# Semiparametric counterfactual density estimation

arXiv: https://arxiv.org/abs/2102.12034 · HTML: https://mlsanity.com/paper/2102.12034/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Feb 2021.

## Built on by (most cited first, 30 of 42)

- [The Causal-Neural Connection: Expressiveness, Learnability, and Inference](https://mlsanity.com/paper/2107.00793/) (2021)
- [Sharp Bounds for Generalized Causal Sensitivity Analysis](https://mlsanity.com/paper/2305.16988/) (2023)
- [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)
- [A Neural Framework for Generalized Causal Sensitivity Analysis](https://mlsanity.com/paper/2311.16026/) (2023)
- [One-Step Estimation of Differentiable Hilbert-Valued Parameters](https://mlsanity.com/paper/2303.16711/) (2023)
- [Multi-Source Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2405.09331/) (2024)
- [Meta-Learners for Partially-Identified Treatment Effects Across Multiple Environments](https://mlsanity.com/paper/2406.02464/) (2024)
- [An Efficient Doubly-Robust Test for the Kernel Treatment Effect](https://mlsanity.com/paper/2304.13237/) (2023)
- [Simplifying debiased inference via automatic differentiation and probabilistic programming](https://mlsanity.com/paper/2405.08675/) (2024)
- [DoubleGen: Debiased Generative Modeling of Counterfactuals](https://mlsanity.com/paper/2509.16842/) (2025)
- [Stateful Offline Contextual Policy Evaluation and Learning](https://mlsanity.com/paper/2110.10081/) (2021)
- [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)
- [Auditing Fairness under Unobserved Confounding](https://mlsanity.com/paper/2403.14713/) (2024)
- [Orthogonal Causal Calibration](https://mlsanity.com/paper/2406.01933/) (2024)
- [Causal K-Means Clustering](https://mlsanity.com/paper/2405.03083/) (2024)
- [Kernel-based estimators for functional causal effects](https://mlsanity.com/paper/2503.05024/) (2025)
- [Moments of Causal Effects](https://mlsanity.com/paper/2505.04971/) (2025)
- [Topological Causal Effects](https://mlsanity.com/paper/2603.02289/) (2026)
- [Predictive Performance Comparison of Decision Policies Under Confounding](https://mlsanity.com/paper/2404.00848/) (2024)
- [Learning Counterfactual Distributions via Kernel Nearest Neighbors](https://mlsanity.com/paper/2410.13381/) (2024)
- [Frugal, Flexible, Faithful: Causal Data Simulation via Frengression](https://mlsanity.com/paper/2508.01018/) (2025)
- [The role of the geometric mean in case-control studies](https://mlsanity.com/paper/2207.09016/) (2022)
- [Treatment Effects in Extreme Regimes](https://mlsanity.com/paper/2306.11697/) (2023)
- [Causal Inference for Genomic Data with Multiple Heterogeneous Outcomes](https://mlsanity.com/paper/2404.09119/) (2024)
- [Semiparametric Inference for Counterfactual Regression under Intervention-Driven Shift](https://mlsanity.com/paper/2504.02694/) (2025)
- [GDR-learners: Orthogonal Learning of Generative Models for Potential Outcomes](https://mlsanity.com/paper/2509.22953/) (2025)
- [Estimating Aleatoric Uncertainty in the Causal Treatment Effect](https://mlsanity.com/paper/2602.08461/) (2026)
- [Debiased Counterfactual Generation via Flow Matching from Observations](https://mlsanity.com/paper/2605.07665/) (2026)

## Abstract (arXiv)

Causal effects are often characterized with averages, which can give an incomplete picture of the underlying counterfactual distributions. Here we consider estimating the entire counterfactual density and generic functionals thereof. We focus on two kinds of target parameters. The first is a density approximation, defined by a projection onto a finite-dimensional model using a generalized distance metric, which includes f-divergences as well as Lp norms. The second is the distance between counterfactual densities, which can be used as a more nuanced effect measure than the mean difference, and as a tool for model selection. We study nonparametric efficiency bounds for these targets, giving results for smooth but otherwise generic models and distances. Importantly, we show how these bounds connect to means of particular non-trivial functions of counterfactuals, linking the problems of density and mean estimation. We go on to propose doubly robust-style estimators for the density approximations and distances, and study their rates of convergence, showing they can be optimally efficient in large nonparametric models. We also give analogous methods for model selection and aggregation, when many models may be available and of interest. Our results all hold for generic models and distances, but throughout we highlight what happens for particular choices, such as L2 projections on linear models, and KL projections on exponential families. Finally we illustrate by estimating the density of CD4 count among patients with HIV, had all been treated with combination therapy versus zidovudine alone, as well as a density effect. Our results suggest combination therapy may have increased CD4 count most for high-risk patients. Our methods are implemented in the freely available R package npcausal on GitHub.