Uncertainty Quantification

Also known as UQ

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44 papers in the last four weeks, up 110% on the four weeks before. 0.4% of all new papers.

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Latest papers 431

Mar 16, 2026cs.LG

Generative Inverse Design with Abstention via Diagonal Flow Matching

Inverse design aims to find design parameters

x

achieving target performance

y^*

. Generative approaches learn bidirectional mappings between designs and labels, enabling diverse solution sampling. However, standard conditional flow matching (CFM), when adapted to inverse problems by pairing labels with design parameters, exhibits strong sensitivity to their arbitrary ordering and scaling, leading to unstable training. We introduce Diagonal Flow Matching (Diag--CFM), which resolves this through a zero-anchoring strategy that pairs design coordinates with noise and labels with zero, making the learning problem provably invariant to coordinate permutations. This yields substantially lower round-trip error than CFM and invertible neural network baselines across design dimensions up to

P{=}784

, including order-of-magnitude gains on several benchmarks. We develop two architecture-intrinsic uncertainty metrics, Zero-Deviation and Self-Consistency, that enable three practical capabilities: selecting the best candidate among multiple generations, abstaining from unreliable predictions, and detecting out-of-distribution targets; consistently outperforming ensemble and general-purpose alternatives across all tasks. We validate on airfoil, gas turbine combustor, scalable analytical benchmarks, a photonics inverse-design task, and an image-statistics benchmark.

Miguel de Campos, Werner Krebs, Hanno Gottschalk

Institute of Mathematics, TU Berlin · Siemens Energy

Flow Matching Uncertainty Quantification Flow-Based Generative Modeling Inverse Design

Mar 10, 2026cs.CV

Evidential Perfusion Physics-Informed Neural Networks with Residual Uncertainty Quantification

Physics-informed neural networks (PINNs) have shown promise in addressing the ill-posed deconvolution problem in computed tomography perfusion (CTP) imaging for acute ischemic stroke assessment. However, existing PINN-based approaches remain deterministic and do not quantify uncertainty associated with violations of physics constraints, limiting reliability assessment. We propose Evidential Perfusion Physics-Informed Neural Networks (EPPINN), a framework that integrates evidential deep learning with physics-informed modeling to enable uncertainty-aware perfusion parameter estimation. EPPINN models arterial input, tissue concentration, and perfusion parameters using coordinate-based networks, and places a Normal--Inverse--Gamma distribution over the physics residual to characterize voxel-wise aleatoric and epistemic uncertainty in physics consistency without requiring Bayesian sampling or ensemble inference. The framework further incorporates physiologically constrained parameterization and stabilization strategies to promote robust per-case optimization. We evaluate EPPINN on digital phantom data, the ISLES 2018 benchmark, and a clinical cohort. On the evaluated datasets, EPPINN achieves lower normalized mean absolute error than classical deconvolution and PINN baselines, particularly under sparse temporal sampling and low signal-to-noise conditions, while providing conservative uncertainty estimates with high empirical coverage. On clinical data, EPPINN attains the highest voxel-level and case-level infarct-core detection sensitivity. These results suggest that evidential physics-informed learning can improve both accuracy and reliability of CTP analysis for time-critical stroke assessment. Source code is available at https://github.com/jhlee0619/EPPINN.

Junhyeok Lee, Minseo Choi, Han Jang +5

Cancer Biology, Seoul National University College of Medicine, Seoul, Korea · Radiology, Seoul National University Hospital, Seoul, Korea · Bioengineering, Seoul National University, Seoul, Korea +3

Evidential Deep Learning Uncertainty Quantification Medical Image Analysis Physics-Informed ML Computed Tomography

Mar 9, 2026cs.LG

Data-Driven Priors for Uncertainty-Aware Risk Prediction of Clinical Deterioration using Multimodal Data

Safe predictions are a crucial requirement for integrating predictive models into clinical decision support systems. One approach to improving trustworthiness is to enable models to express uncertainty about individual predictions. However, current machine learning models frequently lack reliable uncertainty estimation, hindering real-world deployment. This limitation is particularly evident in multimodal settings, where models must effectively integrate heterogeneous information. In this work, we propose MedCertAIn, a predictive uncertainty framework that leverages multimodal clinical data to improve model performance and reliability for in-hospital mortality risk prediction as an indicator of patient deterioration. We design data-driven priors over neural network parameters using a hybrid strategy that considers cross-modal similarity in self-supervised latent representations and modality-specific data corruptions. We train and evaluate the models with such priors using clinical time-series and chest X-ray images from the publicly available datasets MIMIC-IV and MIMIC-CXR. Our results show that MedCertAIn achieves competitive predictive performance and substantial gains in selective prediction compared with the evaluated deterministic and stochastic baselines. These findings highlight the promise of data-driven priors in advancing robust, uncertainty-aware AI tools for high-stakes clinical applications. Our implementation is publicly available at: https://github.com/jlaitue/medcertain.

L. Julián Lechuga López, Tim G. J. Rudner, Farah E. Shamout

Tandon School of Engineering, New York University · New York University Abu Dhabi · University of Toronto

Bayesian Neural Networks Uncertainty Quantification Selective Prediction Mortality Prediction Clinical Risk Prediction Multimodal Learning

Feb 27, 2026stat.ME

VaSST: Variational Inference for Symbolic Regression using Soft Symbolic Trees

Symbolic regression (SR) has gained recent traction in AI-driven scientific discovery for learning closed-form physical laws. Yet existing methods are dominated by heuristic search or data-intensive approaches that often assume low-noise regimes and lack principled uncertainty quantification, while fully probabilistic SR formulations remain scarce. We introduce a scalable probabilistic framework for SR, VaSST, based on variational inference. VaSST uses soft symbolic trees, a continuous relaxation of symbolic expression trees in which discrete operator and feature assignments are replaced by probability distributions over allowable components. This transforms combinatorial symbolic search through an astronomically large expression space into efficient gradient-based optimization while preserving a coherent probabilistic interpretation. The learned soft representations induce posterior distributions over symbolic structures, enabling uncertainty quantification across plausible symbolic forms through posterior-aware symbolic model selection. On simulated experiments and the Feynman Symbolic Regression Database, VaSST achieves strong structural recovery and predictive accuracy compared to state-of-the-art competing SR methods.

