In Bayesian Deep Learning, distributions over the output of classification neural networks are often approximated by first constructing a Gaussian distribution over the weights, then sampling from it to receive a distribution over the softmax outputs. This is costly. We reconsider old work (Laplace Bridge) to construct a Dirichlet approximation of this softmax output distribution, which yields an analytic map between Gaussian distributions in logit space and Dirichlet distributions (the conjugate prior to the Categorical distribution) in the output space. Importantly, the vanilla Laplace Bridge comes with certain limitations. We analyze those and suggest a simple solution that compares favorably to other commonly used estimates of the softmax-Gaussian integral. We demonstrate that the resulting Dirichlet distribution has multiple advantages, in particular, more efficient computation of the uncertainty estimate and scaling to large datasets and networks like ImageNet and DenseNet. We further demonstrate the usefulness of this Dirichlet approximation by using it to construct a lightweight uncertainty-aware output ranking for ImageNet.
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））；Aalto University（阿尔托大学）
We introduce a method, KL-LIME, for explaining predictions of Bayesian predictive models by projecting the information in the predictive distribution locally to a simpler, interpretable explanation model. The proposed approach combines the recent Local Interpretable Model-agnostic Explanations (LIME) method with ideas from Bayesian projection predictive variable selection methods. The information theoretic basis helps in navigating the trade-off between explanation fidelity and complexity. We demonstrate the method in explaining MNIST digit classifications made by a Bayesian deep convolutional…
University of Chicago（芝加哥大学）；University of Toronto（多伦多大学）；Vector Institute（矢量研究所）
While uncertainty estimation is a well-studied topic in deep learning, most such work focuses on marginal uncertainty estimates, i.e. the predictive mean and variance at individual input locations. But it is often more useful to estimate predictive correlations between the function values at different input locations. In this paper, we consider the problem of benchmarking how accurately Bayesian models can estimate predictive correlations. We first consider a downstream task which depends on posterior predictive correlations: transductive active learning (TAL). We find that TAL makes better us…
Principled Bayesian deep learning (BDL) does not live up to its potential when we only focus on marginal predictive distributions (marginal predictives). Recent works have highlighted the importance of joint predictives for (Bayesian) sequential decision making from a theoretical and synthetic perspective. We provide additional practical arguments grounded in real-world applications for focusing on joint predictives: we discuss online Bayesian inference, which would allow us to make predictions while taking into account additional data without retraining, and we propose new challenging evaluat…
Deep neural networks (NNs) are powerful black box predictors that have recently achieved impressive performance on a wide spectrum of tasks. Quantifying predictive uncertainty in NNs is a challenging and yet unsolved problem. Bayesian NNs, which learn a distribution over weights, are currently the state-of-the-art for estimating predictive uncertainty; however these require significant modifications to the training procedure and are computationally expensive compared to standard (non-Bayesian) NNs. We propose an alternative to Bayesian NNs that is simple to implement, readily parallelizable, r…
Evasion Attacks Against Bayesian Predictive Models
Inst. Math. Sciences, Spanish Nat. Research Council（数学科学研究所，西班牙国家研究理事会）；Universidad Autónoma de Madrid, Escuela de Doctorado（马德里自治大学，博士学院）；CUNEF Universidad Madrid, Spain（CUNEF马德里大学，西班牙）
Estimating how uncertain an AI system is in its predictions is important to improve the safety of such systems. Uncertainty in predictive can result from uncertainty in model parameters, irreducible data uncertainty and uncertainty due to distributional mismatch between the test and training data distributions. Different actions might be taken depending on the source of the uncertainty so it is important to be able to distinguish between them. Recently, baseline tasks and metrics have been defined and several practical methods to estimate uncertainty developed. These methods, however, attempt …
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…
University of Oxford（牛津大学）；Novo Nordisk（诺和诺德）；University College London（伦敦大学学院）
Bayesian methods are a popular choice for statistical inference in small-data regimes due to the regularization effect induced by the prior. In the context of density estimation, the standard nonparametric Bayesian approach is to target the posterior predictive of the Dirichlet process mixture model. In general, direct estimation of the posterior predictive is intractable and so methods typically resort to approximating the posterior distribution as an intermediate step. The recent development of quasi-Bayesian predictive copula updates, however, has made it possible to perform tractable predi…
