# Covariate Balancing Propensity Score by Tailored Loss Functions

arXiv: https://arxiv.org/abs/1601.05890 · HTML: https://mlsanity.com/paper/1601.05890/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jan 2016.

## Built on by (most cited first, 30 of 34)

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- [Augmented balancing weights as linear regression](https://mlsanity.com/paper/2304.14545/) (2023)
- [End-to-End Balancing for Causal Continuous Treatment-Effect Estimation](https://mlsanity.com/paper/2107.13068/) (2021)
- [Stabilized Inverse Probability Weighting via Isotonic Calibration](https://mlsanity.com/paper/2411.06342/) (2024)
- [Direct Bias-Correction Term Estimation for Average Treatment Effect Estimation](https://mlsanity.com/paper/2509.22122/) (2025)
- [Kernel Ridge Riesz Representers: Generalization, Mis-specification, and the Counterfactual Effective Dimension](https://mlsanity.com/paper/2102.11076/) (2021)
- [Fair and Robust Estimation of Heterogeneous Treatment Effects for Policy Learning](https://mlsanity.com/paper/2306.03625/) (2023)
- [Sequential Kernel Embedding for Mediated and Time-Varying Dose Response Curves](https://mlsanity.com/paper/2111.03950/) (2021)
- [Outcome Assumptions and Duality Theory for Balancing Weights](https://mlsanity.com/paper/2203.09557/) (2022)
- [Differentially Private Covariate Balancing Causal Inference](https://mlsanity.com/paper/2410.14789/) (2024)
- [Direct Debiased Machine Learning via Bregman Divergence Minimization](https://mlsanity.com/paper/2510.23534/) (2025)
- [Estimating Average Treatment Effects with Support Vector Machines](https://mlsanity.com/paper/2102.11926/) (2021)
- [Statistical inference using Regularized M-estimation in the reproducing kernel Hilbert space for handling missing data](https://mlsanity.com/paper/2107.07371/) (2021)
- [Calibrated and Conformal Propensity Scores for Causal Effect Estimation](https://mlsanity.com/paper/2306.00382/) (2023)
- [Riesz Regression As Direct Density Ratio Estimation](https://mlsanity.com/paper/2511.04568/) (2025)
- [Optimally taming biases in black-box models for efficient semiparametric estimation](https://mlsanity.com/paper/2606.06368/) (2026)
- [CATE Lasso: Conditional Average Treatment Effect Estimation with High-Dimensional Linear Regression](https://mlsanity.com/paper/2310.16819/) (2023)
- [A Unified Theory for Causal Inference: Direct Debiased Machine Learning via Bregman-Riesz Regression](https://mlsanity.com/paper/2510.26783/) (2025)
- [Benign-Overfitting in Conditional Average Treatment Effect Prediction with Linear Regression](https://mlsanity.com/paper/2202.05245/) (2022)
- [Estimating causal effects with optimization-based methods: A review and empirical comparison](https://mlsanity.com/paper/2203.00097/) (2022)
- [Synthetic Principal Component Design: Fast Covariate Balancing with Synthetic Controls](https://mlsanity.com/paper/2211.15241/) (2022)
- [Scalable kernel balancing weights in a nationwide observational study of hospital profit status and heart attack outcomes](https://mlsanity.com/paper/2311.00568/) (2023)
- [Semi-Supervised Treatment Effect Estimation with Unlabeled Covariates for Prediction-Powered Causal Inference](https://mlsanity.com/paper/2511.08303/) (2025)
- [De-paradox Tree: Breaking Down Simpson's Paradox via A Kernel-Based Partition Algorithm](https://mlsanity.com/paper/2603.02174/) (2026)
- [Tailoring Strictly Proper Scoring Rules for Downstream Tasks: An Application to Causal Inference](https://mlsanity.com/paper/2606.03332/) (2026)
- [Optimal use of a black-box learner in semiparametric estimation](https://mlsanity.com/paper/2607.21541/) (2026)
- [Rejoinder: Learning Optimal Distributionally Robust Individualized Treatment Rules](https://mlsanity.com/paper/2110.08936/) (2021)
- [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)
- [Triple/Debiased Lasso for Statistical Inference of Conditional Average Treatment Effects](https://mlsanity.com/paper/2403.03240/) (2024)
- [Penalized Empirical Likelihood for Doubly Robust Causal Inference under Contamination in High Dimensions](https://mlsanity.com/paper/2507.17439/) (2025)

## Abstract (arXiv)

In observational studies, propensity scores are commonly estimated by maxi- mum likelihood but may fail to balance high-dimensional pre-treatment covariates even after specification search. We introduce a general framework that unifies and generalizes several recent proposals to improve covariate balance when designing an observational study. In- stead of the likelihood function, we propose to optimize special loss functions---covariate balancing scoring rules (CBSR)---to estimate the propensity score. A CBSR is uniquely determined by the link function in the GLM and the estimand (a weighted average treatment effect). We show CBSR does not lose asymptotic efficiency to the Bernoulli likelihood in estimating the weighted average treatment effect compared, but CBSR is much more robust in finite sample. Borrowing tools developed in statistical learning, we propose practical strategies to balance covariate functions in rich function classes. This is useful to estimate the maximum bias of the inverse probability weighting (IPW) estimators and construct honest confidence interval in finite sample. Lastly, we provide several numerical examples to demonstrate the trade-off of bias and variance in the IPW-type estimators and the trade-off in balancing different function classes of the covariates.