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A Survey of Uncertainty in Deep Neural Networks

Jakob Gawlikowski, Cedrique Rovile Njieutcheu Tassi, Mohsin Ali, Jongseok Lee, Matthias Humt, Jianxiang Feng, Anna Kruspe, Rudolph Triebel, Peter Jung, Ribana Roscher, Muhammad Shahzad, Wen Yang, Richard Bamler, Xiao Xiang Zhu

arXiv:2107.03342v3cs.LGstat.ML

TL;DR

Confidence estimates matter for real-world neural-network use, where out-of-distribution detection is a major challenge. This survey connects uncertainty sources to applications and concludes that ensemble evaluation is a powerful property for real-world applications, while Bayesian and deterministic approaches also have strengths.

  • Problem

    Real-life uncertainty quantification must address out-of-distribution detection, identified as a major challenge for avoiding unexpected network decisions and recognizing adversarial attacks.

    Source: E. Neural Network Uncertainty… · p. 21

  • Method

    The survey introduces uncertainty sources and compares approaches from uncertainty sources through real-world applications, while identifying current limitations and future challenges.

    Source: §I · p. 2

  • Results

    The survey concludes that ensemble evaluation is a powerful property for real-life applications; Bayesian methods have a strong theoretical foundation, and deterministic prior networks deliver comparable results with less computation.

    Source: E. Neural Network Uncertainty… · p. 21

  • Takeaways & Limitations

    Combining efficient ensemble strategies with Bayesian approaches may capture model-parameter variability while considering multiple prediction modes.

    Source: E. Neural Network Uncertainty… · p. 21

    Available uncertainty-quantification methods provide certainty values without explaining their causes, making transfer across datasets or domain shifts harder and safety-critical use significantly harder.

    Source: tification methods work for real world… · p. 33

Abstract

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Due to their increasing spread, confidence in neural network predictions became more and more important. However, basic neural networks do not deliver certainty estimates or suffer from over or under confidence. Many researchers have been working on understanding and quantifying uncertainty in a neural network's prediction. As a result, different types and sources of uncertainty have been identified and a variety of approaches to measure and quantify uncertainty in neural networks have been proposed. This work gives a comprehensive overview of uncertainty estimation in neural networks, reviews recent advances in the field, highlights current challenges, and identifies potential research opportunities. It is intended to give anyone interested in uncertainty estimation in neural networks a broad overview and introduction, without presupposing prior knowledge in this field. A comprehensive introduction to the most crucial sources of uncertainty is given and their separation into reducible model uncertainty and not reducible data uncertainty is presented. The modeling of these uncertainties based on deterministic neural networks, Bayesian neural networks, ensemble of neural networks, and test-time data augmentation approaches is introduced and different branches of these fields as well as the latest developments are discussed. For a practical application, we discuss different measures of uncertainty, approaches for the calibration of neural networks and give an overview of existing baselines and implementations. Different examples from the wide spectrum of challenges in different fields give an idea of the needs and challenges regarding uncertainties in practical applications. Additionally, the practical limitations of current methods for mission- and safety-critical real world applications are discussed and an outlook on the next steps towards a broader usage of such methods is given.

I. INTRODUCTION · II. UNCERTAINTY IN DEEP NEURAL NETWORKS

The paper surveys uncertainty estimation in deep neural networks, tracing uncertainty from its sources through modeling and application. It distinguishes reducible model uncertainty from irreducible data uncertainty and outlines the steps and factors that shape predictions.

I. INTRODUCTION

  • Deployment in mission- and safety-critical applications remains limited by opaque inference, difficulty detecting out-of-domain samples and domain shifts, unreliable uncertainty estimates, overconfidence, and adversarial vulnerability.These barriers are described as factors limiting deployment in high-risk fields such as medical image analysis and autonomous vehicle control.Source: §I · p. 1
  • Predictive uncertainty separates reducible model (epistemic) uncertainty from irreducible data (aleatoric) uncertainty, modeled with Bayesian inference, ensembles, test-time augmentation, or deterministic networks.Reliable estimates also require calibration, for which re-calibration methods have been proposed.Source: §I · pp. 1–2
  • The survey connects uncertainty sources to real-world applications, reviews current approaches and limitations, and identifies future research challenges.It is designed as an introduction and broad overview for readers incorporating uncertainty estimation into neural-network predictions.Source: §I · p. 2
  • The survey covers uncertainty sources and estimation, uncertainty measures, calibration, evaluation datasets and implementations, real-world applications, and future challenges.It notes that estimation and calibration methods generally apply to regression, classification, and segmentation unless stated otherwise.Source: §I · p. 2

II. UNCERTAINTY IN DEEP NEURAL NETWORKS

  • The section traces uncertainty quantification from raw information to a prediction through data acquisition, DNN design and training, the inference model, and the prediction’s uncertainty model.These four steps can each introduce sources of uncertainty and errors that affect the final prediction.Source: §II · p. 2
  • Five factors are identified as vital sources of predictive uncertainty: real-world variability, measurement errors, architecture-specification errors, training-procedure errors, and unknown-data errors.Source: §II · p. 3
  • The section explains uncertainty sources associated with each step, how uncertainty propagates, and the model and main types of uncertainty used for neural-network predictions.It focuses on mathematical properties relevant to understanding and applying the methodology across fields.Source: §II · p. 3

A. Data Acquisition

Data acquisition represents real-world situations through measurements and targets, both of which can be noisy or erroneous. Uncertainty arises when finite training data fail to cover variable situations or when measurement systems provide limited, noisy, or mislabeled information.

