# Learning Causal Models from Conditional Moment Restrictions by Importance Weighting

arXiv: https://arxiv.org/abs/2108.01312 · HTML: https://mlsanity.com/paper/2108.01312/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 65 references that are in the map.
- First posted on arXiv 3 Aug 2021.

## Builds on (most cited first, 12 of 12)

- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [Deep Generalized Method of Moments for Instrumental Variable Analysis](https://mlsanity.com/paper/1905.12495/) (2019)
- [Minimax Estimation of Conditional Moment Models](https://mlsanity.com/paper/2006.07201/) (2020)
- [Learning Deep Features in Instrumental Variable Regression](https://mlsanity.com/paper/2010.07154/) (2020)
- [Non-Negative Bregman Divergence Minimization for Deep Direct Density Ratio Estimation](https://mlsanity.com/paper/2006.06979/) (2020)
- [Deep Proxy Causal Learning and its Application to Confounded Bandit Policy Evaluation](https://mlsanity.com/paper/2106.03907/) (2021)
- [Adversarial Estimation of Riesz Representers](https://mlsanity.com/paper/2101.00009/) (2020)
- [Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach](https://mlsanity.com/paper/2007.01290/) (2020)
- [The Variational Method of Moments](https://mlsanity.com/paper/2012.09422/) (2020)
- [Mostly Harmless Machine Learning: Learning Optimal Instruments in Linear IV Models](https://mlsanity.com/paper/2011.06158/) (2020)

## Built on by (most cited first, 7 of 7)

- [Deep Learning Methods for Proximal Inference via Maximum Moment Restriction](https://mlsanity.com/paper/2205.09824/) (2022)
- [Minimax Instrumental Variable Regression and L2 Convergence Guarantees without Identification or Closedness](https://mlsanity.com/paper/2302.05404/) (2023)
- [Spectral Representation Learning for Conditional Moment Models](https://mlsanity.com/paper/2210.16525/) (2022)
- [Source Condition Double Robust Inference on Functionals of Inverse Problems](https://mlsanity.com/paper/2307.13793/) (2023)
- [Regularized DeepIV with Model Selection](https://mlsanity.com/paper/2403.04236/) (2024)
- [Direct Debiased Machine Learning via Bregman Divergence Minimization](https://mlsanity.com/paper/2510.23534/) (2025)
- [Conformal Prediction for Nonparametric Instrumental Regression](https://mlsanity.com/paper/2603.25509/) (2026)

## Abstract (arXiv)

We consider learning causal relationships under conditional moment restrictions. Unlike causal inference under unconditional moment restrictions, conditional moment restrictions pose serious challenges for causal inference, especially in high-dimensional settings. To address this issue, we propose a method that transforms conditional moment restrictions to unconditional moment restrictions through importance weighting, using a conditional density ratio estimator. Using this transformation, we successfully estimate nonparametric functions defined under conditional moment restrictions. Our proposed framework is general and can be applied to a wide range of methods, including neural networks. We analyze the estimation error, providing theoretical support for our proposed method. In experiments, we confirm the soundness of our proposed method.