# Hybrid Bernstein Normalizing Flows for Flexible Multivariate Density Regression with Interpretable Marginals

arXiv: https://arxiv.org/abs/2505.14164 · HTML: https://mlsanity.com/paper/2505.14164/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 35 references that are in the map.
- First posted on arXiv 20 May 2025.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [TensorFlow Distributions](https://mlsanity.com/paper/1711.10604/) (2017)
- [Sum-of-Squares Polynomial Flow](https://mlsanity.com/paper/1905.02325/) (2019)
- [Graphical Normalizing Flows](https://mlsanity.com/paper/2006.02548/) (2020)
- [Cubic-Spline Flows](https://mlsanity.com/paper/1906.02145/) (2019)
- [Deep Conditional Transformation Models](https://mlsanity.com/paper/2010.07860/) (2020)
- [Most Likely Transformations](https://mlsanity.com/paper/1508.06749/) (2015)
- [Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas](https://mlsanity.com/paper/1503.03305/) (2015)
- [Deep transformation models: Tackling complex regression problems with neural network based transformation models](https://mlsanity.com/paper/2004.00464/) (2020)
- [Multivariate Conditional Transformation Models](https://mlsanity.com/paper/1906.03151/) (2019)
- [Probabilistic Time Series Forecasts with Autoregressive Transformation Models](https://mlsanity.com/paper/2110.08248/) (2021)
- [Short-Term Density Forecasting of Low-Voltage Load using Bernstein-Polynomial Normalizing Flows](https://mlsanity.com/paper/2204.13939/) (2022)
- [How Inverse Conditional Flows Can Serve as a Substitute for Distributional Regression](https://mlsanity.com/paper/2405.05429/) (2024)
- [Graphical Transformation Models](https://mlsanity.com/paper/2503.17845/) (2025)

## Abstract (arXiv)

Density regression models allow a comprehensive understanding of data by modeling the complete conditional probability distribution. While flexible estimation approaches such as normalizing flows (NF) work particularly well in multiple dimensions, interpreting the input-output relationship of such models is often difficult, due to the black-box character of deep learning models. In contrast, existing statistical methods for multivariate outcomes such as multivariate conditional transformation models (MCTM) are restricted in flexibility and are often not expressive enough to represent complex multivariate probability distributions. In this paper, we combine MCTM with state-of-the-art and autoregressive NF to leverage the transparency of MCTM for modeling interpretable feature effects on the marginal distributions in the first step and the flexibility of neural-network-based NF techniques to account for complex and non-linear relationships in the joint data distribution. We demonstrate our method's versatility in various numerical experiments and compare it with MCTM and other NF models on both simulated and real-world data.