# Combining Observational and Experimental Datasets Using Shrinkage Estimators

arXiv: https://arxiv.org/abs/2002.06708 · HTML: https://mlsanity.com/paper/2002.06708/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Feb 2020.

## Built on by (most cited first, 28 of 28)

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- [Federated Causal Inference in Heterogeneous Observational Data](https://mlsanity.com/paper/2107.11732/) (2021)
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- [Kernel methods for long term dose response curves](https://mlsanity.com/paper/2201.05139/) (2022)
- [Comparison of Methods that Combine Multiple Randomized Trials to Estimate Heterogeneous Treatment Effects](https://mlsanity.com/paper/2303.16299/) (2023)
- [Causal Effect Estimation from Observational and Interventional Data Through Matrix Weighted Linear Estimators](https://mlsanity.com/paper/2306.06002/) (2023)
- [Conditional Cross-Design Synthesis Estimators for Generalizability in Medicaid](https://mlsanity.com/paper/2109.13288/) (2021)
- [Estimating Heterogeneous Treatment Effects by Combining Weak Instruments and Observational Data](https://mlsanity.com/paper/2406.06452/) (2024)
- [The Partial Testimony of Logs: Evaluation of Language Model Generation under Confounded Model Choice](https://mlsanity.com/paper/2605.01311/) (2026)
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- [Combining Incomplete Observational and Randomized Data for Heterogeneous Treatment Effects](https://mlsanity.com/paper/2410.21343/) (2024)
- [Confounding-Valid Conformal Inference for Counterfactual KPIs in Wireless Networks](https://mlsanity.com/paper/2609.05073/) (2026)
- [Combining Interventional and Observational Data Using Causal Reductions](https://mlsanity.com/paper/2103.04786/) (2021)
- [A Cautionary Tale on Integrating Studies with Disparate Outcome Measures for Causal Inference](https://mlsanity.com/paper/2505.11014/) (2025)
- [Transportability without Graphs: A Bayesian Approach to Identifying s-Admissible Backdoor Sets](https://mlsanity.com/paper/2505.12801/) (2025)
- [Deconfounded Warm-Start Thompson Sampling with Applications to Precision Medicine](https://mlsanity.com/paper/2505.17283/) (2025)
- [Testing Effect Homogeneity and Confounding in High-Dimensional Experimental and Observational Studies](https://mlsanity.com/paper/2602.19703/) (2026)
- [Empirical Bayes Rebiasing](https://mlsanity.com/paper/2605.08069/) (2026)

## Abstract (arXiv)

We consider the problem of combining data from observational and experimental sources to make causal conclusions. This problem is increasingly relevant, as the modern era has yielded passive collection of massive observational datasets in areas such as e-commerce and electronic health. These data may be used to supplement experimental data, which is frequently expensive to obtain. In Rosenman et al. (2018), we considered this problem under the assumption that all confounders were measured. Here, we relax the assumption of unconfoundedness. To derive combined estimators with desirable properties, we make use of results from the Stein Shrinkage literature. Our contributions are threefold. First, we propose a generic procedure for deriving shrinkage estimators in this setting, making use of a generalized unbiased risk estimate. Second, we develop two new estimators, prove finite sample conditions under which they have lower risk than an estimator using only experimental data, and show that each achieves a notion of asymptotic optimality. Third, we draw connections between our approach and results in sensitivity analysis, including proposing a method for evaluating the feasibility of our estimators.