A. Data Acquisition

  • Measurements x and targets y represent real-world situations, but measurement noise and errors in target descriptions can affect either.Targets may derive from another measurement or a labeling process.Source: A. Data Acquisition · p. 3
  • Training uses a finite set of N real-world situations, so inadequate coverage of the sample space makes target estimation for new samples uncertain.A new sample generally differs from every training sample.Source: A. Data Acquisition · p. 3
  • Variability in real-world conditions can cause distribution shift, to which neural networks are sensitive, potentially changing their performance significantly.Changes include temperature, illumination, clutter, and physical objects’ size, shape, or expression.Source: A. Data Acquisition · p. 3
  • Limited or noisy measurements, insufficient information modalities, and label noise can make inputs and targets insufficiently informative or reduce confidence in the true class.The measurement system may yield similar measurements or targets for highly different real-world situations.Source: A. Data Acquisition · p. 3

B. Deep Neural Network Design and Training · C. Inference · D. Predictive Uncertainty Model

The paper treats network design, stochastic training, and unfamiliar inputs as distinct sources of prediction error, then frames predictive uncertainty as data, model, and sometimes distributional uncertainty. It emphasizes that both modeling choices and training data shape the resulting uncertainty.

B. Deep Neural Network Design and Training

  • Network structure and training choices form an inductive bias [59], with structural errors identified as a source of prediction uncertainty.The modeler’s structure configuration s includes choices such as parameter count, layers, activation functions, optimization, regularization, and augmentation.Source: B. Deep Neural Network Design and… · p. 3
  • Parameter count can contribute to under- or over-fitting, while deeper networks tend to be overconfident in their softmax outputs.Source: B. Deep Neural Network Design and… · p. 4
  • Stochastic training decisions and settings can lead to different local optima and models whose predictions, and sometimes overall performance, differ.Sources include data order, initialization, regularization, batch size, learning rate, and training epochs.Source: B. Deep Neural Network Design and… · p. 4
  • Imbalanced or low-coverage training data adds uncertainty to learned parameters; augmentation or balancing class and region loss contributions may soften it, while acquisition errors can propagate into training.Source: B. Deep Neural Network Design and… · p. 4

C. Inference

  • Valid samples from an unfamiliar task or domain cause uncertainty at inference, such as a bird image presented to a network trained on cats and dogs.Source: C. Inference · p. 4

D. Predictive Uncertainty Model

  • Predictive uncertainty is the uncertainty propagated to a prediction; its predictive distribution is generally unknown and can only be estimated from the available data.Source: D. Predictive Uncertainty Model · p. 4
  • Predictive uncertainty generally separates into data uncertainty and model uncertainty, with some approaches also explicitly modeling distributional uncertainty for regions uncovered by training data.The paper also calls data uncertainty aleatoric and model uncertainty epistemic uncertainty; distributional uncertainty is associated with examples from regions not covered by training data.Source: D. Predictive Uncertainty Model · p. 4

1) Model- and Data Uncertainty: · 2) Distributional Uncertainty: · E. Uncertainty Classification

The sections distinguish reducible model uncertainty from irreducible data uncertainty, add distributional uncertainty as a predictive component, and classify predictive uncertainty by whether inputs are in-domain, shifted, or out-of-domain.

1) Model- and Data Uncertainty:

  • Model uncertainty reflects model or training shortcomings and missing knowledge, whereas data uncertainty arises from information loss or ambiguity in samples and cannot be explained away.Model uncertainty can theoretically be reduced by improving the architecture, learning process, or training data.Source: 1) Model- and Data Uncertainty · pp. 4–5
  • Bayesian modeling represents model uncertainty as a posterior over parameters and data uncertainty as outputs conditional on parameters, but the predictive distribution remains intractable and needs approximation.Ensembles approximate the parameter posterior by learning and averaging different settings, while Bayesian inference reformulates it using Bayes’ theorem [62].Source: 1) Model- and Data Uncertainty · p. 5

2) Distributional Uncertainty:

  • Some formulations separate predictive uncertainty into data, distributional, and model components; distributional uncertainty concerns input-distribution changes and may model uncertainty over softmax categorical outputs.Model uncertainty affects estimation of distributional uncertainty, which in turn affects data uncertainty; out-of-distribution detection methods often represent distributional uncertainty explicitly,.Source: 2) Distributional Uncertainty · p. 5

E. Uncertainty Classification

  • In-domain uncertainty occurs for inputs from the assumed training distribution and reflects model design errors or task complexity; better training data or training can reduce it,.Source: E. Uncertainty Classification · p. 5
  • Domain-shift uncertainty arises when inputs come from a shifted training distribution, reflecting limited coverage and real-world variability; some shifts, such as occlusions, can be learned, [66].Covering shifted domains in training can reduce it, but errors such as motion noise are difficult or impossible to model.Source: E. Uncertainty Classification · pp. 5–6
  • Out-of-domain uncertainty concerns unknown inputs far from training data, such as a bird for a cat-and-dog classifier, which a DNN cannot explain using in-domain knowledge,,,.Model uncertainty spans in-domain, domain-shift, and out-of-domain cases, while data uncertainty covers in-domain uncertainty arising from overlapping samples or systematic label noise.Source: E. Uncertainty Classification · p. 6

III. UNCERTAINTY ESTIMATION · A. Single Deterministic Methods · 1) Internal Uncertainty Quantification Approaches:

The survey frames uncertainty estimation as necessary because real-world domain mismatch and unknown uncertainty sources make exact prediction-uncertainty computation impossible. It distinguishes single deterministic internal approaches, which affect training and predictions, from external methods, then reviews distributional, classification, and regression techniques.