Somjit Roy, Pritam Dey, Bani K. Mallick

Department of Statistics, Texas A&M University, College Station, Texas, USA

Uncertainty Quantification Symbolic Regression

Feb 4, 2026cs.CL

Semantic Self-Distillation for Language Model Uncertainty

Large language models present challenges for principled uncertainty quantification, in part due to their complexity and the diversity of their outputs. Semantic dispersion, or the variance in the meaning of sampled answers, has been proposed as a useful proxy for model uncertainty, but the associated computational cost prohibits its use in latency-critical applications. We show that sampled semantic distributions can be distilled into lightweight student models which estimate a prompt-conditioned density before the language model generates an answer token. The student model predicts a semantic distribution over possible answers; the entropy of this distribution provides a prompt-level uncertainty signal, and the probability density allows answer-level reliability evaluation. Across experiments on TriviaQA and MMLU, we find our student models perform competitively relative to the teacher's sampled semantic dispersion on a hallucination prediction task, whilst offering additional uncertainty primitives for out-of-domain detection and multiple-choice answer selection. We term this technique Semantic Self-Distillation (SSD), which can serve as a general framework for distilling predictive uncertainty in complex output spaces beyond language.

Edward Phillips, Sean Wu, Fredrik K. Gustafsson +2

Department of Engineering Science, University of Oxford · Oxford Suzhou Centre for Advanced Research, University of Oxford, Suzhou

Uncertainty Quantification Confidence Estimation in Language Models LLM Reliability Language Model Distillation LLM Uncertainty Estimation

Feb 3, 2026stat.ML

Multiparameter Uncertainty Mapping in Quantitative Molecular MRI using a Physics-Structured Variational Autoencoder (PS-VAE)

Quantitative imaging methods, such as magnetic resonance fingerprinting (MRF), aim to extract interpretable pathology biomarkers by estimating biophysical tissue parameters from signal evolutions. However, the pattern-matching algorithms or neural networks used in such inverse problems often lack principled uncertainty quantification, which limits the trustworthiness and transparency, required for clinical acceptance. Here, we describe a physics-structured variational autoencoder (PS-VAE) designed for rapid extraction of voxelwise multi-parameter posterior distributions. Our approach integrates a differentiable spin physics simulator with self-supervised learning, and provides a full covariance that captures the inter-parameter correlations of the latent biophysical space. The method was validated in a multi-proton pool chemical exchange saturation transfer (CEST) and semisolid magnetization transfer (MT) molecular MRF study, across in-vitro phantoms, tumor-bearing mice, healthy human volunteers, and a subject with glioblastoma. The resulting multi-parametric posteriors are in good agreement with those calculated using a brute-force Bayesian analysis, while providing an orders-of-magnitude acceleration in whole brain quantification. In addition, we demonstrate how monitoring the multi-parameter posterior dynamics across progressively acquired signals provides practical insights for protocol optimization and may facilitate real-time adaptive acquisition.

Alex Finkelstein, Ron Moneta, Or Zohar +4

School of Biomedical Engineering, Tel-Aviv University, Tel-Aviv, Israel · Sagol School of Neuroscience, Tel-Aviv University, Tel-Aviv, Israel · School of Biochemistry, Neurobiology, and Biophysics, Tel-Aviv University, Tel-Aviv, Israel +1

Magnetic Resonance Imaging Uncertainty Quantification Physics-Informed Generative Modeling Variational Autoencoders Neural Posterior Estimation

Feb 3, 2026cs.LG

Variational Sparse Paired Autoencoders (vsPAIR) for Inverse Problems and Uncertainty Quantification

Inverse problems are fundamental to many scientific and engineering disciplines; they arise when one seeks to reconstruct hidden, underlying quantities from noisy measurements. Many applications demand not just point estimates but interpretable uncertainty. Providing fast inference alongside uncertainty estimates remains challenging yet desirable in numerous applications. We propose the Variational Sparse Paired Autoencoder (vsPAIR) to address this challenge. The architecture pairs a standard VAE encoding observations with a sparse VAE encoding quantities of interest (QoI), connected through a learned latent mapping. The variational structure enables uncertainty estimation, the paired architecture encourages interpretability by anchoring QoI representations to clean data, and sparse encodings provide structure by concentrating information into identifiable factors rather than diffusing across all dimensions. To validate the effectiveness of our proposed architecture, we conduct experiments on blind inpainting, computed tomography (CT), and initial-condition inference for the heat equation, demonstrating that vsPAIR is a capable inverse problem solver that can provide interpretable and structured uncertainty estimates.

