# Scalable Counterfactual Distribution Estimation in Multivariate Causal Models

arXiv: https://arxiv.org/abs/2311.00927 · HTML: https://mlsanity.com/paper/2311.00927/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 73 references that are in the map.
- First posted on arXiv 2 Nov 2023.

## Builds on (most cited first, 30 of 31)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Causality for Machine Learning](https://mlsanity.com/paper/1911.10500/) (2019)
- [Sliced Wasserstein Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1903.04064/) (2019)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Robust Reinforcement Learning using Offline Data](https://mlsanity.com/paper/2208.05129/) (2022)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions](https://mlsanity.com/paper/1806.08141/) (2018)
- [Sliced Wasserstein Kernels for Probability Distributions](https://mlsanity.com/paper/1511.03198/) (2015)
- [Sliced Wasserstein Generative Models](https://mlsanity.com/paper/1904.05408/) (2019)
- [On the Identifiability and Estimation of Causal Location-Scale Noise Models](https://mlsanity.com/paper/2210.09054/) (2022)
- [Tree-Sliced Variants of Wasserstein Distances](https://mlsanity.com/paper/1902.00342/) (2019)
- [Point-set Distances for Learning Representations of 3D Point Clouds](https://mlsanity.com/paper/2102.04014/) (2021)
- [Semiparametric counterfactual density estimation](https://mlsanity.com/paper/2102.12034/) (2021)
- [Normalizing Flows for Interventional Density Estimation](https://mlsanity.com/paper/2209.06203/) (2022)
- [Differentially Private Sliced Wasserstein Distance](https://mlsanity.com/paper/2107.01848/) (2021)
- [Optimal transport for causal discovery](https://mlsanity.com/paper/2201.09366/) (2022)
- [What's Trending in Difference-in-Differences? A Synthesis of the Recent Econometrics Literature](https://mlsanity.com/paper/2201.01194/) (2022)
- [Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics](https://mlsanity.com/paper/2202.10723/) (2022)
- [Causal effects based on distributional distances](https://mlsanity.com/paper/1806.02935/) (2018)
- [Causal Reasoning in the Presence of Latent Confounders via Neural ADMG Learning](https://mlsanity.com/paper/2303.12703/) (2023)
- [Entropy Partial Transport with Tree Metrics: Theory and Practice](https://mlsanity.com/paper/2101.09756/) (2021)
- [A Meta-Reinforcement Learning Algorithm for Causal Discovery](https://mlsanity.com/paper/2207.08457/) (2022)
- [Scalable Unbalanced Sobolev Transport for Measures on a Graph](https://mlsanity.com/paper/2302.12498/) (2023)
- [Deep Backtracking Counterfactuals for Causally Compliant Explanations](https://mlsanity.com/paper/2310.07665/) (2023)
- [Counterfactual Density Estimation using Kernel Stein Discrepancies](https://mlsanity.com/paper/2309.16129/) (2023)
- [Optimal Transport for Measures with Noisy Tree Metric](https://mlsanity.com/paper/2310.13653/) (2023)
- [When Is Parallel Trends Sensitive to Functional Form?](https://mlsanity.com/paper/2010.04814/) (2020)
- [On the Interventional Kullback-Leibler Divergence](https://mlsanity.com/paper/2302.05380/) (2023)

## Built on by (most cited first, 6 of 6)

- [Quantifying Aleatoric Uncertainty of the Treatment Effect: A Novel Orthogonal Learner](https://mlsanity.com/paper/2411.03387/) (2024)
- [Generalized Sobolev Transport for Probability Measures on a Graph](https://mlsanity.com/paper/2402.04516/) (2024)
- [Tree-Sliced Wasserstein Distance: A Geometric Perspective](https://mlsanity.com/paper/2406.13725/) (2024)
- [Distance-Based Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11050/) (2025)
- [Tree-Sliced Wasserstein Distance with Nonlinear Projection](https://mlsanity.com/paper/2505.00968/) (2025)
- [Causal Discovery via Transformed Low-Rank Quantile Surfaces](https://mlsanity.com/paper/2609.16931/) (2026)

## Abstract (arXiv)

We consider the problem of estimating the counterfactual joint distribution of multiple quantities of interests (e.g., outcomes) in a multivariate causal model extended from the classical difference-in-difference design. Existing methods for this task either ignore the correlation structures among dimensions of the multivariate outcome by considering univariate causal models on each dimension separately and hence produce incorrect counterfactual distributions, or poorly scale even for moderate-size datasets when directly dealing with such multivariate causal model. We propose a method that alleviates both issues simultaneously by leveraging a robust latent one-dimensional subspace of the original high-dimension space and exploiting the efficient estimation from the univariate causal model on such space. Since the construction of the one-dimensional subspace uses information from all the dimensions, our method can capture the correlation structures and produce good estimates of the counterfactual distribution. We demonstrate the advantages of our approach over existing methods on both synthetic and real-world data.