# Causal Estimation with Functional Confounders

arXiv: https://arxiv.org/abs/2102.08533 · HTML: https://mlsanity.com/paper/2102.08533/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 31 references that are in the map.
- First posted on arXiv 17 Feb 2021.

## Builds on (most cited first, 2 of 2)

- [The Blessings of Multiple Causes](https://mlsanity.com/paper/1805.06826/) (2018)
- [Multiple Causal Inference with Latent Confounding](https://mlsanity.com/paper/1805.08273/) (2018)

## Built on by (most cited first, 9 of 9)

- [Desiderata for Representation Learning: A Causal Perspective](https://mlsanity.com/paper/2109.03795/) (2021)
- [Anticipating Performativity by Predicting from Predictions](https://mlsanity.com/paper/2208.07331/) (2022)
- [Invariant Learning via Probability of Sufficient and Necessary Causes](https://mlsanity.com/paper/2309.12559/) (2023)
- [Nuisances via Negativa: Adjusting for Spurious Correlations via Data Augmentation](https://mlsanity.com/paper/2210.01302/) (2022)
- [Black Box Causal Inference: Effect Estimation via Meta Prediction](https://mlsanity.com/paper/2503.05985/) (2025)
- [Specify Robust Causal Representation from Mixed Observations](https://mlsanity.com/paper/2310.13892/) (2023)
- [Computational Identifiability](https://mlsanity.com/paper/2606.19361/) (2026)
- [Causal Inference for Human-Language Model Collaboration](https://mlsanity.com/paper/2404.00207/) (2024)
- [Improving Causal Effect Estimation of Weighted RegressionBased Estimator using Neural Networks](https://mlsanity.com/paper/2110.15075/) (2021)

## Abstract (arXiv)

Causal inference relies on two fundamental assumptions: ignorability and positivity. We study causal inference when the true confounder value can be expressed as a function of the observed data; we call this setting estimation with functional confounders (EFC). In this setting, ignorability is satisfied, however positivity is violated, and causal inference is impossible in general. We consider two scenarios where causal effects are estimable. First, we discuss interventions on a part of the treatment called functional interventions and a sufficient condition for effect estimation of these interventions called functional positivity. Second, we develop conditions for nonparametric effect estimation based on the gradient fields of the functional confounder and the true outcome function. To estimate effects under these conditions, we develop Level-set Orthogonal Descent Estimation (LODE). Further, we prove error bounds on LODE's effect estimates, evaluate our methods on simulated and real data, and empirically demonstrate the value of EFC.