III. UNCERTAINTY ESTIMATION

  • Real-world domain mismatch and unknown uncertainty sources make exact prediction-uncertainty computation impossible, motivating methods that separate model uncertainty from data uncertainty,,.Data uncertainty is commonly represented in the prediction, such as through classification softmax outputs or a regression standard deviation.Source: §III · p. 6
  • The survey groups methods into single deterministic networks, stochastic Bayesian networks, deterministic ensembles, and single networks using test-time input augmentation.The groups differ in network count and determinism; test-time augmentation produces multiple predictions by transforming inputs.Source: §III · p. 7

A. Single Deterministic Methods

  • Single deterministic methods estimate uncertainty from one forward pass; internal approaches train the network to quantify uncertainty, while external methods leave an already-trained network unchanged.Internal methods affect the network’s predictions, whereas external components are generally applied after training and do not alter them.Source: A. Single Deterministic Methods · p. 8

1) Internal Uncertainty Quantification Approaches:

  • Internal methods often predict output distributions rather than point estimates,, but softmax outputs can be overconfident, poorly calibrated, and falsely confident on out-of-distribution inputs,,, [92].Softmax probabilities do not represent model uncertainty, and a network may assign high confidence to an unfamiliar input.Source: 1) Internal Uncertainty Quantification… · pp. 8–9
  • Dirichlet outputs represent high model uncertainty as a flat distribution and high data uncertainty as a sharp, centered one; prior networks train toward sharp in-domain and flat out-of-distribution targets.Prior networks and evidential networks, predict Dirichlet parameters. Reverse-KL training was proposed to avoid undesirable multimodal targets under high data uncertainty and improved uncertainty estimation and adversarial robustness.Source: 1) Internal Uncertainty Quantification… · p. 11, Figure 6 · p. 11
  • Evidential networks derive uncertainty from the evidence in a Dirichlet parameterization, but do not distinguish data from model uncertainty.Extensions distinguish acuity from dissonance and require explicit overlapping-class and out-of-distribution datasets [33]; regression variants learn an evidential normal inverse gamma distribution, while latent-space flows can avoid OOD training data.Source: 1) Internal Uncertainty Quantification… · pp. 11–12
  • Other classification approaches include input perturbation with temperature scaling, inhibited softmax [92], and RBF networks that use centroid distances to estimate uncertainty.RBF networks with Jacobian penalties performed at least equivalently to a five-member deep ensemble for OOD detection and improved the true-positive rate,.Source: 1) Internal Uncertainty Quantification… · p. 12
  • Regression methods use interval bounds for potentially non-symmetric uncertainty [93], quantile intervals and Orthonormal Certificates for data and model uncertainty, or virtual residuals to help avoid overconfident predictions.The interval approach was reported as more robust in the presence of noise; virtual residuals are generated through a cross-validation-like pre-training step and incorporated into training [93],.Source: 1) Internal Uncertainty Quantification… · p. 12

2) External Uncertainty Quantification Approaches: · 3) Summing Up Single Deterministic Methods: · B. Bayesian Neural Networks

The survey contrasts external uncertainty methods and single deterministic approaches with Bayesian neural networks, which model distributions over parameters to support predictive uncertainty. It reviews their inference strategies, benefits, applications, and computational and prior-specification challenges.

2) External Uncertainty Quantification Approaches:

  • External uncertainty methods leave predictions unchanged and can be combined; Raghu et al. proposed a separate uncertainty-prediction network to reduce prediction-task bias.Ramalho and Miranda additionally use training-data representation density to predict whether the main network’s estimate is correct.Source: 2) External Uncertainty Quantification… · p. 12
  • Hsu et al. detect out-of-distribution inputs using low sigmoid-derived total class probabilities, while Oberdiek et al. and Lee et al. use gradient-based sensitivity measures.Lee et al. report state-of-the-art results for out-of-distribution and corrupted-input detection.Source: 2) External Uncertainty Quantification… · p. 12

3) Summing Up Single Deterministic Methods:

  • Single deterministic methods are generally computationally efficient, but relying on one model makes them sensitive to architecture, training procedure, and training data.They typically require one network for training and one or at most two forward passes for evaluation, less than ensemble, Bayesian, or test-time augmentation approaches.Source: 3) Summing Up Single Deterministic… · p. 12

B. Bayesian Neural Networks

  • BNNs model a posterior distribution over network parameters from a prior and training data, then marginalize predictions over that posterior to estimate predictive uncertainty.The posterior is generally intractable, so approximation techniques are typically used.Source: B. Bayesian Neural Networks · p. 13
  • Monte Carlo approximation estimates Bayesian predictions using the mean of deterministic networks parameterized by samples from the posterior distribution of weights.The predictive integral is intractable for common prior-posterior pairs, motivating approximation.Source: B. Bayesian Neural Networks · p. 13
  • BNN marginalization can improve accuracy and calibration, and Bayesian methods also support model selection, compression, active learning, and continual learning.The survey also notes applications to theoretical advances in Bayesian learning.Source: B. Bayesian Neural Networks · p. 13
  • BNN inference lacks a closed-form posterior for complex neural networks, and scaling approximate inference to deep-network data and parameter sizes is computationally difficult.Choosing meaningful priors is another unresolved challenge.Source: B. Bayesian Neural Networks · p. 13
  • The survey groups BNN posterior approximations into variational inference,, sampling approaches, and Laplace approximation,.Related work includes other approximate inference methods, probabilistic programming, prior designs, uncertainty propagation, and aleatoric-uncertainty computation.Source: B. Bayesian Neural Networks · p. 14

1) Variational Inference: · 2) Sampling Methods:

Variational inference approximates neural-network posteriors with a tractable parameterized distribution, while sampling methods represent them through samples without restricting the distribution’s form. Both families have been adapted to deep networks, balancing expressive uncertainty estimates with computational and practical constraints.