High-risk domains require reliable confidence estimates from predictive models. Deep latent variable models provide these, but suffer from the rigid variational distributions used for tractable inference, which err on the side of overconfidence. We propose Stochastic Quantized Activation Distributions (SQUAD), which imposes a flexible yet tractable distribution over discretized latent variables. The proposed method is scalable, self-normalizing and sample efficient. We demonstrate that the model fully utilizes the flexible distribution, learns interesting non-linearities, and provides predicti…
From the statistical learning perspective, complexity control via explicit regularization is a necessity for improving the generalization of over-parameterized models. However, the impressive generalization performance of neural networks with only implicit regularization may be at odds with this conventional wisdom. In this work, we revisit the importance of explicit regularization for obtaining well-calibrated predictive uncertainty. Specifically, we introduce a probabilistic measure of calibration performance, which is lower bounded by the log-likelihood. We then explore explicit regularizat…
The interdependence between nodes in graphs is key to improve class predictions on nodes and utilized in approaches like Label Propagation (LP) or in Graph Neural Networks (GNN). Nonetheless, uncertainty estimation for non-independent node-level predictions is under-explored. In this work, we explore uncertainty quantification for node classification in three ways: (1) We derive three axioms explicitly characterizing the expected predictive uncertainty behavior in homophilic attributed graphs. (2) We propose a new model Graph Posterior Network (GPN) which explicitly performs Bayesian posterior…
Modern machine learning methods including deep learning have achieved great success in predictive accuracy for supervised learning tasks, but may still fall short in giving useful estimates of their predictive {\em uncertainty}. Quantifying uncertainty is especially critical in real-world settings, which often involve input distributions that are shifted from the training distribution due to a variety of factors including sample bias and non-stationarity. In such settings, well calibrated uncertainty estimates convey information about when a model's output should (or should not) be trusted. Ma…
The Bayesian posterior minimizes the "inferential risk" which itself bounds the "predictive risk". This bound is tight when the likelihood and prior are well-specified. However since misspecification induces a gap, the Bayesian posterior predictive distribution may have poor generalization performance. This work develops a multi-sample loss (PAC$^m$) which can close the gap by spanning a trade-off between the two risks. The loss is computationally favorable and offers PAC generalization guarantees. Empirical study demonstrates improvement to the predictive distribution.
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…
Bayesian neural networks (BNNs) augment deep networks with uncertainty quantification by Bayesian treatment of the network weights. However, such models face the challenge of Bayesian inference in a high-dimensional and usually over-parameterized space. This paper investigates a new line of Bayesian deep learning by performing Bayesian inference on network structure. Instead of building structure from scratch inefficiently, we draw inspirations from neural architecture search to represent the network structure. We then develop an efficient stochastic variational inference approach which unifie…
We derive a novel sensitivity analysis of input variables for predictive epistemic and aleatoric uncertainty. We use Bayesian neural networks with latent variables as a model class and illustrate the usefulness of our sensitivity analysis on real-world datasets. Our method increases the interpretability of complex black-box probabilistic models.
Estimating the predictive uncertainty of a Bayesian learning model is critical in various decision-making problems, e.g., reinforcement learning, detecting adversarial attack, self-driving car. As the model posterior is almost always intractable, most efforts were made on finding an accurate approximation the true posterior. Even though a decent estimation of the model posterior is obtained, another approximation is required to compute the predictive distribution over the desired output. A common accurate solution is to use Monte Carlo (MC) integration. However, it needs to maintain a large nu…
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 …
Traditional training of deep classifiers yields overconfident models that are not reliable under dataset shift. We propose a Bayesian framework to obtain reliable uncertainty estimates for deep classifiers. Our approach consists of a plug-in "generator" used to augment the data with an additional class of points that lie on the boundary of the training data, followed by Bayesian inference on top of features that are trained to distinguish these "out-of-distribution" points.
