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A Survey on Uncertainty Quantification Methods for Deep Learning

Wenchong He, Zhe Jiang, Tingsong Xiao, Zelin Xu, Yukun Li

arXiv:2302.13425v7cs.LGstat.ML

TL;DR

DNNs can make incorrect yet overconfident predictions in high-stakes applications, creating a need to quantify predictive uncertainty. This survey organizes UQ methods by uncertainty source, compares their strengths and weaknesses, and connects them to applications and future research directions.

  • Problem

    Existing UQ surveys commonly organize methods by neural-network architecture or Bayesian formulation while overlooking the uncertainty sources each method addresses.

  • Method

    The survey develops a taxonomy of DNN UQ methods covering model uncertainty, data uncertainty, and their combination, while comparing methods and discussing applications.

  • Results

    The survey synthesizes the strengths and weaknesses of UQ approaches across uncertainty sources and identifies applications and future directions including LLMs and scientific simulations.

  • Takeaways & Limitations

    The source-based taxonomy connects UQ method selection with the uncertainty types and application challenges discussed in the survey.

  • Takeaways & Limitations

    Combining data and model uncertainty can require multiple forward passes, extra storage, and post hoc calibration because simple combinations lack a theoretical guarantee.

Abstract

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Deep neural networks (DNNs) have achieved tremendous success in computer vision, natural language processing, and scientific and engineering domains. However, DNNs can make unexpected, incorrect, yet overconfident predictions, leading to serious consequences in high-stakes applications such as autonomous driving, medical diagnosis, and disaster response. Uncertainty quantification (UQ) estimates the confidence of DNN predictions in addition to their accuracy. In recent years, many UQ methods have been developed for DNNs. It is valuable to systematically categorize these methods and compare their strengths and limitations. Existing surveys mostly categorize UQ methodologies by neural network architecture or Bayesian formulation, while overlooking the uncertainty sources each method addresses, making it difficult to select an appropriate approach in practice. To fill this gap, this paper presents a taxonomy of UQ methods for DNNs based on uncertainty sources (e.g., data versus model uncertainty). We summarize the advantages and disadvantages of each category, and illustrate how UQ can be applied to machine learning problems (e.g., active learning, out-of-distribution robustness, and deep reinforcement learning). We also identify future research directions, including UQ for large language models (LLMs), AI-driven scientific simulations, and deep neural networks with structured outputs.

1 INTRODUCTION

DNNs can produce incorrect yet overconfident predictions in high-stakes settings, motivating uncertainty quantification. This survey addresses limitations of existing architecture- and Bayesian-perspective surveys by organizing UQ methods around the uncertainty sources they capture.

  • Motivation: DNNs can make incorrect, overconfident predictions with serious consequences in autonomous driving, medical diagnosis, and disaster response.UQ assigns uncertainty scores to predictions so models can recognize cases requiring additional scrutiny.
  • Uncertainty sources: Data uncertainty reflects inherent randomness or conflicting labels, whereas model uncertainty reflects limited knowledge from training, architecture, optimization, or out-of-distribution data.Data uncertainty is generally irreducible with more training samples, while model uncertainty can be reduced with additional evidence.
  • Research gap: Existing UQ surveys commonly categorize methods by neural-network architecture or Bayesian modeling without connecting those categories to the uncertainty types addressed.This gap makes practical method selection more difficult.
  • Survey contribution: The survey introduces a taxonomy based on model uncertainty, data uncertainty, and methods addressing both.It compares category-specific advantages and disadvantages and connects UQ with major deep-learning topics and applications.

2 TYPES OF UNCERTAINTY SOURCE

The paper distinguishes model uncertainty from data uncertainty according to their sources in supervised learning. Model uncertainty concerns imperfect knowledge about models and distributions, while data uncertainty arises from inherent randomness, noise, or class confusion.