1) Variational Inference:

  • Variational inference fits a parameterized distribution q(θ) to the posterior p(θ|x, y) by minimizing KL divergence, typically optimizing the equivalent ELBO instead.The ELBO is used because the posterior makes direct KL-divergence minimization infeasible.Source: 1) Variational Inference · p. 14
  • Early neural-network methods used diagonal Gaussian or full-covariance variational families; later stochastic methods optimized the ELBO on mini-batches.Bayes By Backprop extended stochastic variational inference to non-Gaussian priors and demonstrated unbiased stochastic gradients; the local reparameterization trick reduces gradient variance.Source: 1) Variational Inference · p. 14
  • Modern variational methods address gradient and initialization fragility and expand posterior expressiveness beyond mean-field, while mean-field can reduce computational complexity.Methods include hierarchical priors, matrix-normal and Kronecker or low-rank covariance families,,, [147],, and normalizing flows,; whether layer-wise correlations matter remains open [137]. Variational inference has also been scaled to ImageNet-size data and architectures using multiple GPUs.Source: 1) Variational Inference · p. 14
  • MC Dropout treats dropout as Bernoulli-distributed variables and estimates predictive uncertainty by keeping dropout active at test time [61],,.DropConnect drops incoming node activations [37]; it was found more robust for uncertainty representation, while combining it with dropout can improve test accuracy and robustness [154].Source: 1) Variational Inference · p. 15

2) Sampling Methods:

  • Sampling methods represent distributions nonparametrically through drawn hypotheses, allowing multimodal or non-Gaussian forms; MCMC is commonly used for neural networks because alternatives can be inefficient in high dimensions [62].MCMC iteratively proposes states and accepts or rejects them according to rules, with each Markov-chain state depending only on the current state.Source: 2) Sampling Methods · p. 15
  • Hamiltonian Monte Carlo uses momentum and Hamiltonian dynamics rather than a random walk, followed by leapfrog integration and Metropolis acceptance,,,,.Despite being regarded as a gold standard,,, HMC processes the full dataset each iteration, making it expensive for datasets of millions or billions.Source: 2) Sampling Methods · p. 15
  • Stochastic-gradient MCMC makes sampling more scalable: Welling and Teh combined mini-batch SGD with Langevin dynamics, showing Bayesian inference can use noisy SGD.Extensions use preconditioning and second-order information,,,,,, parameter-dependent diffusion and thermostats [172],,,,,,, distributed computing, sample distillation, and variance- or bias-reduction techniques,,,.Source: 2) Sampling Methods · pp. 15–16
  • SG-MCMC theory has established convergence, consistency, and step-size-dependent bias-variance results, while applications include shape classification, uncertainty quantification, cold-posterior studies, and generalization,,,,,,.Higher-order integrators can yield faster convergence and more accurate invariant measures than first-order Euler integration.Source: 2) Sampling Methods · p. 16

3) Laplace Approximation: … 1) Principles of Ensemble Methods:

The section explains how Laplace approximation estimates model uncertainty with a Gaussian posterior around a local mode, reviews Bayesian neural-network inference and open challenges, and introduces ensembles as a way to combine predictions and expose model uncertainty.

3) Laplace Approximation:

  • Laplace approximation fits a Gaussian posterior around the MAP estimate, with covariance (H + τI)^-1, and applies to trained networks using common losses and piece-wise linear activations.A Gaussian prior with scalar precision τ corresponds to L2 regularization.Source: 3) Laplace Approximation · p. 16
  • Because full Hessians are infeasible for modern networks, methods approximate diagonal or off-diagonal structure, including scalable layer-wise Kronecker factors,,,.Eigenvalue rescaling improves diagonal variance accuracy, while correcting parameter-space diagonals addresses inaccurate eigenvectors.Source: 3) Laplace Approximation · p. 16
  • Diagonal Fisher or Gauss–Newton approximations support pruning or continual learning, while Kronecker and low-rank approaches capture within-layer correlations and scale Laplace to ImageNet,,,.Low-rank sparsification makes correlation-modeling memory comparable to diagonal approximations; a last-layer Gaussian can mitigate ReLU overconfidence,.Source: 3) Laplace Approximation · p. 16
  • Extensions address the bell-shaped-posterior assumption and underfitting through Bayesian-optimized hyperparameters for calibration and uncertainty units that reshape loss geometry, [205], [206].These are presented as routes to improved calibration and more accurate inference.Source: 3) Laplace Approximation · p. 17
  • Applications include autonomous driving, contextual and semi-supervised domain adaptation for image classification, and real-time linearized Laplace and Laplace-bridge methods [209],, [211], [212].The image-classification approach uses unary potentials in a Conditional Random Field; subnetwork inference aims to increase covariance expressivity while retaining tractability, [212].Source: 3) Laplace Approximation · p. 17

4) Sum Up Bayesian Methods:

  • Bayesian deep-learning advances include stochastic mini-batch approximate inference and scalable tools such as MC-dropout and Laplace, enabling posterior inference in large-scale settings,,.Open challenges include meaningful priors,, efficient parameter marginalization,, [211], benchmarks and tools [214],,, [215], and deeper methodological understanding and applications [137],, [216],.Source: 4) Sum Up Bayesian Methods · p. 17

1) Principles of Ensemble Methods:

  • Ensembles combine predictions from multiple members, often by averaging, to exploit group decision-making synergy,.The approach is applied across practical tasks, including bioinformatics,,, remote sensing,,, and reinforcement learning,.Source: 1) Principles of Ensemble Methods · p. 17
  • Ensemble prediction diversity offers an intuitive measure of model uncertainty, but ensembles were originally developed to improve accuracy and many works do not explicitly quantify uncertainty,.The section notes that ensemble methods have relatively few fundamental differences across application types and are well suited to uncertainty estimation.Source: 1) Principles of Ensemble Methods · p. 17

2) Single- and Multi-Mode Evaluation: … 5) Making Ensemble Methods more Efficient:

These sections explain how ensembles combine models converging to different local optima, describe ways to induce useful diversity, and review uncertainty-estimation results and efficiency trade-offs.

