A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges

1 INTRODUCTION

• Aleatoric uncertainty is irreducible uncertainty inherent in data, whereas epistemic uncertainty arises from inadequate knowledge or data and is harder to estimate.The introduction also notes that class overlap and data noise can produce data uncertainty.Source: §1 · p. 2
• Trustworthy AI requires effective uncertainty handling across data selection and quality, model limitations, and operational performance, amid theoretical, causal-model, data-quality, and computational challenges.The introduction describes predictions as vulnerable to noise, incorrect inference, and inductive assumptions.Source: §1 · pp. 1–2

1.1 Research Objectives and Outline

• The authors present this as the first comprehensive review of UQ methods in ML and DL and survey newly proposed methods, while stating that cross-method performance comparison is beyond scope because methods target different data and tasks.They aim to help readers obtain reliable results and inform future research.Source: §1.1 · p. 3
• The review identifies important application areas for UQ and points out research gaps in existing methods.These are listed among the study's main contributions.Source: §1.1 · p. 3
• The review discusses future research directions for UQ in ML and DL.The authors describe these directions as intended to guide future researchers in the field.Source: §1.1 · p. 3

2.1 Feed-forward neural network

• A single-hidden-layer feed-forward network maps inputs through a linear transform and nonlinearity to outputs, using softmax loss for classification and Euclidean loss for regression.Source: §2.1 · p. 3

2.2 Uncertainty Modeling

• Predictive uncertainty comprises epistemic (model) and aleatoric (data) uncertainty, with aleatoric uncertainty classified as homoscedastic or heteroscedastic.Source: §2.2 · p. 3
• Bayesian modeling places a distribution over model parameters and defines classification and regression likelihoods, then uses Bayes’ theorem to obtain a posterior over parameters.The passages specify softmax likelihood for classification and Gaussian likelihood for regression.Source: §2.2 · p. 3
• Because the parameter posterior p(ω|X,Y) cannot be computed analytically, variational inference approximates the predictive distribution using qθ(ω), minimizing KL divergence or equivalently maximizing the ELBO.Dropout VI is described as one of the most common approaches, widely used to approximate inference in complex models [22].Source: §2.2 · pp. 3–4
• Data-dependent uncertainty can be modeled by making precision a function of the data and combining predictive mean with model precision in a Gaussian likelihood.The passages also state that predictive variance can then be obtained.Source: §2.2 · p. 4

3.1 Bayesian Deep Learning/Bayesian Neural Networks

• BDL/BNNs [24], [25], [26],,, [29], [30], can interpret model parameters to address standard deep learning’s inability to convey prediction reliability.Source: §3.1 · p. 4
• BNNs are described as robust to overfitting and trainable on both small and big datasets.Source: §3.1 · p. 4

3.2 Monte Carlo (MC) dropout

• MC dropout uses dropout as a regularizer to estimate prediction uncertainty, avoiding the slow, computationally expensive integration of Monte Carlo into deep architectures.During training, dropout randomly removes network units to reduce overfitting.Source: §3.2 · p. 4
• For each input, dropout samples binary variables for network units in every non-output layer with layer-specific probability p_i, reusing those values in the backward pass.Figure 4 depicts variational distributions on a simple neural network.Source: §3.2 · p. 4, Figure 4 · p. 5
• Applications include estimating epistemic, aleatoric, transformation, and segmentation uncertainty in medical imaging, and uncertainty under universal adversarial perturbations [37].Wang et al. used test-time image augmentation and MC sampling for pixel- and structure-level segmentation uncertainty; Liu et al. combined output-neuron Bernoulli dropout with texture bias; Nasir et al. considered MC-sample variance, predictive entropy, and mutual information for MRI lesion segmentation.Source: §3.2 · p. 4
• For autonomous-vehicle control, [41] used element-wise and spatial Bernoulli dropout; McClure and Kriegeskorte proposed spike-and-slab sampling after studying weight-sampling alternatives.Their reported outcomes suggest Bernoulli or Gaussian dropout can improve CNN classification accuracy, while sampling weights may depict uncertainty more accurately than sampling units.Source: §3.2 · p. 5
• Other approaches add dropout to U-Net for myocardial arterial spin labeling segmentation, treat batch normalization as approximate Bayesian inference, or use MC-dropout uncertainty in semi-supervised LA segmentation.In UA-MT, a student learns from a teacher using segmentation and consistency losses, with teacher uncertainty informing the student; Table 1 lists direct MC-dropout applications.Source: §3.2 · p. 5, Figure 5 · p. 5

