# Efficient Online Estimation of Causal Effects by Deciding What to Observe

arXiv: https://arxiv.org/abs/2108.09265 · HTML: https://mlsanity.com/paper/2108.09265/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 47 references that are in the map.
- First posted on arXiv 20 Aug 2021.

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

- [Inference for Batched Bandits](https://mlsanity.com/paper/2002.03217/) (2020)
- [Active Structure Learning of Causal DAGs via Directed Clique Tree](https://mlsanity.com/paper/2011.00641/) (2020)
- [Causal Effect Identification from Multiple Incomplete Data Sources: A General Search-based Approach](https://mlsanity.com/paper/1902.01073/) (2019)
- [Efficient adjustment sets in causal graphical models with hidden variables](https://mlsanity.com/paper/2004.10521/) (2020)
- [Active Feature Acquisition with Supervised Matrix Completion](https://mlsanity.com/paper/1802.05380/) (2018)
- [Estimating Treatment Effects with Observed Confounders and Mediators](https://mlsanity.com/paper/2003.11991/) (2020)
- [Efficient adjustment sets for population average treatment effect estimation in non-parametric causal graphical models](https://mlsanity.com/paper/1912.00306/) (2019)
- [Active Matrix Factorization for Surveys](https://mlsanity.com/paper/1902.07634/) (2019)
- [Causal Inference With Selectively Deconfounded Data](https://mlsanity.com/paper/2002.11096/) (2020)
- [Confounding Feature Acquisition for Causal Effect Estimation](https://mlsanity.com/paper/2011.08753/) (2020)

## Built on by (most cited first, 11 of 11)

- [Active Adaptive Experimental Design for Treatment Effect Estimation with Covariate Choices](https://mlsanity.com/paper/2403.03589/) (2024)
- [Adaptive Instrument Design for Indirect Experiments](https://mlsanity.com/paper/2312.02438/) (2023)
- [Optimal Best Arm Identification in Two-Armed Bandits with a Fixed Budget under a Small Gap](https://mlsanity.com/paper/2201.04469/) (2022)
- [Targeted Sequential Indirect Experiment Design](https://mlsanity.com/paper/2405.19985/) (2024)
- [Best Arm Identification with Contextual Information under a Small Gap](https://mlsanity.com/paper/2209.07330/) (2022)
- [Asymptotically Optimal Fixed-Budget Best Arm Identification with Variance-Dependent Bounds](https://mlsanity.com/paper/2302.02988/) (2023)
- [Worst-Case Optimal Multi-Armed Gaussian Best Arm Identification with a Fixed Budget](https://mlsanity.com/paper/2310.19788/) (2023)
- [Adaptive Experimental Design for Policy Learning](https://mlsanity.com/paper/2401.03756/) (2024)
- [Efficient Adaptive Experimentation with Noncompliance](https://mlsanity.com/paper/2505.17468/) (2025)
- [Confounder Balancing for Instrumental Variable Regression with Latent Variable](https://mlsanity.com/paper/2211.10008/) (2022)
- [Online Data Collection for Efficient Semiparametric Inference](https://mlsanity.com/paper/2411.03195/) (2024)

## Abstract (arXiv)

Researchers often face data fusion problems, where multiple data sources are available, each capturing a distinct subset of variables. While problem formulations typically take the data as given, in practice, data acquisition can be an ongoing process. In this paper, we aim to estimate any functional of a probabilistic model (e.g., a causal effect) as efficiently as possible, by deciding, at each time, which data source to query. We propose online moment selection (OMS), a framework in which structural assumptions are encoded as moment conditions. The optimal action at each step depends, in part, on the very moments that identify the functional of interest. Our algorithms balance exploration with choosing the best action as suggested by current estimates of the moments. We propose two selection strategies: (1) explore-then-commit (OMS-ETC) and (2) explore-then-greedy (OMS-ETG), proving that both achieve zero asymptotic regret as assessed by MSE. We instantiate our setup for average treatment effect estimation, where structural assumptions are given by a causal graph and data sources may include subsets of mediators, confounders, and instrumental variables.