Jack Michael Solomon, Rishi Leburu, Matthias Chung

Emory University Department of Mathematics

Variational Autoencoders Uncertainty Quantification Autoencoders Sparse Autoencoders Inverse Problems

Jan 19, 2026eess.IV

Pixelwise Uncertainty Quantification of Accelerated MRI Reconstruction

Parallel imaging techniques reduce magnetic resonance imaging (MRI) scan time but image quality degrades as the acceleration factor increases. In clinical practice, conservative acceleration factors are chosen because no mechanism exists to automatically assess the diagnostic quality of undersampled reconstructions. This work introduces a general framework for pixel-wise uncertainty quantification in parallel MRI reconstructions, enabling automatic identification of unreliable regions without access to any ground-truth reference image. Our method integrates conformal quantile regression with image reconstruction methods to estimate statistically rigorous pixel-wise uncertainty intervals. We trained and evaluated our model on Cartesian undersampled brain and knee data obtained from the fastMRI dataset using acceleration factors ranging from 2 to 10. An end-to-end Variational Network was used for image reconstruction. Quantitative experiments demonstrate strong agreement between predicted uncertainty maps and true reconstruction error. Using our method, the corresponding Pearson correlation coefficient was higher than 90% at acceleration levels at and above four-fold; whereas it dropped to less than 70% when the uncertainty was computed using a simpler a heuristic notion (magnitude of the residual). Qualitative examples further show the uncertainty maps based on quantile regression capture the magnitude and spatial distribution of reconstruction errors across acceleration factors, with regions of elevated uncertainty aligning with pathologies and artifacts. The proposed framework enables evaluation of reconstruction quality without access to fully-sampled ground-truth reference images. It represents a step toward adaptive MRI acquisition protocols that may be able to dynamically balance scan time and diagnostic reliability.

Ilias I. Giannakopoulos, Lokesh B Gautham Muthukumar, Yvonne W. Lui +1

Bernard and Irene Schwartz Center for Biomedical Imaging, Department of Radiology, NYU Grossman School of Medicine, 10016, New York, NY, United States of America · Courant Institute of Mathematical Sciences, NYU, 10012, New York, NY, United States of America · Center for Advanced Imaging Innovation and Research (CAI2R), Department of Radiology, NYU Grossman School of Medicine, 10016, New York, NY, United States of America

Uncertainty Quantification Accelerated MRI Reconstruction

Jan 12, 2026stat.ML

Neural Architectures for Amortized Bayesian Inference: Statistical Foundations and Empirical Assessments

Since the turn of the century, approximate Bayesian inference has steadily evolved as new computational techniques have been incorporated to handle increasingly complex, large-scale predictive problems. The recent success of deep neural networks and foundation models has now given rise to a new paradigm in statistical modeling, in which Bayesian inference can be amortized through large-scale learned predictors. In amortized inference, substantial computation is required at the beginning to train a neural network, but it can subsequently produce approximate posteriors or predictions at much lower computational cost across a wide range of tasks. While the typical Bayesian inference procedures are computationally expensive due to repeated likelihood calculations and Monte Carlo steps for each new dataset, amortized inference provides a much lower computational cost at deployment. Despite the growing popularity of amortized inference, its statistical interpretation and position within Bayesian inference remain poorly explored. In this paper, we present a statistical perspective on several major neural architectures, including feedforward networks, Deep Sets, and Transformers, and examine how they naturally support amortized Bayesian inference. We explore how these models perform structured approximation and also probabilistic reasoning in ways that yield controlled generalization error throughout a wide range of deployment scenarios, and how these properties can be harnessed for Bayesian computation. Via simulation studies, we evaluate the accuracy, robustness, and uncertainty quantification of amortized inference across varying sample sizes, varying noise distributional families, varying sparsity levels, and multimodality, highlighting its strengths and limitations.

Roy Shivam Ram Shreshtth, Arnab Hazra, Gourab Mukherjee

Department of Statistics and Data Science, Indian Institute of Technology Kanpur, Kanpur, India 208016 · Marshall School of Business, University of Southern California, Los Angeles, United States 90089-0809

Uncertainty Quantification Amortized Inference Bayesian Inference

Jan 7, 2026cs.LG

Disentangling Aleatoric and Epistemic Uncertainty in Physics-Informed Neural Networks. Application to Insulation Material Degradation Prognostics

Physics-Informed Neural Networks (PINNs) provide a framework for integrating physical laws with data. However, their application to Prognostics and Health Management (PHM) remains constrained by the limited uncertainty quantification (UQ) capabilities. Most existing PINN-based prognostics approaches are deterministic or account only for epistemic uncertainty, limiting their suitability for risk-aware decision-making. This work introduces a heteroscedastic Bayesian Physics-Informed Neural Network (B-PINN) framework that jointly models epistemic and aleatoric uncertainty, yielding full predictive posteriors for spatiotemporal insulation material ageing estimation. The approach integrates Bayesian Neural Networks (BNNs) with physics-based residual enforcement and prior distributions, enabling probabilistic inference within a physics-informed learning architecture. The framework is evaluated on transformer insulation ageing application, validated with a finite-element thermal model and field measurements from a solar power plant, and benchmarked against deterministic PINNs, dropout-based PINNs (d-PINNs), and alternative B-PINN variants. Results show that the proposed B-PINN provides improved predictive accuracy and better-calibrated uncertainty estimates than competing approaches. A systematic sensitivity study further analyzes the impact of boundary-condition, initial-condition, and residual sampling strategies on accuracy, calibration, and generalization, and the influence of measurement noise on aleatoric uncertainty. Overall, the findings highlight the capability of Bayesian physics-informed learning to support uncertainty-aware prognostics and informed decision-making in transformer asset management by tracking aleatoric and epistemic sources of uncertainty.

