# The Variational Method of Moments

arXiv: https://arxiv.org/abs/2012.09422 · HTML: https://mlsanity.com/paper/2012.09422/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Dec 2020.

## Built on by (most cited first, 23 of 23)

- [Causal Inference Under Unmeasured Confounding With Negative Controls: A Minimax Learning Approach](https://mlsanity.com/paper/2103.14029/) (2021)
- [Proximal Reinforcement Learning: Efficient Off-Policy Evaluation in Partially Observed Markov Decision Processes](https://mlsanity.com/paper/2110.15332/) (2021)
- [A Minimax Learning Approach to Off-Policy Evaluation in Confounded Partially Observable Markov Decision Processes](https://mlsanity.com/paper/2111.06784/) (2021)
- [Long-term Causal Inference Under Persistent Confounding via Data Combination](https://mlsanity.com/paper/2202.07234/) (2022)
- [Robust and Agnostic Learning of Conditional Distributional Treatment Effects](https://mlsanity.com/paper/2205.11486/) (2022)
- [Inference on Strongly Identified Functionals of Weakly Identified Functions](https://mlsanity.com/paper/2208.08291/) (2022)
- [Instrumental Variables in Causal Inference and Machine Learning: A Survey](https://mlsanity.com/paper/2212.05778/) (2022)
- [Falsification of Internal and External Validity in Observational Studies via Conditional Moment Restrictions](https://mlsanity.com/paper/2301.13133/) (2023)
- [Source Condition Double Robust Inference on Functionals of Inverse Problems](https://mlsanity.com/paper/2307.13793/) (2023)
- [Functional Generalized Empirical Likelihood Estimation for Conditional Moment Restrictions](https://mlsanity.com/paper/2207.04771/) (2022)
- [Learning Causal Models from Conditional Moment Restrictions by Importance Weighting](https://mlsanity.com/paper/2108.01312/) (2021)
- [Regularized DeepIV with Model Selection](https://mlsanity.com/paper/2403.04236/) (2024)
- [Estimation Beyond Data Reweighting: Kernel Method of Moments](https://mlsanity.com/paper/2305.10898/) (2023)
- [Towards Principled Causal Effect Estimation by Deep Identifiable Models](https://mlsanity.com/paper/2109.15062/) (2021)
- [Quasi-Bayesian Dual Instrumental Variable Regression](https://mlsanity.com/paper/2106.08750/) (2021)
- [Causal Inference with Complex Treatments: A Survey](https://mlsanity.com/paper/2407.14022/) (2024)
- [Adversarial Estimators](https://mlsanity.com/paper/2204.10495/) (2022)
- [Personalized Pricing with Invalid Instrumental Variables: Identification, Estimation, and Policy Learning](https://mlsanity.com/paper/2302.12670/) (2023)
- [Heterogeneous Transfer Learning for Building High-Dimensional Generalized Linear Models with Disparate Datasets](https://mlsanity.com/paper/2312.12786/) (2023)
- [Geometry-Aware Instrumental Variable Regression](https://mlsanity.com/paper/2405.11633/) (2024)
- [Detecting clinician implicit biases in diagnoses using proximal causal inference](https://mlsanity.com/paper/2501.16399/) (2025)
- [Adaptive Estimation and Inference in Conditional Moment Models via the Discrepancy Principle](https://mlsanity.com/paper/2603.01337/) (2026)
- [Conformal Prediction for Nonparametric Instrumental Regression](https://mlsanity.com/paper/2603.25509/) (2026)

## Abstract (arXiv)

The conditional moment problem is a powerful formulation for describing structural causal parameters in terms of observables, a prominent example being instrumental variable regression. A standard approach reduces the problem to a finite set of marginal moment conditions and applies the optimally weighted generalized method of moments (OWGMM), but this requires we know a finite set of identifying moments, can still be inefficient even if identifying, or can be theoretically efficient but practically unwieldy if we use a growing sieve of moment conditions. Motivated by a variational minimax reformulation of OWGMM, we define a very general class of estimators for the conditional moment problem, which we term the variational method of moments (VMM) and which naturally enables controlling infinitely-many moments. We provide a detailed theoretical analysis of multiple VMM estimators, including ones based on kernel methods and neural nets, and provide conditions under which these are consistent, asymptotically normal, and semiparametrically efficient in the full conditional moment model. We additionally provide algorithms for valid statistical inference based on the same kind of variational reformulations, both for kernel- and neural-net-based varieties. Finally, we demonstrate the strong performance of our proposed estimation and inference algorithms in a detailed series of synthetic experiments.