mlsanityData to 14 Oct 2026

Normalizing Flows for Interventional Density Estimation

arXiv 2209.06203 ↗First posted 13 Sep 2022cs.LG cs.AI stat.MEcited by 21

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Abstract (arXiv)

Existing machine learning methods for causal inference usually estimate quantities expressed via the mean of potential outcomes (e.g., average treatment effect). However, such quantities do not capture the full information about the distribution of potential outcomes. In this work, we estimate the density of potential outcomes after interventions from observational data. For this, we propose a novel, fully-parametric deep learning method called Interventional Normalizing Flows. Specifically, we combine two normalizing flows, namely (i) a nuisance flow for estimating nuisance parameters and (ii) a target flow for parametric estimation of the density of potential outcomes. We further develop a tractable optimization objective based on a one-step bias correction for efficient and doubly robust estimation of the target flow parameters. As a result, our Interventional Normalizing Flows offer a properly normalized density estimator. Across various experiments, we demonstrate that our Interventional Normalizing Flows are expressive and highly effective, and scale well with both sample size and high-dimensional confounding. To the best of our knowledge, our Interventional Normalizing Flows are the first proper fully-parametric, deep learning method for density estimation of potential outcomes.

Key facts. Cited by 21 papers in the map (data through 14 Oct 2026). Builds on 43 of its 96 references that are in the map. First posted on arXiv 13 Sep 2022.

Builds on

43 of 96 references.

References · 43

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2013Auto-Encoding Variational Bayescited by 4,053
  3. 2015Variational Inference with Normalizing Flowscited by 2,170
  4. 2016Density estimation using Real NVPcited by 2,042
  5. 2014NICE: Non-linear Independent Components Estimationcited by 1,058
  6. 2016HyperNetworkscited by 1,027
  7. 2015Estimation and Inference of Heterogeneous Treatment Effects using Random Forestscited by 600
  8. 2019Neural Spline Flowscited by 596
  9. 2017Meta-learners for Estimating Heterogeneous Treatment Effects using Machine Learningcited by 478
  10. 2016Learning Representations for Counterfactual Inferencecited by 329
  11. 2019Adapting Neural Networks for the Estimation of Treatment Effectscited by 287
  12. 2020Differentiable Causal Discovery from Interventional Datacited by 218
31 more
  1. 2018Neural Autoregressive Flowscited by 208
  2. 2017Bayesian Inference of Individualized Treatment Effects using Multi-task Gaussian Processescited by 158
  3. 2021Nonparametric Estimation of Heterogeneous Treatment Effects: From Theory to Learning Algorithmscited by 144
  4. 2022Semiparametric doubly robust targeted double machine learning: a reviewcited by 133
  5. 2017Automated versus do-it-yourself methods for causal inference: Lessons learned from a data analysis competitioncited by 106
  6. 2020Causal Autoregressive Flowscited by 99
  7. 2019Learning Counterfactual Representations for Estimating Individual Dose-Response Curvescited by 90
  8. 2020Estimating the Effects of Continuous-valued Interventions using Generative Adversarial Networkscited by 78
  9. 2021VCNet and Functional Targeted Regularization For Learning Causal Effects of Continuous Treatmentscited by 61
  10. 2020Identifying Causal-Effect Inference Failure with Uncertainty-Aware Modelscited by 57
  11. 2018Conditional Density Estimation with Bayesian Normalising Flowscited by 54
  12. 2021Quantifying Ignorance in Individual-Level Causal-Effect Estimates under Hidden Confoundingcited by 50
  13. 2021Semiparametric counterfactual density estimationcited by 42
  14. 2020Invertible Generative Modeling using Linear Rational Splinescited by 40
  15. 2020Graphical Normalizing Flowscited by 37
  16. 2018Benchmarking Framework for Performance-Evaluation of Causal Inference Analysiscited by 33
  17. 2018Counterfactual Mean Embeddingscited by 32
  18. 2022Combining Observational and Randomized Data for Estimating Heterogeneous Treatment Effectscited by 32
  19. 2021Estimating Average Treatment Effects via Orthogonal Regularizationcited by 25
  20. 2019Noise Regularization for Conditional Density Estimationcited by 23
  21. 2020Targeting for long-term outcomescited by 23
  22. 2022Scalable Sensitivity and Uncertainty Analysis for Causal-Effect Estimates of Continuous-Valued Interventionscited by 20
  23. 2020Learning Robust Models Using The Principle of Independent Causal Mechanismscited by 18
  24. 2021Policy Analysis using Synthetic Controls in Continuous-Timecited by 18
  25. 2017Bayesian Nonparametric Causal Inference: Information Rates and Learning Algorithmscited by 16
  26. 2018Causal effects based on distributional distancescited by 16
  27. 2022Personalized Public Policy Analysis in Social Sciences using Causal-Graphical Normalizing Flowscited by 16
  28. 2021Harmonization with Flow-based Causal Inferencecited by 14
  29. 2014Deep Exponential Familiescited by 12
  30. 2022Estimating Conditional Average Treatment Effects with Missing Treatment Informationcited by 7
  31. 2022A Free Lunch with Influence Functions? Improving Neural Network Estimates with Concepts from Semiparametric Statisticscited by 4

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 7; 2024: 5; 2025: 6; 2026: 3.

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  1. 2024Causal machine learning for predicting treatment outcomescited by 74
  2. 2023Bayesian Neural Controlled Differential Equations for Treatment Effect Estimationcited by 26
  3. 2023Sharp Bounds for Generalized Causal Sensitivity Analysiscited by 24
  4. 2023Counterfactual Generative Models for Time-Varying Treatmentscited by 17
  5. 2023A Neural Framework for Generalized Causal Sensitivity Analysiscited by 17
  6. 2023Reliable Off-Policy Learning for Dosage Combinationscited by 15
  7. 2023Structured Neural Networks for Density Estimation and Causal Inferencecited by 13
  8. 2024IGC-Net for conditional average potential outcome estimation over timecited by 10
  9. 2024Stabilized Neural Prediction of Potential Outcomes in Continuous Timecited by 7
  10. 2023Scalable Counterfactual Distribution Estimation in Multivariate Causal Modelscited by 6
  11. 2024Deep Autoregressive Models as Causal Inference Enginescited by 6
  12. 2024chiSPN: Characteristic Interventional Sum-Product Networks for Causal Inference in Hybrid Domainscited by 3
9 more
  1. 2025Flow-based Generative Modeling of Potential Outcomes and Counterfactualscited by 3
  2. 2025A Diffusion-Based Method for Learning the Multi-Outcome Distribution of Medical Treatmentscited by 3
  3. 2025PrivATE: Differentially Private Confidence Intervals for Average Treatment Effectscited by 2
  4. 2025ExOSITO: Explainable Off-Policy Learning with Side Information for Intensive Care Unit Blood Test Orderscited by 1
  5. 2025GDR-learners: Orthogonal Learning of Generative Models for Potential Outcomescited by 1
  6. 2025Overlap-Adaptive Regularization for Conditional Average Treatment Effect Estimationcited by 1
  7. 2026Debiased Counterfactual Generation via Flow Matching from Observationscited by 1
  8. 2026Benchmarking Counterfactual Prediction in Epidemic Time Series with Time-Varying Interventionscited by 0
  9. 2026Recovering Weak Signals with Normalizing Flowscited by 0