# Rank-Based Causal Discovery for Post-Nonlinear Models

arXiv: https://arxiv.org/abs/2302.12341 · HTML: https://mlsanity.com/paper/2302.12341/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 29 references that are in the map.
- First posted on arXiv 23 Feb 2023.

## Builds on (most cited first, 4 of 4)

- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [CAM: Causal additive models, high-dimensional order search and penalized regression](https://mlsanity.com/paper/1310.1533/) (2013)
- [Optimal transport for causal discovery](https://mlsanity.com/paper/2201.09366/) (2022)
- [Maximum pairwise-rank-likelihood-based inference for the semiparametric transformation model](https://mlsanity.com/paper/2103.13435/) (2021)

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

- [Distinguishing Cause from Effect with Causal Velocity Models](https://mlsanity.com/paper/2502.05122/) (2025)
- [Towards Identifiable Latent Additive Noise Models](https://mlsanity.com/paper/2403.15711/) (2024)
- [Learned Causal Method Prediction](https://mlsanity.com/paper/2311.03989/) (2023)
- [Optimal Kernel Choice for Score Function-based Causal Discovery](https://mlsanity.com/paper/2407.10132/) (2024)
- [Enabling Causal Discovery in Post-Nonlinear Models with Normalizing Flows](https://mlsanity.com/paper/2407.04980/) (2024)
- [MXMap: A Multivariate Cross Mapping Framework for Causal Discovery in Dynamical Systems](https://mlsanity.com/paper/2502.03802/) (2025)
- [DAGAF: A directed acyclic generative adversarial framework for joint structure learning and tabular data synthesis](https://mlsanity.com/paper/2604.04290/) (2026)

## Abstract (arXiv)

Learning causal relationships from empirical observations is a central task in scientific research. A common method is to employ structural causal models that postulate noisy functional relations among a set of interacting variables. To ensure unique identifiability of causal directions, researchers consider restricted subclasses of structural causal models. Post-nonlinear (PNL) causal models constitute one of the most flexible options for such restricted subclasses, containing in particular the popular additive noise models as a further subclass. However, learning PNL models is not well studied beyond the bivariate case. The existing methods learn non-linear functional relations by minimizing residual dependencies and subsequently test independence from residuals to determine causal orientations. However, these methods can be prone to overfitting and, thus, difficult to tune appropriately in practice. As an alternative, we propose a new approach for PNL causal discovery that uses rank-based methods to estimate the functional parameters. This new approach exploits natural invariances of PNL models and disentangles the estimation of the non-linear functions from the independence tests used to find causal orientations. We prove consistency of our method and validate our results in numerical experiments.