2) Single- and Multi-Mode Evaluation:

  • Single-mode methods evaluate one local optimum, whereas ensembles seek different optima whose varied strengths can produce synergy when their predictions are combined [227].Deterministic networks use a point estimate; Bayesian networks also account for a region around one solution, while ensembles learn several parameter settings.Source: 2) Single- and Multi-Mode Evaluation · pp. 17–18, Figure 7 · p. 18

3) Bringing Variety into Ensembles:

  • Ensemble diversity can come from random initialization and data shuffling, bagging or boosting, member-specific augmentation, and different network architectures,.Random initialization and shuffling were reported sufficient in some settings; bagging could weaken performance, and augmentation could improve accuracy yet harm calibration, [230],,. Attention masks and learning-rate schedules that find multiple optima within one run are other approaches,,.Source: 3) Bringing Variety into Ensembles · p. 18

4) Ensemble Methods and Uncertainty Quantification:

  • Deep ensembles performed at least as well as compared Bayesian baselines, and were reported as reliable for real-life use, better calibrated in active learning, and robust to distribution shifts,,,.In the distribution-shift evaluation, ensembles of five already appeared to perform best among the compared methods.Source: 4) Ensemble Methods and Uncertainty… · pp. 18–19
  • Vyas et al. improved out-of-distribution detection by training ensemble members to distinguish in-domain data from member-specific out-of-distribution subsets.Their loss seeks a positive minimum margin between the average entropy of the in-domain and out-of-distribution subsets.Source: 4) Ensemble Methods and Uncertainty… · p. 19

5) Making Ensemble Methods more Efficient:

  • Ensembles increase computation and memory use, so pruning methods remove redundant members using diversity measures while aiming to preserve performance,,,,,.These costs can bottleneck deployment when computing resources or reaction time are limited.Source: 5) Making Ensemble Methods more… · p. 19
  • Ensemble distillation transfers an ensemble into one student; higher-order distribution targets can retain model uncertainty that averaging softmax outputs loses,,.Some approaches match Dirichlet distributions by minimizing KL divergence, and tests found distilled models could distinguish data from model uncertainty; they cannot fully reproduce an ensemble but sometimes achieve comparable results,,.Source: 5) Making Ensemble Methods more… · p. 19
  • Sub-ensembles share a fixed trunk for general features while independently training each member’s task network, reducing repeated trunk computations.Sharing can constrain architecture choice and prevent fully independent training, so training time may not fall as much as computation does.Source: 5) Making Ensemble Methods more… · pp. 19–20
  • Batch-ensembles share a weight matrix and represent each member’s weights with an individual rank-one factor, adding n + m parameters per member.Members must be trained in parallel, but the authors report that this parallelization can resemble mini-batch optimization on one unit.Source: 5) Making Ensemble Methods more… · p. 20

6) Sum Up Ensemble Methods: … 1) Measuring Data Uncertainty in Classification Tasks:

The paper surveys uncertainty estimation methods and their practical limitations, then explains how uncertainty estimates can be evaluated, including classification measures that distinguish data uncertainty from model uncertainty.

6) Sum Up Ensemble Methods:

  • Ensembles are straightforward to apply and parallelize, but their memory and computation grow linearly with member count, limiting deployment under resource or time constraints.Random initialization, data shuffling, and augmentations can provide diversity in many applications,; further work should examine uncertainty-aware training and ensemble behavior, including loss landscapes [227].Source: 6) Sum Up Ensemble Methods · p. 20

D. Test Time Augmentation

  • Test-time augmentation estimates uncertainty by augmenting each test sample and aggregating predictions without changing the underlying model or requiring additional data [252], [91],.It is commonly used in medical imaging, including cell segmentation, where augmented inputs to UNet or Mask R-CNN are combined by majority voting,,,,.Source: D. Test Time Augmentation · p. 20
  • Test-time augmentation requires valid, in-distribution transformations, and its effect on uncertainty—including the needed number and type of augmentations—remains an open question [252].Reflection may capture less uncertainty than domain-specific stretching or shearing, a concern for resource-limited global inference such as earth observation.Source: D. Test Time Augmentation · pp. 20–21

E. Neural Network Uncertainty Quantification Approaches for Real Life Applications

  • For real-life applications, deterministic networks are efficient, while Bayesian methods offer parameter-uncertainty descriptions and theoretical grounding; Laplace approximations can be applied to pretrained networks with less computation than sampling.Method choice depends on resource limits and implementation needs, which vary substantially among Bayesian approaches.Source: E. Neural Network Uncertainty… · p. 21
  • In real-world tests, ensembles outperformed the considered Bayesian approaches, while deterministic prior networks were comparable with lower computation but may require separate in- and out-of-distribution training sets [56],,,, [33].The authors identify ensemble multi-mode evaluation as a useful property and suggest combining efficient ensemble strategies with Bayesian approaches as a research direction, [211], [6],.Source: E. Neural Network Uncertainty… · p. 21

IV. UNCERTAINTY MEASURES AND QUALITY

  • Uncertainty quality is difficult to evaluate because estimates depend on the method, ground-truth uncertainty is lacking, and no unified metric covers task-specific definitions [256],, [257].Regression commonly uses prediction intervals or standard deviations, whereas classification and segmentation use entropy and related measures.Source: §IV · p. 21

A. Evaluating Uncertainty in Classification Tasks

1) Measuring Data Uncertainty in Classification Tasks:

  • For classification, maximal class probability directly represents certainty, while entropy measures average information; a single prediction cannot reveal how much model uncertainty affects its data-uncertainty estimate.Source: 1) Measuring Data Uncertainty in… · p. 22

2) Measuring Model Uncertainty in Classification Tasks: … 1) Measuring Data Uncertainty in Regression Predictions:

The section describes uncertainty measures for classification predictions, set-level evaluations of their ability to separate cases, and methods for measuring data uncertainty in regression. It also notes that prediction variability can miss distributional shifts and out-of-distribution examples.