Ibai Ramirez, Jokin Alcibar, Joel Pino +2

Department of Electronics and Computer Science Mondragon University Arrasate-Mondragon, 20500, Spain · CARTIF Technology Centre Valladolid, 47151, Spain · Department of Physical Chemistry University of the Basque Country (EHU)[cs.LG] Basque Centre for Applied Mathematics (BCAM) Leioa, 20018, Spain +1

Bayesian Neural Networks Uncertainty Quantification Predictive Maintenance Physics-Informed ML Aleatoric Uncertainty Uncertainty Calibration

Dec 30, 2025hep-ph

Quantitative Understanding of PDF Fits and their Uncertainties

Parton Distribution Functions (PDFs) play a central role in describing experimental data at colliders and provide insight into the structure of nucleons. As the LHC enters an era of high-precision measurements, a robust PDF determination with a reliable uncertainty quantification has become mandatory in order to match the experimental precision. The NNPDF collaboration has pioneered the use of Machine Learning (ML) techniques for PDF determinations, using Neural Networks (NNs) to parametrise the unknown PDFs in a flexible and unbiased way. The NNs are then trained on experimental data by means of stochastic gradient descent algorithms. The statistical robustness of the results is validated by extensive closure tests using synthetic data. In this work, we develop a theoretical framework based on the Neural Tangent Kernel (NTK) to analyse the training dynamics of neural networks. This approach allows us to derive, under precise assumptions, an analytical description of the neural network evolution during training, enabling a quantitative understanding of the training process. Having an analytical handle on the training dynamics allows us to clarify the role of the NN architecture and the impact of the experimental data in a transparent way. Similarly, we are able to describe the evolution of the covariance of the NN output during training, providing a quantitative description of how uncertainties are propagated from the data to the fitted function. While our results are not a substitute for PDF fitting, they do provide a powerful diagnostic tool to assess the robustness of current fitting methodologies. Beyond its relevance for particle physics phenomenology, our analysis of PDF determinations provides a testbed to apply theoretical ideas about the learning process developed in the ML community.

Amedeo Chiefa, Luigi Del Debbio, Richard Kenway

The Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom

Uncertainty Quantification High-Energy Physics Neural Network Training Dynamics

Dec 16, 2025cs.CV

Improving Semantic Uncertainty Quantification in LVLMs with Semantic Gaussian Processes

Large Vision-Language Models (LVLMs) often produce plausible but unreliable outputs, making robust uncertainty estimation essential. Recent work on semantic uncertainty estimates relies on external models to cluster multiple sampled responses and measure their semantic consistency. However, these clustering methods are often fragile, highly sensitive to minor phrasing variations, and can incorrectly group or separate semantically similar answers, leading to unreliable uncertainty estimates. We propose Semantic Gaussian Process Uncertainty (SGPU), a Bayesian framework that quantifies semantic uncertainty by analyzing the geometric structure of answer embeddings, avoiding brittle clustering. SGPU maps generated answers into a dense semantic space, computes the Gram matrix of their embeddings, and summarizes their semantic configuration via the eigenspectrum. This spectral representation is then fed into a Gaussian Process Classifier that learns to map patterns of semantic consistency to predictive uncertainty, and that can be applied in both black-box and white-box settings. Across six LLMs and LVLMs on eight datasets spanning VQA, image classification, and textual QA, SGPU consistently achieves state-of-the-art calibration (ECE) and discriminative (AUROC, AUARC) performance. We further show that SGPU transfers across models and modalities, indicating that its spectral representation captures general patterns of semantic uncertainty.

Joseph Hoche, Andrei Bursuc, David Brellmann +4

AMIAD, Pôle Recherche, Palaiseau · Safran Tech · University of Liège +3

Vision-Language Models Uncertainty Quantification Large Vision-Language Models Uncertainty Calibration

Nov 19, 2025stat.ML

Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings

Conformal prediction (CP) constructs uncertainty sets for model outputs with finite-sample coverage guarantees. Yet ranking scores is straightforward only when they are scalar-valued, limiting CP to real-valued scores or ad-hoc one-dimensional reductions. Vector-valued scores arise naturally in multi-output regression and model aggregation, where each predictor in an ensemble provides its own score. Optimal transport (OT) defines vector ranks and center-outward multivariate quantile regions, though generally with asymptotic coverage guarantees. Applying a fixed transport map learned from calibration data to a new point introduces an uncontrolled approximation error. We restore finite-sample, distribution-free coverage by conformalizing vector-valued OT quantile regions. Each candidate's rank is defined by transporting the calibration scores augmented with that candidate's score, preserving the symmetry needed for validity. This appears to require a continuum of OT problems. However, we prove that the optimal assignment is piecewise constant across a fixed polyhedral partition of score space. This lets us characterize the entire prediction set in

O(n^3)

time, matching the cost of a single assignment solve. It also addresses a limitation of prediction sets: they indicate which outcomes are plausible, but not their relative likelihood. In one dimension, conformal predictive distributions (CPDs) fill this gap by producing a predictive distribution with finite-sample calibration. Extending CPDs beyond one dimension remained an open problem. We construct, to our knowledge, the first multivariate CPDs with finite-sample calibration: a center-outward predictive distribution whose derived uncertainty regions have conformal coverage. We present both conservative and exact randomized versions; the latter generalizes the classical Dempster-Hill procedure.

Eugene Ndiaye

Apple

Uncertainty Quantification Conformal Prediction Optimal Transport

Nov 7, 2025cs.LG

QiVC-Net: Quantum-Inspired Variational Convolutional Network, with Application to Biosignal Classification

In this paper, a learning framework is introduced which incorporates principles of probabilistic inference, variational optimization, and geometry-preserving operations inspired by quantum transformations. The central innovation of this quantum-inspired variational convolution (QiVC) lies in its quantum-inspired rotated ensemble (QiRE) mechanism. QiRE performs differentiable low-dimensional subspace rotations of convolutional weights. By drawing a mathematical analogy from unitary evolution, this approach enables structured uncertainty modeling that respects the intrinsic geometry of the parameter space. To demonstrate its practical potential, the concept is instantiated in a QiVC-based convolutional network (QiVC-Net) and evaluated in the context of biosignal classification, focusing on phonocardiogram (PCG) recordings. The proposed QiVC-Net integrates an architecture in which the QiVC layer does not introduce additional parameters, instead performing an ensemble rotation of the convolutional weights through a structured mechanism ensuring robustness without added highly computational burden. Experiments on two benchmark datasets, PhysioNet CinC 2016 and PhysioNet CirCor DigiScope 2022, show that QiVC-Net achieves state-of-the-art performance, reaching accuracies of 97.84% and 97.89%, respectively. These findings highlight the versatility of the QiVC framework and its promise for advancing uncertainty-aware modeling in real-world biomedical signal analysis. The implementation of the QiVConv layer is available in GitHub for public use.