3.2.1 Comparison of MC dropout with other UQ methods

• Comparative studies examined MC dropout against bootstrap ensembling and Gaussian mixtures for vehicle control, as well as in regression and astronomical observing conditions [68].Foong et al. also empirically and theoretically studied MC dropout; the supplied passage does not specify their comparison target or report comparative outcomes.Source: §3.2.1 · p. 5

• MCMC repeatedly transitions a random draw from q(z0) or q(z0|x) toward the exact posterior; Salakhutdinov et al. used it for movie-rating predictive distributions, but iteration needs are unknown and convergence is slow.The transition is repeated T times, with its outcome converging in distribution to the exact posterior.Source: §3.3 · p. 5
• Salimans et al. expanded the state space with auxiliary random variables and interpreted the stochastic Markov chain as a variational approximation.Source: §3.3 · p. 6
• SG-MCMC, trains DNNs with mini-batch gradient estimates and can approach the true posterior as step sizes decrease,, but short-run error and multilayer-network performance remain concerns,.Gong et al. combined amortized inference with SG-MCMC to improve generalization, while Li et al. proposed an accelerated version.Source: §3.3 · p. 6
• Zhang et al. [80] developed cyclical SG-MCMC (cSG-MCMC), using alternating large and small step sizes to promote large sampler moves and local-mode exploration.SG-MCMC's mini-batch subsets add noise to the model and increase system uncertainty.Source: §3.3 · p. 6
• To handle mini-batch noise, Luo et al. introduced thermostat-assisted continuously tempered Hamiltonian Monte Carlo; dropout HMC was later proposed for uncertainty estimation and compared with SG-MCMC and SGLD.The thermostat-assisted method uses Nosé-Hoover thermostats,; HMC is a MCMC method.Source: §3.3 · p. 6
• MCMC was also applied to GAN-learned stochastic object models to approximate the ideal observer and incorporated into a VAE-based visual tracking system (VAE-MCMC).Source: §3.3 · p. 6

3.4 Variational Inference (VI)

• VI approximates posterior distributions over neural-network weights by formulating Bayesian inference as an optimization problem trained with SGD.The passage also refers to a loss and an equation for computing it, but the supplied text does not provide their expressions.Source: §3.4 · pp. 6–7
• Posch et al. [90] used diagonal-covariance Gaussian variational distributions, later replaced with traditional covariance matrices to model correlated parameters.MOPED used deterministic weights from a pretrained network to select priors; a parametric empirical-Bayes extension used fully factorized Gaussians and was evaluated on diabetic retinopathy diagnosis.Source: §3.4 · p. 7
• VI applications include multi-modal Bayesian fusion for activity recognition, stochastic uncertainty modeling, and latent-space prediction of superconducting transition temperature.The activity-recognition framework combines deterministic and variational layers; the superconductivity model is generative-based and uses VI to compute a latent-space distribution.Source: §3.4 · p. 7
• Stochastic-gradient VI estimates weight posteriors; another stochastic VI jointly models parameter and model uncertainty using latent binary variables to include or exclude weights [100].VI was also integrated with a spatial-temporal network to estimate posterior parameters and prediction probabilities, and with a graph convolutional network for molecular-property prediction.Source: §3.4 · p. 7
• A low-rank factorization of variational parameters yielded a more compact Gaussian mean-field approximation and improved stochastic-gradient SNR for lower-bound estimation [89].Franquhar et al. argued that deeper linear mean-field networks can yield a function-space distribution analogous to shallow full-covariance networks.Source: §3.4 · p. 7

3.5 Bayesian Active Learning (BAL)

• AL queries an oracle for labels, but choosing an acquisition function to identify informative samples is challenging, especially given existing frameworks’ limited scalability to high-dimensional data.Bayesian approaches can be integrated into deep learning to represent uncertainty and guide acquisition of uncertain samples.Source: §3.5 · p. 7
• DBAL uses BALD to select top-n samples, while BatchBALD uses greedy batch selection to reduce runtime; other methods use MOPED [108], MC dropout, or compare entropy and BALD.BatchBALD selects a batch in linear time; MOPED uses BALD to evaluate uncertainty, and MC dropout estimates model uncertainty.Source: §3.5 · p. 7
• ActiveHARNet links BNNs with a GP using dropout to estimate mean and variance, while DeepBASS pairs expectation-maximization with active learning and MC dropout.ActiveHARNet applies dropout before each fully connected layer.Source: §3.5 · p. 7
• For biomedical image segmentation, Scandalea et al. use MC-sample posterior-probability standard deviation to suggest annotations; Zheng et al. [115] find few Bayesian layers near the output suffice for MNIST uncertainty estimation.Zheng et al. varied the number and positions of Bayesian layers.Source: §3.5 · p. 7
• Bayesian batch AL selects samples for posterior inference at each iteration in large-scale problems; active user training addresses complex, high-dimensional classification, and Bayesian inference models annotator confusion-matrix uncertainty.The batch method was inspired by.Source: §3.5 · pp. 7–8
• Generative AL includes a semi-supervised BNN model [120], BGADL [121], which combines DBAL selection with VAE-ACGAN sample generation, and a VAE-based activity-recognition framework for aleatoric and epistemic uncertainty.The activity-recognition framework extracts time-series features with an unsupervised model and applies dropout after dense layers and at test time.Source: §3.5 · p. 8