2) Measuring Model Uncertainty in Classification Tasks:

  • An approximate posterior over model parameters makes softmax outputs stochastic, allowing their variation to quantify model uncertainty beyond a single, often poorly calibrated prediction.The passage notes that a single softmax prediction does not indicate the model’s own certainty about that output.Source: 2) Measuring Model Uncertainty in… · p. 22
  • Mutual information, expected Kullback-Leibler divergence, and predictive variance measure variation among stochastic softmax outputs; mutual information is interpreted as model uncertainty.Mutual information compares the entropy of the expected softmax output with the expected entropy of softmax outputs; expected KL divergence captures divergence among possible outputs, while predictive variance evaluates their variance.Source: 2) Measuring Model Uncertainty in… · p. 22
  • Monte Carlo approximations, ensembles, and test-time augmentation provide M prediction samples, enabling expectations in the uncertainty measures to be approximated with averages.This approach is used when posterior propagation is intractable or the underlying distribution is undefined.Source: 2) Measuring Model Uncertainty in… · p. 22

3) Measuring Distributional Uncertainty in Classification Tasks:

  • Prediction variability can miss distribution shifts and out-of-distribution inputs when all predictors favor the same false class, producing falsely low uncertainty.Proposed approaches use logit magnitude as evidence, interpreting summed exponentials as Dirichlet precision or comparing accumulated evidence with a constant,,,, [92].Source: 3) Measuring Distributional… · p. 22

4) Performance Measure on Complete Data Set:

  • ROC and PR curves show how uncertainty measures separate sample groups across thresholds, while AUC summarizes ranking; test-set-derived thresholds can fail under distribution shift,,,.The passage gives correct-versus-false classification and in-domain-versus-out-of-distribution detection as example groupings.Source: 4) Performance Measure on Complete… · p. 23

1) Measuring Data Uncertainty in Regression Predictions:

  • Regression models can predict a probability distribution’s parameters, such as a mean and standard deviation, to provide data uncertainty and quantify a likely region for the true value [31],.The region is defined under the assumption that the predicted distribution is correct.Source: 1) Measuring Data Uncertainty in… · p. 23
  • Prediction intervals directly specify a range expected to contain the prediction; Mean Prediction Interval Width measures average interval size, and Prediction Interval Coverage Probability measures captured test values [259], [260].The passage describes interval size as a measure of certainty and coverage as a measure of interval correctness.Source: 1) Measuring Data Uncertainty in… · p. 23

2) Measuring Model Uncertainty in Regression Predictions: · C. Evaluating Uncertainty in Segmentation Tasks · V. CALIBRATION

The sections describe how uncertainty is measured in regression and segmentation, and define calibration as agreement between predictive confidence and observed outcomes. They distinguish methods that reduce model uncertainty from recalibration methods that instead propagate it into reported data uncertainty.

2) Measuring Model Uncertainty in Regression Predictions:

  • Regression model uncertainty can be measured as in classification by averaging predictions and measuring divergence among them.The passage attributes model uncertainty mainly to architecture, training, and underrepresented training data.Source: 2) Measuring Model Uncertainty in… · p. 23

C. Evaluating Uncertainty in Segmentation Tasks

  • Segmentation uncertainty is estimated using approximate Bayesian inference,, [4],,, [262],, or test-time augmentation, and pixelwise measures include intervals, variance, entropy, and mutual information.The cited measures are confidence intervals [4],, predictive variance [262],, predictive entropy,,,, and mutual information.Source: C. Evaluating Uncertainty in… · pp. 23–24
  • Structure uncertainty is obtained by averaging pixelwise estimates, while volume uncertainty is assessed with coefficient of variation, average Dice score, or intersection over union to measure pairwise area overlap,,,.The cited volume metrics assess agreement in area overlap between pairs of estimates.Source: C. Evaluating Uncertainty in… · p. 24
  • To test whether false segmentations increase uncertainty, evaluations use pixel-level true positive and false detection rates across uncertainty thresholds, ROC curves, and, in related work, area under the ROC curve,.The passage attributes the true-positive rate, false-detection rate, and retained-pixel ROC analysis to Nair et al., and AUC analysis to McClure et al..Source: C. Evaluating Uncertainty in… · p. 24

V. CALIBRATION

  • Calibration means confidence approximates correctness: for example, 70%-confidence classification predictions should be 30% false, while regression intervals should match empirical coverage; reliability diagrams visualize calibration,.The regression definition compares predicted intervals with intervals empirically computed from the dataset.Source: §V · p. 24
  • Calibration errors are generally linked to model uncertainty, and several works report that deeper networks tend to be more overconfident than shallower ones,,.The passage contrasts model uncertainty with data uncertainty, which it describes as underlying uncertainty in the real-world information represented by an input and target.Source: §V · p. 24
  • Some uncertainty-estimation methods improve calibration by separately quantifying model and data uncertainty and reducing model uncertainty in predictions,.These methods are distinguished from methods that explicitly reduce calibration errors.Source: §V · p. 24
  • Explicit recalibration methods do not reduce model uncertainty; they propagate it into predicted data uncertainty, so reliable confidence is not necessarily equivalent to data uncertainty.For example, recalibration may map an overfitted classifier’s probability-1 predictions to 0.5 when half the test samples belong to the other class.Source: §V · p. 24

A. Calibration Methods · 1) Regularization Methods:

Calibration methods are applied during training, after training, or through uncertainty-estimation approaches. Regularization methods modify training to produce inherently calibrated networks, with techniques targeting confidence, model uncertainty, or calibration under data augmentation.

