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

Jun 19, 2026physics.geo-ph

Bayesian three-dimensional seismic travel-time tomography for active- and passive-source seismic data using physics-informed neural network

Accurate 3D seismic velocity modeling through seismic travel-time tomography using both active- and passive-source data provides critical underpinning models for seismicity monitoring and hazard assessment. Because travel-time tomography is an inherently ill-posed inverse problem, UQ of the estimated models using Bayesian methods is also important for reliable downstream interpretations and analyses. However, Bayesian inference for 3D tomography based on conventional grid-based representations faces the ``curse of dimensionality'' and severe computational bottlenecks. Consequently, rigorous Bayesian UQ for margin-wide 3D travel-time tomography has remained largely unexplored. In this study, we propose a meshless 3D Bayesian travel-time tomography method that combines PINNs with a neural representation of the velocity structure, enabling tractable and data-efficient Bayesian inference through function-space particle-based variational inference. To efficiently integrate passive-source data into the Bayesian estimation of the velocity structure, we conduct analytical marginalization treating uncertain source parameters as nuisance parameters, with passive-source relocation carried out in post-processing. We validated the capability of our approach for 3D problems through synthetic experiments. Furthermore, we applied the method to a real-world dataset from marine active-source surveys and natural earthquakes off the Kii Peninsula, Nankai Trough. Our probabilistic 3D ensemble successfully resolves key geological features and provides data-consistent uncertainty maps. The posterior mean hypocenters shifted mainly in the vertical direction by 10-15 km, consistent with a previous relocation result. Finally, the neural representation drastically reduces storage requirements for the entire ensemble velocity model, highlighting the scalability and data efficiency of the proposed framework.

Ryoichiro Agata, Kazuya Shiraishi, Gou Fujie +1

Japan Agency for Marine-Earth Science and Technology · Earth Science New Zealand

Uncertainty Quantification Bayesian Inverse Problems Seismology Inverse Problems

Jun 19, 2026cs.LG

Embedding Linear Equality Constraints in Probabilistic Neural Networks for Dynamic Modelling

Machine learning models are increasingly used to model chemical process systems, yet they often lack principled uncertainty quantification and mechanisms to enforce physical constraints. We propose a probabilistic neural network framework that guarantees satisfaction of linear equality constraints within a given tolerance, while capturing aleatoric uncertainty. Compared to state-of-the-art methods, our formulation demonstrates improved predictive accuracy, uncertainty calibration, and adherence to constraints on reduced data. It also demonstrates competitive performance, but with significantly faster training times when evaluated on large data regimes. We evaluated this on two batch reactor case studies, enforcing mass balances.

Matthew Marsh, Benoit Chachuat, Antonio del Rio Chanona

The Sargent Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, UK

Neural Surrogate Modeling Bayesian Neural Networks Uncertainty Quantification Uncertainty Calibration

Jun 19, 2026cs.LG

Rejections Based on Predictive Uncertainty Enable Reliable Routine Soil Spectroscopy

Soil properties relevant to agricultural and environmental applications are conventionally measured using elaborate laboratory methods involving physical and chemical processing. While highly accurate, these conventional methods are costly and time-consuming. In contrast, optical spectroscopy paired with machine learning enables rapid and cost-effective predictions of multiple soil properties. However, spectroscopic modelling is often considered unreliable, as the predictive accuracy varies between soil properties and individual samples. To balance this trade-off between cost and reliability, we introduce reject-to-remeasure: an AI-based measurement framework that combines probabilistic modelling with uncertainty-guided rejection. In this framework, soil samples are first analysed using spectroscopy, after which predictions are rejected if their predictive uncertainty exceeds predefined quality constraints. Rejected samples are subsequently remeasured using conventional laboratory procedures. On a regional visible-near-infrared spectral soil library from Québec, we demonstrate that reject-to-remeasure with modern foundation models (TabPFNv2.5 and TabICLv2) can facilitate the integration of optical spectroscopy into routine laboratory workflows while meeting user-defined accuracy requirements and reducing measurement costs.

Jonas Schmidinger, Robin Gebbers, Marc-Olivier Gasser +3

Joint Lab Artificial Intelligence and Data Science, Osnabrück University, Osnabrück, Germany. · Department of Agromechatronics, Leibniz Institute for Agricultural Engineering and Bioeconomy (ATB), Potsdam, Germany. · Research and Development Institute for the Agri-environment (IRDA), Qu´ebec City, Canada. +1

Uncertainty Quantification Selective Prediction

Jun 19, 2026cs.LG

Continuous-Time Probabilistic Correctors for Uncertainty-Aware Physics-Based Spacecraft Trajectory Forecasting

Long-horizon spacecraft trajectory forecasting suffers from error accumulation due to the absence of corrective observations in the forecast regime, making reliable uncertainty estimation crucial for safety-critical decision-making such as space domain awareness and conjunction assessment. While high-fidelity physics-based orbit propagators provide accurate deterministic forecasts, they typically lack calibrated uncertainty estimates over long horizons. We introduce a Predictor--Corrector framework in which a physics-based continuous-time

\textit{deterministic}

forecaster is augmented with a learned continuous-time

\textit{probabilistic}

Corrector that models forecast errors. The proposed Corrector can be wrapped around an existing deterministic propagator to improve forecast accuracy while producing sharp and calibrated full-covariance uncertainty estimates. The Corrector is based on Latent Neural Controlled Differential Equations (Latent NCDEs) and models the probabilistic temporal evolution of forecast errors in continuous time, naturally supporting irregular sampling and missing features. We further introduce a loss function that promotes calibration and sharpness in long-horizon uncertainty propagation. We evaluate the proposed framework on long-horizon spacecraft trajectory forecasting using real-world data from NASA's Crustal Dynamics Data Information System (CDDIS), wrapping the Corrector around NASA's General Mission Analysis Tool (GMAT). Across forecast horizons of 2--4 days without observations and six rolling test windows, the proposed approach consistently improves accuracy and uncertainty calibration compared to deterministic baselines and Latent ODE-based correctors, demonstrating the effectiveness of the continuous-time probabilistic Corrector for trajectory forecasting.

