# Energy Balancing of Covariate Distributions

arXiv: https://arxiv.org/abs/2004.13962 · HTML: https://mlsanity.com/paper/2004.13962/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2020.

## Built on by (most cited first, 9 of 9)

- [Covariate-Balancing-Aware Interpretable Deep Learning models for Treatment Effect Estimation](https://mlsanity.com/paper/2203.03185/) (2022)
- [Differentially Private Covariate Balancing Causal Inference](https://mlsanity.com/paper/2410.14789/) (2024)
- [Estimating Average Treatment Effects with Support Vector Machines](https://mlsanity.com/paper/2102.11926/) (2021)
- [Robust Sample Weighting to Facilitate Individualized Treatment Rule Learning for a Target Population](https://mlsanity.com/paper/2105.00581/) (2021)
- [What can the millions of random treatments in nonexperimental data reveal about causes?](https://mlsanity.com/paper/2105.01152/) (2021)
- [Meta-analysis of individualized treatment rules via sign-coherency](https://mlsanity.com/paper/2211.15476/) (2022)
- [Differentially Private Two-Stage Empirical Risk Minimization with Applications to Individualized Treatment Rule](https://mlsanity.com/paper/2602.12604/) (2026)
- [Causal Generalization of Continuous Treatment Effects under Covariate Shift](https://mlsanity.com/paper/2608.19383/) (2026)
- [Weighted Spline-Expanded Networks with Distributional Balancing for Continuous Treatment Effects](https://mlsanity.com/paper/2609.33740/) (2026)

## Abstract (arXiv)

Bias in causal comparisons has a direct correspondence with distributional imbalance of covariates between treatment groups. Weighting strategies such as inverse propensity score weighting attempt to mitigate bias by either modeling the treatment assignment mechanism or balancing specified covariate moments. This paper introduces a new weighting method, called energy balancing, which instead aims to balance weighted covariate distributions. By directly targeting distributional imbalance, the proposed weighting strategy can be flexibly utilized in a wide variety of causal analyses, including the estimation of average treatment effects and individualized treatment rules. Our energy balancing weights (EBW) approach has several advantages over existing weighting techniques. First, it offers a model-free and robust approach for obtaining covariate balance that does not require tuning parameters, obviating the need for modeling decisions of secondary nature to the scientific question at hand. Second, since this approach is based on a genuine measure of distributional balance, it provides a means for assessing the balance induced by a given set of weights for a given dataset. Finally, the proposed method is computationally efficient and has desirable theoretical guarantees under mild conditions. We demonstrate the effectiveness of this EBW approach in a suite of simulation experiments, and in studies on the safety of right heart catheterization and the effect of indwelling arterial catheters.