Daily Papers

Modeling dynamic 3D environments from LiDAR sequences is central to building reliable 4D worlds for autonomous driving and embodied AI. Existing generative frameworks, however, often treat all spatial regions uniformly, overlooking the varying uncertainty across real-world scenes. This uniform generation leads to artifacts in complex or ambiguous regions, limiting realism and temporal stability. In this work, we present U4D, an uncertainty-aware framework for 4D LiDAR world modeling. Our approach first estimates spatial uncertainty maps from a pretrained segmentation model to localize semantically challenging regions. It then performs generation in a "hard-to-easy" manner through two sequential stages: (1) uncertainty-region modeling, which reconstructs high-entropy regions with fine geometric fidelity, and (2) uncertainty-conditioned completion, which synthesizes the remaining areas under learned structural priors. To further ensure temporal coherence, U4D incorporates a mixture of spatio-temporal (MoST) block that adaptively fuses spatial and temporal representations during diffusion. Extensive experiments show that U4D produces geometrically faithful and temporally consistent LiDAR sequences, advancing the reliability of 4D world modeling for autonomous perception and simulation.

Collaborative perception, fusing information from multiple agents, can extend perception range so as to improve perception performance. However, temporal asynchrony in real-world environments, caused by communication delays, clock misalignment, or sampling configuration differences, can lead to information mismatches. If this is not well handled, then the collaborative performance is patchy, and what's worse safety accidents may occur. To tackle this challenge, we propose CoDynTrust, an uncertainty-encoded asynchronous fusion perception framework that is robust to the information mismatches caused by temporal asynchrony. CoDynTrust generates dynamic feature trust modulus (DFTM) for each region of interest by modeling aleatoric and epistemic uncertainty as well as selectively suppressing or retaining single-vehicle features, thereby mitigating information mismatches. We then design a multi-scale fusion module to handle multi-scale feature maps processed by DFTM. Compared to existing works that also consider asynchronous collaborative perception, CoDynTrust combats various low-quality information in temporally asynchronous scenarios and allows uncertainty to be propagated to downstream tasks such as planning and control. Experimental results demonstrate that CoDynTrust significantly reduces performance degradation caused by temporal asynchrony across multiple datasets, achieving state-of-the-art detection performance even with temporal asynchrony. The code is available at https://github.com/CrazyShout/CoDynTrust.

This paper presents the Entropy-Driven Unified Process Reward Model (EDU-PRM), a novel framework that approximates state-of-the-art performance in process supervision while drastically reducing training costs. EDU-PRM introduces an entropy-guided dynamic step partitioning mechanism, using logit distribution entropy to pinpoint high-uncertainty regions during token generation dynamically. This self-assessment capability enables precise step-level feedback without manual fine-grained annotation, addressing a critical challenge in process supervision. Experiments on the Qwen2.5-72B model with only 7,500 EDU-PRM-generated training queries demonstrate accuracy closely approximating the full Qwen2.5-72B-PRM (71.1% vs. 71.6%), achieving a 98% reduction in query cost compared to prior methods. This work establishes EDU-PRM as an efficient approach for scalable process reward model training.

Reliable uncertainty estimation is critical for medical image segmentation, where automated contours feed downstream quantification and clinical decision support. Many strong uncertainty methods require repeated inference, while efficient single-forward-pass alternatives often provide weaker failure ranking or rely on restrictive feature-space assumptions. We present SegWithU, a post-hoc framework that augments a frozen pretrained segmentation backbone with a lightweight uncertainty head. SegWithU taps intermediate backbone features and models uncertainty as perturbation energy in a compact probe space using rank-1 posterior probes. It produces two voxel-wise uncertainty maps: a calibration-oriented map for probability tempering and a ranking-oriented map for error detection and selective prediction. Across ACDC, BraTS2024, and LiTS, SegWithU is the strongest and most consistent single-forward-pass baseline, achieving AUROC/AURC of 0.9838/2.4885, 0.9946/0.2660, and 0.9925/0.8193, respectively, while preserving segmentation quality. These results suggest that perturbation-based uncertainty modeling is an effective and practical route to reliability-aware medical segmentation. Source code is available at https://github.com/ProjectNeura/SegWithU.

Predicting attention regions of interest is an important yet challenging task for self-driving systems. Existing methodologies rely on large-scale labeled traffic datasets that are labor-intensive to obtain. Besides, the huge domain gap between natural scenes and traffic scenes in current datasets also limits the potential for model training. To address these challenges, we are the first to introduce an unsupervised way to predict self-driving attention by uncertainty modeling and driving knowledge integration. Our approach's Uncertainty Mining Branch (UMB) discovers commonalities and differences from multiple generated pseudo-labels achieved from models pre-trained on natural scenes by actively measuring the uncertainty. Meanwhile, our Knowledge Embedding Block (KEB) bridges the domain gap by incorporating driving knowledge to adaptively refine the generated pseudo-labels. Quantitative and qualitative results with equivalent or even more impressive performance compared to fully-supervised state-of-the-art approaches across all three public datasets demonstrate the effectiveness of the proposed method and the potential of this direction. The code will be made publicly available.