Deep neural networks (DNN) are versatile parametric models utilised successfully in a diverse number of tasks and domains. However, they have limitations---particularly from their lack of robustness and over-sensitivity to out of distribution samples. Bayesian Neural Networks, due to their formulation under the Bayesian framework, provide a principled approach to building neural networks that address these limitations. This paper describes a study that empirically evaluates and compares Bayesian Neural Networks to their equivalent point estimate Deep Neural Networks to quantify the predictive …
University of Stuttgart（斯图加特大学）；Heidelberg University（海德堡大学）
Bayesian model comparison (BMC) offers a principled probabilistic approach to study and rank competing models. In standard BMC, we construct a discrete probability distribution over the set of possible models, conditional on the observed data of interest. These posterior model probabilities (PMPs) are measures of uncertainty, but -- when derived from a finite number of observations -- are also uncertain themselves. In this paper, we conceptualize distinct levels of uncertainty which arise in BMC. We explore a fully probabilistic framework for quantifying meta-uncertainty, resulting in an appli…
Neural Linear Models (NLM) are deep Bayesian models that produce predictive uncertainty by learning features from the data and then performing Bayesian linear regression over these features. Despite their popularity, few works have focused on formally evaluating the predictive uncertainties of these models. Furthermore, existing works point out the difficulties of encoding domain knowledge in models like NLMs, making them unsuitable for applications where interpretability is required. In this work, we show that traditional training procedures for NLMs can drastically underestimate uncertainty …
German Research Center for Artificial Intelligence（德国人工智能研究中心）
Estimating predictive uncertainty is crucial for many computer vision tasks, from image classification to autonomous driving systems. Hamiltonian Monte Carlo (HMC) is an sampling method for performing Bayesian inference. On the other hand, Dropout regularization has been proposed as an approximate model averaging technique that tends to improve generalization in large scale models such as deep neural networks. Although, HMC provides convergence guarantees for most standard Bayesian models, it does not handle discrete parameters arising from Dropout regularization. In this paper, we present a r…
Despite numerous studies of deep autoencoders (AEs) for unsupervised anomaly detection, AEs still lack a way to express uncertainty in their predictions, crucial for ensuring safe and trustworthy machine learning systems in high-stake applications. Therefore, in this work, the formulation of Bayesian autoencoders (BAEs) is adopted to quantify the total anomaly uncertainty, comprising epistemic and aleatoric uncertainties. To evaluate the quality of uncertainty, we consider the task of classifying anomalies with the additional option of rejecting predictions of high uncertainty. In addition, we…
Neural Linear Models (NLM) are deep Bayesian models that produce predictive uncertainties by learning features from the data and then performing Bayesian linear regression over these features. Despite their popularity, few works have focused on methodically evaluating the predictive uncertainties of these models. In this work, we demonstrate that traditional training procedures for NLMs drastically underestimate uncertainty on out-of-distribution inputs, and that they therefore cannot be naively deployed in risk-sensitive applications. We identify the underlying reasons for this behavior and p…
University of Victoria（维多利亚大学）；Department of Civil Engineering（土木工程系）；Energy and Cities Group（能源与城市小组）
Fast machine learning-based surrogate models are trained to emulate slow, high-fidelity engineering simulation models to accelerate engineering design tasks. This introduces uncertainty as the surrogate is only an approximation of the original model. Bayesian methods can quantify that uncertainty, and deep learning models exist that follow the Bayesian paradigm. These models, namely Bayesian neural networks and Gaussian process models, enable us to give predictions together with an estimate of the model's uncertainty. As a result we can derive uncertainty-aware surrogate models that can automa…
One of the main challenges of deep learning tools is their inability to capture model uncertainty. While Bayesian deep learning can be used to tackle the problem, Bayesian neural networks often require more time and computational power to train than deterministic networks. Our work explores whether fully Bayesian networks are needed to successfully capture model uncertainty. We vary the number and position of Bayesian layers in a network and compare their performance on active learning with the MNIST dataset. We found that we can fully capture the model uncertainty by using only a few Bayesian…
In this study, we present a multi-class graphical Bayesian predictive classifier that incorporates the uncertainty in the model selection into the standard Bayesian formalism. For each class, the dependence structure underlying the observed features is represented by a set of decomposable Gaussian graphical models. Emphasis is then placed on the Bayesian model averaging which takes full account of the class-specific model uncertainty by averaging over the posterior graph model probabilities. An explicit evaluation of the model probabilities is well known to be infeasible. To address this issue…
