# Distributional Anchor Regression

arXiv: https://arxiv.org/abs/2101.08224 · HTML: https://mlsanity.com/paper/2101.08224/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jan 2021.

## Built on by (most cited first, 11 of 11)

- [Regularizing towards Causal Invariance: Linear Models with Proxies](https://mlsanity.com/paper/2103.02477/) (2021)
- [Evaluating Robustness to Dataset Shift via Parametric Robustness Sets](https://mlsanity.com/paper/2205.15947/) (2022)
- [Model-based causal feature selection for general response types](https://mlsanity.com/paper/2309.12833/) (2023)
- [Achievable distributional robustness when the robust risk is only partially identified](https://mlsanity.com/paper/2502.02710/) (2025)
- [Domain Generalization and Adaptation in Intensive Care with Anchor Regression](https://mlsanity.com/paper/2507.21783/) (2025)
- [Nonlinear Causal Discovery via Kernel Anchor Regression](https://mlsanity.com/paper/2210.16775/) (2022)
- [Invariant Probabilistic Prediction](https://mlsanity.com/paper/2309.10083/) (2023)
- [Out-of-distribution robustness for multivariate analysis via causal regularisation](https://mlsanity.com/paper/2403.01865/) (2024)
- [Anti-causal domain generalization: Leveraging unlabeled data](https://mlsanity.com/paper/2602.17187/) (2026)
- [How Useful is Causal Invariance for Domain Adaptation in Finite-Sample Settings?](https://mlsanity.com/paper/2606.12680/) (2026)
- [CVaR anchor regression protects against rare shifts](https://mlsanity.com/paper/2609.27034/) (2026)

## Abstract (arXiv)

Prediction models often fail if train and test data do not stem from the same distribution. Out-of-distribution (OOD) generalization to unseen, perturbed test data is a desirable but difficult-to-achieve property for prediction models and in general requires strong assumptions on the data generating process (DGP). In a causally inspired perspective on OOD generalization, the test data arise from a specific class of interventions on exogenous random variables of the DGP, called anchors. Anchor regression models, introduced by Rothenhaeusler et al. (2021), protect against distributional shifts in the test data by employing causal regularization. However, so far anchor regression has only been used with a squared-error loss which is inapplicable to common responses such as censored continuous or ordinal data. Here, we propose a distributional version of anchor regression which generalizes the method to potentially censored responses with at least an ordered sample space. To this end, we combine a flexible class of parametric transformation models for distributional regression with an appropriate causal regularizer under a more general notion of residuals. In an exemplary application and several simulation scenarios we demonstrate the extent to which OOD generalization is possible.