# Multi-Source Causal Inference Using Control Variates

arXiv: https://arxiv.org/abs/2103.16689 · HTML: https://mlsanity.com/paper/2103.16689/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 67 references that are in the map.
- First posted on arXiv 30 Mar 2021.

## Builds on (most cited first, 11 of 11)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Recommendations as Treatments: Debiasing Learning and Evaluation](https://mlsanity.com/paper/1602.05352/) (2016)
- [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 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)
- [Propensity Score Methods for Merging Observational and Experimental Datasets](https://mlsanity.com/paper/1804.07863/) (2018)
- [Principal stratification analysis using principal scores](https://mlsanity.com/paper/1602.01196/) (2016)
- [Quantifying Confounding Bias in Neuroimaging Datasets with Causal Inference](https://mlsanity.com/paper/1907.04102/) (2019)
- [The correlation-assisted missing data estimator](https://mlsanity.com/paper/1911.01859/) (2019)
- [Predicting Counterfactuals from Large Historical Data and Small Randomized Trials](https://mlsanity.com/paper/1610.07667/) (2016)

## Built on by (most cited first, 13 of 13)

- [Prediction-powered Generalization of Causal Inferences](https://mlsanity.com/paper/2406.02873/) (2024)
- [Multiply Robust Federated Estimation of Targeted Average Treatment Effects](https://mlsanity.com/paper/2309.12600/) (2023)
- [Constructing Confidence Intervals for Average Treatment Effects from Multiple Datasets](https://mlsanity.com/paper/2412.11511/) (2024)
- [Efficient and robust transfer learning of optimal individualized treatment regimes with right-censored survival data](https://mlsanity.com/paper/2301.05491/) (2023)
- [Heterogeneous Treatment Effect with Trained Kernels of the Nadaraya-Watson Regression](https://mlsanity.com/paper/2207.09139/) (2022)
- [iMTSP: Solving Min-Max Multiple Traveling Salesman Problem with Imperative Learning](https://mlsanity.com/paper/2405.00285/) (2024)
- [Causal Inference from Small High-dimensional Datasets](https://mlsanity.com/paper/2205.09281/) (2022)
- [Deep Causal Reasoning for Recommendations](https://mlsanity.com/paper/2201.02088/) (2022)
- [BENK: The Beran Estimator with Neural Kernels for Estimating the Heterogeneous Treatment Effect](https://mlsanity.com/paper/2211.10793/) (2022)
- [Optimal Ensemble Construction for Multi-Study Prediction with Applications to COVID-19 Excess Mortality Estimation](https://mlsanity.com/paper/2109.09164/) (2021)
- [Accounting for Missing Covariates in Heterogeneous Treatment Estimation](https://mlsanity.com/paper/2410.15655/) (2024)
- [Uncovering Bias Mechanisms in Observational Studies](https://mlsanity.com/paper/2506.01191/) (2025)
- [Data Fusion for High-Resolution Estimation](https://mlsanity.com/paper/2508.14858/) (2025)

## Abstract (arXiv)

While many areas of machine learning have benefited from the increasing availability of large and varied datasets, the benefit to causal inference has been limited given the strong assumptions needed to ensure identifiability of causal effects; these are often not satisfied in real-world datasets. For example, many large observational datasets (e.g., case-control studies in epidemiology, click-through data in recommender systems) suffer from selection bias on the outcome, which makes the average treatment effect (ATE) unidentifiable. We propose a general algorithm to estimate causal effects from multiple data sources, where the ATE may be identifiable only in some datasets but not others. The key idea is to construct control variates using the datasets in which the ATE is not identifiable. We show theoretically that this reduces the variance of the ATE estimate. We apply this framework to inference from observational data under outcome selection bias, assuming access to an auxiliary small dataset from which we can obtain a consistent estimate of the ATE. We construct a control variate by taking the difference of the odds ratio estimates from the two datasets. Across simulations and two case studies with real data, we show that this control variate can significantly reduce the variance of the ATE estimate.