# Causal effects based on distributional distances

arXiv: https://arxiv.org/abs/1806.02935 · HTML: https://mlsanity.com/paper/1806.02935/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2018.

## Built on by (most cited first, 16 of 16)

- [Conditional Distributional Treatment Effect with Kernel Conditional Mean Embeddings and U-Statistic Regression](https://mlsanity.com/paper/2102.08208/) (2021)
- [Normalizing Flows for Interventional Density Estimation](https://mlsanity.com/paper/2209.06203/) (2022)
- [Counterfactual Generative Models for Time-Varying Treatments](https://mlsanity.com/paper/2305.15742/) (2023)
- [An Efficient Doubly-Robust Test for the Kernel Treatment Effect](https://mlsanity.com/paper/2304.13237/) (2023)
- [Quantifying Aleatoric Uncertainty of the Treatment Effect: A Novel Orthogonal Learner](https://mlsanity.com/paper/2411.03387/) (2024)
- [Fair and Robust Estimation of Heterogeneous Treatment Effects for Policy Learning](https://mlsanity.com/paper/2306.03625/) (2023)
- [Scalable Counterfactual Distribution Estimation in Multivariate Causal Models](https://mlsanity.com/paper/2311.00927/) (2023)
- [Hierarchical and Density-based Causal Clustering](https://mlsanity.com/paper/2411.01250/) (2024)
- [Counterfactual Probabilistic Diffusion with Expert Models](https://mlsanity.com/paper/2508.13355/) (2025)
- [Moments of Causal Effects](https://mlsanity.com/paper/2505.04971/) (2025)
- [Learning Counterfactual Distributions via Kernel Nearest Neighbors](https://mlsanity.com/paper/2410.13381/) (2024)
- [GDR-learners: Orthogonal Learning of Generative Models for Potential Outcomes](https://mlsanity.com/paper/2509.22953/) (2025)
- [Debiased Counterfactual Generation via Flow Matching from Observations](https://mlsanity.com/paper/2605.07665/) (2026)
- [A Bayesian Semiparametric Method For Estimating Causal Quantile Effects](https://mlsanity.com/paper/2211.01591/) (2022)
- [Distributional Causal Mediation via Conditional Generative Modeling](https://mlsanity.com/paper/2605.01765/) (2026)
- [Geometry Adaptive Counterfactual Distribution Learning with Diffusion-Guided Smoothing](https://mlsanity.com/paper/2605.25811/) (2026)

## Abstract (arXiv)

Comparing counterfactual distributions can provide more nuanced and valuable measures for causal effects, going beyond typical summary statistics such as averages. In this work, we consider characterizing causal effects via distributional distances, focusing on two kinds of target parameters. The first is the counterfactual outcome density. We propose a doubly robust-style estimator for the counterfactual density and study its rates of convergence and limiting distributions. We analyze asymptotic upper bounds on the Lq and the integrated Lq risks of the proposed estimator, and propose a bootstrap-based confidence band. The second is a novel distributional causal effect defined by the L1 distance between different counterfactual distributions. We study three approaches for estimating the proposed distributional effect: smoothing the counterfactual density, smoothing the L1 distance, and imposing a margin condition. For each approach, we analyze asymptotic properties and error bounds of the proposed estimator, and discuss potential advantages and disadvantages. We go on to present a bootstrap approach for obtaining confidence intervals, and propose a test of no distributional effect. We conclude with a numerical illustration and a real-world example.