Muhammad Bilal Shahid, Zhanhong Jiang, Soumik Sarkar +1

Iowa State University, Ames, Iowa, 50011, USA

Neural Controlled Differential Equations Uncertainty Quantification Probabilistic Forecasting Uncertainty Propagation Trajectory Prediction Uncertainty Calibration

Jun 18, 2026cs.CV

Stochastic Signed Distance Processes

Multi-view surface reconstruction is a core problem in computer vision. One prominent line of work represents the surface implicitly as a signed distance field (SDF), optimizing it based on the photometric loss between rendered and observed pixel colors. These approaches typically employ SDF-based volume rendering to obtain a differentiable relaxation of discontinuous visibility along rays, thereby reducing reliance on silhouette supervision. In this paper, we reformulate SDF-based volume rendering as probabilistic surface rendering, where each pixel color is modeled as a mixture distribution induced by the random first ray-surface intersection. To this end, we introduce Stochastic Signed Distance Processes (SSDP), which model the SDF along each ray as a stochastic process, inducing a first-passage-time distribution for each ray. We then derive the first-passage probability for each sampling interval based on Bayesian filtering, together with its practical approximation for parallel rendering. We further show that NeuS, an existing SDF-based volume rendering method, arises as a special case of our formulation. Experiments on the DTU and MobileBrick datasets demonstrate that our method outperforms baselines in both surface reconstruction and uncertainty quantification, supporting the effectiveness of our first-passage formulation. Our code is available at https://github.com/skmhrk1209/SSDP.

Hiroki Sakuma, Masatoshi Okutomi

Department of Systems and Control Engineering Institute of Science Tokyo

Uncertainty Quantification Neural Rendering Multi-View 3D Reconstruction

Jun 18, 2026cs.LG

Neural network surrogates with uncertainty quantification for inverse problems in partial differential equations

Inverse problems for differential equations arise throughout science and engineering, where one seeks to infer unknown model parameters from noisy or incomplete observations. Traditional numerical methods for these problems are often computationally expensive, particularly in Bayesian settings where evaluating the likelihood becomes costly for complex forward models and high-dimensional parameter spaces. To address this challenge, we introduce DeepGaLA, a neural-network surrogate for differential equation solvers that provides uncertainty-aware predictions, reducing overconfident inference when training data are limited. To evaluate the fidelity of the surrogate-induced posterior approximations in practice, we show that a short run of delayed-acceptance Markov chain Monte Carlo can serve as an effective diagnostic. Across a range of numerical experiments, DeepGaLA delivers forward-model approximations with accuracy comparable to established Gaussian-process surrogates, while better maintaining efficiency as parameter dimension grows. Moreover, it can incorporate differential-equation constraints, including in nonlinear settings. Overall, these results indicate that uncertainty-quantified neural surrogates can enable scalable and reliable Bayesian inference for inverse problems in complex systems.

Christian Jimenez-Beltran, Aretha L. Teckentrup, Antonio Vergari +1

School of Mathematics and Maxwell Institute for Mathematical Sciences University of Edinburgh · School of Informatics University of Edinburgh

Neural Surrogate Modeling Uncertainty Quantification Bayesian Inverse Problems PDE Surrogate Modeling

Jun 18, 2026cs.LG

Evidential Fusion Network for Multimodal Survival Prediction under Missing Modalities

Recent multimodal survival prediction models have demonstrated strong predictive performance by leveraging complementary information across modalities. However, such models generally assume data completeness and exhibit limited robustness toward missing modalities, which are frequently encountered in real-world clinical settings. We propose the Evidential Missing Modality Survival Fusion (EMMS) model for multimodal survival prediction under missing modalities. EMMS offers a straightforward, computationally effective approach to survival analysis without requiring a generative phase for missing data. By employing Dempster-Shafer theory and Gaussian Random Fuzzy Numbers for multimodal decision fusion, it considers both aleatoric and epistemic uncertainty alongside modality reliability for fusion. Moreover, the model treats missing modalities as vacuous evidence, preventing interference with available inputs and naturally reflecting increased uncertainty and calibrated predictions. Extensive experiments on four cancer datasets demonstrate state-of-the-art performance while providing calibrated and interpretable uncertainty estimates under incomplete multimodal observations, without introducing additional computational overhead.

Yucheng Xing, Hailan Mo, Zi Wang +2

National University of Singapore, Singapore · Imperial College London, London, UK

Missing-Modality Learning Uncertainty Quantification Cancer Survival Prediction Survival Analysis Dempster-Shafer Theory Uncertainty-Aware Multimodal Fusion+3

Jun 18, 2026cs.AI

A Systematic Evaluation of Black-Box Uncertainty Estimation Methods for Large Language Models

Although large language models (LLMs) have shown strong capabilities across a wide range of tasks, their outputs often remain unreliable and may contain hallucinations, making uncertainty estimation (UE) essential for building trustworthy LLMs. In practice, many mainstream LLMs are only accessible through restricted APIs, where internal signals such as logits and hidden states are unavailable, making black-box UE especially important. However, existing work on black-box UE for LLMs remains fragmented in methodology and lacks a unified empirical comparison. To address this gap, we present a systematic review of black-box UE methods and organize them into five categories: verbalization-based, sampling-based, explanation-based, multi-agent, and hybrid methods. We further build a unified evaluation framework and benchmark 24 representative methods across 4 models and 4 dataset settings. Our results show that no single method consistently dominates across all settings. Nevertheless, methods that reason over and compare candidates in the answer space are generally effective, and hybrid methods that combine multiple uncertainty signals perform well under most conditions. By releasing the benchmark data and a unified evaluation framework, we aim to facilitate reproducible comparisons and support future research, while our empirical findings provide practical guidance for developing future black-box UE methods for LLMs.