A. Calibration Methods

  • Calibration approaches comprise training-time regularization, [269],,, [271], post-processing with held-out data,, and uncertainty estimation using Bayesian methods, [209], [273], [274], or deep ensembles,.Post-processing assumes the calibration set and inference data have equivalent distributions, and its size can affect calibration.Source: A. Calibration Methods · p. 24

1) Regularization Methods:

  • Regularization changes the training objective or data and, unlike uncertainty-estimation methods, aims to minimize model uncertainty that is unavailable at inference.The paper distinguishes these calibration methods from approaches that quantify model and data uncertainty during a single forward pass.Source: 1) Regularization Methods · pp. 24–25
  • Label smoothing redistributes a small share of the true class probability across false classes, reducing calibration error and often increasing accuracy despite higher training loss,.It avoids the impractical hard-label optimum, where logits for the correct class grow much larger than those for incorrect classes.Source: 1) Regularization Methods · p. 25
  • Seo et al. use stochastic training-time predictions to derive a variance-weighted confidence-integrated loss that pushes high-variance examples toward uniform predictions and permits single-pass inference.The adaptive smoothing parameter increases prediction scores for samples with low model uncertainty while reducing prediction variance.Source: 1) Regularization Methods · p. 25
  • Pereyra et al. [269] add a negative-entropy penalty to the standard loss, with its strength controlled by a parameter computed as for the VWCI loss.The penalty increases with predicted confidence.Source: 1) Regularization Methods · p. 25
  • Mixup trains on convex combinations of random training pairs and can provide inherent calibration through label smoothing, but mixed-input uncertainty and distribution shift can harm calibration, [279],.Maroñas et al. [279] proposed accounting for calibration on unmixed samples with a loss combining mixed and unmixed training losses and a batchwise ECE measure weighted by β.Source: 1) Regularization Methods · pp. 25–26
  • On-manifold adversarial training improved uncertainty calibration over mixup, and training with out-of-distribution examples can also improve classifier calibration,.The adversarial training generates out-of-domain samples, whereas mixup combines training samples.Source: 1) Regularization Methods · p. 26

2) Post-Processing Methods: · 3) Calibration with Uncertainty Estimation Approaches: · B. Evaluating Calibration Quality

The sections describe post-hoc and uncertainty-based approaches to improve calibration, then explain visual and quantitative ways to evaluate it. They also note limitations in existing methods and calibration measures.

2) Post-Processing Methods:

  • Post-hoc methods learn a recalibration function from held-out calibration data, but no existing approach meets all three requirements of accuracy preservation, data efficiency, and expressiveness.The requirements call for leaving only a small fraction of training data for calibration and approximating the correct recalibration map when enough data is available.Source: 2) Post-Processing Methods · p. 27
  • Temperature scaling adjusts softmax confidence through T, while an ensemble of scalers improves calibration, data efficiency, and expressiveness without reducing classification accuracy,.T = 1 leaves softmax unchanged; T > 1 increases entropy and lowers confidence, while T ∈ (0, 1) decreases entropy and raises confidence. Basic temperature scaling preserves accuracy but has limited expressiveness.Source: 2) Post-Processing Methods · p. 27
  • Other post-hoc options include classwise linear recalibration, regression standard-deviation scaling [285], and Gaussian-process calibration.Classwise recalibration and Gaussian-process calibration do not guarantee preservation of original predictions; std-scaling rescales predicted standard deviations and can use Gaussian log-likelihood loss.Source: 2) Post-Processing Methods · p. 27

3) Calibration with Uncertainty Estimation Approaches:

  • Deep ensembles improved calibration over single networks,, but can worsen it with mixup; temperature scaling averaged predictions helps, last-layer Bayesian approximation helps, and MC dropout can be poorly calibrated [234],, [273], [274].Structured dropout, which drops channels, blocks, or layers to promote model diversity, was proposed to reduce calibration errors [273].Source: 3) Calibration with Uncertainty… · p. 27

B. Evaluating Calibration Quality

  • Reliability diagrams compare average confidence with accuracy across confidence bins; well-calibrated predictions have matching values and plot near the diagonal,.Multidimensional reliability diagrams distinguish classes and can improve calibration-error interpretability.Source: B. Evaluating Calibration Quality · pp. 27–28
  • The ECE measures top-label calibration, whereas the SCE assesses all classes; all-label measures such as SCE were found more effective than top-label measures such as ECE.ECE averages bin-wise calibration errors across equally spaced confidence bins; SCE computes errors by class within bins and averages across bins.Source: B. Evaluating Calibration Quality · p. 28
  • Adaptive measures aECE and aSCE use equal-count bins of varying width and are more robust to bin count than equal-width ECE and SCE,,.The adaptive measures extend ECE and SCE, respectively.Source: B. Evaluating Calibration Quality · p. 28
  • In multiclass evaluation, class imbalance can let larger classes’ weighted errors hide poor calibration in smaller classes.This can happen even when calibration is computed classwise, because errors are weighted by the number of samples in each class.Source: B. Evaluating Calibration Quality · p. 28

VI. DATA SETS AND BASELINES

The section surveys commonly used tasks, data sets, and baselines for evaluating uncertainty estimation, offering researchers and practitioners a concise guide to benchmark choices. Its overview emphasizes frequently used methodological comparisons rather than architecture choices.

VI. DATA SETS AND BASELINES

  • The overview organizes frequently used tasks, data sets, and baselines by methodological category, and does not compare architecture choices despite their performance impact.Its displayed elements are ranked by frequency of use in the reviewed literature.Source: §VI · p. 28
  • Regression, classification calibration, out-of-distribution detection, and adversarial robustness are common evaluation tasks; medical studies predominantly calibrate semantic segmentation.Regression studies examine samples near to and far from the training distribution.Source: §VI · p. 28
  • Toy and UCI data sets are common for regression, while MNIST, CIFAR10/100, SVHN, and ImageNet variants feature in calibration and out-of-distribution studies.OOD evaluations often pair CIFAR-trained models with SVHN and vice versa, or MNIST with notMNIST and FashionMNIST; corrupted classification data sets blur OOD and adversarial evaluations.Source: §VI · p. 28
  • MC Dropout and deep ensembles are the most common baselines, with deterministic-model softmax outputs almost always serving as a surrogate baseline.Within categories, BNNs are often compared with variational inference methods such as BBB or PBP, while deterministic-model OOD comparisons more often use distance-based methods; BNN studies cover more diverse tasks but less often use ImageNet-scale data.Source: §VI · pp. 28–29

VII. APPLICATIONS OF UNCERTAINTY ESTIMATES … 2) Robotics:

Uncertainty estimates help characterize prediction reliability and support better-informed decisions across applications. The sections cover active learning, reinforcement learning, and real-world fields, with robotics posing distinctive safety and deployment challenges.

