# Permutation Weighting

arXiv: https://arxiv.org/abs/1901.01230 · HTML: https://mlsanity.com/paper/1901.01230/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jan 2019.

## Built on by (most cited first, 14 of 14)

- [Diagnosing failures of fairness transfer across distribution shift in real-world medical settings](https://mlsanity.com/paper/2202.01034/) (2022)
- [Causal Effect Inference for Structured Treatments](https://mlsanity.com/paper/2106.01939/) (2021)
- [End-to-End Balancing for Causal Continuous Treatment-Effect Estimation](https://mlsanity.com/paper/2107.13068/) (2021)
- [Outcome Assumptions and Duality Theory for Balancing Weights](https://mlsanity.com/paper/2203.09557/) (2022)
- [Mining Stable Preferences: Adaptive Modality Decorrelation for Multimedia Recommendation](https://mlsanity.com/paper/2306.14179/) (2023)
- [Who's Gaming the System? A Causally-Motivated Approach for Detecting Strategic Adaptation](https://mlsanity.com/paper/2412.02000/) (2024)
- [Deep Stable Representation Learning on Electronic Health Records](https://mlsanity.com/paper/2209.01321/) (2022)
- [Multiply-Robust Causal Change Attribution](https://mlsanity.com/paper/2404.08839/) (2024)
- [Causal Inference with Complex Treatments: A Survey](https://mlsanity.com/paper/2407.14022/) (2024)
- [Off-Policy Evaluation in Embedded Spaces](https://mlsanity.com/paper/2203.02807/) (2022)
- [Deep Stable Multi-Interest Learning for Out-of-distribution Sequential Recommendation](https://mlsanity.com/paper/2304.05615/) (2023)
- [Towards Representation Learning for Weighting Problems in Design-Based Causal Inference](https://mlsanity.com/paper/2409.16407/) (2024)
- [Debugging Concept Bottleneck Models through Removal and Retraining](https://mlsanity.com/paper/2509.21385/) (2025)
- [Learning density ratios in causal inference using Bregman-Riesz regression](https://mlsanity.com/paper/2510.16127/) (2025)

## Abstract (arXiv)

In observational causal inference, in order to emulate a randomized experiment, weights are used to render treatments independent of observed covariates. This property is known as balance; in its absence, estimated causal effects may be arbitrarily biased. In this work we introduce permutation weighting, a method for estimating balancing weights using a standard binary classifier (regardless of cardinality of treatment). A large class of probabilistic classifiers may be used in this method; the choice of loss for the classifier implies the particular definition of balance. We bound bias and variance in terms of the excess risk of the classifier, show that these disappear asymptotically, and demonstrate that our classification problem directly minimizes imbalance. A wide variety of existing balancing weights may be estimated through this regime, allowing for direct comparison between methods based on classifier loss, as well as hyper-parameter tuning using cross-validation. Empirical evaluations indicate that permutation weighting provides favorable performance in comparison to existing methods.