3.6 Bayes by Backprop (BBB)

• Blundell et al. [123] proposed Bayes by Backprop (BBB), which quantifies uncertainty over neural-network weights by minimizing variational free energy, the expected lower bound of marginal likelihood.BBB uses unbiased gradient estimates of its cost function to learn a distribution over network weights.Source: §3.6 · p. 8
• Fortunato et al. [124] used BBB with truncated back-propagation through time to develop a Bayesian recurrent neural network.Source: §3.6 · p. 8
• Ebrahimi et al. [125] proposed uncertainty-guided continual learning with BNNs, using posterior predictive uncertainty to modify important parameters through hard thresholds or soft adjustments.The approach aims to learn new tasks while retaining previously acquired knowledge.Source: §3.6 · p. 8
• De la Riva and Mettes extended BBB to 3D CNNs, using Gaussian distributions parameterized by means and standard deviations to model uncertainty in convolution weights for action recognition from few videos.Source: §3.6 · p. 8
• MC dropout and BBB showed almost similar performance in cardiac MRI segmentation using a U-Net model.Source: §3.6 · p. 8

3.7 Variational Autoencoders

• VAEs model posterior distributions through variational inference, using an encoder to map inputs to latent variables and a decoder to reconstruct the inputs.The encoder and decoder are represented by qφ(z|x) and pθ(x|z), respectively, and training uses the evidence lower bound.Source: §3.7 · pp. 8–9
• Zamani et al. [128] developed a discrete Bernoulli-latent VAE and pairwise supervised hashing that uses weighted KL regularization and pairwise loss to learn informative binary codes.The loss rewards within-class similarity and between-class dissimilarity; stochastic gradients are used to learn the model.Source: §3.7 · p. 9, Figure 11 · p. 9
• For inverse problems, Bohm et al. [129] trained a mean-field Gaussian-posterior VAE under the ELBO and used EL2O to approximate the posterior; Edupuganti et al. [131] used a VAE-GAN and SURE for MRI recovery uncertainty.The VAE-GAN maps low-quality images to high-quality ones, with a CNN generator and discriminator; SURE serves as a proxy to predict error and estimate uncertainty.Source: §3.7 · p. 9
• A variational U-Net was proposed for uncertainty quantification in reservoir simulations, while Cosmo VAE used variational Bayes to restore missing cosmic microwave background map observations.Cosmo VAE determines the ELBO of the reconstructed image likelihood.Source: §3.7 · p. 9
• For sequential and anomaly-detection tasks, APP VAE uses two LSTMs to estimate prior and posterior distributions, while Sato et al. estimate the posterior by MC sampling.Source: §3.7 · p. 9
• Because VAEs encode only finite-dimensional priors, Mishra et al. developed πVAE, a stochastic process inspired by Gaussian processes [138] that learns distributions over functions.Its encoder maps locations to a high-dimensional feature space, uses a linear mapping to outputs, and recreates that mapping from a lower-dimensional probabilistic embedding for reconstruction.Source: §3.7 · p. 9
• Guo et al. used variable-wise Gaussian latent features and KL divergence to select samples relevant to a new sample for data uncertainty, while Daxberger et al. used a Bayesian VAE with SG-MCMC decoder-posterior estimation for test-time OoD detection.Source: §3.7 · p. 9

