# You say Normalizing Flows I see Bayesian Networks

arXiv: https://arxiv.org/abs/2006.00866 · HTML: https://mlsanity.com/paper/2006.00866/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2020.

## Built on by (most cited first, 5 of 5)

- [Simulation-based Inference for Cardiovascular Models](https://mlsanity.com/paper/2307.13918/) (2023)
- [Deep Learning Hamiltonian Monte Carlo](https://mlsanity.com/paper/2105.03418/) (2021)
- [Leveraging Cardiovascular Simulations for In-Vivo Prediction of Cardiac Biomarkers](https://mlsanity.com/paper/2412.17542/) (2024)
- [LeapfrogLayers: A Trainable Framework for Effective Topological Sampling](https://mlsanity.com/paper/2112.01582/) (2021)
- [Graphical Residual Flows](https://mlsanity.com/paper/2204.11846/) (2022)

## Abstract (arXiv)

Normalizing flows have emerged as an important family of deep neural networks for modelling complex probability distributions. In this note, we revisit their coupling and autoregressive transformation layers as probabilistic graphical models and show that they reduce to Bayesian networks with a pre-defined topology and a learnable density at each node. From this new perspective, we provide three results. First, we show that stacking multiple transformations in a normalizing flow relaxes independence assumptions and entangles the model distribution. Second, we show that a fundamental leap of capacity emerges when the depth of affine flows exceeds 3 transformation layers. Third, we prove the non-universality of the affine normalizing flow, regardless of its depth.