# On counterfactual inference with unobserved confounding

arXiv: https://arxiv.org/abs/2211.08209 · HTML: https://mlsanity.com/paper/2211.08209/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 26 references that are in the map.
- First posted on arXiv 14 Nov 2022.

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

- [Synthetic Difference in Differences](https://mlsanity.com/paper/1812.09970/) (2018)
- [Synthetic Interventions](https://mlsanity.com/paper/2006.07691/) (2020)
- [Efficient Learning of Discrete Graphical Models](https://mlsanity.com/paper/1902.00600/) (2019)
- [Optimal Learning of Mallows Block Model](https://mlsanity.com/paper/1906.01009/) (2019)
- [A Computationally Efficient Method for Learning Exponential Family Distributions](https://mlsanity.com/paper/2110.15397/) (2021)
- [On Learning Continuous Pairwise Markov Random Fields](https://mlsanity.com/paper/2010.15031/) (2020)
- [Statistical Estimation from Dependent Data](https://mlsanity.com/paper/2107.09773/) (2021)

## Built on by (most cited first, 10 of 10)

- [Counterfactual Identifiability of Bijective Causal Models](https://mlsanity.com/paper/2302.02228/) (2023)
- [Two-way Deconfounder for Off-policy Evaluation in Causal Reinforcement Learning](https://mlsanity.com/paper/2412.05783/) (2024)
- [Towards Characterizing Domain Counterfactuals For Invertible Latent Causal Models](https://mlsanity.com/paper/2306.11281/) (2023)
- [On Computationally Efficient Learning of Exponential Family Distributions](https://mlsanity.com/paper/2309.06413/) (2023)
- [Learning Counterfactual Distributions via Kernel Nearest Neighbors](https://mlsanity.com/paper/2410.13381/) (2024)
- [Causal Inference out of Control: Estimating the Steerability of Consumption](https://mlsanity.com/paper/2302.04989/) (2023)
- [Teleporter Theory: A General and Simple Approach for Modeling Cross-World Counterfactual Causality](https://mlsanity.com/paper/2406.11501/) (2024)
- [Learning Counterfactual Outcomes Under Rank Preservation](https://mlsanity.com/paper/2502.06398/) (2025)
- [Causal Inference for Sequential Settings under Interference and Latent Confounding](https://mlsanity.com/paper/2607.14940/) (2026)
- [Causal Inference with Unobserved Confounding: A Mixture Learning Perspective](https://mlsanity.com/paper/2609.23219/) (2026)

## Abstract (arXiv)

Given an observational study with n independent but heterogeneous units, our goal is to learn the counterfactual distribution for each unit using only one p-dimensional sample per unit containing covariates, interventions, and outcomes. Specifically, we allow for unobserved confounding that introduces statistical biases between interventions and outcomes as well as exacerbates the heterogeneity across units. Modeling the conditional distribution of the outcomes as an exponential family, we reduce learning the unit-level counterfactual distributions to learning n exponential family distributions with heterogeneous parameters and only one sample per distribution. We introduce a convex objective that pools all n samples to jointly learn all n parameter vectors, and provide a unit-wise mean squared error bound that scales linearly with the metric entropy of the parameter space. For example, when the parameters are s-sparse linear combination of k known vectors, the error is O(slog k/p). En route, we derive sufficient conditions for compactly supported distributions to satisfy the logarithmic Sobolev inequality. As an application of the framework, our results enable consistent imputation of sparsely missing covariates.