4.1 Deep Gaussian Processes

• Deep Gaussian processes (DGPs),,, [144], [145],, [147] extend Gaussian processes into multilayer models; inducing samples address conventional GPs’ difficulty scaling to large datasets.The variational formulation uses inducing-sample locations Z and an approximating distribution q(fx) for the latent function.Source: §4.1 · p. 9
• HIB [150] uses stochastic embeddings and MC sampling; SGHMC and SWA-G approximate Bayesian posteriors, while flipout [155] decorrelates per-input perturbations to reduce gradient variance.Flipout samples pseudo-independent weight perturbations for each input, reducing variance and training time with multiplicative Gaussian perturbations.Source: §4.1 · p. 10
• To address adversarial uncertainty, GPDNNs [157] combine GPs with DNNs, while Choi et al. model YOLOv3 bounding-box coordinates with Gaussian means and variances.Source: §4.1 · p. 10
• Weight-uncertainty methods include Khan et al.’s [160] natural-gradient Gaussian mean-field VI and matrix-variate Gaussian posteriors,; convolutional structure was also introduced into GPs,,, [167],.The matrix-variate approaches model structured correlations or treat the weight matrix as a whole rather than modeling each weight independently.Source: §4.1 · pp. 10–11
• For failure prediction, Corbière et al. found TCP more suitable than MCP and introduced a normalized TCP confidence criterion.Source: §4.1 · p. 11
• Stochastic BN approximates inefficient inference, while variance layers model activations as zero-mean distributions and were reported to defend against adversarial attacks and support RL exploration.Source: §4.1 · p. 11

4.2 Laplace approximations

• Laplace approximations form a Gaussian around the MAP; variants address neural networks, CRF-based semi-supervision, KITTI driving, ImageNet inference, and hyperparameter tuning, where BO used fewer iterations than random search.The KITTI study found uncertainty meaningful but called for further experimental evaluation; the ImageNet application required considerable hyperparameter tuning.Source: §4.2 · p. 11

5 UNCERTAINTY QUANTIFICATION IN REINFORCEMENT LEARNING

• Lee et al. [180] framed model uncertainty as a Bayes-Adaptive Markov Decision Process (BAMDP), with beliefs updated over latent state.The BAMDP includes observable states, latent variables, actions, transitions, rewards, an initial belief, and a discount factor.Source: §5 · p. 11
• Bayesian Policy Optimization (BPO) applies a Bayes filter to POMDPs to compute beliefs about hidden states [180].The passage presents a general view of BPO reproduced based on [180].Source: §5 · p. 11, Figure 16 · p. 12
• O’Donoghue et al. [181] proposed an uncertainty Bellman equation that propagates posterior uncertainty, described here as variance.Source: §5 · p. 12
• Kahn et al. [182] used bootstrapping and dropout to estimate neural-network uncertainty for a mobile-robot collision-prediction model.Source: §5 · p. 12
• Tschantz et al. [184] used ensembles of point-estimate parameters to capture both aleatoric and epistemic uncertainty in RL.Bayesian approaches for parameter uncertainty have also been used in RL, and Ghavamzadeh et al. reviewed Bayesian RL; Table 2 summarizes methods across RL subjects.Source: §5 · p. 12

6 ENSEMBLE TECHNIQUES

• Predictive entropy estimates total uncertainty, while mutual-information disagreement distinguishes knowledge uncertainty from expected data uncertainty.In class-overlap regions, high entropy with low mutual information indicates data uncertainty; for out-of-domain inputs, model disagreement indicates high knowledge uncertainty.Source: §6 · pp. 13–14
• Wilson and Izmailov argue deep ensembles approximate Bayesian model averaging better than standard Bayesian methods; MultiSWAG supports epistemic uncertainty and robustness and can alleviate double descent.They also report an induced correlation function for varied images, useful features, support for structured versus corrupted datasets through marginal likelihood, and improved performance over standard training.Source: §6 · p. 14
• The deep-ensemble-equivalent score [212] makes comparisons interpretable by asking how many deep-ensemble samples match another ensembling method’s calibrated log-likelihood performance.Deep ensembles are presented as a strong baseline under a fixed test-time compute budget.Source: §6 · p. 14

6.1 Deep Ensemble

• Deep ensembles improve test predictions and estimate model uncertainty on OoD inputs through variance reduction; D-EESN uses bootstrap or hierarchical Bayesian frameworks for spatio-temporal forecasting.The hierarchical Bayesian version accommodates multiple uncertainty levels and non-Gaussian data.Source: §6.1 · p. 15

6.2 Deep Ensemble Bayesian

• Deep Ensemble BAL [217] targets variational mode collapse in DBAL, while anchored ensembling regularizes network parameters toward values sampled from a distribution.Fersini et al. used Bayesian model averaging that considers each model’s reliability and uncertainty.Source: §6.2 · pp. 15, 17

