# Infinite Time Horizon Safety of Bayesian Neural Networks

arXiv: https://arxiv.org/abs/2111.03165 · HTML: https://mlsanity.com/paper/2111.03165/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 57 references that are in the map.
- First posted on arXiv 4 Nov 2021.

## Builds on (most cited first, 22 of 22)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [Constrained Policy Optimization](https://mlsanity.com/paper/1705.10528/) (2017)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [Deep Learning for Medical Image Analysis](https://mlsanity.com/paper/1708.08987/) (2017)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [A Lyapunov-based Approach to Safe Reinforcement Learning](https://mlsanity.com/paper/1805.07708/) (2018)
- [Learning to Drive in a Day](https://mlsanity.com/paper/1807.00412/) (2018)
- [Towards Stable and Efficient Training of Verifiably Robust Neural Networks](https://mlsanity.com/paper/1906.06316/) (2019)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [An approach to reachability analysis for feed-forward ReLU neural networks](https://mlsanity.com/paper/1706.07351/) (2017)
- [Reachability Analysis of Deep Neural Networks with Provable Guarantees](https://mlsanity.com/paper/1805.02242/) (2018)
- [Verisig: verifying safety properties of hybrid systems with neural network controllers](https://mlsanity.com/paper/1811.01828/) (2018)
- [Uncertainty Quantification with Statistical Guarantees in End-to-End Autonomous Driving Control](https://mlsanity.com/paper/1909.09884/) (2019)
- [Probabilistic Safety for Bayesian Neural Networks](https://mlsanity.com/paper/2004.10281/) (2020)
- [Statistical Guarantees for the Robustness of Bayesian Neural Networks](https://mlsanity.com/paper/1903.01980/) (2019)
- [Scalable Verification of Quantized Neural Networks (Technical Report)](https://mlsanity.com/paper/2012.08185/) (2020)
- [Global Optimization of Objective Functions Represented by ReLU Networks](https://mlsanity.com/paper/2010.03258/) (2020)
- [Chance-Constrained Trajectory Optimization for Safe Exploration and Learning of Nonlinear Systems](https://mlsanity.com/paper/2005.04374/) (2020)
- [Lagrangian Reachtubes: The Next Generation](https://mlsanity.com/paper/2012.07458/) (2020)
- [Automated and Formal Synthesis of Neural Barrier Certificates for Dynamical Models](https://mlsanity.com/paper/2007.03251/) (2020)

## Built on by (most cited first, 13 of 13)

- [Stability Verification in Stochastic Control Systems via Neural Network Supermartingales](https://mlsanity.com/paper/2112.09495/) (2021)
- [Learning Control Policies for Stochastic Systems with Reach-avoid Guarantees](https://mlsanity.com/paper/2210.05308/) (2022)
- [BNN-DP: Robustness Certification of Bayesian Neural Networks via Dynamic Programming](https://mlsanity.com/paper/2306.10742/) (2023)
- [Revisiting the Adversarial Robustness-Accuracy Tradeoff in Robot Learning](https://mlsanity.com/paper/2204.07373/) (2022)
- [Adversarial Robustness Certification for Bayesian Neural Networks](https://mlsanity.com/paper/2306.13614/) (2023)
- [Provably Safe Neural Network Controllers via Differential Dynamic Logic](https://mlsanity.com/paper/2402.10998/) (2024)
- [Learning Provably Stabilizing Neural Controllers for Discrete-Time Stochastic Systems](https://mlsanity.com/paper/2210.05304/) (2022)
- [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)
- [Bayesian inference for data-efficient, explainable, and safe robotic motion planning: A review](https://mlsanity.com/paper/2307.08024/) (2023)
- [Combining Bayesian Inference and Reinforcement Learning for Agent Decision Making: A Review](https://mlsanity.com/paper/2505.07911/) (2025)
- [Hybrid Robustness Verification for Spatio-Temporal Neural Networks](https://mlsanity.com/paper/2606.09746/) (2026)
- [A GHOST in Long-Horizon Agents: Governance Hazard from Overlooked Safety Constraints across Turns](https://mlsanity.com/paper/2610.02664/) (2026)

## Abstract (arXiv)

Bayesian neural networks (BNNs) place distributions over the weights of a neural network to model uncertainty in the data and the network's prediction. We consider the problem of verifying safety when running a Bayesian neural network policy in a feedback loop with infinite time horizon systems. Compared to the existing sampling-based approaches, which are inapplicable to the infinite time horizon setting, we train a separate deterministic neural network that serves as an infinite time horizon safety certificate. In particular, we show that the certificate network guarantees the safety of the system over a subset of the BNN weight posterior's support. Our method first computes a safe weight set and then alters the BNN's weight posterior to reject samples outside this set. Moreover, we show how to extend our approach to a safe-exploration reinforcement learning setting, in order to avoid unsafe trajectories during the training of the policy. We evaluate our approach on a series of reinforcement learning benchmarks, including non-Lyapunovian safety specifications.