  • Model uncertainty: Model uncertainty includes uncertainty from model-family choice, parameter learning, and distribution differences between training and inference.Out-of-distribution samples and sparse regions of feature space can produce higher uncertainty.
  • Data uncertainty: Data uncertainty is irreducible variability caused by noisy measurements, natural randomness, or conflicting class evidence.It can be represented through conditional class probabilities, entropy, or input-dependent noise models.
  • Data uncertainty representation: Greater overlap between class clusters corresponds to higher data uncertainty, whereas sharper class boundaries produce lower uncertainty.In the illustrated classification example, entropy is highest near confused class boundaries.
  • Data uncertainty representation: Heteroscedastic noise models allow observation-noise variance to vary across inputs, unlike homoscedastic models with constant variance.This is useful when different samples have different noise levels.
  • Summary: The predominant uncertainty source depends on the application, because data properties and model or distribution shortcomings contribute differently across settings.The paper summarizes these distinctions as separate sources rather than treating uncertainty as a single quantity.

3 APPLICATION DOMAINS

The survey frames UQ as important across medical diagnosis, geoscience, transportation, and natural language processing. These domains combine high-impact predictions with uncertainty from both data and models, while LLM hallucinations create a prominent NLP challenge.

  • Cross-domain perspective: Across applications, the survey discusses motivation, uncertainty sources, and challenges associated with developing uncertainty-aware models.The application coverage includes medical diagnosis, geoscience, transportation, and NLP.
  • Medical diagnosis: Medical diagnosis requires uncertainty estimates because incorrect overconfident predictions can have serious consequences, with both data and model uncertainty present.Sources include noisy measurements, ambiguous labels, and model-related uncertainty.
  • Geoscience: Geoscience uses growing spatiotemporal data and deep learning for Earth-system tasks such as river-flow, temperature, and hurricane-related prediction.The application discussion motivates UQ by the complexity of these scientific prediction settings.
  • Transportation: Transportation applies deep learning to sensor and video data for traffic monitoring, pattern analysis, flow prediction, and decision-making.Temporal graph neural networks and physical principles are among the approaches discussed.
  • Natural Language Processing: NLP models, including LLMs, can hallucinate misleading or fabricated content, making uncertainty quantification important for trustworthiness.NLP uncertainty can arise from both data and model sources, including ambiguity and noise.

4 A TAXONOMY OF UQ METHODOLOGIES FOR DNNS

The survey’s taxonomy groups DNN UQ methods by whether they address model uncertainty, data uncertainty, or both. It also compares the underlying intuitions, strengths, and weaknesses of methods within these categories.

  • Taxonomy: The taxonomy organizes existing DNN UQ literature into model uncertainty, data uncertainty, and combined-uncertainty approaches.This source-based organization is presented as the survey’s central classification framework.
  • Comparison criteria: Methods are compared by their underlying intuitions and by the advantages and disadvantages associated with the uncertainty sources they address.The taxonomy is intended to support systematic comparison across method categories.

4.1 Model Uncertainty

Model uncertainty methods address uncertainty arising from parameter estimation, model choices, and distribution shifts. Bayesian, ensemble, and sample-distribution approaches differ in what they capture and in their computational and modeling limitations.

  • 4.1 Model Uncertainty: Model uncertainty methods include Bayesian neural networks, ensemble models, and sample distribution-based models, each targeting different uncertainty sources.The survey organizes these methods by the source of uncertainty they capture rather than by architecture alone.
  • 4.1.1 Bayesian Neural Networks: Bayesian neural networks model a posterior over neural-network parameters, but most existing methods primarily capture parameter uncertainty rather than total uncertainty.Total uncertainty would require modeling both the predictive distribution and the parameter posterior; existing work often treats predictions as deterministic.
  • 4.1.1 Bayesian Neural Networks: Variational inference and Laplace approximation make posterior inference tractable, but approximation quality, covariance representation, and computational cost remain limitations.Mean-field variational inference restricts posterior correlations, while Laplace approximation is local and large Hessian inversions are infeasible for modern DNNs.
  • 4.1.1 Bayesian Neural Networks: MCMC can converge to the exact posterior after sufficient iterations, yet high-dimensional DNN parameter spaces cause slow exploration, uncertain convergence time, and excessive resource use.The survey highlights random-walk behavior and high rejection rates as practical obstacles.
  • 4.1.2 Ensemble Models: Ensembles quantify model uncertainty through prediction variability across bootstrapped datasets, architectures, initializations, or other model variations.They can represent uncertainty from limited data, model misspecification, and hyperparameter choices, but require high computational cost.
  • 4.1.3 Sample Distribution-related Methods: Sample distribution-based methods capture uncertainty from distribution shifts or sparse samples, but learning distance-aware feature spaces and imposing neural-network constraints are difficult.Gaussian-process inference gives lower uncertainty in dense training regions and higher uncertainty near sparse or boundary regions.