Jiayi Wang, Xu-Yao Zhang

School of Advanced Interdisciplinary Sciences, University of Chinese Academy of Sciences, Beijing 100049, China · State Key Laboratory of Multimodal Artificial Intelligence Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China

LLM Evaluation Uncertainty Quantification LLM Reliability LLM Uncertainty Estimation

Jun 17, 2026cs.LG

On the QUEST for Uncertainty Quantification via Highest Density Regions

Uncertainty quantification (UQ) is essential for reliable decision-making in safety-critical applications in probabilistic machine learning. For regression problems, dominant scalar UQ approaches - notably, those based on proper scoring rules - measure uncertainty via pointwise predictive risk. This can lead to counterintuitive results when the target statistic is not the conditional expectation. We propose an alternative framework, in which uncertainty is characterised by the volume of the most probable subset of a distribution's support. QUEST (Quantifying Uncertainty via highest dEnSiTy regions) is a novel approach to UQ based on the concentration of Lebesgue measure at a distribution's peak(s), evaluated at one or more values of a robustness parameter

α

. We establish connections between our measures and classical statistics from information theory and economics. We show that, unlike popular alternatives based on proper scoring rules, QUEST measures of epistemic and aleatoric uncertainty satisfy a set of axioms adapted from the UQ literature, including monotonicity under distributional spread and invariance to location shifts. Selective prediction benchmarks confirm that QUEST performs favourably against standard measures such as variance and differential entropy.

Sam Goring, Tom Kuipers, Nicola Paoletti +1

King’s College London · Northeastern University London

Uncertainty Quantification Aleatoric Uncertainty Epistemic Uncertainty

Jun 17, 2026cs.CV

Quantification of Uncertainty with Adversarial Models in Medical Image Segmentation

Reliable pixel-level uncertainty quantification holds the potential to transform clinical workflows by enabling high-fidelity longitudinal monitoring and distinguishing true pathological changes from artifacts. Ideally, these models provide the stability required for critical treatment planning and surgical intervention. However, standard deep learning models often suffer from miscalibration, yielding overconfident predictions that mask underlying vulnerabilities at subtle pathological boundaries. To address this, we propose QUAM-SM, a post-hoc framework using targeted adversarial search to identify "adversarially fragile" pixels. By actively seeking perturbations that expose predictive instability, our method highlights regions where decisions are most vulnerable to being flipped. Importantly, the framework disentangles epistemic uncertainty from aleatoric uncertainty. Experiments on two public datasets with multiple expert annotations demonstrate that QUAM-SM outperforms both standard and recent uncertainty estimation approaches in terms of reliability and boundary sensitivity. Code is available at https://github.com/HanaJebril/quam_sm

Hana Jebril, Thomas Pinetz, Günter Klambauer +1

Institute of Artificial Intelligence, Center for Medical Data Science, Medical University of Vienna, Austria · Comprehensive Center for AI in Medicine, Medical University of Vienna, Austria · ELLIS Unit Linz, LIT AI Lab and Institute for Machine Learning, Johannes Kepler University Linz, Austria +1

Uncertainty Quantification Medical Image Segmentation Segmentation Uncertainty Estimation

Jun 16, 2026cs.LG

Geometry-Aware Post-Hoc Uncertainty Quantification in Operator Learning

Neural operators provide fast surrogates for PDEs but their deterministic predictions limit their use in tasks requiring uncertainty quantification (UQ), especially under geometric variability. Existing approaches primarily model uncertainty in network parameters, largely overlooking the geometry-aware representations learned by the operator itself. We propose REEF-GP (Residual on Embedded Features Gaussian Process), a post-hoc UQ framework that fits a GP to the residuals of a frozen neural operator whose internal embeddings define the kernel feature space. Rather than learning a separate feature map, REEF-GP adapts the operator's intrinsic coordinate-feature representations to construct geometry-aware uncertainties. To ensure stability and scalability on unstructured domains, REEF-GP incorporates spectral-normalized projections, heteroscedastic geometry-aware noise, and efficient subset-based training that avoids restrictive low-rank approximations. Across five PDE benchmarks with varying geometries, REEF-GP preserves predictive accuracy while achieving calibrated uncertainty estimates competitive with deep ensembles but at a fraction of their cost. Our approach remains robust under geometric distribution shift, with uncertainty concentrating in physically meaningful regions (e.g., shock fronts). Our results demonstrate that accurate and scalable post-hoc UQ for neural operators can be achieved directly in their learned feature space, offering a practical alternative to parameter-centric approaches.

