# Training on Test Data with Bayesian Adaptation for Covariate Shift

arXiv: https://arxiv.org/abs/2109.12746 · HTML: https://mlsanity.com/paper/2109.12746/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 58 references that are in the map.
- First posted on arXiv 27 Sep 2021.

## Builds on (most cited first, 30 of 38)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [FixMatch: Simplifying Semi-Supervised Learning with Consistency and Confidence](https://mlsanity.com/paper/2001.07685/) (2020)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [The Many Faces of Robustness: A Critical Analysis of Out-of-Distribution Generalization](https://mlsanity.com/paper/2006.16241/) (2020)
- [MixMatch: A Holistic Approach to Semi-Supervised Learning](https://mlsanity.com/paper/1905.02249/) (2019)
- [Self-training with Noisy Student improves ImageNet classification](https://mlsanity.com/paper/1911.04252/) (2019)
- [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)
- [Unsupervised Data Augmentation for Consistency Training](https://mlsanity.com/paper/1904.12848/) (2019)
- [Virtual Adversarial Training: A Regularization Method for Supervised and Semi-Supervised Learning](https://mlsanity.com/paper/1704.03976/) (2017)
- [AugMix: A Simple Data Processing Method to Improve Robustness and Uncertainty](https://mlsanity.com/paper/1912.02781/) (2019)
- [Do We Really Need to Access the Source Data? Source Hypothesis Transfer for Unsupervised Domain Adaptation](https://mlsanity.com/paper/2002.08546/) (2020)
- [Deep Visual Domain Adaptation: A Survey](https://mlsanity.com/paper/1802.03601/) (2018)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Measuring Robustness to Natural Distribution Shifts in Image Classification](https://mlsanity.com/paper/2007.00644/) (2020)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [Regularization With Stochastic Transformations and Perturbations for Deep Semi-Supervised Learning](https://mlsanity.com/paper/1606.04586/) (2016)
- [Semi-supervised Domain Adaptation via Minimax Entropy](https://mlsanity.com/paper/1904.06487/) (2019)
- [A Baseline for Few-Shot Image Classification](https://mlsanity.com/paper/1909.02729/) (2019)
- [Evaluating Prediction-Time Batch Normalization for Robustness under Covariate Shift](https://mlsanity.com/paper/2006.10963/) (2020)
- [Universal Source-Free Domain Adaptation](https://mlsanity.com/paper/2004.04393/) (2020)
- [Efficient and Scalable Bayesian Neural Nets with Rank-1 Factors](https://mlsanity.com/paper/2005.07186/) (2020)
- [Functional Variational Bayesian Neural Networks](https://mlsanity.com/paper/1903.05779/) (2019)
- [Adversarial Examples Are a Natural Consequence of Test Error in Noise](https://mlsanity.com/paper/1901.10513/) (2019)
- [AutoDIAL: Automatic DomaIn Alignment Layers](https://mlsanity.com/paper/1704.08082/) (2017)
- [Calibrated Prediction with Covariate Shift via Unsupervised Domain Adaptation](https://mlsanity.com/paper/2003.00343/) (2020)

## Built on by (most cited first, 8 of 8)

- [Continual Test-Time Domain Adaptation](https://mlsanity.com/paper/2203.13591/) (2022)
- [A Probabilistic Framework for Lifelong Test-Time Adaptation](https://mlsanity.com/paper/2212.09713/) (2022)
- [Adversarial Unlearning: Reducing Confidence Along Adversarial Directions](https://mlsanity.com/paper/2206.01367/) (2022)
- [Energy-Based Test Sample Adaptation for Domain Generalization](https://mlsanity.com/paper/2302.11215/) (2023)
- [Federated Learning under Covariate Shifts with Generalization Guarantees](https://mlsanity.com/paper/2306.05325/) (2023)
- [Towards Unified and Effective Domain Generalization](https://mlsanity.com/paper/2310.10008/) (2023)
- [Source-Free Test-Time Adaptation For Online Surface-Defect Detection](https://mlsanity.com/paper/2408.09494/) (2024)
- [Efficient Learning Under Density Shift in Incremental Settings Using Cram'er-Rao-Based Regularization](https://mlsanity.com/paper/2502.12949/) (2025)

## Abstract (arXiv)

When faced with distribution shift at test time, deep neural networks often make inaccurate predictions with unreliable uncertainty estimates. While improving the robustness of neural networks is one promising approach to mitigate this issue, an appealing alternate to robustifying networks against all possible test-time shifts is to instead directly adapt them to unlabeled inputs from the particular distribution shift we encounter at test time. However, this poses a challenging question: in the standard Bayesian model for supervised learning, unlabeled inputs are conditionally independent of model parameters when the labels are unobserved, so what can unlabeled data tell us about the model parameters at test-time? In this paper, we derive a Bayesian model that provides for a well-defined relationship between unlabeled inputs under distributional shift and model parameters, and show how approximate inference in this model can be instantiated with a simple regularized entropy minimization procedure at test-time. We evaluate our method on a variety of distribution shifts for image classification, including image corruptions, natural distribution shifts, and domain adaptation settings, and show that our method improves both accuracy and uncertainty estimation.