# Graphical Normalizing Flows

arXiv: https://arxiv.org/abs/2006.02548 · HTML: https://mlsanity.com/paper/2006.02548/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2020.

## Built on by (most cited first, 30 of 37)

- [Differentiable DAG Sampling](https://mlsanity.com/paper/2203.08509/) (2022)
- [Normalizing Flows for Interventional Density Estimation](https://mlsanity.com/paper/2209.06203/) (2022)
- [A deep generative model for probabilistic energy forecasting in power systems: normalizing flows](https://mlsanity.com/paper/2106.09370/) (2021)
- [Personalized Public Policy Analysis in Social Sciences using Causal-Graphical Normalizing Flows](https://mlsanity.com/paper/2202.03281/) (2022)
- [Concept Graph Neural Networks for Surgical Video Understanding](https://mlsanity.com/paper/2202.13402/) (2022)
- [Structured Neural Networks for Density Estimation and Causal Inference](https://mlsanity.com/paper/2311.02221/) (2023)
- [Simulation-based Inference for Cardiovascular Models](https://mlsanity.com/paper/2307.13918/) (2023)
- [Deep Learning With DAGs](https://mlsanity.com/paper/2401.06864/) (2024)
- [On the Role of Entropy-based Loss for Learning Causal Structures with Continuous Optimization](https://mlsanity.com/paper/2106.02835/) (2021)
- [Dimension-independent rates for structured neural density estimation](https://mlsanity.com/paper/2411.15095/) (2024)
- [Leveraging Cardiovascular Simulations for In-Vivo Prediction of Cardiac Biomarkers](https://mlsanity.com/paper/2412.17542/) (2024)
- [ADAVI: Automatic Dual Amortized Variational Inference Applied To Pyramidal Bayesian Models](https://mlsanity.com/paper/2106.12248/) (2021)
- [Individual Survival Curves with Conditional Normalizing Flows](https://mlsanity.com/paper/2107.12825/) (2021)
- [Neuro-Causal Factor Analysis](https://mlsanity.com/paper/2305.19802/) (2023)
- [Permutation-based Inference for Variational Learning of Directed Acyclic Graphs](https://mlsanity.com/paper/2402.02644/) (2024)
- [Flow-based Spatio-Temporal Structured Prediction of Motion Dynamics](https://mlsanity.com/paper/2104.04391/) (2021)
- [Embedded-model flows: Combining the inductive biases of model-free deep learning and explicit probabilistic modeling](https://mlsanity.com/paper/2110.06021/) (2021)
- [Counterfactual Analysis of the Impact of the IMF Program on Child Poverty in the Global-South Region using Causal-Graphical Normalizing Flows](https://mlsanity.com/paper/2202.09391/) (2022)
- [Breaking the curse of dimensionality in structured density estimation](https://mlsanity.com/paper/2410.07685/) (2024)
- [MACAW: A Causal Generative Model for Medical Imaging](https://mlsanity.com/paper/2412.02900/) (2024)
- [Generating Directed Graphs with Dual Attention and Asymmetric Encoding](https://mlsanity.com/paper/2506.16404/) (2025)
- [Graphical Residual Flows](https://mlsanity.com/paper/2204.11846/) (2022)
- [ρ-GNF: A Copula-based Sensitivity Analysis to Unobserved Confounding Using Normalizing Flows](https://mlsanity.com/paper/2209.07111/) (2022)
- [Variational Flow Graphical Model](https://mlsanity.com/paper/2207.02722/) (2022)
- [Probabilistic Forecasting of Irregular Time Series via Conditional Flows](https://mlsanity.com/paper/2402.06293/) (2024)
- [Causal Posterior Estimation](https://mlsanity.com/paper/2505.21468/) (2025)
- [Causal Mediation Analysis with Multiple Mediators: A Simulation Approach](https://mlsanity.com/paper/2506.14019/) (2025)
- [Intervening to Learn and Compose Causally Disentangled Representations](https://mlsanity.com/paper/2507.04754/) (2025)
- [Sensitivity Analysis to Unobserved Confounding with Copula-based Normalizing Flows](https://mlsanity.com/paper/2508.08752/) (2025)
- [SIReN-VAE: Leveraging Flows and Amortized Inference for Bayesian Networks](https://mlsanity.com/paper/2204.11847/) (2022)

## Abstract (arXiv)

Normalizing flows model complex probability distributions by combining a base distribution with a series of bijective neural networks. State-of-the-art architectures rely on coupling and autoregressive transformations to lift up invertible functions from scalars to vectors. In this work, we revisit these transformations as probabilistic graphical models, showing they reduce to Bayesian networks with a pre-defined topology and a learnable density at each node. From this new perspective, we propose the graphical normalizing flow, a new invertible transformation with either a prescribed or a learnable graphical structure. This model provides a promising way to inject domain knowledge into normalizing flows while preserving both the interpretability of Bayesian networks and the representation capacity of normalizing flows. We show that graphical conditioners discover relevant graph structure when we cannot hypothesize it. In addition, we analyze the effect of ℓ1-penalization on the recovered structure and on the quality of the resulting density estimation. Finally, we show that graphical conditioners lead to competitive white box density estimators. Our implementation is available at https://github.com/AWehenkel/DAG-NF.