Oriol Vendrell-Gallart, Nima Negarandeh, Ramin Bostanabad

Department of Mechanical and Aerospace Engineering University of California, Irvine Irvine, CA 92617

Uncertainty Quantification PDE Surrogate Modeling Neural Operators Gaussian Process Regression

Jun 16, 2026stat.ML

Bounded Difference Concentration for Infinitely Exchangeable Sequences with Applications to AI Benchmark Uncertainty

We consider the concentration properties of functions of infinitely exchangeable random variables. By conditioning on the de Finetti directing measure, we show that the deviation of any function with bounded-difference constants

c_1, \dots, c_n

decomposes into a conditional sampling fluctuation and a latent mixture fluctuation. When this latent mixture is

σ_{\mathrm{mix}}^2

-subgaussian, we establish a concentration inequality with an effective variance proxy of

\frac{1}{4}\sum_i c_i^2 + σ_{\mathrm{mix}}^2

. Crucially, we demonstrate that for zero-sum linear contrasts, such as the difference between a subsample mean and a full population mean, the latent mixture term cancels exactly. This cancellation yields a tight, mixture-free Hoeffding-type bound that provides a direct de Finetti mechanism for the infinite-extendibility limit of recent finite-exchangeable concentration results. We apply this framework to quantify uncertainty in composite AI benchmarks, such as MMLU, where question items naturally exhibit exchangeable dependence across domains. Our results provide both a domain-stratified hierarchical model for bounding the uncertainty of accuracy scores, and a distribution-free, cost-saving statistical guarantee for accurately estimating full benchmark scores from random subsets.

Fangyuan Lin, Spencer Frei, Victor H. de la Pena

Department of Statistics, Columbia University · Google DeepMind

Uncertainty Quantification

Jun 15, 2026cs.CV

Trustworthy MRI Reconstruction via Bayesian Uncertainty Quantification with Sparsity Prior Models

We propose a novel Bayesian framework for joint image reconstruction and uncertainty quantification from compressed sensing magnetic resonance imaging data. The problem is formulated as a linear inverse problem, where prior distributions are assigned to the unknown image parameters. Specifically, the image is assumed to be sparse in a given transform domain. We develop a general framework applicable to any sparsifying transform and demonstrate its performance using (1) a total variation transform based on image spatial gradients and (2) a wavelet-domain transform. Bayesian inference is performed using a split-and-augmented Gibbs sampler, while the resulting non-differentiable conditional distributions are efficiently sampled using a proximal Markov chain Monte Carlo method. The proposed algorithms are validated on both single-coil and multi-coil datasets using various k-space sampling patterns and acceleration factors. The results demonstrate that the proposed Bayesian methods consistently outperform their optimisation-based counterparts in image reconstruction while providing uncertainty estimates for the reconstructed images. Furthermore, the estimated uncertainty maps show a strong correlation with the true reconstruction errors and substantially outperformed deep learning-based uncertainty estimation methods.

Ahmed Karam Eldaly, Matteo Figini, Daniel C. Alexander

Department of Computer Science, University of Exeter, Exeter EX4 4QF, U.K. · UCL Hawkes Institute, Department of Computer Science, University College London, London, U.K.

Sparse Recovery Uncertainty Quantification Bayesian Inverse Problems Accelerated MRI Reconstruction Magnetic Resonance Imaging

Jun 15, 2026cs.LG

Uncertainty Quantification of Engineering Structures by Polynomial Chaos Expansion and Multivariate Active Learning

In many engineering applications, a single high-fidelity model produces multiple quantities of interest (QoIs) under the same input parameters, e.g. finite element models of complex physical systems. To alleviate the high computational cost of direct model evaluations, surrogate models are widely used to construct efficient approximations of model responses. Naturally, the accuracy of surrogates strongly depends on the quality of the experimental design (ED). However, a single ED may not provide an adequate representation for all outputs simultaneously, especially when different outputs exhibit varying sensitivities to the input variables. A straightforward solution is to perform separate sampling for each output, but this results in increased sampling complexity and computational cost. From a statistical perspective, such an approach also ignores potential correlations among all outputs and may compromise data consistency. To address this issue, an adaptive sequential sampling method for constructing polynomial chaos expansion surrogate models is generalized for vector valued QoIs. The method sequentially selects new samples from a candidate pool based on their local contribution to the output variance, while balancing distance-based exploration of the input space and exploitation of aggregated variance information across all outputs. Its performance is compared with non-sequential Latin Hypercube Sampling through several numerical examples from engineering problems. Numerical results demonstrate that the proposed strategy improves both surrogate accuracy and stability, and provides a more reliable estimation of second-order statistics.

Qitian Lu, Jafar Jafari-Asl, Panagiotis Spyridis +1

Brno University of Technology, Brno, Czech Republic · University of Rostock, Rostock, Germany

Surrogate Modeling Uncertainty Quantification Adaptive Sampling

Jun 15, 2026cs.LG

Simulation-Augmented Multi-Step Split Conformal Prediction for Aggregated Forecasts

We study uncertainty quantification for aggregated forecasting tasks such as annual totals and year-over-year growth rates. We propose SA-MSCP, a simulation-augmented multi-step split conformal method that generates future paths from cross-validated residuals using a block bootstrap and constructs prediction intervals from empirical quantiles. Experiments show that SA-MSCP improves empirical coverage over a simulated-path baseline for aggregated and growth-rate targets. Our results demonstrate that simulation-enhanced conformal calibration is an effective and general framework for uncertainty quantification in aggregated time-series forecasting.