6.3 Uncertainty Quantification in Traditional Machine Learning domain using Ensemble Techniques

• Uncertainty Weighted Accuracy improved heterogeneous-ensemble pruning accuracy over baseline and state-of-the-art measures, while bootstrap ensemble width approximated uncertainty in atomistic predictions.SVM-GSU handles input uncertainty for classification; Pereira et al. examined Venn-ABERS, conformal, Platt scaling, and isotonic-regression methods for probabilistic uncertainty estimates.Source: §6.3 · p. 17

7 FURTHER STUDIES OF UQ METHODS

• NADS searches distributions of architectures while recognizing shared blocks, and other methods target nonconvergent prediction-interval training and biased uncertainty estimates for highly confident predictions [237] [238] [239].Kabir et al. [238] used a smooth customizable cost function and evaluated it on electricity-demand and wind-power data; Geifman et al. [239] used model snapshots before approximations were jittered.Source: §7 · p. 17
• SQR and Orthonormal Certificates estimate aleatoric and epistemic uncertainty, while DUQ uses a single test-time forward pass and gradient penalties to detect out-of-distribution data [227],, [242].MC-DropConnect approximates Bayesian inference with Bernoulli-distributed weights and improved uncertainty estimation and prediction accuracy over prior approaches [243].Source: §7 · p. 18
• Neighborhood rough-set measures address real-valued data, while regression calibration, natural-gradient analysis, visual-attention certainty estimates, noise contrastive priors, and uncertainty decomposition support varied prediction and decision tasks [245] [248], [14].Kuleshov et al. [245] calibrated regression uncertainty; Depeweg et al. [248] decomposed uncertainty for active learning and risk-sensitive reinforcement learning.Source: §7 · pp. 18–19
• Further methods model causal structure and class imbalance, use hypernetworks or prediction cascades, and distinguish distributional from data uncertainty,, [251], [252],,.SLANG approximates posteriors with a diagonal-plus-low-rank structure, while Prior Networks explicitly model distributional uncertainty and distinguish it from data uncertainty.Source: §7 · pp. 19, 21
• Other applications include physics-informed uncertainty propagation, calibrated 3D scene flow, mixup calibration, uncertainty in LIME explanations, adversarial detection, certainty-driven consistency learning, and Bayesian subspace inference [256] [257] [259] [261] [262],.Mixup-trained networks were better calibrated and less prone to overconfident predictions on random-noise and OoD data; pre-training improved uncertainty estimates and robustness, with adversarial pre-training showing approximately 10% enhancement over existing methods [261].Source: §7 · pp. 19–21

