- [Score matching enables causal discovery of nonlinear additive noise models](https://mlsanity.com/paper/2203.04413/) (2022)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [Diffusion Models for Causal Discovery via Topological Ordering](https://mlsanity.com/paper/2210.06201/) (2022)
- [Gradient Estimators for Implicit Models](https://mlsanity.com/paper/1705.07107/) (2017)
- [On the Identifiability and Estimation of Causal Location-Scale Noise Models](https://mlsanity.com/paper/2210.09054/) (2022)
- [Learning latent causal graphs via mixture oracles](https://mlsanity.com/paper/2106.15563/) (2021)
- [Latent Hierarchical Causal Structure Discovery with Rank Constraints](https://mlsanity.com/paper/2210.01798/) (2022)
- [A Spectral Approach to Gradient Estimation for Implicit Distributions](https://mlsanity.com/paper/1806.02925/) (2018)
- [Scalable Causal Discovery with Score Matching](https://mlsanity.com/paper/2304.03382/) (2023)
- [Causal Discovery with Score Matching on Additive Models with Arbitrary Noise](https://mlsanity.com/paper/2304.03265/) (2023)
- [Learning Linear Non-Gaussian Causal Models in the Presence of Latent Variables](https://mlsanity.com/paper/1908.03932/) (2019)
- [Identifying Patient-Specific Root Causes with the Heteroscedastic Noise Model](https://mlsanity.com/paper/2205.13085/) (2022)
- [Structure Learning with Continuous Optimization: A Sober Look and Beyond](https://mlsanity.com/paper/2304.02146/) (2023)
- [A Versatile Causal Discovery Framework to Allow Causally-Related Hidden Variables](https://mlsanity.com/paper/2312.11001/) (2023)
- [Recursive Causal Structure Learning in the Presence of Latent Variables and Selection Bias](https://mlsanity.com/paper/2110.12036/) (2021)
- [Assumption violations in causal discovery and the robustness of score matching](https://mlsanity.com/paper/2310.13387/) (2023)
- [Nonparametric Score Estimators](https://mlsanity.com/paper/2005.10099/) (2020)
- [Causal Inference via Kernel Deviance Measures](https://mlsanity.com/paper/1804.04622/) (2018)
- [Effective Causal Discovery under Identifiable Heteroscedastic Noise Model](https://mlsanity.com/paper/2312.12844/) (2023)
- [Heteroscedastic Causal Structure Learning](https://mlsanity.com/paper/2307.07973/) (2023)

## Built on by (most cited first, 7 of 7)

- [Causal Discovery via Quantile Partial Effect](https://mlsanity.com/paper/2509.12981/) (2025)
- [Identifying Causal Direction via Variational Bayesian Compression](https://mlsanity.com/paper/2505.07503/) (2025)
- [DAG-FM: A Foundation Model for Causal Discovery under Heterogeneous Causal Mechanisms](https://mlsanity.com/paper/2607.11510/) (2026)
- [Skewness-Robust Causal Discovery in Location-Scale Noise Models](https://mlsanity.com/paper/2511.14441/) (2025)
- [Beyond Additivity: Causal Discovery in Location-Scale Noise Models with Hidden Variables](https://mlsanity.com/paper/2606.08196/) (2026)
- [Causal Direction from Convergence Time: Faster Training in the True Causal Direction](https://mlsanity.com/paper/2602.22254/) (2026)
- [Moment Matters: Mean and Variance Causal Graph Discovery from Heteroscedastic Observational Data](https://mlsanity.com/paper/2602.23602/) (2026)

## Abstract (arXiv)

Real-world data often violates the equal-variance assumption (homoscedasticity), making it essential to account for heteroscedastic noise in causal discovery. In this work, we explore heteroscedastic symmetric noise models (HSNMs), where the effect Y is modeled as Y = f(X) + σ(X)N, with X as the cause and N as independent noise following a symmetric distribution. We introduce a novel criterion for identifying HSNMs based on the skewness of the score (i.e., the gradient of the log density) of the data distribution. This criterion establishes a computationally tractable measurement that is zero in the causal direction but nonzero in the anticausal direction, enabling the causal direction discovery. We extend this skewness-based criterion to the multivariate setting and propose SkewScore, an algorithm that handles heteroscedastic noise without requiring the extraction of exogenous noise. We also conduct a case study on the robustness of SkewScore in a bivariate model with a latent confounder, providing theoretical insights into its performance. Empirical studies further validate the effectiveness of the proposed method.