Andro Sabashvili

Uncertainty Quantification Time Series Forecasting Conformal Prediction Regression Prediction Intervals

Jun 15, 2026cs.LG

Calibrated Sampling-Free Uncertainty Estimation in Bayesian Deep Learning

Modern deep learning models remain notoriously prone to overconfidence, limiting their reliability in high-stakes applications. Bayesian methods aim to counter this by learning a distribution over model parameters, and recent advances now make this feasible for large-scale architectures at costs comparable to AdamW. However, a challenge remains at test time: predictions must be averaged across many forward passes with weights sampled from the posterior, which is prohibitively expensive. Variance propagation offers an efficient alternative, computing layer-wise analytical approximations of uncertainty in a single forward pass. While such techniques are effective for MLPs, their extension to modern architectures remains challenging, due to increased depth and diversity of layer types. To fill this gap, we propose Calibrated Variance Propagation (CVP), which introduces a new propagation method for normalization layers, combines it with recent techniques for handling activation functions, and absorbs residual error through a light calibration step. CVP yields comparably accurate uncertainty estimates to MC sampling across transformers and CNNs, at a fraction of the cost. Against prior variance propagation work, CVP improves coverage at

0.5\%

risk from

8.2\%

to

14.6\%

with BEiT-3 on Visual Reasoning (NLVR2) and from

2.6\%

to

10.8\%

with ViLT on VQAv2, with gains extending to convolutional architectures.

Tobias Jan Wieczorek, Leon de Andrade, Thomas Möllenhoff +1

TU Darmstadt & hessian.AI, Darmstadt, Germany · RIKEN Center for Advanced Intelligence Project, Tokyo, Japan

Bayesian Neural Networks Uncertainty Quantification Uncertainty Propagation Uncertainty Calibration

Jun 14, 2026cs.LG

Visualizing Uncertainty: Spatial Maps of Missing and Conflicting Evidence in Deep Learning

Understanding when and why deep neural networks are uncertain is crucial for deploying reliable machine learning systems in safety-critical domains. While existing uncertainty quantification methods provide scalar measures of model confidence, they offer limited insight into which spatial regions of an input contribute to different types of uncertainty. We propose a novel visualization framework, Uncertainty Activation Map (UAM), that combines Evidential Deep Learning (EDL) with Full-Gradient Class Activation Mapping (FullGrad) to generate interpretable spatial uncertainty activation maps. Our approach distinguishes between two fundamental types of uncertainty: vacuity, representing lack of evidence, and dissonance, capturing conflicting evidence between competing hypotheses. By leveraging the complete gradient decomposition property of FullGrad and the principled uncertainty quantification of Subjective Logic, our method produces theoretically grounded visualizations that highlight specific image regions responsible for model uncertainty. With this framework, vacuity and dissonance activation maps are generated by computing belief-weighted attributions, enabling identification of where models lack knowledge versus where they encounter ambiguous evidence. Extensive evaluations across multiple benchmark datasets demonstrate that the proposed framework effectively addresses the critical gap between uncertainty quantification and explainability, providing intuitive visual feedback to assess model reliability in complex visual recognition tasks.

Dong Hyun Jeong, Feng Chen, Jin-Hee Cho +3

University of the District of Columbia Department of Computer Science Washington, DC, USA · University of Texas at Dallas Department of Computer Science Richardson, TX, USA · Virginia Tech Department of Computer Science Blacksburg, VA, USA +3

Uncertainty Quantification

Jun 14, 2026cs.LG

Multi-Agent Framework for Audit Risk Assessment with Explicit Uncertainty and Evidence Conflict Modeling

Audit risk assessment increasingly benefits from combining heterogeneous evidence sources, yet existing approaches typically produce point predictions without quantifying how well different evidence streams agree. We propose UMAR (Uncertainty-Aware Multi-Agent Risk Assessment), a framework that employs three specialized agents: an MD&A Text Agent, a Financial Ratio Agent, and a CAM Agent, each producing independent risk scores with calibrated uncertainty estimates. An Uncertainty Aggregator based on Dempster-Shafer evidence theory fuses these scores while explicitly measuring inter-agent conflict. We evaluate UMAR on a U.S. dataset of 3,200 firm-year observations from SEC 10-K filings (2019-2023), with financial restatement as the target label. Experimental results show that UMAR achieves an AUROC of 0.782 and a PR-AUC of 0.341, outperforming logistic regression, XGBoost, FinBERT, and single-agent and dual-agent LLM baselines. UMAR attains the lowest expected calibration error (ECE = 0.052) among all methods and identifies evidence-conflict patterns that correlate with actual restatement risk, offering auditors potentially actionable and interpretable risk signals.

Yuhan Wang, Manqing Wang, Yixuan Lu +2

Columbia University New York, USA · Trine University Phoenix, USA · University of Sofia Palo Alto, USA +2

Uncertainty Quantification Dempster-Shafer Theory Decision-Making under Uncertainty

Jun 13, 2026cs.LG

Bayesian 3D Steerable CNNs: Enabling Equivariance and Uncertainty Quantification Simultaneously

Steerable convolutional neural networks (Steerable-CNNs) guarantee SE(3)-equivariance by parameterizing kernels as linear combinations of steerable basis functions, but their deterministic nature precludes uncertainty quantification - limiting their use in settings where confidence estimates are essential. We propose a Bayesian Steerable-CNN that places posterior distributions over the basis coefficients, yielding stochastic kernels while preserving equivariance exactly. The loss function of the model is obtained via variational inference and minimized by Bayes-by-Backpropagation. The framework admits a decomposition of predictive uncertainty into epistemic and aleatoric components. Empirically, the model attains competitive classification accuracy alongside an expected calibration error of 0.0263 and outperforms its deterministic counterpart by up to 6.17% under distributional shift induced by additive Gaussian noise. Furthermore, we leverage the model's uncertainty estimates to enhance its performance significantly, achieving a notable gain - approximately 4% higher accuracy across 84% of the test dataset. A statistically significant negative correlation between epistemic uncertainty and prediction error confirms that the learned posterior variance is semantically meaningful. The framework unifies Bayesian uncertainty quantification with the inductive bias of equivariant CNNs.