4.2 Data Uncertainty

Data uncertainty is modeled through predictive distributions or intervals that capture inherent noise and variation in targets. The survey contrasts discriminative and generative approaches, highlighting trade-offs between flexibility, structured-output modeling, computational cost, and distributional assumptions.

  • Data uncertainty is modeled as p(y|x, θ), with methods divided into deep discriminative and deep generative models.Discriminative models directly predict distributions or intervals, while generative models represent output distributions through learned samples.
  • Deep discriminative model: Parametric discriminative models predict distribution parameters, but softmax probabilities can remain overconfident despite temperature scaling.Temperature scaling adds a calibration parameter, yet the resulting uncertainty may still be overconfident because the approach lacks constraints.
  • Deep discriminative model: Heteroscedastic regression predicts sample-dependent means and variances, while mixture-density models represent multiple prediction modes at higher computational cost.Choosing an appropriate parameterized distribution remains essential and can be difficult when prior information is unavailable.
  • Deep discriminative model: Non-parametric prediction intervals bound regression targets at a prescribed confidence level without assuming an explicit output distribution.Interval-learning objectives can minimize width while enforcing coverage through prediction-interval losses or constrained optimization.
  • Deep generative model: Generative models better capture structured output uncertainty, but repeated sampling or iterative generation increases computational demands.VAE frameworks model joint structure, GANs avoid explicit distribution formats but are harder to train, and diffusion models provide expressive distributions while suffering slow training and inference.

4.3 Model and data uncertainty

Methods that jointly quantify data and model uncertainty combine complementary estimators or use evidential deep learning to avoid repeated inference. These approaches trade computational cost and theoretical guarantees against efficiency and distributional assumptions.

  • Combination approaches: MC dropout combines Gaussian predictive distributions with sampled weights, using average predicted variance for data uncertainty and disagreement across passes for model uncertainty.The combined predictive uncertainty is approximated through the law of total variance.
  • Combination approaches: Deep ensembles represent model uncertainty through disagreement among models while modifying each output into a parameterized distribution to represent data uncertainty.Models may differ in parameters or architecture choices.
  • Combination approaches: Ensemble prediction intervals can reflect both uncertainty types, but independently aggregating lower and upper bounds lacks theoretical justification.A split-normal aggregation method is proposed to address this limitation.
  • Limitations: Joint uncertainty methods can require multiple forward passes and extra storage, while simple combinations may lack theoretical guarantees and need post hoc calibration.Efficiency and calibration remain central limitations of these approaches.
  • Evidential deep learning: Evidential deep learning uses one deterministic network to predict class-wise evidence and capture both uncertainty types without multiple forward passes.The evidence parameterizes a Dirichlet distribution over categorical probabilities.
  • Summary: Combining data- and model-uncertainty methods is simple but computationally expensive, whereas evidential deep learning and conformal prediction are more efficient but require stronger distributional assumptions.The survey summarizes these trade-offs in Table 4.

4.4 Evaluation Metrics

UQ evaluation uses calibration, ranking, proper scoring, and multiclass calibration metrics to assess whether uncertainty corresponds to prediction correctness. The survey describes both binned and smoothing-based calibration measures.

