# Statistical Guarantees for the Robustness of Bayesian Neural Networks

arXiv: https://arxiv.org/abs/1903.01980 · HTML: https://mlsanity.com/paper/1903.01980/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Mar 2019.

## Built on by (most cited first, 30 of 35)

- [Making Substitute Models More Bayesian Can Enhance Transferability of Adversarial Examples](https://mlsanity.com/paper/2302.05086/) (2023)
- [Bayesian Inference with Certifiable Adversarial Robustness](https://mlsanity.com/paper/2102.05289/) (2021)
- [Make Sure You're Unsure: A Framework for Verifying Probabilistic Specifications](https://mlsanity.com/paper/2102.09479/) (2021)
- [Infinite Time Horizon Safety of Bayesian Neural Networks](https://mlsanity.com/paper/2111.03165/) (2021)
- [Towards Practical Robustness Analysis for DNNs based on PAC-Model Learning](https://mlsanity.com/paper/2101.10102/) (2021)
- [Structured Stochastic Gradient MCMC](https://mlsanity.com/paper/2107.09028/) (2021)
- [On the Robustness of Bayesian Neural Networks to Adversarial Attacks](https://mlsanity.com/paper/2207.06154/) (2022)
- [Distributionally Robust Statistical Verification with Imprecise Neural Networks](https://mlsanity.com/paper/2308.14815/) (2023)
- [Certification of Iterative Predictions in Bayesian Neural Networks](https://mlsanity.com/paper/2105.10134/) (2021)
- [BNN-DP: Robustness Certification of Bayesian Neural Networks via Dynamic Programming](https://mlsanity.com/paper/2306.10742/) (2023)
- [TrajPAC: Towards Robustness Verification of Pedestrian Trajectory Prediction Models](https://mlsanity.com/paper/2308.05985/) (2023)
- [Adversarial Robustness Certification for Bayesian Neural Networks](https://mlsanity.com/paper/2306.13614/) (2023)
- [Evaluating the Robustness of Bayesian Neural Networks Against Different Types of Attacks](https://mlsanity.com/paper/2106.09223/) (2021)
- [Adversarial Robustness Guarantees for Gaussian Processes](https://mlsanity.com/paper/2104.03180/) (2021)
- [Probabilistic Reach-Avoid for Bayesian Neural Networks](https://mlsanity.com/paper/2310.01951/) (2023)
- [Tight Verification of Probabilistic Robustness in Bayesian Neural Networks](https://mlsanity.com/paper/2401.11627/) (2024)
- [Robustness of Bayesian Neural Networks to White-Box Adversarial Attacks](https://mlsanity.com/paper/2111.08591/) (2021)
- [Solving Probabilistic Verification Problems of Neural Networks using Branch and Bound](https://mlsanity.com/paper/2405.17556/) (2024)
- [The Effect of Prior Lipschitz Continuity on the Adversarial Robustness of Bayesian Neural Networks](https://mlsanity.com/paper/2101.02689/) (2021)
- [Individual Fairness in Bayesian Neural Networks](https://mlsanity.com/paper/2304.10828/) (2023)
- [Improving Transferability of Adversarial Examples via Bayesian Attacks](https://mlsanity.com/paper/2307.11334/) (2023)
- [A Survey of Neural Network Robustness Assessment in Image Recognition](https://mlsanity.com/paper/2404.08285/) (2024)
- [Variational Routing: A Scalable Bayesian Framework for Calibrated Mixture-of-Experts Transformers](https://mlsanity.com/paper/2603.09453/) (2026)
- [Fast Falsification of Neural Networks using Property Directed Testing](https://mlsanity.com/paper/2104.12418/) (2021)
- [PopSkipJump: Decision-Based Attack for Probabilistic Classifiers](https://mlsanity.com/paper/2106.07445/) (2021)
- [Incremental Satisfiability Modulo Theory for Verification of Deep Neural Networks](https://mlsanity.com/paper/2302.06455/) (2023)
- [Probabilistic Inverse Modeling: An Application in Hydrology](https://mlsanity.com/paper/2210.06213/) (2022)
- [Learning Beyond Euclid: Curvature-Adaptive Generalization for Neural Networks on Manifolds](https://mlsanity.com/paper/2507.02999/) (2025)
- [Robustness of Control Design via Bayesian Learning](https://mlsanity.com/paper/2205.06896/) (2022)
- [Uncertainty Quantification in Inverse Models in Hydrology](https://mlsanity.com/paper/2310.02193/) (2023)

## Abstract (arXiv)

We introduce a probabilistic robustness measure for Bayesian Neural Networks (BNNs), defined as the probability that, given a test point, there exists a point within a bounded set such that the BNN prediction differs between the two. Such a measure can be used, for instance, to quantify the probability of the existence of adversarial examples. Building on statistical verification techniques for probabilistic models, we develop a framework that allows us to estimate probabilistic robustness for a BNN with statistical guarantees, i.e., with a priori error and confidence bounds. We provide experimental comparison for several approximate BNN inference techniques on image classification tasks associated to MNIST and a two-class subset of the GTSRB dataset. Our results enable quantification of uncertainty of BNN predictions in adversarial settings.