Abhishek Keripale, Ponkrshnan Thiagarajan, Susanta Ghosh

Department of Mechanical and Aerospace Engineering, Michigan Technological University, Houghton, MI 49931 · Hopkins Extreme Materials Institute, Johns Hopkins University, Baltimore, MD 21218 · The Center for Artificial Intelligence at the Institute of Computing and Cybersystems, Michigan Technological University, USA

Bayesian Neural Networks Uncertainty Quantification Equivariant Neural Networks 3D CNNs Uncertainty Calibration

Jun 12, 2026stat.ML

Audited Conformal Prediction for Classification under Unknown Distribution Shift

We consider the problem of uncertainty quantification for a pretrained classification model deployed under unknown distribution shift. We propose Audited Conformal Prediction (ACP), a method that leverages a small labeled dataset from the target population to train an auxiliary audit model identifying inputs where the legacy model is likely to fail. By integrating the audit model's outputs into the conformal prediction framework, ACP produces prediction sets that guarantee marginal coverage while achieving substantially higher conditional coverage in practice than existing approaches. We develop and analyze two complementary integration strategies -- one targeting marginal coverage with improved conditional performance, the other providing explicit group-conditional coverage guarantees -- and establish theoretical guarantees for both. Experiments on synthetic and real-world datasets validate the method and illustrate trade-offs between prediction set size and conditional coverage.

Yanfei Zhou, Rizal Fathony, Nam H. Nguyen +1

Department of Data Sciences and Operations University of Southern California Los Angeles, CA 90089 · AI Foundations, Capital One New York, NY 10017 · Department of Data Sciences and Operations Thomas Lord Department of Computer Science University of Southern California Los Angeles, CA 90089

Distribution Shift Robustness Uncertainty Quantification Group-Conditional Conformal Prediction Conformal Prediction

Jun 12, 2026stat.ML

Hybrid Uncertainty Sensitivity Analysis Based on the HSIC for High-Dimensional Responses with Aleatory--Epistemic Separation

Quantifying the influence of hybrid aleatory and epistemic uncertainties on high-dimensional system responses remains a major challenge in global sensitivity analysis (GSA). Existing Hilbert--Schmidt Independence Criterion (HSIC)-based approaches are primarily restricted to single-output settings and lack a rigorous decomposition of heterogeneous uncertainty sources and their interactions. To address this limitation, a novel double-space tensor-product RKHS framework is proposed for sensitivity analysis under hybrid uncertainty. By constructing factorized kernels over both the latent input space and the multidimensional output space, a concurrent double Möbius inversion is derived to orthogonally decompose the global dependence measure into pure aleatory effects, pure epistemic effects, and their interaction contributions. The resulting dimension-wise sensitivity indices preserve the uncertainty attribution structure across all output dimensions. To satisfy the independence assumptions required by the decomposition, an auxiliary-variable representation based on the inverse probability integral transform is introduced, enabling the treatment of hierarchical uncertainties and Copula-induced correlations within a unified latent space. A fully vectorized single-loop implementation is further developed to avoid the computational burden of nested Monte Carlo simulation. Statistical significance and estimation uncertainty are quantified through permutation testing and Bootstrap confidence intervals. Numerical studies on a modified multi-output Ishigami function and an aerodynamic pressure-field problem demonstrate the accuracy, scalability, and practical applicability of the proposed framework.

Shijie Zhong, Jiangfeng Fu, Pengfei Wei

School of Power and Energy Northwestern Polytechnical University Xi’an, Shanxi 710129

Uncertainty Quantification Aleatoric Uncertainty Sensitivity Analysis Epistemic Uncertainty

Jun 11, 2026cs.LG

Uncertainty Estimation and Generalization Bounds for Modern Deep Learning

This thesis investigates how Bayesian principles can deepen our understanding of modern deep learning systems. While neural networks achieve remarkable predictive performance, their ability to generalize and to quantify uncertainty remains only partly understood. This thesis approaches this challenge from both methodological and theoretical angles: unifying Bayesian inference, function-space modeling, and large-deviation theory under a common probabilistic perspective. On the methodological side, the thesis introduces the Deep Variational Implicit Process (DVIP), a scalable Bayesian framework that extends implicit processes to deep architectures. Complementing this, two post-hoc methods -- the Variational Linearized Laplace Approximation (VaLLA) and the Fixed-Mean Gaussian Process (FMGP) -- are proposed to equip pretrained deterministic networks with calibrated uncertainty estimates. The theoretical contributions focus on one of the central open questions in modern machine learning: why do large, over-parameterized neural networks generalize so well? To address this, the thesis develops a unified probabilistic framework that connects three key mechanisms -- diversity, smoothness, and stochasticity -- within the language of PAC-Bayesian and large-deviation theory.

Luis A. Ortega

Department of Computer Science Machine Learning Group Madrid

Bayesian Neural Networks Neural Network Generalization PAC-Bayesian Generalization Bounds Uncertainty Quantification Variational Inference Uncertainty Calibration

Jun 10, 2026stat.ML

Epistemic Uncertainty Is Not the Reducible Kind

The standard taxonomy of predictive uncertainty defines epistemic uncertainty as the part removable by collecting more data, while the standard measure identifies it with a mutual-information term. We prove the definition and the measure are extensionally inconsistent. On an explicit construction, the measure assigns all uncertainty to the epistemic class, yet no quantity of training data reduces it. Reducibility is instead a property of the pair (uncertainty, acquisition class), and the dichotomy resolves into three parts: aleatoric, sample-reducible epistemic, and mechanism-reducible epistemic uncertainty. An exact identity for the value of an observation shows that in-distribution data never reduces mechanism-irreducible uncertainty and generically increases it. Ensemble disagreement, the deployed epistemic estimate, tracks the training procedure rather than the epistemic term. It collapses to zero beneath a positive truth under consistent training, and equals hyperparameter-scaled initialization noise under interpolation. A finite-sample falsification test and seed-swept experiments confirm the theory.

