Recent work has observed that one can outperform exact inference in Bayesian neural networks by tuning the "temperature" of the posterior on a validation set (the "cold posterior" effect). To help interpret this phenomenon, we argue that commonly used priors in Bayesian neural networks can significantly overestimate the aleatoric uncertainty in the labels on many classification datasets. This problem is particularly pronounced in academic benchmarks like MNIST or CIFAR, for which the quality of the labels is high. For the special case of Gaussian process regression, any positive temperature corresponds to a valid posterior under a modified prior, and tuning this temperature is directly analogous to empirical Bayes. On classification tasks, there is no direct equivalence between modifying the prior and tuning the temperature, however reducing the temperature can lead to models which better reflect our belief that one gains little information by relabeling existing examples in the training set. Therefore although cold posteriors do not always correspond to an exact inference procedure, we believe they may often better reflect our true prior beliefs.
Aleatoric uncertainty captures the inherent randomness of the data, such as measurement noise. In Bayesian regression, we often use a Gaussian observation model, where we control the level of aleatoric uncertainty with a noise variance parameter. By contrast, for Bayesian classification we use a categorical distribution with no mechanism to represent our beliefs about aleatoric uncertainty. Our work shows that explicitly accounting for aleatoric uncertainty significantly improves the performance of Bayesian neural networks. We note that many standard benchmarks, such as CIFAR, have essentially…
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；University of Bristol（布里斯托大学）
Typical Bayesian approaches to OOD detection use epistemic uncertainty. Surprisingly from the Bayesian perspective, there are a number of methods that successfully use aleatoric uncertainty to detect OOD points (e.g. Hendryks et al. 2018). In addition, it is difficult to use outlier exposure to improve a Bayesian OOD detection model, as it is not clear whether it is possible or desirable to increase posterior (epistemic) uncertainty at outlier points. We show that a generative model of data curation provides a principled account of aleatoric uncertainty for OOD detection. In particular, aleato…
On Cold Posteriors of Probabilistic Neural Networks: Understanding the Cold Posterior Effect and A New Way to Learn Cold Posteriors with Tight Generalization Guarantees
A Ground-Truth Framework for Uncertainty Disentanglement with Posterior Risk
University of Tübingen（蒂宾根大学）；Hertie Institute for AI in Brain Health（赫蒂脑健康人工智能研究所）；Tübingen AI Center（蒂宾根人工智能中心）；Charité–Universitätsmedizin Berlin（柏林夏里特医学院）；Bernstein Center for Computational Neuroscience Berlin（柏林伯恩斯坦计算神经科学中心）
Imperial College London（帝国理工学院）；University of Bristol（布里斯托大学）；University of Cambridge（剑桥大学）；ETH Zürich（苏黎世联邦理工学院）
Bayesian neural networks that incorporate data augmentation implicitly use a ``randomly perturbed log-likelihood [which] does not have a clean interpretation as a valid likelihood function'' (Izmailov et al. 2021). Here, we provide several approaches to developing principled Bayesian neural networks incorporating data augmentation. We introduce a ``finite orbit'' setting which allows likelihoods to be computed exactly, and give tight multi-sample bounds in the more usual ``full orbit'' setting. These models cast light on the origin of the cold posterior effect. In particular, we find that the …
To get Bayesian neural networks to perform comparably to standard neural networks it is usually necessary to artificially reduce uncertainty using a "tempered" or "cold" posterior. This is extremely concerning: if the prior is accurate, Bayes inference/decision theory is optimal, and any artificial changes to the posterior should harm performance. While this suggests that the prior may be at fault, here we argue that in fact, BNNs for image classification use the wrong likelihood. In particular, standard image benchmark datasets such as CIFAR-10 are carefully curated. We develop a generative m…
Accurate estimation of aleatoric and epistemic uncertainty is crucial to build safe and reliable systems. Traditional approaches, such as dropout and ensemble methods, estimate uncertainty by sampling probability predictions from different submodels, which leads to slow uncertainty estimation at inference time. Recent works address this drawback by directly predicting parameters of prior distributions over the probability predictions with a neural network. While this approach has demonstrated accurate uncertainty estimation, it requires defining arbitrary target parameters for in-distribution …
Within the last few years, there has been a move towards using statistical models in conjunction with neural networks with the end goal of being able to better answer the question, "what do our models know?". From this trend, classical metrics such as Prediction Interval Coverage Probability (PICP) and new metrics such as calibration error have entered the general repertoire of model evaluation in order to gain better insight into how the uncertainty of our model compares to reality. One important component of uncertainty modeling is model uncertainty (epistemic uncertainty), a measurement of …