We describe Bayesian Layers, a module designed for fast experimentation with neural network uncertainty. It extends neural network libraries with drop-in replacements for common layers. This enables composition via a unified abstraction over deterministic and stochastic functions and allows for scalability via the underlying system. These layers capture uncertainty over weights (Bayesian neural nets), pre-activation units (dropout), activations ("stochastic output layers"), or the function itself (Gaussian processes). They can also be reversible to propagate uncertainty from input to output. W…
Deep neural networks (DNNs) provide state-of-the-art results for a multitude of applications, but the approaches using DNNs for multimodal audiovisual applications do not consider predictive uncertainty associated with individual modalities. Bayesian deep learning methods provide principled confidence and quantify predictive uncertainty. Our contribution in this work is to propose an uncertainty aware multimodal Bayesian fusion framework for activity recognition. We demonstrate a novel approach that combines deterministic and variational layers to scale Bayesian DNNs to deeper architectures. O…
VERSES AI Research Lab（VERSES AI研究实验室）；School of Engineering and Informatics, University of Sussex（萨塞克斯大学工程与信息学院）；Max Planck Institute of Animal Behavior（马克斯·普朗克动物行为研究所）；Duke University（杜克大学）
University of Campinas (Unicamp)（坎皮纳斯州立大学 (Unicamp)）
We evaluate the uncertainty quality in neural networks using anomaly detection. We extract uncertainty measures (e.g. entropy) from the predictions of candidate models, use those measures as features for an anomaly detector, and gauge how well the detector differentiates known from unknown classes. We assign higher uncertainty quality to candidate models that lead to better detectors. We also propose a novel method for sampling a variational approximation of a Bayesian neural network, called One-Sample Bayesian Approximation (OSBA). We experiment on two datasets, MNIST and CIFAR10. We compare …
University of Cambridge（剑桥大学）；Microsoft Research（微软研究院）；Alan Turing Institute（艾伦·图灵研究所）
We describe a limitation in the expressiveness of the predictive uncertainty estimate given by mean-field variational inference (MFVI), a popular approximate inference method for Bayesian neural networks. In particular, MFVI fails to give calibrated uncertainty estimates in between separated regions of observations. This can lead to catastrophically overconfident predictions when testing on out-of-distribution data. Avoiding such overconfidence is critical for active learning, Bayesian optimisation and out-of-distribution robustness. We instead find that a classical technique, the linearised L…
When using the K-nearest neighbors method, one often ignores uncertainty in the choice of K. To account for such uncertainty, Holmes and Adams (2002) proposed a Bayesian framework for K-nearest neighbors (KNN). Their Bayesian KNN (BKNN) approach uses a pseudo-likelihood function, and standard Markov chain Monte Carlo (MCMC) techniques to draw posterior samples. Holmes and Adams (2002) focused on the performance of BKNN in terms of misclassification error but did not assess its ability to quantify uncertainty. We present some evidence to show that BKNN still significantly underestimates model u…
We show that training a deep network using batch normalization is equivalent to approximate inference in Bayesian models. We further demonstrate that this finding allows us to make meaningful estimates of the model uncertainty using conventional architectures, without modifications to the network or the training procedure. Our approach is thoroughly validated by measuring the quality of uncertainty in a series of empirical experiments on different tasks. It outperforms baselines with strong statistical significance, and displays competitive performance with recent Bayesian approaches.
Explaining Predictive Uncertainty by Exposing Second-Order Effects
Technische Universität Berlin（柏林工业大学）；BIFOLD – Berlin Institute for the Foundations of Learning and Data（BIFOLD——柏林学习与数据基础研究所）；Fraunhofer Heinrich Hertz Institute（弗劳恩霍夫海因里希·赫兹研究所）；Freie Universität Berlin（柏林自由大学）
Institute of Computer Graphics and Vision（计算机图形与视觉研究所）；Graz University of Technology（格拉茨理工大学）；School of Computer Science（计算机科学学院）；University of Auckland（奥克兰大学）
Additionally to the extensive use in clinical medicine, biological age (BA) in legal medicine is used to assess unknown chronological age (CA) in applications where identification documents are not available. Automatic methods for age estimation proposed in the literature are predicting point estimates, which can be misleading without the quantification of predictive uncertainty. In our multi-factorial age estimation method from MRI data, we used the Variational Inference approach to estimate the uncertainty of a Bayesian CNN model. Distinguishing model uncertainty from data uncertainty, we in…
One-step ahead prediction for the multinomial model is considered. The performance of a predictive density is evaluated by the average Kullback-Leibler divergence from the true density to the predictive density. Asymptotic approximations of risk functions of Bayesian predictive densities based on Dirichlet priors are obtained. It is shown that a Bayesian predictive density based on a specific Dirichlet prior is asymptotically minimax. The asymptotically minimax prior is different from known objective priors such as the Jeffreys prior or the uniform prior.
A Bayesian framework is attractive in the context of prediction, but a fast recursive update of the predictive distribution has apparently been out of reach, in part because Monte Carlo methods are generally used to compute the predictive. This paper shows that online Bayesian prediction is possible by characterizing the Bayesian predictive update in terms of a bivariate copula, making it unnecessary to pass through the posterior to update the predictive. In standard models, the Bayesian predictive update corresponds to familiar choices of copula but, in nonparametric problems, the appropriate…
