# Many Experiments, Few Repetitions, Unpaired Data, and Sparse Effects: Is Causal Inference Possible?

arXiv: https://arxiv.org/abs/2601.15254 · HTML: https://mlsanity.com/paper/2601.15254/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 53 references that are in the map.
- First posted on arXiv 21 Jan 2026.

## Builds on (most cited first, 7 of 7)

- [Large-Scale Differentiable Causal Discovery of Factor Graphs](https://mlsanity.com/paper/2206.07824/) (2022)
- [Deep Learning of Causal Structures in High Dimensions](https://mlsanity.com/paper/2212.04866/) (2022)
- [Regularization Methods for High-Dimensional Instrumental Variables Regression With an Application to Genetical Genomics](https://mlsanity.com/paper/1304.7829/) (2013)
- [Identifiability of Sparse Causal Effects using Instrumental Variables](https://mlsanity.com/paper/2203.09380/) (2022)
- [A Two-Stage Penalized Least Squares Method for Constructing Large Systems of Structural Equations](https://mlsanity.com/paper/1511.00370/) (2015)
- [The synthetic instrument: From sparse association to sparse causation](https://mlsanity.com/paper/2304.01098/) (2023)
- [Sparse Causal Effect Estimation using Two-Sample Summary Statistics in the Presence of Unmeasured Confounding](https://mlsanity.com/paper/2410.12300/) (2024)

## Abstract (arXiv)

We study the problem of estimating causal effects under hidden confounding in the following unpaired data setting: we observe some covariates X and an outcome Y under different experimental conditions (environments) but do not observe them jointly; we either observe X or Y. Under appropriate regularity conditions, the problem can be cast as an instrumental variable (IV) regression with the environment acting as a (possibly high-dimensional) instrument. When there are many environments but only a few observations per environment, standard two-sample IV estimators fail to be consistent. We propose a GMM-type estimator based on cross-fold sample splitting of the instrument-covariate sample and prove that it is consistent as the number of environments grows but the sample size per environment remains constant. We further extend the method to sparse causal effects via ℓ1-regularized estimation and post-selection refitting.