Amin Golnari, Jamileh Yousefi, Reza Moheimani +1

College of Engineering and Computer Science VinUniversity Hanoi, Vietnam · Shannon School of Business Cape Breton University Sydney, NS, Canada · Faculty of Computer Science Chemnitz University of Technology Chemnitz, Germany +2

Uncertainty Quantification Quantum-Inspired ML

Oct 29, 2025cs.LG

Uncertainty Quantification for Regression: A Unified Framework based on kernel scores

Regression tasks, notably in safety-critical domains, require reliable uncertainty quantification, yet the literature remains largely classification-focused. To address this, we introduce a family of measures for total, aleatoric, and epistemic uncertainty in multivariate regression based on strictly proper kernel scores. The framework provides a principled recipe for designing new uncertainty measures whose behavior, such as tail sensitivity or out-of-distribution responsiveness, is governed by the choice of the underlying kernel, while also encompassing existing measures under a joint analysis. We prove explicit correspondences between properties of the kernel and behavior of resulting uncertainty measures, yielding concrete design guidelines for practitioners. Extensive experiments across structured regression tasks, including spatial and functional domains, demonstrate effectiveness on downstream tasks such as out-of-distribution detection and active learning, and reveal that different kernel choices lead to distinct trade-offs, offering practitioners guidance for task-specific selection.

Christopher Bülte, Yusuf Sale, Gitta Kutyniok +1

LMU Munich · 2Munich Center for Machine Learning (MCML) · 3DLR-German Aerospace Center +2

Uncertainty Quantification Aleatoric Uncertainty Uncertainty Decomposition Epistemic Uncertainty

Oct 5, 2025quant-ph

Toward Uncertainty-Aware and Generalizable Neural Decoding for Quantum LDPC Codes

Quantum error correction (QEC) is essential for scalable quantum computing, yet decoding errors via conventional algorithms result in limited accuracy (i.e., suppression of logical errors) and high overheads, both of which can be alleviated by inference-based decoders. To date, such machine-learning (ML) decoders lack two key properties crucial for practical fault tolerance: reliable uncertainty quantification and robust generalization to previously unseen QEC codes. To address this gap, we propose a Quantum Bayesian graph Attention decoder \textbf{(QuBA)} that enables expressive error-pattern recognition alongside calibrated uncertainty estimates. Building on QuBA, we further develop a multi-phase training framework with enhanced cross-domain robustness enabling decoding beyond the training set called Sequential Aggregate Generalization under Uncertainty \textbf{(SAGU)}. Experiments on bivariate bicycle (BB) codes and their coprime variants demonstrate that (i) both QuBA and SAGU consistently outperform the classical baseline belief propagation (BP), achieving up to a \emph{two orders of magnitude} reduction in logical error rate (LER) under confident-decision bounds on the coprime BB code

[[154,6,16]]

; (ii) SAGU achieves decoding performance comparable to or even outperforming QuBA's domain-specific training approach.

Xiangjun Mi, Frank Mueller

North Carolina State University, Raleigh, NC 27695, USA

OOD Generalization Uncertainty Quantification Graph Neural Networks

Sep 17, 2025cs.CL

Geometric Uncertainty for Detecting and Correcting Hallucinations in LLMs

Large language models are known to hallucinate, generating linguistically plausible but incorrect answers to questions. Uncertainty quantification has been proposed as a strategy to detect such behaviour, but existing methods lack a unified framework to assess reliability at both the prompt and answer level. We introduce a geometric framework which quantifies language model uncertainty at both levels by explicitly modelling a prompt-conditioned semantic distribution in answer embedding space. Our approach is black-box and sampling-based; we generate multiple answers per prompt, and use archetypal analysis to estimate a geometric support for the answer distribution. At the prompt level, we approximate the distribution entropy to quantify uncertainty; for each individual answer, we then use notions of atypicality to assess its reliability relative to the batch. We employ our framework to not only detect hallucinations but correct them, by selecting the batch example deemed most reliable. Experiments show that our framework performs comparably to or better than prior methods on short form question-answering datasets, and achieves superior results on medical datasets where hallucinations carry particularly critical risks. Beyond pure performance, we suggest the theoretical grounding of our work provides support for semantic distributions as useful objects of study for language model uncertainty.

Edward Phillips, Sean Wu, Soheila Molaei +3

Department of Engineering Science, University of Oxford · GlaxoSmithKline · Oxford Suzhou Centre for Advanced Research

Uncertainty Quantification LLM Hallucination Mitigation Hallucination in Language Models LLM Reliability LLM Hallucination Detection LLM Uncertainty Estimation

Aug 28, 2025cs.RO

Uncertainty-Aware Ankle Exoskeleton Control

Lower limb exoskeletons show promise to assist human movement, but their utility is limited by controllers designed for discrete, predefined actions in controlled environments, restricting their real-world applicability. We present an uncertainty-aware control framework that enables ankle exoskeletons to operate safely across diverse scenarios by automatically disengaging when encountering unfamiliar movements. Our approach uses an uncertainty estimator to classify movements as similar (in-distribution) or different (out-of-distribution) relative to actions in the training set. We evaluated three architectures (model ensembles, autoencoders, and generative adversarial networks) on an offline dataset and tested the strongest performing architecture (ensemble of gait phase estimators) online. The online test demonstrated the ability of our uncertainty estimator to turn assistance on and off as the user transitioned between in-distribution and out-of-distribution tasks (F1: 89.2). This new framework provides a path for exoskeletons to safely and autonomously support human movement in unstructured, everyday environments.