VII. APPLICATIONS OF UNCERTAINTY ESTIMATES

  • Uncertainty quantification supports classifying predictions and making more confident decisions, with application concepts intended to transfer across domains.The overview presents active and reinforcement learning alongside domain applications as representative examples.Source: §VII · p. 29

1) Active Learning:

  • Uncertainty sampling selects the most uncertain unlabeled examples for annotation, enabling sequential training on an expanding labeled dataset [292],.Uncertainty-based active learning has been applied in deep learning, [24], [294], [25],. Figure 10 illustrates the acquisition, labeling, and retraining cycle.Source: 1) Active Learning · pp. 29–30, Figure 10 · p. 30

2) Reinforcement Learning:

  • Uncertainty estimates can balance exploration of unknown environments against exploitation of known knowledge and help robots avoid catastrophic failures.Approaches include bootstrapped model ensembles, dropout-based Bayesian approximation, and mixtures of Bayesian networks with dropout sampling over bootstrapped ensembles [28],.Source: 2) Reinforcement Learning · p. 30, Figure 11 · p. 30

A. Uncertainty in Real-World Applications

  • Risk minimization and difficult-to-control data, which complicate trustworthy ground truth, motivate uncertainty quantification across fields including autonomous driving, medical imaging, and earth observation.Source: A. Uncertainty in Real-World… · p. 30

1) Medical Analysis:

  • In medical imaging, uncertainty helps assess segmentation reliability, identify false areas, guide expert refinement, and judge whether automated diagnoses can be trusted,.Methods include Monte Carlo, spike-and-slab, and spatial dropout, as well as test-time augmentation for data-dependent uncertainty,,,,,, [4],, [6],, [262],,.Source: 1) Medical Analysis · p. 30

2) Robotics:

  • Robotics combines safety-critical mistakes and train-test distribution shifts with a longstanding role for probabilistic reasoning in perception, planning, control, and safe interaction,,,,.Earlier uncertainty-aware work includes localization,,, SLAM,,,, factor graphs,, POMDPs,, and advances in semantics, active perception, and active learning,,,,,,,.Source: 2) Robotics · p. 31
  • Robotics-specific deep-learning approaches include autoencoder novelty detection, rotational uncertainty [324], uncertainty-aware reinforcement learning, [28],,, spatial information with MC-dropout,, and uncertain localization,,.Other approaches learn from robots’ past failures or detect predictor inconsistencies,.Source: 2) Robotics · p. 31
  • Robotics uncertainty methods must address limited real-time computation, sequential spatial and temporal information, online sample selection and learning, and purposeful scene manipulation.These challenges arise because robots are physically situated systems.Source: 2) Robotics · p. 31

3) Earth Observation(EO): … B. Outlook •

Uncertainty estimation is especially important in Earth observation and other real-world, safety-critical applications, yet current methods face gaps in validation, evaluation, ground truth, and explainability. The outlook calls for standardized benchmarks, systematic real-world comparisons, uncertainty ground truths, explainable methods, and integration of physical knowledge.

3) Earth Observation(EO):

  • Earth observation uncertainty is crucial amid variable sensors, regions, and conditions, while real-world samples may be out of distribution and uncertainty research remains limited.The passage highlights obscured or snow-covered images and the difficulty of defining out-of-distribution boundaries across the wide range of house types.Source: 3) Earth Observation(EO) · pp. 31–32
  • Earth observation fusion must quantify uncertainty contributions across channels and sensors and favor trustworthy sources; physical sensor and atmospheric effects also matter.For some applications, derived geophysical variables must include probability distributions for assimilation into process models; physically guided models can incorporate domain knowledge and uncertainty representations, [344],. Explainable estimates could clarify decisions and support risk-averse deployment, with Antorán et al. identified as an early work.Source: 3) Earth Observation(EO) · p. 32, B. Outlook • · p. 34

tification methods work for real world applications?

  • Real-world adoption remains limited because uncertainty methods lack broad validation on real-world problems and standardized evaluation protocols, making direct comparisons and method selection difficult.Many models are developed on standard datasets rather than complex settings such as noisy, low-resolution satellite data.Source: tification methods work for real world… · p. 32
  • Dataset-level uncertainty metrics can miss the reliability of individual predictions, while absent ground-truth uncertainties and distribution shifts hinder validation and transfer.These gaps are especially consequential in mission- and safety-critical use and in fields such as Earth observation, where training data are costly to prepare.Source: tification methods work for real world… · p. 32
  • Unexplained uncertainty estimates and poorly understood methods make domain transfer and safety-critical deployment harder, motivating research into how methods capture uncertainty.The passage identifies understanding deterministic approaches, ensembles, and Bayesian methods as an ongoing research direction [227].Source: tification methods work for real world… · p. 33

B. Outlook •

  • A generic evaluation protocol should combine baseline datasets and metrics covering uncertainty types with risk-averse, worst-case tests in which highly uncertain predictions never fail.Such a standard would support comparisons across methods and real-world datasets, and conferences and journals should encourage its adoption.Source: B. Outlook • · p. 33
  • Domain experts should compare methods and expose their weaknesses, while a central platform could collect cross-domain real-world evaluations and challenges.The authors suggest Google’s repository on uncertainty baselines as a possible step toward this platform [340].Source: B. Outlook • · pp. 33–34
  • An uncertainty ground truth could enable ImageNet-like comparisons and single-sample evaluation; studying data generation and labeling could help establish it.Source: B. Outlook • · p. 34

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