Addressing the Inconsistency in Bayesian Deep Learning via Generalized Laplace Approximation
School of Engineering Deakin University Melbourne, VIC, 3216, Australia（工程学院 德堪大学 澳大利亚墨尔本 VIC 3216）；Faculty of Mathematics（数学与计算机科学学院）；Department of Computing Technologies Swinburne University of Technology Hawthorn, VIC, 3122, Australia（计算技术系 斯winburne技术大学 澳大利亚霍斯顿 VIC 3122）
Quantifying uncertainty in medical image segmentation applications is essential, as it is often connected to vital decision-making. Compelling attempts have been made in quantifying the uncertainty in image segmentation architectures, e.g. to learn a density segmentation model conditioned on the input image. Typical work in this field restricts these learnt densities to be strictly Gaussian. In this paper, we propose to use a more flexible approach by introducing Normalizing Flows (NFs), which enables the learnt densities to be more complex and facilitate more accurate modeling for uncertainty…
Google Research（谷歌研究院）；ETH Zurich（苏黎世联邦理工学院）；University of Amsterdam（阿姆斯特丹大学）；University of Warsaw（华沙大学）；Imperial College London（帝国理工学院）；University of California, Irvine（加利福尼亚大学欧文分校）；Microsoft Research（微软研究院）
During the past five years the Bayesian deep learning community has developed increasingly accurate and efficient approximate inference procedures that allow for Bayesian inference in deep neural networks. However, despite this algorithmic progress and the promise of improved uncertainty quantification and sample efficiency there are---as of early 2020---no publicized deployments of Bayesian neural networks in industrial practice. In this work we cast doubt on the current understanding of Bayes posteriors in popular deep neural networks: we demonstrate through careful MCMC sampling that the po…
Beyond the Loss Curve: Scaling Laws, Active Learning, and the Limits of Learning from Exact Posteriors
Mila-Quebec AI Institute, Canada（魁北克AI研究院）；School of Computer Science and Engineering, Vellore Institute of Technology, India（印度维洛雷理工学院计算机科学与工程学院）；Plasma Physics Lab, Princeton University, USA（普林斯顿大学等离子物理实验室）；High School Student, John P. Stevens High School, USA（约翰·P·斯蒂文斯高中学生）；Center of Data Science, New York University, New York, USA（纽约大学数据科学中心）
TU Kaiserslautern（凯泽斯劳滕工业大学）；MindGarage；Imperial College London（帝国理工学院）；NeuralSpace；Luleå University of Technology（吕勒奥理工大学）
We introduce a novel uncertainty estimation for classification tasks for Bayesian convolutional neural networks with variational inference. By normalizing the output of a Softplus function in the final layer, we estimate aleatoric and epistemic uncertainty in a coherent manner. The intractable posterior probability distributions over weights are inferred by Bayes by Backprop. Firstly, we demonstrate how this reliable variational inference method can serve as a fundamental construct for various network architectures. On multiple datasets in supervised learning settings (MNIST, CIFAR-10, CIFAR-1…
Brown University（布朗大学）；Pacific Northwest National Laboratory（西北太平洋国家实验室）
We propose a Bayesian physics-informed neural network (B-PINN) to solve both forward and inverse nonlinear problems described by partial differential equations (PDEs) and noisy data. In this Bayesian framework, the Bayesian neural network (BNN) combined with a PINN for PDEs serves as the prior while the Hamiltonian Monte Carlo (HMC) or the variational inference (VI) could serve as an estimator of the posterior. B-PINNs make use of both physical laws and scattered noisy measurements to provide predictions and quantify the aleatoric uncertainty arising from the noisy data in the Bayesian framewo…
ETH Zürich（苏黎世联邦理工学院）；University of Cambridge（剑桥大学）；Google AI Berlin（谷歌柏林人工智能研究院）；Imperial College London（伦敦帝国理工学院）；University of Bristol（布里斯托尔大学）
Isotropic Gaussian priors are the de facto standard for modern Bayesian neural network inference. However, it is unclear whether these priors accurately reflect our true beliefs about the weight distributions or give optimal performance. To find better priors, we study summary statistics of neural network weights in networks trained using stochastic gradient descent (SGD). We find that convolutional neural network (CNN) and ResNet weights display strong spatial correlations, while fully connected networks (FCNNs) display heavy-tailed weight distributions. We show that building these observatio…
Temperature Optimization for Bayesian Deep Learning
Department of Econometrics and Business Statistics（经济计量学与商业统计系）；Monash University（墨尔本大学）；Department of Electrical and Electronic Engineering（电气与电子工程系）；University of Melbourne（墨尔本大学）；School of Mathematics and Statistics（数学与统计学学院）
We exhibit examples of high-dimensional unimodal posterior distributions arising in non-linear regression models with Gaussian process priors for which MCMC methods can take an exponential run-time to enter the regions where the bulk of the posterior measure concentrates. Our results apply to worst-case initialised (`cold start') algorithms that are local in the sense that their step-sizes cannot be too large on average. The counter-examples hold for general MCMC schemes based on gradient or random walk steps, and the theory is illustrated for Metropolis-Hastings adjusted methods such as pCN a…
