# Causal Effect Estimation from Observational and Interventional Data Through Matrix Weighted Linear Estimators

arXiv: https://arxiv.org/abs/2306.06002 · HTML: https://mlsanity.com/paper/2306.06002/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 62 references that are in the map.
- First posted on arXiv 9 Jun 2023.

## Builds on (most cited first, 10 of 10)

- [Permutation-based Causal Inference Algorithms with Interventions](https://mlsanity.com/paper/1705.10220/) (2017)
- [External Validity: From Do-Calculus to Transportability Across Populations](https://mlsanity.com/paper/1503.01603/) (2015)
- [Removing Hidden Confounding by Experimental Grounding](https://mlsanity.com/paper/1810.11646/) (2018)
- [Causal inference methods for combining randomized trials and observational studies: a review](https://mlsanity.com/paper/2011.08047/) (2020)
- [Combining Observational and Randomized Data for Estimating Heterogeneous Treatment Effects](https://mlsanity.com/paper/2202.12891/) (2022)
- [Combining multiple observational data sources to estimate causal effects](https://mlsanity.com/paper/1801.00802/) (2018)
- [Combining Observational and Experimental Datasets Using Shrinkage Estimators](https://mlsanity.com/paper/2002.06708/) (2020)
- [Adaptive Combination of Randomized and Observational Data](https://mlsanity.com/paper/2111.15012/) (2021)
- [Spectral Deconfounding via Perturbed Sparse Linear Models](https://mlsanity.com/paper/1811.05352/) (2018)
- [Propensity Score Methods for Merging Observational and Experimental Datasets](https://mlsanity.com/paper/1804.07863/) (2018)

## Built on by (most cited first, 3 of 3)

- [Identifiable Causal Representation Learning: Unsupervised, Multi-View, and Multi-Environment](https://mlsanity.com/paper/2406.13371/) (2024)
- [Transferring Causal Effects using Proxies](https://mlsanity.com/paper/2510.25924/) (2025)
- [Are Good Generators Good Decision-Makers? Policy Learning for General Interventions via Retargeted Counterfactual Generation](https://mlsanity.com/paper/2606.07399/) (2026)

## Abstract (arXiv)

We study causal effect estimation from a mixture of observational and interventional data in a confounded linear regression model with multivariate treatments. We show that the statistical efficiency in terms of expected squared error can be improved by combining estimators arising from both the observational and interventional setting. To this end, we derive methods based on matrix weighted linear estimators and prove that our methods are asymptotically unbiased in the infinite sample limit. This is an important improvement compared to the pooled estimator using the union of interventional and observational data, for which the bias only vanishes if the ratio of observational to interventional data tends to zero. Studies on synthetic data confirm our theoretical findings. In settings where confounding is substantial and the ratio of observational to interventional data is large, our estimators outperform a Stein-type estimator and various other baselines.