Fatima Mumtaza Tourk, Bishoy Galoaa, Sanat Shajan +3

Department of Mechanical Engineering, Northeastern University, Boston, MA 02115 USA

Uncertainty Quantification Ankle Exoskeleton Control Robot Locomotion Assistive Robotics OOD Detection

Aug 16, 2025cs.LG

DE-VAE: Revealing Uncertainty in Parametric and Inverse Projections with Variational Autoencoders using Differential Entropy

Recently, autoencoders (AEs) have gained interest for creating parametric and invertible projections of multidimensional data. Parametric projections make it possible to embed new, unseen samples without recalculating the entire projection, while invertible projections allow the synthesis of new data instances. However, existing methods perform poorly when dealing with out-of-distribution samples in either the data or embedding space. Thus, we propose DE-VAE, an uncertainty-aware variational AE using differential entropy (DE) to improve the learned parametric and invertible projections. Given a fixed projection, we train DE-VAE to learn a mapping into 2D space and an inverse mapping back to the original space. We conduct quantitative and qualitative evaluations on four well-known datasets, using UMAP and t-SNE as baseline projection methods. Our findings show that DE-VAE can create parametric and inverse projections with comparable accuracy to other current AE-based approaches while enabling the analysis of embedding uncertainty.

Frederik L. Dennig, Daniel A. Keim

University of Konstanz

Variational Autoencoders Uncertainty Quantification OOD Generalization Inverse Dimensionality Reduction Dimensionality Reduction

Jul 31, 2025stat.ME

A General Approach to Visualizing Uncertainty in Statistical Graphics

We present a general approach to visualizing uncertainty in static 2-D statistical graphics. If we treat a visualization as a function of its underlying quantities, uncertainty in those quantities induces a distribution over images. We show how to aggregate these images into a single visualization that represents the uncertainty. The approach can be viewed as a generalization of sample-based approaches that use overlay. Notably, standard representations, such as confidence intervals and bands, emerge with their usual coverage guarantees without being explicitly quantified or visualized. As a proof of concept, we implement our approach in the IID setting using resampling, provided as an open-source Python library. Because the approach operates directly on images, the user needs only to supply the data and the code for visualizing the quantities of interest without uncertainty. Through several examples, we show how both familiar and novel forms of uncertainty visualization can be created. The implementation is not only a practical validation of the underlying theory but also an immediately usable tool that can complement existing uncertainty-visualization libraries.

Bernarda Petek, David Nabergoj, Erik Štrumbelj

Scientific Visualization Data Visualization Uncertainty Quantification

Jul 11, 2025cs.LG

Last Layer Hamiltonian Monte Carlo

We explore the use of Hamiltonian Monte Carlo (HMC) sampling as a probabilistic last layer approach for deep neural networks (DNNs). While HMC is widely regarded as a gold standard for uncertainty estimation, the computational demands limit its application to large-scale datasets and large DNN architectures. Although the predictions from the sampled DNN parameters can be parallelized, the computational cost still scales linearly with the number of samples (similar to an ensemble). Last layer HMC (LL-HMC) reduces the required computations by restricting the HMC sampling to the final layer of a DNN, making it applicable to more data-intensive scenarios with limited computational resources. In this paper, we compare LL-HMC against five last layer probabilistic deep learning (LL-PDL) methods across three real-world video datasets for driver action and intention. We evaluate the in-distribution classification performance, calibration, and out-of-distribution (OOD) detection. Due to the stochastic nature of the probabilistic evaluations, we performed five grid searches for different random seeds to avoid being reliant on a single initialization for the hyperparameter configurations. The results show that LL-HMC achieves competitive in-distribution classification and OOD detection performance. Additional sampled last layer parameters do not improve the classification performance, but can improve the OOD detection. Multiple chains or starting positions did not yield consistent improvements.

Koen Vellenga, H. Joe Steinhauer, Göran Falkman +2

University of Skövde & Volvo Car Corporation, Sweden · University of Skövde, Sweden · Volvo Car Corporation, Sweden

Bayesian Neural Networks Markov Chain Monte Carlo Uncertainty Quantification OOD Detection

Jul 10, 2025stat.ML

CLEAR: Calibrated Learning for Epistemic and Aleatoric Risk

Accurate uncertainty quantification is critical for reliable predictive modeling. Existing methods typically address either aleatoric uncertainty due to measurement noise or epistemic uncertainty resulting from limited data, but not both in a balanced manner. We propose CLEAR, a calibration method with two distinct parameters,

γ_1

and

γ_2

, to combine the two uncertainty components and improve the conditional coverage of predictive intervals for regression tasks. CLEAR is compatible with any pair of aleatoric and epistemic estimators; we show how it can be used with (i) quantile regression for aleatoric uncertainty and (ii) ensembles drawn from the Predictability-Computability-Stability (PCS) framework for epistemic uncertainty. Across 17 diverse real-world datasets, CLEAR achieves an average improvement of 28.3% and 17.5% in the interval width compared to the two individually calibrated baselines while maintaining nominal coverage. Similar improvements are observed when applying CLEAR to Deep Ensembles (epistemic) and Simultaneous Quantile Regression (aleatoric). The benefits are especially evident in scenarios dominated by high aleatoric or epistemic uncertainty. Project page: https://unco3892.github.io/clear/

Ilia Azizi, Juraj Bodik, Jakob Heiss +1

Department of Operations, HEC, University of Lausanne, Switzerland · BegooAI, Switzerland · Department of Statistics, University of California, Berkeley, USA +1

Uncertainty Quantification Aleatoric Uncertainty Uncertainty Calibration Regression Prediction Intervals Epistemic Uncertainty