Progressive Bayesian Confidence Architectures for Cold-Start Personal Health Analytics: Formalizing Early Insight Through Posterior Contraction and Risk-Aware Interpretation
Mila - Quebec AI Institute（米拉-魁北克人工智能研究所）；Université de Montréal（蒙特利尔大学）；Massachusetts Institute of Technology（麻省理工学院）
Epistemic Uncertainty is a measure of the lack of knowledge of a learner which diminishes with more evidence. While existing work focuses on using the variance of the Bayesian posterior due to parameter uncertainty as a measure of epistemic uncertainty, we argue that this does not capture the part of lack of knowledge induced by model misspecification. We discuss how the excess risk, which is the gap between the generalization error of a predictor and the Bayes predictor, is a sound measure of epistemic uncertainty which captures the effect of model misspecification. We thus propose a principl…
Department of Industrial Engineering and Operations Research and Data Science Institute, Columbia University（工业工程与运筹学系及数据科学研究所，哥伦比亚大学）；Decision, Risk, and Operations Division, Columbia Business School（决策、风险与运营部门，哥伦比亚商学院）
Uncertainty Modeling in Graph Neural Networks via Stochastic Differential Equations
Department of Engineering, University of Cambridge（剑桥大学工程系）；Mathematical Institute, University of Oxford（牛津大学数学系）；Oxford-Man Institute of Quantitative Finance, University of Oxford（牛津大学量化金融研究所）；Department of Computer Science and Technology, University of Cambridge（剑桥大学计算机科学与技术系）
The University of Manchester（曼彻斯特大学）；University of British Columbia（不列颠哥伦比亚大学）；University of Pennsylvania（宾夕法尼亚大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
MindGarage（思维车库）；Technical University Kaiserslautern（凯泽斯劳滕工业大学）；NeuralSpace（神经空间）；Imperial College London（帝国理工学院）；Luleå University of Technology（吕勒奥理工大学）
Artificial Neural Networks are connectionist systems that perform a given task by learning on examples without having prior knowledge about the task. This is done by finding an optimal point estimate for the weights in every node. Generally, the network using point estimates as weights perform well with large datasets, but they fail to express uncertainty in regions with little or no data, leading to overconfident decisions. In this paper, Bayesian Convolutional Neural Network (BayesCNN) using Variational Inference is proposed, that introduces probability distribution over the weights. Further…
We investigate the cold posterior effect through the lens of PAC-Bayes generalization bounds. We argue that in the non-asymptotic setting, when the number of training samples is (relatively) small, discussions of the cold posterior effect should take into account that approximate Bayesian inference does not readily provide guarantees of performance on out-of-sample data. Instead, out-of-sample error is better described through a generalization bound. In this context, we explore the connections between the ELBO objective from variational inference and the PAC-Bayes objectives. We note that, whi…
The Bayesian brain hypothesis postulates that the brain accurately operates on statistical distributions according to Bayes' theorem. The random failure of presynaptic vesicles to release neurotransmitters may allow the brain to sample from posterior distributions of network parameters, interpreted as epistemic uncertainty. It has not been shown previously how random failures might allow networks to sample from observed distributions, also known as aleatoric or residual uncertainty. Sampling from both distributions enables probabilistic inference, efficient search, and creative or generative p…
Fine-Grained Uncertainty Quantification via Collisions
Department of Electrical and Computer Engineering, The University of Arizona（电气与计算机工程系，亚利桑那大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）；Marvell Technology Inc.（Marvell技术公司）
ETH Zürich（苏黎世联邦理工学院）；University of Cambridge（剑桥大学）；Imperial College London（帝国理工学院）；University of Bristol（布里斯托大学）
Bayesian neural networks have shown great promise in many applications where calibrated uncertainty estimates are crucial and can often also lead to a higher predictive performance. However, it remains challenging to choose a good prior distribution over their weights. While isotropic Gaussian priors are often chosen in practice due to their simplicity, they do not reflect our true prior beliefs well and can lead to suboptimal performance. Our new library, BNNpriors, enables state-of-the-art Markov Chain Monte Carlo inference on Bayesian neural networks with a wide range of predefined priors, …
Monotonicity and Double Descent in Uncertainty Estimation with Gaussian Processes
University of Melbourne（墨尔本大学）；University of Queensland（昆士兰大学）；International Computer Science Institute（国际计算机科学研究所）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of California at Berkeley（加州大学伯克利分校）
The posterior over Bayesian neural network (BNN) parameters is extremely high-dimensional and non-convex. For computational reasons, researchers approximate this posterior using inexpensive mini-batch methods such as mean-field variational inference or stochastic-gradient Markov chain Monte Carlo (SGMCMC). To investigate foundational questions in Bayesian deep learning, we instead use full-batch Hamiltonian Monte Carlo (HMC) on modern architectures. We show that (1) BNNs can achieve significant performance gains over standard training and deep ensembles; (2) a single long HMC chain can provide…