Robin Young

University of Cambridge

Uncertainty Quantification Uncertainty Decomposition Epistemic Uncertainty

Jun 10, 2026cs.LG

What Uncertainties Do We Need for Dynamical Systems?

The distinction between aleatoric and epistemic uncertainty has received considerable attention in machine learning research, mainly in the context of supervised learning but also in other settings such as generative modeling. In this paper, we offer a machine learning perspective on uncertainty modeling for dynamical systems, which has been studied much less so far. In particular, we ask: what uncertainties do we need for dynamical systems? We discuss sources of uncertainty, clarify their nature (aleatoric or epistemic), and consider how the objectives of representing and quantifying uncertainty vary across different tasks.

Yusuf Sale, Christopher Bülte, Felix Czaja +2

Institute of Computer Science, LMU Munich · Munich · Department of Mathematics, LMU Munich +1

Dynamical Systems Uncertainty Quantification Aleatoric Uncertainty Epistemic Uncertainty

Jun 10, 2026cond-mat.mtrl-sci

Modelling magnetic material properties with uncertainty-aware neural networks

Machine learning is increasingly used to accelerate materials discovery across large compositional and structural design spaces, but limited and heterogeneous data make reliable uncertainty estimation essential. In this work, we investigate uncertainty quantification for permanent magnet modeling across three complementary studies. First, on a public Curie temperature surrogate dataset, we benchmark Gaussian process regression, random forest bagging, and dropout-based Bayesian neural networks and show that calibration, sharpness, and confidence curve diagnostics reveal differences in uncertainty quality that are not visible from point-prediction metrics alone. Second, we transfer this framework to the prediction of intrinsic magnetic properties in Nd2 Fe14 B-based magnets. Third, we extend the same uncertainty-aware perspective to coercivity prediction from microstructural information using a graph neural network. Together, these studies show that uncertainty quantification improves the trustworthiness of magnetic material property predictions and can be transferred from composition-based surrogate models to more complex structure-sensitive learning tasks.

Clemens Wager, Heisam Moustafa, Alexander Kovacs +10

Christian Doppler Laboratory for magnet design through physics informed machine learning, University for Continuing Education Krems, Wr. Neustadt, 2700, Austria · Vienna Doctoral School in Physics, Vienna, Austria · Department for integrated sensor systems, University for Continuing Education Krems, Wr. Neustadt, 2700, Austria +1

Uncertainty Quantification Materials Property Prediction Uncertainty Calibration

Jun 10, 2026cs.LG

Structure-Preserving Neural Surrogates with Tractable Uncertainty Quantification

Recent advances in scientific machine learning provide a means of near-real-time solution to partial differential equations (PDEs), but lack the theoretical underpinnings of conventional simulators that support contemporary verification and validation. In this work, we construct data-driven reduced-order models that serve as structure-preserving, real-time surrogates. Remarkably, the exterior calculus that imposes physical conservation structure also exposes topological structure that we use to build a Gaussian process (GP) representation of uncertainty in state-flux relationships, ultimately yielding a Dirichlet-to-Neumann map for quantities of interest with closed-form expressions for posterior uncertainty. We specifically propose structure-preserving

H(\mathrm{div})

--

L^2

subspaces of conventional Raviart--Thomas and

dgP_0

elements prescribed by a lightweight transformer. Reduced-order dynamics consistent with this subspace are learned by posing a conservation law in which a GP describes the fluxes between volumes. This work hinges on a novel interface between mixed FEM spaces and GP regression; when training is posed as the optimal recovery problem (ORP), the resulting GP regression can be written as an optimization problem with equality constraints that impose a conservation structure, amenable to a fast Schur-complement training strategy. The trained model can then be solved in real time with closed-form estimators for boundary fluxes driven by prescribed Dirichlet data. The paper includes RKHS posterior error bounds for linear functionals to support uncertainty quantification, as well as numerical experiments demonstrating the accuracy of the posterior distribution as a surrogate for error estimation.

Handi Zhang, Adrienne M. Propp, Brooks Kinch +2

Applied Mathematics and Computational Science, University of Pennsylvania, Philadelphia, PA, USA · Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA · Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA, USA +1

Uncertainty Quantification PDE Surrogate Modeling Reduced-Order Modeling Finite Element Method Gaussian Process Regression

Jun 9, 2026cs.AI

Confidence Laundering in Agent Systems: Why Uncertainty Needs a Latent Carrier

Modern agent systems can turn uncertainty into overconfidence. Fragile upstream decisions are often exposed to downstream components as clean intermediate artifacts, while the uncertainty behind those decisions is lost at the interface. As a result, local ambiguity can become system-level error amplification. We argue that this reveals an interface bottleneck in agent uncertainty propagation: uncertainty does not propagate simply because a trajectory contains uncertain steps; it propagates only when it survives the handoff between components. We define uncertain decision handoff as the transfer of an intermediate decision made under uncertainty, and identify confidence laundering as a failure mode in which fragile upstream states are repackaged as procedurally valid artifacts that downstream agents over-trust. To address this bottleneck, we propose latent uncertainty as an uncertainty-bearing carrier attached to decision handoffs. Rather than replacing text with hidden states, latent uncertainty aims to preserve pre-commitment fragility in a form that downstream components can use. This position shifts agent uncertainty propagation from step-wise uncertainty estimation toward uncertainty-preserving interface design for more recoverable agent systems.

