# Multi-Domain Empirical Bayes for Linearly-Mixed Causal Representations

arXiv: https://arxiv.org/abs/2603.18404 · HTML: https://mlsanity.com/paper/2603.18404/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 59 references that are in the map.
- First posted on arXiv 19 Mar 2026.

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

- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning](https://mlsanity.com/paper/1805.08651/) (2018)
- [VAE with a VampPrior](https://mlsanity.com/paper/1705.07120/) (2017)
- [Deep Unsupervised Clustering with Gaussian Mixture Variational Autoencoders](https://mlsanity.com/paper/1611.02648/) (2016)
- [Interventional Causal Representation Learning](https://mlsanity.com/paper/2209.11924/) (2022)
- [Weakly supervised causal representation learning](https://mlsanity.com/paper/2203.16437/) (2022)
- [Nonparametric Identifiability of Causal Representations from Unknown Interventions](https://mlsanity.com/paper/2306.00542/) (2023)
- [Independent mechanism analysis, a new concept?](https://mlsanity.com/paper/2106.05200/) (2021)
- [Learning Linear Causal Representations from Interventions under General Nonlinear Mixing](https://mlsanity.com/paper/2306.02235/) (2023)
- [Linear Causal Disentanglement via Interventions](https://mlsanity.com/paper/2211.16467/) (2022)
- [Identifiability Guarantees for Causal Disentanglement from Soft Interventions](https://mlsanity.com/paper/2307.06250/) (2023)
- [Causal Representation Learning from Multiple Distributions: A General Setting](https://mlsanity.com/paper/2402.05052/) (2024)
- [Estimating High Order Gradients of the Data Distribution by Denoising](https://mlsanity.com/paper/2111.04726/) (2021)
- [Causal Component Analysis](https://mlsanity.com/paper/2305.17225/) (2023)
- [Causal Discovery in Heterogeneous Environments Under the Sparse Mechanism Shift Hypothesis](https://mlsanity.com/paper/2206.02013/) (2022)
- [Modelling Cellular Perturbations with the Sparse Additive Mechanism Shift Variational Autoencoder](https://mlsanity.com/paper/2311.02794/) (2023)
- [General Identifiability and Achievability for Causal Representation Learning](https://mlsanity.com/paper/2310.15450/) (2023)
- [Unifying Causal Representation Learning with the Invariance Principle](https://mlsanity.com/paper/2409.02772/) (2024)
- [Embrace the Gap: VAEs Perform Independent Mechanism Analysis](https://mlsanity.com/paper/2206.02416/) (2022)
- [Score-based Causal Representation Learning: Linear and General Transformations](https://mlsanity.com/paper/2402.00849/) (2024)
- [Two Modeling Strategies for Empirical Bayes Estimation](https://mlsanity.com/paper/1409.2677/) (2014)
- [Towards Interpretable Deep Generative Models via Causal Representation Learning](https://mlsanity.com/paper/2504.11609/) (2025)
- [Identifiable Causal Representation Learning: Unsupervised, Multi-View, and Multi-Environment](https://mlsanity.com/paper/2406.13371/) (2024)
- [Causality-oriented robustness: exploiting general noise interventions](https://mlsanity.com/paper/2307.10299/) (2023)
- [Bayesian Empirical Bayes: Simultaneous Inference from Probabilistic Symmetries](https://mlsanity.com/paper/2512.16239/) (2025)

## Built on by (most cited first, 1 of 1)

- [Environment-Robust Representation Learning with Empirical Bayes](https://mlsanity.com/paper/2606.05365/) (2026)

## Abstract (arXiv)

Causal representation learning (CRL) aims to learn low-dimensional causal latent variables from high-dimensional observations. While identifiability has been extensively studied for CRL, estimation has been less explored. In this paper, we explore the use of empirical Bayes (EB) to estimate causal representations. In particular, we consider the problem of learning from data from multiple domains, where differences between domains are modeled by interventions in a shared underlying causal model. Multi-domain CRL naturally poses a simultaneous inference problem that EB is designed to tackle. Here, we propose an EB f-modeling algorithm that improves the quality of learned causal variables by exploiting invariant structure within and across domains. Specifically, we consider a linear measurement model and interventional priors arising from a shared acyclic SCM. When the graph and intervention targets are known, we develop an EM-style algorithm based on causally structured score matching. We further discuss EB g-modeling in the context of existing CRL approaches. In experiments on synthetic data, our proposed method achieves more accurate estimation than other methods for CRL.