# Adaptive normalization for IPW estimation

arXiv: https://arxiv.org/abs/2106.07695 · HTML: https://mlsanity.com/paper/2106.07695/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jun 2021.

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

- [Layer-Neighbor Sampling -- Defusing Neighborhood Explosion in GNNs](https://mlsanity.com/paper/2210.13339/) (2022)
- [A Two-Part Machine Learning Approach to Characterizing Network Interference in A/B Testing](https://mlsanity.com/paper/2308.09790/) (2023)
- [Prediction-Powered Inference with Inverse Probability Weighting](https://mlsanity.com/paper/2508.10149/) (2025)
- [Robust Policy Expansion for Offline-to-Online RL under Diverse Data Corruption](https://mlsanity.com/paper/2509.24748/) (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)
- [Near-Optimal Experimental Design Under the Budget Constraint in Online Platforms](https://mlsanity.com/paper/2302.05005/) (2023)
- [Scalable Expectation Estimation with Subtractive Mixture Models](https://mlsanity.com/paper/2503.21346/) (2025)

## Abstract (arXiv)

Inverse probability weighting (IPW) is a general tool in survey sampling and causal inference, used both in Horvitz-Thompson estimators, which normalize by the sample size, and H'ajek/self-normalized estimators, which normalize by the sum of the inverse probability weights. In this work we study a family of IPW estimators, first proposed by Trotter and Tukey in the context of Monte Carlo problems, that are normalized by an affine combination of these two terms. We show how selecting an estimator from this family in a data-dependent way to minimize asymptotic variance leads to an iterative procedure that converges to an estimator with connections to regression control methods. We refer to this estimator as an adaptively normalized estimator. For mean estimation in survey sampling, this estimator has asymptotic variance that is never worse than the Horvitz--Thompson or H'ajek estimators, and is smaller except in edge cases. Going further, we show that adaptive normalization can be used to propose improvements of the augmented IPW (AIPW) estimator, average treatment effect (ATE) estimators, and policy learning objectives. Appealingly, these proposals preserve both the asymptotic efficiency of AIPW and the regret bounds for policy learning with IPW objectives, and deliver consistent finite sample improvements in simulations for all three of mean estimation, ATE estimation, and policy learning.