7.1 Other UQ Techniques

• Other UQ techniques use shrinkage priors for BNN node selection, α-divergence inference, probabilistic visual dialogue, and Radial BNNs designed to scale and avoid mean-field sampling pathologies [447].Radial BNNs were reported as robust to hyperparameters and effective on real-world tasks without intensive tuning.Source: §7.1 · pp. 21, 23
• Additional approaches include neural tangents and probabilistic latent ODEs, plus BNN methods for adversarial robustness, calibrated uncertainty, and strong out-of-distribution performance [449],, [450], [451], [294].Rank-1 BNNs with mixture approximate posteriors reported state-of-the-art performance across OoD variants, test-set calibration, accuracy, and log-likelihood [294].Source: §7.1 · p. 23
• A unified hierarchical model targets open-set, intrinsic-data, and model-capacity uncertainty, while trust scores assess classification reliability and outperformed confidence scores and other baselines [452],.The section also describes a framework combining bootstrap and constraint-based learning for model selection and averaging, though its discussion continues beyond this supplied part.Source: §7.1 · p. 23
• A graph-based algorithm reused stable low-order independencies to improve MAP models, scale to hundreds of variables, and yield more reliable causal relationships; Radev et al. [454] proposed Bayesian model comparison and an epistemic-uncertainty measure for intractable models.Belakaria et al. [395] introduced USeMO, which uses surrogate models and uncertainty to select evaluations for expensive multi-objective optimization, with theoretical analysis of its efficiency.Source: §7.1 · p. 24
• Ariafar et al. [455] used importance sampling to accelerate Bayesian optimization, while Meng et al. [456] used a Bayesian learning rule for binary-weight networks to estimate uncertainty in continual learning and avoid catastrophic forgetting.Importance sampling leveraged the relative information supplied by training examples to improve black-box evaluations and runtime; the binary-weight algorithm achieved state-of-the-art performance.Source: §7.1 · p. 24
• For semi-supervised 3D learning, Xia et al. combined multi-view co-training with BDL-based uncertainty-weighted label fusion, while CADA-VAE aligned image and class-embedding distributions for generalized zero-shot learning.CADA-VAE’s learned latent features contained multimodal information for unseen classes, and were reported to achieve state-of-the-art results on several benchmarks; Fig. 23 depicts the model.Source: §7.1 · p. 24, Figure 23 · p. 24
• Tseng et al. [460] used Gaussian- or Bernoulli-sampled gradient dropout to mitigate overfitting and improve generalization in gradient-based meta-learning; Wu et al. [461] developed robust, efficient variational Bayesian inference for BNNs.Wu et al.’s methods approximate neural-network moments without gradient variance and use hierarchical priors with automated prior-variance selection. Pearce et al. [462] explored BNN architectures that mirror combined and periodic GP kernels, demonstrating practical value in reinforcement and supervised settings.Source: §7.1 · pp. 24–25
• Atanov et al. [463] introduced deep weight priors that improved BNN performance with little training data and accelerated CNN training, while CBLN addressed continual-learning forgetting without requiring past data [464].CBLN uses task-specific Gaussian posterior mixtures and uncertainty-based weight selection at test time; discriminative jackknife is described as applicable across a wide range of deep-learning methods.Source: §7.1 · p. 25
• Shekhovtsov et al. attributed improved generalization with batch normalization partly to randomness in batch statistics, and RevPHiSeg used reversible blocks to reduce memory for segmentation-uncertainty training [467].RevPHiSeg computes activations from later-layer outputs during backpropagation, enabling training with restricted GPU memory and processing larger fields of view.Source: §7.1 · p. 25
• Other applications included Bayesian DeepLab-v3+ [10], deep-learning image registration, physics-constrained UQ without labeled data, and BDL combining epistemic with input-dependent aleatoric uncertainty.The registration work predicted initial momentum for LDDMM in multi- and uni-modal settings; prognostics UQ addressed noisy sensors, unknown future operating conditions, and scarce failure labels [218]. Begoli et al. highlighted adoption challenges, including the lack of sound underlying theory, and called for further theoretical and practical work in AI-assisted medical decision making.Source: §7.1 · pp. 25–26
• Further work reported robust compound-density-network uncertainty on out-of-distribution and adversarial data [471], lightweight Dirichlet output ranking for ImageNet [472], and diverse ensemble predictions for multimodal modeling under dataset shift [219].Kendall et al. used a Bayesian CNN to estimate uncertainty in real-time six-degree-of-freedom camera relocalization and improved outdoor localization accuracy; the review also notes the need for metrics to compare BDL uncertainty estimates [471].Source: §7.1 · p. 26

8.1 Image Processing and Computer Vision

• Across computer-vision tasks, MC dropout was reported as the most effective uncertainty technique in nine studies, [48], [269],,,, [10], [487], [494]; deep ensembles performed better on 3D semantic segmentation [500].Other approaches include sampling-free epistemic uncertainty estimation and Bayesian CNN adaptation for hand segmentation in a new domain.Source: §8.1 · pp. 26–27

8.2 Medical Applications

• In medical applications, uncertainty-aware methods improved segmentation or prediction in several settings, including UMCT, deep spectral learning for sO2, and ensemble-based left-ventricular segmentation [511].RNNs with Bayesian embeddings were reported as a competent approach to model uncertainty [504]; selecting regions rather than full images for annotation led to better-calibrated models [516].Source: §8.2 · pp. 27–28

8.3 Natural Language Processing and Text Mining

• NLP uncertainty detection matters for distinguishing uncertain from factual propositions, but corpora lack unified annotation principles and detectors transfer poorly across domains,.The review identifies a need for adaptable, preferably language-independent approaches that reduce the cost of creating domain-specific resources.Source: §8.3 · pp. 28–29

8.4 Summary of some applied UQ methods in NLP

• For clinical named-entity recognition, uncertainty sampling outperformed other active-learning techniques on area under the learning curve, while ANFU outperformed state-of-the-art methods in F1 across four social-media datasets.Bayesian data-noising variants also outperformed prevailing data-noising techniques on two language-modeling datasets [520].Source: §8.4 · p. 29
• NLP applications also include uncertainty measures for long generated sentences, translation-data uncertainty and calibration [524], and Bayesian techniques for prediction and trading [522].Bayesian Layers [121] and a black-box uncertainty wrapper with rejection [526] are among the described tools; Bayesian RNN variants also improved image-captioning and language-modeling results over traditional RNNs [124].Source: §8.4 · pp. 29–30

