# Dangers of Bayesian Model Averaging under Covariate Shift

arXiv: https://arxiv.org/abs/2106.11905 · HTML: https://mlsanity.com/paper/2106.11905/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 69 references that are in the map.
- First posted on arXiv 22 Jun 2021.

## Builds on (most cited first, 30 of 37)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [End to End Learning for Self-Driving Cars](https://mlsanity.com/paper/1604.07316/) (2016)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [What is being transferred in transfer learning?](https://mlsanity.com/paper/2008.11687/) (2020)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Variational Dropout Sparsifies Deep Neural Networks](https://mlsanity.com/paper/1701.05369/) (2017)
- [Efficient and Scalable Bayesian Neural Nets with Rank-1 Factors](https://mlsanity.com/paper/2005.07186/) (2020)
- [Understanding the Failure Modes of Out-of-Distribution Generalization](https://mlsanity.com/paper/2010.15775/) (2020)
- [MNIST-C: A Robustness Benchmark for Computer Vision](https://mlsanity.com/paper/1906.02337/) (2019)
- [Bayesian Neural Network Priors Revisited](https://mlsanity.com/paper/2102.06571/) (2021)
- [Priors in Bayesian Deep Learning: A Review](https://mlsanity.com/paper/2105.06868/) (2021)
- [There Are Many Consistent Explanations of Unlabeled Data: Why You Should Average](https://mlsanity.com/paper/1806.05594/) (2018)
- [Loss Surface Simplexes for Mode Connecting Volumes and Fast Ensembling](https://mlsanity.com/paper/2102.13042/) (2021)
- [Neural Arithmetic Logic Units](https://mlsanity.com/paper/1808.00508/) (2018)
- [Does Your Dermatology Classifier Know What It Doesn't Know? Detecting the Long-Tail of Unseen Conditions](https://mlsanity.com/paper/2104.03829/) (2021)
- [Excessive Invariance Causes Adversarial Vulnerability](https://mlsanity.com/paper/1811.00401/) (2018)
- [A Systematic Comparison of Bayesian Deep Learning Robustness in Diabetic Retinopathy Tasks](https://mlsanity.com/paper/1912.10481/) (2019)
- [Uncertainty Quantification with Statistical Guarantees in End-to-End Autonomous Driving Control](https://mlsanity.com/paper/1909.09884/) (2019)

## Built on by (most cited first, 30 of 31)

- [Prior and Posterior Networks: A Survey on Evidential Deep Learning Methods For Uncertainty Estimation](https://mlsanity.com/paper/2110.03051/) (2021)
- [Do Bayesian Neural Networks Need To Be Fully Stochastic?](https://mlsanity.com/paper/2211.06291/) (2022)
- [Position: Bayesian Deep Learning is Needed in the Age of Large-Scale AI](https://mlsanity.com/paper/2402.00809/) (2024)
- [On Uncertainty, Tempering, and Data Augmentation in Bayesian Classification](https://mlsanity.com/paper/2203.16481/) (2022)
- [Pre-Train Your Loss: Easy Bayesian Transfer Learning with Informative Priors](https://mlsanity.com/paper/2205.10279/) (2022)
- [A Primer on Bayesian Neural Networks: Review and Debates](https://mlsanity.com/paper/2309.16314/) (2023)
- [When are ensembles really effective?](https://mlsanity.com/paper/2305.12313/) (2023)
- [Beyond Deep Ensembles: A Large-Scale Evaluation of Bayesian Deep Learning under Distribution Shift](https://mlsanity.com/paper/2306.12306/) (2023)
- [Reparameterization invariance in approximate Bayesian inference](https://mlsanity.com/paper/2406.03334/) (2024)
- [How Reliable is Your Regression Model's Uncertainty Under Real-World Distribution Shifts?](https://mlsanity.com/paper/2302.03679/) (2023)
- [Towards out of distribution generalization for problems in mechanics](https://mlsanity.com/paper/2206.14917/) (2022)
- [Test-Time Alignment via Hypothesis Reweighting](https://mlsanity.com/paper/2412.08812/) (2024)
- [Approximately Bayes-Optimal Pseudo Label Selection](https://mlsanity.com/paper/2302.08883/) (2023)
- [Robust Bayesian Learning for Reliable Wireless AI: Framework and Applications](https://mlsanity.com/paper/2207.00300/) (2022)
- [Incorporating Unlabelled Data into Bayesian Neural Networks](https://mlsanity.com/paper/2304.01762/) (2023)
- [Tackling covariate shift with node-based Bayesian neural networks](https://mlsanity.com/paper/2206.02435/) (2022)
- [Input-gradient space particle inference for neural network ensembles](https://mlsanity.com/paper/2306.02775/) (2023)
- [Improving robustness to corruptions with multiplicative weight perturbations](https://mlsanity.com/paper/2406.16540/) (2024)
- [On out-of-distribution detection with Bayesian neural networks](https://mlsanity.com/paper/2110.06020/) (2021)
- [Robustness to corruption in pre-trained Bayesian neural networks](https://mlsanity.com/paper/2206.12361/) (2022)
- [Conformalizing Machine Translation Evaluation](https://mlsanity.com/paper/2306.06221/) (2023)
- [Can a Confident Prior Replace a Cold Posterior?](https://mlsanity.com/paper/2403.01272/) (2024)
- [BayTTA: Uncertainty-aware medical image classification with optimized test-time augmentation using Bayesian model averaging](https://mlsanity.com/paper/2406.17640/) (2024)
- [Perspectives on the State and Future of Deep Learning - 2023](https://mlsanity.com/paper/2312.09323/) (2023)
- [Learning to Explore for Stochastic Gradient MCMC](https://mlsanity.com/paper/2408.09140/) (2024)
- [Uncertainty Calibration in Bayesian Neural Networks via Distance-Aware Priors](https://mlsanity.com/paper/2207.08200/) (2022)
- [Synthetic Model Combination: An Instance-wise Approach to Unsupervised Ensemble Learning](https://mlsanity.com/paper/2210.05320/) (2022)
- [Uncertainty quantification in neural network classifiers -- a local linear approach](https://mlsanity.com/paper/2303.07114/) (2023)
- [Understanding and mitigating difficulties in posterior predictive evaluation](https://mlsanity.com/paper/2405.19747/) (2024)
- [Domain Generalization Guided by Large-Scale Pre-Trained Priors](https://mlsanity.com/paper/2406.05628/) (2024)

## Abstract (arXiv)

Approximate Bayesian inference for neural networks is considered a robust alternative to standard training, often providing good performance on out-of-distribution data. However, Bayesian neural networks (BNNs) with high-fidelity approximate inference via full-batch Hamiltonian Monte Carlo achieve poor generalization under covariate shift, even underperforming classical estimation. We explain this surprising result, showing how a Bayesian model average can in fact be problematic under covariate shift, particularly in cases where linear dependencies in the input features cause a lack of posterior contraction. We additionally show why the same issue does not affect many approximate inference procedures, or classical maximum a-posteriori (MAP) training. Finally, we propose novel priors that improve the robustness of BNNs to many sources of covariate shift.