# Uncertainty Quantification via Stable Distribution Propagation

arXiv: https://arxiv.org/abs/2402.08324 · HTML: https://mlsanity.com/paper/2402.08324/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 52 references that are in the map.
- First posted on arXiv 13 Feb 2024.

## Builds on (most cited first, 30 of 35)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Selective Classification for Deep Neural Networks](https://mlsanity.com/paper/1705.08500/) (2017)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [On Evaluating Adversarial Robustness](https://mlsanity.com/paper/1902.06705/) (2019)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [Bayesian Neural Networks: An Introduction and Survey](https://mlsanity.com/paper/2006.12024/) (2020)
- [A General Framework for Uncertainty Estimation in Deep Learning](https://mlsanity.com/paper/1907.06890/) (2019)
- [A Review and Comparative Study on Probabilistic Object Detection in Autonomous Driving](https://mlsanity.com/paper/2011.10671/) (2020)
- [Lightweight Probabilistic Deep Networks](https://mlsanity.com/paper/1805.11327/) (2018)
- [Sampling-free Epistemic Uncertainty Estimation Using Approximated Variance Propagation](https://mlsanity.com/paper/1908.00598/) (2019)
- [Deep Differentiable Logic Gate Networks](https://mlsanity.com/paper/2210.08277/) (2022)
- [Natural-Parameter Networks: A Class of Probabilistic Neural Networks](https://mlsanity.com/paper/1611.00448/) (2016)
- [Amodal Segmentation through Out-of-Task and Out-of-Distribution Generalization with a Bayesian Model](https://mlsanity.com/paper/2010.13175/) (2020)
- [Monotonic Differentiable Sorting Networks](https://mlsanity.com/paper/2203.09630/) (2022)
- [Quasi-Monte Carlo Variational Inference](https://mlsanity.com/paper/1807.01604/) (2018)
- [A Survey on Epistemic (Model) Uncertainty in Supervised Learning: Recent Advances and Applications](https://mlsanity.com/paper/2111.01968/) (2021)
- [Autoregressive Quantile Flows for Predictive Uncertainty Estimation](https://mlsanity.com/paper/2112.04643/) (2021)
- [Differentiable Clustering with Perturbed Spanning Forests](https://mlsanity.com/paper/2305.16358/) (2023)
- [Robust Convolutional Neural Networks under Adversarial Noise](https://mlsanity.com/paper/1511.06306/) (2015)
- [GenDR: A Generalized Differentiable Renderer](https://mlsanity.com/paper/2204.13845/) (2022)

## Built on by (most cited first, 10 of 10)

- [Consensus Entropy: Harnessing Multi-VLM Agreement for Self-Verifying and Self-Improving OCR](https://mlsanity.com/paper/2504.11101/) (2025)
- [Enhancing Uncertainty Modeling with Semantic Graph for Hallucination Detection](https://mlsanity.com/paper/2501.02020/) (2025)
- [Streamlining Prediction in Bayesian Deep Learning](https://mlsanity.com/paper/2411.18425/) (2024)
- [Generalizing Stochastic Smoothing for Differentiation and Gradient Estimation](https://mlsanity.com/paper/2410.08125/) (2024)
- [Low-Order Flow Reconstruction and Uncertainty Quantification in Disturbed Aerodynamics Using Sparse Pressure Measurements](https://mlsanity.com/paper/2501.03406/) (2025)
- [Assumed Density Filtering and Smoothing with Neural Network Surrogate Models](https://mlsanity.com/paper/2511.09016/) (2025)
- [A Start To End Machine Learning Approach To Maximize Scientific Throughput From The LCLS-II-HE](https://mlsanity.com/paper/2505.23858/) (2025)
- [Calibrated Sampling-Free Uncertainty Estimation in Bayesian Deep Learning](https://mlsanity.com/paper/2606.16214/) (2026)
- [Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fields](https://mlsanity.com/paper/2608.09071/) (2026)
- [Uncertainty propagation in auto-regressive random neural network models](https://mlsanity.com/paper/2608.20483/) (2026)

## Abstract (arXiv)

We propose a new approach for propagating stable probability distributions through neural networks. Our method is based on local linearization, which we show to be an optimal approximation in terms of total variation distance for the ReLU non-linearity. This allows propagating Gaussian and Cauchy input uncertainties through neural networks to quantify their output uncertainties. To demonstrate the utility of propagating distributions, we apply the proposed method to predicting calibrated confidence intervals and selective prediction on out-of-distribution data. The results demonstrate a broad applicability of propagating distributions and show the advantages of our method over other approaches such as moment matching.