8.5 Further Applications

• Outside those areas, discrete-continuous Bayesian deep learning outperformed Gaussian models in climate uncertainty calibration and predictive accuracy, and cascaded multi-task loss improved satellite segmentation over state-of-the-art techniques by 8.3%.For object detection, a method accounting for noise in ground-truth labels improved detection performance and learned-distribution accuracy [532]; Bayesian RNNs were applied to nonlinear spatiotemporal forecasting.Source: §8.5 · pp. 30–31
• Other applications include rough ELM, which showed comparable accuracy and repeatability on most classification tasks, and Bayesian CNNs that produced reliable, well-calibrated galaxy-label posteriors.Active learning with the Bayesian CNNs required up to 35–60% fewer labelled galaxies, depending on morphological features; a Bayesian RNNLM study also reported improvements in word error rate and perplexity over conventional RNNLMs.Source: §8.5 · p. 31
• A driving-policy method penalized both task cost and divergence from training states while unrolling learned dynamics over multiple steps [538].It was examined using a large observational dataset of driving behavior recorded from traffic cameras.Source: §8.5 · p. 31
• Bayesian semisupervised graph convolutional networks used posterior sampling for UQ and active learning in small-molecule property and bioactivity prediction.Separating regression from representation learning supported active learning from little training data while maintaining accurate uncertainty estimates in the low-data limit.Source: §8.5 · p. 31
• BNNs used predictive uncertainty to detect out-of-domain ICU patients and avert probable mortality-prediction errors [540], while label distributions modeled annotation uncertainty for DensePose.For DensePose, understanding uncertainty improved benchmark accuracy to a new state of the art, and fusing multiple models’ uncertainty estimates could further enhance accuracy.Source: §8.5 · p. 31
• A continuous relaxation of dropout masks provided better performance and calibrated uncertainties across tasks, with larger dropout probabilities often used in deeper layers.The variant addressed the costly or impossible probability grid search required for RL and large vision models.Source: §8.5 · p. 31
• A fully convolutional network modeled uncertainty in obstacle distances with a Laplace distribution and produced maps containing distance estimates and their uncertainty.The application addresses indoor robot navigation and mapping with 2D laser scanners that cannot measure full occupancy or detect transparent surfaces.Source: §8.5 · pp. 31–32
• Deep counterparts to shallow data-reduction methods used multiple layers, dropout, and SGD for variable selection and estimation, with Airbnb bookings as an illustration.The examined methods provided predictive performance gains over traditional high-dimensional reduction approaches.Source: §8.5 · p. 32
• 87% overall classification accuracy was reported for CNN-based land-cover mapping focused on urban areas and small objects.The method applied UQ to remote-sensing images.Source: §8.5 · p. 32
• BDL-based day-ahead energy forecasting provided robust out-of-sample uncertainty indications, while PV-aware net-load forecasting captured aleatoric and epistemic uncertainty.The net-load method used residential rooftop PV features and subprofile clustering, proving effective with high PV visibility and subprofile clustering.Source: §8.5 · p. 32

9 LITERATURE GAPS AND OPEN ISSUES

• The review highlights gaps in semisupervised UQ, uncertainty analysis for traditional machine-learning methods, and the use of fusion and ensemble methods to quantify uncertainty.It notes that traditional ML can perform well on small data, while fusion and ensemble methods show potential but remain underused for UQ.Source: §9 · p. 32
• Further gaps include decision-making theories for UQ, limited uncertainty-aware active learning despite costly labeling, and uncertainty quantification for transfer learning.The review recommends using decision-making theories during model decisions and pursuing UQ research for active learning and transfer learning.Source: §9 · p. 33
• The review identifies sparse UQ research for neural architecture search, self-supervised learning [261], [567], and attention mechanisms,.It also notes neural ensemble search methods [566] as a developed technique for uncertainty calibration.Source: §9 · p. 33
• Other open UQ challenges include out-of-distribution detection, hypernetworks, continual learning, GNNs, Bayesian optimization, and uncertainty calibration.The review notes limited studies or calls for further investigation and more efficient methods across these areas.Source: §9 · pp. 33–34