Jun 23, 2025cs.CV

Latent Space Analysis for Interpretable Uncertainty in Melanoma Classification

Melanoma is a highly aggressive skin cancer, making early and accurate diagnosis critical. While deep learning excels in skin lesion classification, standard ``black-box" models struggle to explain diagnostic uncertainty, limiting clinical trust. This work introduces a hybrid framework combining a class-aware adversarial Variational Autoencoder and an XGBoost classifier, transcending simple binary classification by leveraging a generative latent space for interpretable decision support. Guided by adversarial training, the model learns the visual characteristics of skin lesions and projects them into a continuous latent space, ensuring that similar images are grouped closely together. Trained on this latent space, the XGBoost classifier achieves a robust AUC of 0.868, competing closely with state-of-the-art models. For borderline cases, the framework enables clinicians to leverage the latent topology through Content-Based Image Retrieval. This provides a dual benefit: it allows the clinician to visually compare an ambiguous lesion against biopsy-confirmed precedents and acts as an early warning sign since a borderline classification can indicate that a lesion shares features of both nevi and melanomas, potentially requiring close monitoring. Our approach translates algorithmic hesitation into transparent, evidence-based visual support, bridging the gap between predictive performance and clinical trust.

Ciro Listone, Aniello Murano

University of Naples Federico II, Naples, Italy.

Variational Autoencoders Skin Lesion Classification Uncertainty Quantification Medical Image Classification Explainable Medical Image Analysis Medical Image Representation Learning

May 26, 2025cs.LG

Representation Learning for Equivariant Inference with Guarantees

In many real-world applications of regression, conditional probability estimation, and uncertainty quantification, exploiting symmetries rooted in physics or geometry can dramatically improve generalization and sample efficiency. While geometric deep learning has made empirical advances by incorporating symmetry and geometry priors, less attention has been given to statistical learning guarantees. In this paper, we introduce an equivariant representation learning framework that simultaneously addresses regression, conditional probability estimation, and uncertainty quantification while providing first-of-its-kind non-asymptotic statistical learning guarantees. Grounded in operator and group representation theory, our framework approximates the spectral decomposition of the conditional expectation operator, building representations that are both equivariant and disentangled along independent symmetry quotient groups. Empirical evaluations on synthetic datasets and real-world robotics applications confirm the potential of our approach, matching or outperforming existing equivariant baselines in regression while providing well-calibrated uncertainty estimates.

Daniel Ordoñez-Apraez, Vladimir Kostić, Alek Fröhlich +3

CSML, Italian Institute of Technology, Genova, Italy · Universit`a di Genova, Genova, Italy · University of Novi Sad, Novi Sad, Serbia +2

Disentangled Representation Learning Representation Learning Uncertainty Quantification Conditional Distribution Estimation Equivariant Representation Learning Statistical Learning Theory

Apr 25, 2025cs.LG

An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression

Uncertainty quantification is crucial in machine learning, yet most (axiomatic) studies of uncertainty measures focus on classification, leaving a gap in regression settings with limited formal justification and evaluations. In this work, we provide a formal way of representing uncertainty in continuous space, using a general parametric formulation, allowing for tractable analysis and evaluation of uncertainty measures. Within this framework, we propose a set of axioms that enable rigorous assessment of total, aleatoric, and epistemic uncertainty measures. Together, this allows for a theoretical examination of uncertainty measures and their corresponding properties. As a specific example, we compare the widely used entropy- and variance-based measures with respect to established predictive models and analyze their limitations and challenges in uncertainty quantification. Our work provides a principled way to understand and develop uncertainty measures in supervised regression, offering theoretical insights and practical guidelines for reliable uncertainty assessment.

Christopher Bülte, Yusuf Sale, Timo Löhr +3

1LMU Munich · 2Munich Center for Machine Learning (MCML) · 3DLR-German Aerospace Center +2

Uncertainty Quantification Aleatoric Uncertainty Epistemic Uncertainty

Apr 2, 2025cs.LG

Bifidelity Parameter Estimation Using Conditional Diffusion Models

We present a bifidelity method for uncertainty quantification of parameter estimates in complex systems, leveraging generative models trained to sample the target conditional distribution. In the Bayesian inference setting, traditional parameter estimation methods rely on repeated simulations of potentially expensive forward models to determine the posterior distribution of the parameter values, which may result in computationally intractable workflows. Furthermore, methods such as Markov Chain Monte Carlo (MCMC) necessitate rerunning the entire algorithm for each new data observation, further increasing the computational burden. Hence, we propose a novel method for efficiently obtaining posterior distributions of parameter estimates for high-fidelity models given data observations of interest. The method first constructs a low-fidelity, conditional generative model capable of amortized Bayesian inference and hence rapid posterior density approximation over a wide-range of data observations. When higher accuracy is needed for a specific data observation, the method employs adaptive refinement of the density approximation. It uses outputs from the low-fidelity generative model to refine the parameter sampling space, ensuring efficient use of the computationally expensive high-fidelity solver. Subsequently, a high-fidelity, unconditional generative model is trained to achieve greater accuracy in the target posterior distribution. Both low- and high- fidelity generative models enable efficient sampling from the target posterior and do not require repeated simulation of the high-fidelity forward model. We demonstrate the effectiveness of the proposed method on several numerical examples, including cases with multi-modal densities, as well as an application in plasma physics for a runaway electron simulation model.