Kaiwen Shi, Zheyuan Zhang, Han Bao +2

University of Notre Dame

Agent Reliability Uncertainty Quantification Uncertainty Propagation

Jun 9, 2026cs.CV

An Uncertainty Estimation Framework for Dose Accumulation in Adaptive Radiotherapy: Application to CBCT-Guided Radiotherapy for Cervical Cancer

Background and purpose: oART enables daily plan adaptation to interfraction anatomical variations, but cumulative dose estimation remains limited by DIR, segmentation, and anatomical uncertainties. We introduce IMPACT-DoseAcc, an uncertainty-aware dose accumulation framework, within IMPACT for semantic feature-driven image analysis. The framework is modality- and disease-agnostic and is applied to CBCT-guided oART for cervical cancer (LACC). Material and Methods: Nine LACC patients were retrospectively analyzed using daily CBCT-derived virtual CTs for dose recalculation. IMPACT-DoseAcc focuses on uncertainty from DIR, without modeling vCT-generation uncertainty. Two DIR uncertainty strategies were tested within IMPACT-Reg: a Bayesian segmentation-guided approach using one probabilistic model to quantify anatomical uncertainty, and an ensemble of segmentation models targeting structures to capture epistemic variability. Voxel-wise uncertainty maps were propagated through dose warping and accumulation to generate probabilistic dose-volume histograms. Ensemble uncertainty was quantified from voxel-wise standard deviation across deformation fields, and geometric error was assessed using surface distance between warped and validated contours. Anatomical-variability weighting refined aggregation. Results: Ensemble DIR uncertainty correlated with geometric error, with Pearson coefficients of 0.63 for CTVt and 0.66 for bladder. For CTVt, pDVHs achieved 96.3 +/- 3.9% coverage, showing calibration of propagated uncertainty. Weighting stabilized estimates across fractions and organs. Conclusions: IMPACT-DoseAcc propagates registration-driven uncertainty to cumulative dose metrics, improving interpretation of accumulated dose under anatomical variations. Its 3DSlicer integration supports reproducible, uncertainty-informed ART workflows.

Cedric Hemon, Delphine Lebret, Jean-Claude Nunes +8

Univ. Rennes, CLCC Eugène Marquis, INSERM, LTSI - UMR 1099, F-35000 Rennes, France · Department of Radiation Oncology, Centre Georges-Francois Leclerc, F-21000 Dijon, France · Institut de Cancérologie de l’Ouest–Site Paul Papin, F-49000 Angers, France +1

Uncertainty Quantification Oncology Deformable Image Registration

Jun 9, 2026cs.LG

Can we trust our models? Epistemic calibration in second-order classification

Uncertainty estimation is critical for deploying machine learning models in high-stakes settings. However, classical calibration only assesses the reliability of predicted probabilities and does not evaluate whether epistemic uncertainty estimates are themselves trustworthy. This limitation is particularly relevant for second-order classification models. We introduce epistemic calibration, a principled criterion that measures whether reported epistemic uncertainty faithfully reflects the dispersion of model predictions around the ground truth. We show that epistemic calibration is a strictly stronger notion than classical calibration and captures failure modes invisible to standard metrics. We relate this work to the existing literature through an impossibility theorem that holds under the epistemic calibration hypothesis. To operationalize this concept, we propose the Expected Epistemic Calibration Error (EECE), which we prove to be a consistent estimator of a True Epistemic Calibration Error (TECE). Experiments across a broad range of uncertainty quantification methods show that epistemic calibration is a coherent and meaningful criterion and reveal substantial differences across methods, despite similar predictive performance.

Arthur Hoarau

Université de Lorraine, CentraleSupélec · Loria, CNRS, Metz, France

Uncertainty Quantification Uncertainty Calibration Epistemic Uncertainty

Jun 8, 2026astro-ph.CO

Learning the Universe: Posterior Reliability of Neural Generative Models in High-Dimensional Field-Level Inference of Cosmic Initial Conditions

Accurate posterior estimation is central to scientific inference, as uncertainties determine what can be reliably learned from observational data. While Markov chain Monte Carlo methods provide asymptotic convergence guarantees, they are computationally demanding in high-dimensional settings. Neural network-based generative models for entire discretized 3D fields enable fast amortized inference but often lack convergence guarantees and principled accuracy assessment. Using Hamiltonian Monte Carlo to obtain reference posterior samples, we conduct a controlled field-level evaluation of an implicit generative model (Stochastic Interpolants) and an explicit likelihood-based model (GLOW normalizing flows). This comparison, unavailable in typical applications, enables the detection of posterior geometry failures that standard metrics cannot capture. As a case study, we consider the cosmological inverse problem of inferring cosmic initial conditions from present-day large-scale structure. To match the precision of modern cosmological data, this problem increasingly relies on complex, non-linear, and non-differentiable simulators, which are incompatible with gradient-based inference frameworks. Generative models offer a route to address these challenges, provided their inferred posteriors are reliable. In this work, we show that matching posterior means, marginal distributions, or achieving high cross-correlation does not imply correct uncertainty structure, as revealed by posterior variance fields and sample-based evaluations. Through this work, we aim to raise awareness of the challenges of uncertainty estimation in high-dimensional field-level settings, highlighting the importance of careful design and validation of neural generative approaches for scientific applications.

Ludvig Doeser, Jens Jasche

The Oskar Klein Centre, Department of Physics, Stockholm University, AlbaNova University Centre, SE 106 91 Stockholm, Sweden · Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA

Simulation-Based Inference Normalizing Flows Uncertainty Quantification Bayesian Inverse Problems Conditional Generative Modeling