# Inference on Strongly Identified Functionals of Weakly Identified Functions

arXiv: https://arxiv.org/abs/2208.08291 · HTML: https://mlsanity.com/paper/2208.08291/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 96 references that are in the map.
- First posted on arXiv 17 Aug 2022.

## Builds on (most cited first, 30 of 38)

- [Orthogonal Statistical Learning](https://mlsanity.com/paper/1901.09036/) (2019)
- [Identifying Causal Effects With Proxy Variables of an Unmeasured Confounder](https://mlsanity.com/paper/1609.08816/) (2016)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [An Introduction to Proximal Causal Learning](https://mlsanity.com/paper/2009.10982/) (2020)
- [Deep Generalized Method of Moments for Instrumental Variable Analysis](https://mlsanity.com/paper/1905.12495/) (2019)
- [Off-Policy Evaluation via the Regularized Lagrangian](https://mlsanity.com/paper/2007.03438/) (2020)
- [Minimax Estimation of Conditional Moment Models](https://mlsanity.com/paper/2006.07201/) (2020)
- [Assessing Algorithmic Fairness with Unobserved Protected Class Using Data Combination](https://mlsanity.com/paper/1906.00285/) (2019)
- [Double/Debiased/Neyman Machine Learning of Treatment Effects](https://mlsanity.com/paper/1701.08687/) (2017)
- [Deep Neural Networks for Estimation and Inference](https://mlsanity.com/paper/1809.09953/) (2018)
- [Semiparametric proximal causal inference](https://mlsanity.com/paper/2011.08411/) (2020)
- [Finite Sample Analysis of Minimax Offline Reinforcement Learning: Completeness, Fast Rates and First-Order Efficiency](https://mlsanity.com/paper/2102.02981/) (2021)
- [Proximal Causal Learning with Kernels: Two-Stage Estimation and Moment Restriction](https://mlsanity.com/paper/2105.04544/) (2021)
- [Balanced Policy Evaluation and Learning](https://mlsanity.com/paper/1705.07384/) (2017)
- [Causal Inference Under Unmeasured Confounding With Negative Controls: A Minimax Learning Approach](https://mlsanity.com/paper/2103.14029/) (2021)
- [RieszNet and ForestRiesz: Automatic Debiased Machine Learning with Neural Nets and Random Forests](https://mlsanity.com/paper/2110.03031/) (2021)
- [Generalized Optimal Matching Methods for Causal Inference](https://mlsanity.com/paper/1612.08321/) (2016)
- [Bounds on the conditional and average treatment effect with unobserved confounding factors](https://mlsanity.com/paper/1808.09521/) (2018)
- [Robust Inference on Average Treatment Effects with Possibly More Covariates than Observations](https://mlsanity.com/paper/1309.4686/) (2013)
- [A Kernel Loss for Solving the Bellman Equation](https://mlsanity.com/paper/1905.10506/) (2019)
- [The Influence Function of Semiparametric Estimators](https://mlsanity.com/paper/1508.01378/) (2015)
- [A confounding bridge approach for double negative control inference on causal effects](https://mlsanity.com/paper/1808.04945/) (2018)
- [Adversarial Estimation of Riesz Representers](https://mlsanity.com/paper/2101.00009/) (2020)
- [De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers](https://mlsanity.com/paper/1802.08667/) (2018)
- [Adversarial Generalized Method of Moments](https://mlsanity.com/paper/1803.07164/) (2018)
- [Augmented Minimax Linear Estimation](https://mlsanity.com/paper/1712.00038/) (2017)
- [Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach](https://mlsanity.com/paper/2007.01290/) (2020)
- [Optimal Sup-norm Rates and Uniform Inference on Nonlinear Functionals of Nonparametric IV Regression](https://mlsanity.com/paper/1508.03365/) (2015)
- [Kernel Methods for Unobserved Confounding: Negative Controls, Proxies, and Instruments](https://mlsanity.com/paper/2012.10315/) (2020)
- [Doubly-Valid/Doubly-Sharp Sensitivity Analysis for Causal Inference with Unmeasured Confounding](https://mlsanity.com/paper/2112.11449/) (2021)

## Built on by (most cited first, 12 of 12)

- [Spectral Representation for Causal Estimation with Hidden Confounders](https://mlsanity.com/paper/2407.10448/) (2024)
- [Falsification of Internal and External Validity in Observational Studies via Conditional Moment Restrictions](https://mlsanity.com/paper/2301.13133/) (2023)
- [Targeted Sequential Indirect Experiment Design](https://mlsanity.com/paper/2405.19985/) (2024)
- [Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regression](https://mlsanity.com/paper/2501.04898/) (2025)
- [Outcome-Aware Spectral Feature Learning for Instrumental Variable Regression](https://mlsanity.com/paper/2512.00919/) (2025)
- [Detecting clinician implicit biases in diagnoses using proximal causal inference](https://mlsanity.com/paper/2501.16399/) (2025)
- [Stable Probability Weighting: Large-Sample and Finite-Sample Estimation and Inference Methods for Heterogeneous Causal Effects of Multivalued Treatments Under Limited Overlap](https://mlsanity.com/paper/2301.05703/) (2023)
- [Addressing Dynamic and Sparse Qualitative Data: A Hilbert Space Embedding of Categorical Variables](https://mlsanity.com/paper/2308.11781/) (2023)
- [Learning Treatment Representations for Downstream Instrumental Variable Regression](https://mlsanity.com/paper/2506.02200/) (2025)
- [Adaptive Estimation and Inference in Conditional Moment Models via the Discrepancy Principle](https://mlsanity.com/paper/2603.01337/) (2026)
- [Point-Identification of a Robust Predictor Under Latent Shift with Imperfect Proxies](https://mlsanity.com/paper/2603.15158/) (2026)
- [Nonparametric Instrumental Variable Analysis Without Structural Equations: Debiased Inference on Functionals of Inverse Problems with No Solutions](https://mlsanity.com/paper/2604.24660/) (2026)

## Abstract (arXiv)

In a variety of applications, including nonparametric instrumental variable (NPIV) analysis, proximal causal inference under unmeasured confounding, and missing-not-at-random data with shadow variables, we are interested in inference on a continuous linear functional (e.g., average causal effects) of nuisance function (e.g., NPIV regression) defined by conditional moment restrictions. These nuisance functions are generally weakly identified, in that the conditional moment restrictions can be severely ill-posed as well as admit multiple solutions. This is sometimes resolved by imposing strong conditions that imply the function can be estimated at rates that make inference on the functional possible. In this paper, we study a novel condition for the functional to be strongly identified even when the nuisance function is not; that is, the functional is amenable to asymptotically-normal estimation at √n-rates. The condition implies the existence of debiasing nuisance functions, and we propose penalized minimax estimators for both the primary and debiasing nuisance functions. The proposed nuisance estimators can accommodate flexible function classes, and importantly they can converge to fixed limits determined by the penalization regardless of the identifiability of the nuisances. We use the penalized nuisance estimators to form a debiased estimator for the functional of interest and prove its asymptotic normality under generic high-level conditions, which provide for asymptotically valid confidence intervals. We also illustrate our method in a novel partially linear proximal causal inference problem and a partially linear instrumental variable regression problem.