# Comparing Normalizing Flows with Kernel Density Estimation in Estimating Risk of Automated Driving Systems

arXiv: https://arxiv.org/abs/2507.22429 · HTML: https://mlsanity.com/paper/2507.22429/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 36 references that are in the map.
- First posted on arXiv 30 Jul 2025.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Tutorial on Variational Autoencoders](https://mlsanity.com/paper/1606.05908/) (2016)
- [How Does Batch Normalization Help Optimization?](https://mlsanity.com/paper/1805.11604/) (2018)
- [The highD Dataset: A Drone Dataset of Naturalistic Vehicle Trajectories on German Highways for Validation of Highly Automated Driving Systems](https://mlsanity.com/paper/1810.05642/) (2018)
- [A Survey on Safety-Critical Driving Scenario Generation -- A Methodological Perspective](https://mlsanity.com/paper/2202.02215/) (2022)
- [Flexible Tails for Normalizing Flows](https://mlsanity.com/paper/2406.16971/) (2024)
- [1001 Ways of Scenario Generation for Testing of Self-driving Cars: A Survey](https://mlsanity.com/paper/2304.10850/) (2023)
- [Real-World Scenario Mining for the Assessment of Automated Vehicles](https://mlsanity.com/paper/2006.00483/) (2020)
- [Diffusion Density Estimators](https://mlsanity.com/paper/2410.06986/) (2024)
- [Exploratory data analysis for moderate extreme values using non-parametric kernel methods](https://mlsanity.com/paper/1602.08807/) (2016)
- [Scenario-based assessment of automated driving systems: How (not) to parameterize scenarios?](https://mlsanity.com/paper/2409.01117/) (2024)

## Abstract (arXiv)

The development of safety validation methods is essential for the safe deployment and operation of Automated Driving Systems (ADSs). One of the goals of safety validation is to prospectively evaluate the risk of an ADS dealing with real-world traffic. Scenario-based assessment is a widely-used approach, where test cases are derived from real-world driving data. To allow for a quantitative analysis of the system performance, the exposure of the scenarios must be accurately estimated. The exposure of scenarios at parameter level is expressed using a Probability Density Function (PDF). However, assumptions about the PDF, such as parameter independence, can introduce errors, while avoiding assumptions often leads to oversimplified models with limited parameters to mitigate the curse of dimensionality. This paper considers the use of Normalizing Flows (NF) for estimating the PDF of the parameters. NF are a class of generative models that transform a simple base distribution into a complex one using a sequence of invertible and differentiable mappings, enabling flexible, high-dimensional density estimation without restrictive assumptions on the PDF's shape. We demonstrate the effectiveness of NF in quantifying risk and risk uncertainty of an ADS, comparing its performance with Kernel Density Estimation (KDE), a traditional method for non-parametric PDF estimation. While NF require more computational resources compared to KDE, NF is less sensitive to the curse of dimensionality. As a result, NF can improve risk uncertainty estimation, offering a more precise assessment of an ADS's safety. This work illustrates the potential of NF in scenario-based safety. Future work involves experimenting more with using NF for scenario generation and optimizing the NF architecture, transformation types, and training hyperparameters to further enhance their applicability.