# Causal Discovery via Quantile Partial Effect

arXiv: https://arxiv.org/abs/2509.12981 · HTML: https://mlsanity.com/paper/2509.12981/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 63 references that are in the map.
- First posted on arXiv 16 Sep 2025.

## Builds on (most cited first, 30 of 31)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [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-learn: Causal Discovery in Python](https://mlsanity.com/paper/2307.16405/) (2023)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)
- [Diffusion Models for Causal Discovery via Topological Ordering](https://mlsanity.com/paper/2210.06201/) (2022)
- [gCastle: A Python Toolbox for Causal Discovery](https://mlsanity.com/paper/2111.15155/) (2021)
- [On the Identifiability and Estimation of Causal Location-Scale Noise Models](https://mlsanity.com/paper/2210.09054/) (2022)
- [Causal normalizing flows: from theory to practice](https://mlsanity.com/paper/2306.05415/) (2023)
- [Causal Discovery with Score Matching on Additive Models with Arbitrary Noise](https://mlsanity.com/paper/2304.03265/) (2023)
- [Causal de Finetti: On the Identification of Invariant Causal Structure in Exchangeable Data](https://mlsanity.com/paper/2203.15756/) (2022)
- [Identifying Patient-Specific Root Causes with the Heteroscedastic Noise Model](https://mlsanity.com/paper/2205.13085/) (2022)
- [Directed Cyclic Graphical Representations of Feedback Models](https://mlsanity.com/paper/1302.4982/) (2013)
- [A Scale-Invariant Sorting Criterion to Find a Causal Order in Additive Noise Models](https://mlsanity.com/paper/2303.18211/) (2023)
- [Optimal transport for causal discovery](https://mlsanity.com/paper/2201.09366/) (2022)
- [Standardizing Structural Causal Models](https://mlsanity.com/paper/2406.11601/) (2024)
- [Ordering-Based Causal Discovery for Linear and Nonlinear Relations](https://mlsanity.com/paper/2410.05890/) (2024)
- [Cause-Effect Inference in Location-Scale Noise Models: Maximum Likelihood vs. Independence Testing](https://mlsanity.com/paper/2301.12930/) (2023)
- [Heteroscedastic Causal Structure Learning](https://mlsanity.com/paper/2307.07973/) (2023)
- [A Skewness-Based Criterion for Addressing Heteroscedastic Noise in Causal Discovery](https://mlsanity.com/paper/2410.06407/) (2024)
- [Distinguishing Cause from Effect with Causal Velocity Models](https://mlsanity.com/paper/2502.05122/) (2025)
- [Multi-Domain Causal Discovery in Bijective Causal Models](https://mlsanity.com/paper/2504.21261/) (2025)

## Built on by (most cited first, 4 of 4)

- [DAG-FM: A Foundation Model for Causal Discovery under Heterogeneous Causal Mechanisms](https://mlsanity.com/paper/2607.11510/) (2026)
- [Causal Discovery via Transformed Low-Rank Quantile Surfaces](https://mlsanity.com/paper/2609.16931/) (2026)
- [Decoupled Causal Discovery](https://mlsanity.com/paper/2609.23535/) (2026)
- [Local Fisher Information Enables Sparse Causal Discovery](https://mlsanity.com/paper/2610.04291/) (2026)

## Abstract (arXiv)

Quantile Partial Effect (QPE) is a statistic associated with conditional quantile regression, measuring the effect of covariates at different levels. Our theory demonstrates that when the QPE of cause on effect is assumed to lie in a finite linear span, cause and effect are identifiable from their observational distribution. This generalizes previous identifiability results based on Functional Causal Models (FCMs) with additive, heteroscedastic noise, etc. Meanwhile, since QPE resides entirely at the observational level, this parametric assumption does not require considering mechanisms, noise, or even the Markov assumption, but rather directly utilizes the asymmetry of shape characteristics in the observational distribution. By performing basis function tests on the estimated QPE, causal directions can be distinguished, which is empirically shown to be effective in experiments on a large number of bivariate causal discovery datasets. For multivariate causal discovery, leveraging the close connection between QPE and score functions, we find that Fisher Information is sufficient as a statistical measure to determine causal order when assumptions are made about the second moment of QPE. We validate the feasibility of using Fisher Information to identify causal order on multiple synthetic and real-world multivariate causal discovery datasets.