Caroline Tatsuoka, Minglei Yang, Dongbin Xiu +1

Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA · Fusion Energy Science Division, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA · Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA

Simulation-Based Inference Surrogate Modeling Uncertainty Quantification Bayesian Inverse Problems Multifidelity Modeling Conditional Diffusion Models

Mar 13, 2025stat.ML

Explainable Bayesian deep learning through input-skip Latent Binary Bayesian Neural Networks

Modeling natural phenomena with artificial neural networks (ANNs) often provides highly accurate predictions. However, ANNs often suffer from over-parameterization, complicating interpretation and raising uncertainty issues. Bayesian neural networks (BNNs) address the latter by representing weights as probability distributions, allowing for predictive uncertainty evaluation. Latent binary Bayesian neural networks (LBBNNs) further handle structural uncertainty and sparsify models by removing redundant weights. This article advances LBBNNs by enabling covariates to skip to any succeeding layer or be excluded, simplifying networks and clarifying input impacts on predictions. This further allows us to learn simpler structures (e.g., linear or even constant intercept only models) when appropriate. Furthermore, the input-skip LBBNN (ISLaB) approach reduces network density significantly compared to standard LBBNNs, achieving over 99% reduction for small networks and over 99.9% for larger ones, while still maintaining high predictive accuracy and uncertainty quantification. For example, on MNIST, we reached 97% accuracy and great calibration with just 935 weights, reaching state-of-the-art for compression of neural networks. Furthermore, the proposed method accurately identifies the true covariates and adjusts for system non-linearity. The main contribution is the introduction of active paths, enhancing directly designed global and local explanations within the LBBNN framework. The latter are exact with theoretical guarantees and do not require post hoc external tools.

Eirik Høyheim, Lars Skaaret-Lund, Solve Sæbø +1

Bioinformatics and applied statistics (BIAS), Norwegian University of Life Sciences, Ås, Norway

Bayesian Neural Networks Uncertainty Quantification Neural Network Interpretability Neural Network Compression Neural Network Pruning

Feb 1, 2025cs.LG

Active Learning with Bayesian Multi-Fidelity Laplace Neural Operators for Oscillatory Parametric PDEs

Surrogate models of parametric dynamical systems are essential for many-query and real-time predictions in engineering applications such as design optimization and digital twins. However, generating high-fidelity (HF) training data over a broad range of parameters and operating conditions remains computationally expensive. To address this challenge, we propose a Bayesian multi-fidelity Laplace neural operator (MF-LNO) for uncertainty-aware active learning of oscillatory parametric PDEs. Specifically, the proposed Bayesian MF-LNO iteratively calibrates the discrepancy between low- and high-fidelity data, where predictive uncertainty guides the adaptive acquisition of informative HF trajectories. Such predictive uncertainty is quantified via replica-exchange stochastic gradient Langevin dynamics (reSGLD), whose broad posterior exploration enables uncertainty to serve as an error indicator for adaptive HF sample acquisition. Numerical experiments on the Lorenz system, Duffing oscillator, and beam dynamics demonstrate that uncertainty-guided HF sample acquisition consistently outperforms random sampling, while the proposed Bayesian MF-LNO achieves higher prediction accuracy than MF-DeepONet with predictive uncertainty quantification. These results demonstrate that Bayesian multi-fidelity LNOs, combined with uncertainty-guided active learning, provide a data-efficient framework for operator learning in engineering dynamical systems.

Bongseok Kim, Haoyang Zheng, Michael Penwarden +1

School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, 47907, IN, USA. · Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, 47907, IN, USA.

Dynamical Systems Surrogate Modeling Uncertainty Quantification Active Learning PDE Operator Learning Neural Operators

Dec 10, 2024cs.CL

Label-Confidence-Aware Uncertainty Estimation in Natural Language Generation

Large Language Models (LLMs) demonstrate remarkable capabilities in generative tasks but pose potential risks due to their tendency to generate hallucinatory responses. Therefore, Uncertainty Quantification (UQ), which aims to distinguish the validity of answers, is crucial for ensuring the safety and robustness of AI systems. However, existing methods primarily rely on measuring the entropy of multiple stochastic samples to represent uncertainty, often overlooking the specific uncertainty information associated with the candidate answer under evaluation. This oversight can lead to biased classification outcomes. In this paper, we investigate the discrepancy between global entropy from multiple samples and local confidence of candidate answer, and propose a Label-Confidence-Aware Uncertainty Quantification (LCA-UQ) method based on Pointwise Kullback-Leibler (PKL) divergence. Our method effectively bridges the gap between the consistency of sampled outputs and the calibration of the candidate answer, thereby enhancing the reliability and stability of uncertainty assessments. Empirical evaluations across a range of popular LLMs and NLP datasets reveal that label sources significantly impact classification. Furthermore, our approach effectively captures the nuances between sampling results and label sources, demonstrating superior performance in uncertainty estimation.

Qinhong Lin, Yinglun Feng, Yuhao Zhang +2

School of Cyberspace Security, Beijing University of Posts and Telecommunications, Beijing, China · QuanCheng Laboratory, Jinan, China

Uncertainty Quantification LLM Uncertainty Estimation Uncertainty Calibration

Nov 2, 2024cs.LG

Uncertainty measurement for complex event prediction in safety-critical systems

Complex events originate from other primitive events combined according to defined patterns and rules. Instead of using specialists' manual work to compose the model rules, we use machine learning (ML) to self-define these patterns and regulations based on incoming input data to produce the desired complex event. Complex events processing (CEP) uncertainty is critical for embedded and safety-critical systems. This paper exemplifies how we can measure uncertainty for the perception and prediction of events, encompassing embedded systems that can also be critical to safety. Then, we propose an approach (ML_CP) incorporating ML and sensitivity analysis that verifies how the output varies according to each input parameter. Furthermore, our model also measures the uncertainty associated with the predicted complex event. Therefore, we use conformal prediction to build prediction intervals, as the model itself has uncertainties, and the data has noise. Also, we tested our approach with classification (binary and multi-level) and regression problems test cases. Finally, we present and discuss our results, which are very promising within our field of research and work.

Maria J. P. Peixoto, Akramul Azim

Department of Electrical, Computer and Software Engineering, Ontario Tech University, Oshawa, Ontario, Canada.

Uncertainty Quantification Sensitivity Analysis