  • Calibration metrics: Expected calibration error compares the fraction of correct predictions with mean predicted confidence across uncertainty bins.A model is calibrated when correctness probability equals the uncertainty measure.
  • Calibration metrics: Calibration is formally achieved when P(Y = ŷ) equals the uncertainty measure p̂ for a prediction and its ground truth.This relation underlies the calibration-error formulation.
  • Other metrics: The survey also covers smECE, AUROC, Brier score, and ACE for smoothing calibration, ranking uncertainty, measuring probability error, and multiclass calibration.smECE avoids binning with an RBF kernel, while AUROC evaluates uncertainty-based ranking of correctness.

5 UNCERTAINTY ESTIMATION IN VARIOUS MACHINE LEARNING PROBLEMS

UQ supports out-of-distribution detection, active learning, and deep reinforcement learning by matching uncertainty sources to each problem. Model uncertainty is especially useful for detecting distribution shift and selecting informative labels, while both uncertainty types matter for reinforcement learning.

  • Active learning: Active learning prioritizes samples with high model uncertainty because additional data can reduce it, whereas high data uncertainty is generally irreducible.BNNs, deep ensembles, and MC dropout can quantify model uncertainty for selection.
  • Deep reinforcement learning: Deep reinforcement learning involves both data and model uncertainty because environment randomness affects transitions and rewards while limited training states create model uncertainty.Data uncertainty concerns intrinsic interaction randomness, whereas model uncertainty concerns insufficiently learned states.
  • Out-of-distribution detection: UQ methods address OOD detection by treating domain-shift uncertainty as model uncertainty, with BNNs, ensembles, and distance-based methods used to detect unfamiliar samples.OOD samples can cause unstable predictions and performance drops when they differ from training data.
  • Deep reinforcement learning: The survey recommends combined uncertainty approaches for reinforcement learning because model uncertainty supports exploration while data uncertainty characterizes environmental randomness.This reflects the exploitation–exploration trade-off in policy learning.

6 FUTURE DIRECTION

The survey identifies future UQ directions spanning LLMs, scientific simulation, explainability, and structured outputs, with emphasis on semantic calibration, evaluation, control, physical constraints, and structured dependencies.

  • 6.1 UQ for Large Language Models: Future LLM research should develop semantic-level and hidden-state uncertainty measures, while improving calibration for long-form and multi-step generation.Token-level confidence can be poorly aligned with semantic correctness in these settings.
  • 6.1 UQ for Large Language Models: LLM UQ evaluation should use span-level, answer-level, and decision-oriented metrics tailored to open-ended outputs and hallucination detection.Evaluation should separate sampling-strategy effects from uncertainty-signal quality.
  • 6.1 UQ for Large Language Models: LLM agents could use uncertainty as a control signal for tool calling, retrieval, escalation, planning, abstention, clarification, and verification.Propagating uncertainty across retrieval, reasoning, and action remains an open problem.
  • 6.2 UQ for deep learning in scientific simulations: Scientific-simulation UQ should decompose physical and modeling uncertainties, handle rare sensitive extreme events, incorporate physical knowledge, and improve computational efficiency.The survey highlights model misspecification, stochasticity, incomplete physical knowledge, initial and boundary conditions, and external forcings.
  • 6.3 Combine UQ with DNN explainability: Combining UQ with explainability could filter untrustworthy explanations and identify uncertain input features or model layers.The survey notes that few techniques explain why predictions are uncertain.
  • 6.4 UQ for DNNs with structured outputs: Structured-output UQ must account for correlations and domain constraints in inverse imaging, non-independent spatiotemporal data, and graph predictions with structural dependencies.The survey points to physics-informed models for imaging and graph-specific treatment of topology shifts and structured predictions.

7 CONCLUSION

The survey organizes DNN UQ methods by the uncertainty sources they address and examines their strengths, weaknesses, applications, challenges, and future directions.

  • 7 CONCLUSION: The survey categorizes DNN UQ methods into model uncertainty, data uncertainty, and their combination, while analyzing approach-specific strengths and weaknesses.It also summarizes uncertainty sources and application-specific challenges across machine-learning problems.
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