9.1 Future Directions

• For computer vision and image processing, the review calls for resilient UQ, semi-supervised active sampling, automated labeling, and simpler UQ-integrated cascade methods.It also proposes directions including dynamic or multimodal restoration, pose prediction, adversarial detection, probabilistic embeddings, online Bayesian networks, continual learning, and multitask transfer learning.Source: §9.1 · p. 34
• For medical image analysis, the review recommends closer medical-AI collaboration, larger datasets, uncertainty visualization and fusion, and further UQ research for segmentation.It also identifies semi-supervised segmentation, transfer learning, multitask learning, and practical probabilistic methods as directions, noting sampling can be too slow.Source: §9.1 · pp. 34–35
• NLP directions include UQ for machine translation, image captioning, medical visual question answering, and classification of data-stream text.Other proposed directions span UQ for meta-reinforcement learning, stochastic optimization, synthetic data, digital healthcare, sequential decisions, and multi-agent systems.Source: §9.1 · p. 35

9.2 Lack of Data and Code Availability

• The review urges researchers, especially in medical case studies, to share data and code so others can inspect and improve prior methods.The authors acknowledge that some cases may prevent public release, but argue that sharing can accelerate method optimization.Source: §9.2 · p. 35

10 CONCLUSION

• The study provides a comprehensive review and comparative analysis of state-of-the-art uncertainty quantification methods in traditional machine learning and deep learning.It frames uncertainty quantification as important to decision making and increasingly integral to machine learning.Source: §10 · p. 36
• The review discusses uncertainty quantification applications, research gaps, and future research directions.It presents the review as a potential guideline for using uncertainty quantification in deep learning-based applications.Source: §10 · p. 36

Malinin and Gales [365]

• The Uncertainty-Gated Graph propagates identities between tracklets represented as graph nodes for video-based face recognition.The table lists this method alongside the video-based face-recognition task.Source: Malinin and Gales [365] · p. 58
• The reviewed approach combines feature-uncertainty averaging with model averaging over posterior uncertainty.The table also lists COCO and Flickr30k datasets for image-caption embedding and retrieval.Source: Malinin and Gales [365] · p. 58
• Depth estimation for 3D scene reconstruction is among the surveyed computer-vision applications.Source: Malinin and Gales [365] · p. 59
• The survey includes 3D pedestrian localization as an application of uncertainty quantification.Source: Malinin and Gales [365] · p. 60
• For segmenting woven-composite CT volumes, the listed method applies MC dropout during both training and inference.The table describes dividing each 1001×1150×1150 volume into 48 training sub-volumes and 8 each for validation and testing.Source: Malinin and Gales [365] · p. 60
• UQ methods include noise-injection sampling, multivariate Gaussian modeling, HydraNet, probabilistic regression over SO(3), and Distribution Net with random feature vectors.The entries describe methods rather than comparative results.Source: Malinin and Gales [365] · p. 60
• Other approaches use CVAE loss for aleatoric UQ, probabilistic residual learning, HMC and VI for BNNs, relative-entropy objectives for adversarial attacks, and Riemannian VAEs.The adversarial-attack entry specifies a relative entropy objective; the VAE entry names both VAE and Riemannian VAE.Source: Malinin and Gales [365] · p. 61
• Further methods include teacher-student uncertainty-aware consistency regularization, Graph CNN and GAN mesh decoding, Self-Cure Networks, and data uncertainty learning.The mesh approach is identified as an Uncertainty-Aware Mesh Decoder.Source: Malinin and Gales [365] · pp. 61–62
• Medical segmentation and classification entries use weighted or entropy-based uncertainty, MC dropout, ensembles, variational inference, test-time augmentation, and horizontal stacked ensembles.The listed tasks include brain-tumor segmentation and diabetic-retinopathy classification, among others.Source: Malinin and Gales [365] · pp. 63–66
• The medical-application table also lists MC samples for voxel-wise models, distributed weight consolidation, probabilistic U-Net, and uncertainty-aware segmentation modules.The entries span brain segmentation, brain tumors, and cardiovascular and retinal OCT segmentation.Source: Malinin and Gales [365] · pp. 64–65
• DRLA is listed for ResNet classification of lung disease and diabetic retinopathy, while another ResNet entry concerns EGD classification.Source: Malinin and Gales [365] · p. 66
• Bayesian uncertainty estimates and ensembles are listed for DNN-based brain-cell classification.The table fragment also names Bayesian uncertainty for this task.Source: Malinin and Gales [365] · p. 66
• Gaussian sampling is listed with CycleGAN for prostate-lesion segmentation.Source: Malinin and Gales [365] · p. 66
• Li et al. list ResUNet for organ and skin-lesion segmentation, while Yu et al. list FusionBranch for glaucoma classification.The table also names test-time augmentation and test-time dropout among the methods.Source: Malinin and Gales [365] · p. 66