# An Expectation-Maximization Algorithm for Domain Adaptation in Gaussian Causal Models

arXiv: https://arxiv.org/abs/2601.03459 · HTML: https://mlsanity.com/paper/2601.03459/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 42 references that are in the map.
- First posted on arXiv 6 Jan 2026.

## Builds on (most cited first, 13 of 13)

- [Detecting and Correcting for Label Shift with Black Box Predictors](https://mlsanity.com/paper/1802.03916/) (2018)
- [A review of domain adaptation without target labels](https://mlsanity.com/paper/1901.05335/) (2019)
- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Domain Adaptation by Using Causal Inference to Predict Invariant Conditional Distributions](https://mlsanity.com/paper/1707.06422/) (2017)
- [Invariant Information Bottleneck for Domain Generalization](https://mlsanity.com/paper/2106.06333/) (2021)
- [External Validity: From Do-Calculus to Transportability Across Populations](https://mlsanity.com/paper/1503.01603/) (2015)
- [Invariant Causal Prediction for Sequential Data](https://mlsanity.com/paper/1706.08058/) (2017)
- [Few-shot Domain Adaptation by Causal Mechanism Transfer](https://mlsanity.com/paper/2002.03497/) (2020)
- [Domain adaptation under structural causal models](https://mlsanity.com/paper/2010.15764/) (2020)
- [A Unifying Causal Framework for Analyzing Dataset Shift-stable Learning Algorithms](https://mlsanity.com/paper/1905.11374/) (2019)
- [Do causal predictors generalize better to new domains?](https://mlsanity.com/paper/2402.09891/) (2024)
- [Accelerating Recursive Partition-Based Causal Structure Learning](https://mlsanity.com/paper/2102.11545/) (2021)
- [On Causality in Domain Adaptation and Semi-Supervised Learning: an Information-Theoretic Analysis for Parametric Models](https://mlsanity.com/paper/2205.04641/) (2022)

## Abstract (arXiv)

We study the problem of imputing a designated target variable that is systematically missing in a shifted deployment domain, when a Gaussian causal DAG is available from a fully observed source domain. We propose a unified EM-based framework that combines source and target data through the DAG structure to transfer information from observed variables to the missing target. On the methodological side, we formulate a population EM operator in the DAG parameter space and introduce a first-order (gradient) EM update that replaces the costly generalized least-squares M-step with a single projected gradient step. Under standard local strong-concavity and smoothness assumptions and a BWY-style citeBalakrishnan2017EM gradient-stability (bounded missing-information) condition, we show that this first-order EM operator is locally contractive around the true target parameters, yielding geometric convergence and finite-sample guarantees on parameter error and the induced target-imputation error in Gaussian SEMs under covariate shift and local mechanism shifts. Algorithmically, we exploit the known causal DAG to freeze source-invariant mechanisms and re-estimate only those conditional distributions directly affected by the shift, making the procedure scalable to higher-dimensional models. In experiments on a synthetic seven-node SEM, the 64-node MAGIC-IRRI genetic network, and the Sachs protein-signaling data, the proposed DAG-aware first-order EM algorithm improves target imputation accuracy over a fit-on-source Bayesian network and a Kiiveri-style EM baseline, with the largest gains under pronounced domain shift.