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1. Generalized median of means principle for Bayesian inference

   https://doi.org/10.1007/s10994-025-06754-9  

   Minsker, Stanislav; Yao, Shunan (April 2025, Machine Learning)

   The topic of robustness is experiencing a resurgence of interest in the statistical and machine learning communities. In particular, robust algorithms making use of the so-called median of means estimator were shown to satisfy strong performance guarantees for many problems, including estimation of the mean, covariance structure as well as linear regression. In this work, we propose an extension of the median of means principle to the Bayesian framework, leading to the notion of the robust posterior distribution. In particular, we (a) quantify robustness of this posterior to outliers, (b) show that it satisfies a version of the Bernstein-von Mises theorem that connects Bayesian credible sets to the traditional confidence intervals, and (c) demonstrate that our approach performs well in applications. 

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2. Covariances, robustness and variational Bayes

   Giordano, Ryan; Broderick, Tamara; Jordan, Michael I (January 2018, Journal of machine learning research)

   Mean-field Variational Bayes (MFVB) is an approximate Bayesian posterior inference technique that is increasingly popular due to its fast runtimes on large-scale data sets. However, even when MFVB provides accurate posterior means for certain parameters, it often mis-estimates variances and covariances. Furthermore, prior robustness measures have remained undeveloped for MFVB. By deriving a simple formula for the effect of infinitesimal model perturbations on MFVB posterior means, we provide both improved covariance estimates and local robustness measures for MFVB, thus greatly expanding the practical usefulness of MFVB posterior approximations. The estimates for MFVB posterior covariances rely on a result from the classical Bayesian robustness literature that relates derivatives of posterior expectations to posterior covariances and includes the Laplace approximation as a special case. Our key condition is that the MFVB approximation provides good estimates of a select subset of posterior means---an assumption that has been shown to hold in many practical settings. In our experiments, we demonstrate that our methods are simple, general, and fast, providing accurate posterior uncertainty estimates and robustness measures with runtimes that can be an order of magnitude faster than MCMC. 

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3. Attacking Bayes: Are Bayesian Neural Networks Inherently Robust?

   Feng, Yunzhen; Rudner, Tim G; Tsilivis, Nikolaos; Kempe, Julia (July 2024, Transactions on Machine Learning Research)

   Adversarial examples have been shown to cause neural networks to fail on a wide range of vision and language tasks, but recent work has claimed that {\em Bayesian} neural networks (BNNs) are inherently robust to adversarial perturbations. In this work, we examine this claim. To study the adversarial robustness of BNNs, we investigate whether it is possible to successfully break state-of-the-art BNN inference methods and prediction pipelines using even relatively unsophisticated attacks for three tasks: (1) label prediction under the posterior predictive mean, (2) adversarial example detection with Bayesian predictive uncertainty, and (3) semantic shift detection. We find that BNNs trained with state-of-the-art approximate inference methods, and even BNNs trained with Hamiltonian Monte Carlo, are highly susceptible to adversarial attacks. We also identify various conceptual and experimental errors in previous works that claimed inherent adversarial robustness of BNNs and conclusively demonstrate that BNNs and uncertainty-aware Bayesian prediction pipelines are {\em not} inherently robust against adversarial attacks. 

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4. Bayes-SAR Net: Robust SAR Image Classification with Uncertainty Estimation Using Bayesian Convolutional Neural Network

   Dera, D; Rasool, G; Bouaynaya, N; Eicheny, A; Shankoy, S; Cammeratay, J; Arnold, S (July 2020, IEEE International Radar Conference)

   Synthetic aperture radar (SAR) image classification is a challenging problem due to the complex imaging mechanism as well as the random speckle noise, which affects radar image interpretation. Recently, convolutional neural networks (CNNs) have been shown to outperform previous state-of-the-art techniques in computer vision tasks owing to their ability to learn relevant features from the data. However, CNNs in particular and neural networks, in general, lack uncertainty quantification and can be easily deceived by adversarial attacks. This paper proposes Bayes-SAR Net, a Bayesian CNN that can perform robust SAR image classification while quantifying the uncertainty or confidence of the network in its decision. Bayes-SAR Net propagates the first two moments (mean and covariance) of the approximate posterior distribution of the network parameters given the data and obtains a predictive mean and covariance of the classification output. Experiments, using the benchmark datasets Flevoland and Oberpfaffenhofen, show superior performance and robustness to Gaussian noise and adversarial attacks, as compared to the SAR-Net homologue. Bayes-SAR Net achieves a test accuracy that is around 10% higher in the case of adversarial perturbation (levels > 0.05). 

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5. Dangers of Bayesian Model Averaging under Covariate Shift

   Izmailov, Pavel; Nicholson, Patrick; Lotfi, Sanae; Wilson, Andrew G. (January 2021, Advances in neural information processing systems)

   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. 

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