The accessibility of spatially distributed data, enabled by affordable sensors, field, and numerical experiments, has facilitated the development of data-driven solutions for scientific problems, including climate change, weather prediction, and urban planning. Neural Partial Differential Equations (Neural PDEs), which combine deep learning (DL) techniques with domain expertise (e.g., governing equations) for parameterization, have proven to be effective in capturing valuable correlations within spatiotemporal datasets. However, sparse and noisy measurements coupled with modeling approximation introduce aleatoric and epistemic uncertainties. Therefore, quantifying uncertainties propagated from model inputs to outputs remains a challenge and an essential goal for establishing the trustworthiness of Neural PDEs. This work evaluates various Uncertainty Quantification (UQ) approaches for both Forward and Inverse Problems in scientific applications. Specifically, we investigate the effectiveness of Bayesian methods, such as Hamiltonian Monte Carlo (HMC) and Monte-Carlo Dropout (MCD), and a more conventional approach, Deep Ensembles (DE). To illustrate their performance, we take two canonical PDEs: Burger's equation and the Navier-Stokes equation. Our results indicate that Neural PDEs can effectively reconstruct flow systems and predict the associated unknown parameters. However, it is noteworthy that the results derived from Bayesian methods, based on our observations, tend to display a higher degree of certainty in their predictions as compared to those obtained using the DE. This elevated certainty in predictions suggests that Bayesian techniques might underestimate the true underlying uncertainty, thereby appearing more confident in their predictions than the DE approach.

Uncertainty estimation is an essential and heavily-studied component for the reliable application of semantic segmentation methods. While various studies exist claiming methodological advances on the one hand, and successful application on the other hand, the field is currently hampered by a gap between theory and practice leaving fundamental questions unanswered: Can data-related and model-related uncertainty really be separated in practice? Which components of an uncertainty method are essential for real-world performance? Which uncertainty method works well for which application? In this work, we link this research gap to a lack of systematic and comprehensive evaluation of uncertainty methods. Specifically, we identify three key pitfalls in current literature and present an evaluation framework that bridges the research gap by providing 1) a controlled environment for studying data ambiguities as well as distribution shifts, 2) systematic ablations of relevant method components, and 3) test-beds for the five predominant uncertainty applications: OoD-detection, active learning, failure detection, calibration, and ambiguity modeling. Empirical results on simulated as well as real-world data demonstrate how the proposed framework is able to answer the predominant questions in the field revealing for instance that 1) separation of uncertainty types works on simulated data but does not necessarily translate to real-world data, 2) aggregation of scores is a crucial but currently neglected component of uncertainty methods, 3) While ensembles are performing most robustly across the different downstream tasks and settings, test-time augmentation often constitutes a light-weight alternative. Code is at: https://github.com/IML-DKFZ/values

Accurate ocean modeling and coastal hazard prediction depend on high-resolution bathymetric data; yet, current worldwide datasets are too coarse for exact numerical simulations. While recent deep learning advances have improved earth observation data resolution, existing methods struggle with the unique challenges of producing detailed ocean floor maps, especially in maintaining physical structure consistency and quantifying uncertainties. This work presents a novel uncertainty-aware mechanism using spatial blocks to efficiently capture local bathymetric complexity based on block-based conformal prediction. Using the Vector Quantized Variational Autoencoder (VQ-VAE) architecture, the integration of this uncertainty quantification framework yields spatially adaptive confidence estimates while preserving topographical features via discrete latent representations. With smaller uncertainty widths in well-characterized areas and appropriately larger bounds in areas of complex seafloor structures, the block-based design adapts uncertainty estimates to local bathymetric complexity. Compared to conventional techniques, experimental results over several ocean regions show notable increases in both reconstruction quality and uncertainty estimation reliability. This framework increases the reliability of bathymetric reconstructions by preserving structural integrity while offering spatially adaptive uncertainty estimates, so opening the path for more solid climate modeling and coastal hazard assessment.

Neural Radiance Fields (NeRFs) have shown promise in applications like view synthesis and depth estimation, but learning from multiview images faces inherent uncertainties. Current methods to quantify them are either heuristic or computationally demanding. We introduce BayesRays, a post-hoc framework to evaluate uncertainty in any pre-trained NeRF without modifying the training process. Our method establishes a volumetric uncertainty field using spatial perturbations and a Bayesian Laplace approximation. We derive our algorithm statistically and show its superior performance in key metrics and applications. Additional results available at: https://bayesrays.github.io.

Semi-supervised learning relaxes the need of large pixel-wise labeled datasets for image segmentation by leveraging unlabeled data. A prominent way to exploit unlabeled data is to regularize model predictions. Since the predictions of unlabeled data can be unreliable, uncertainty-aware schemes are typically employed to gradually learn from meaningful and reliable predictions. Uncertainty estimation methods, however, rely on multiple inferences from the model predictions that must be computed for each training step, which is computationally expensive. Moreover, these uncertainty maps capture pixel-wise disparities and do not consider global information. This work proposes a novel method to estimate segmentation uncertainty by leveraging global information from the segmentation masks. More precisely, an anatomically-aware representation is first learnt to model the available segmentation masks. The learnt representation thereupon maps the prediction of a new segmentation into an anatomically-plausible segmentation. The deviation from the plausible segmentation aids in estimating the underlying pixel-level uncertainty in order to further guide the segmentation network. The proposed method consequently estimates the uncertainty using a single inference from our representation, thereby reducing the total computation. We evaluate our method on two publicly available segmentation datasets of left atria in cardiac MRIs and of multiple organs in abdominal CTs. Our anatomically-aware method improves the segmentation accuracy over the state-of-the-art semi-supervised methods in terms of two commonly used evaluation metrics.

Uncertainty is almost ubiquitous in safety-critical autonomous systems due to dynamic environments and the integration of learning-based components. Quantifying this uncertainty--particularly for time-series predictions in multi-stage optimization--is essential for safe control and verification tasks. Conformal Prediction (CP) is a distribution-free uncertainty quantification tool with rigorous finite-sample guarantees, but its performance relies on the design of the nonconformity measure, which remains challenging for time-series data. Existing methods either overfit on small datasets, or are computationally intensive on long-time-horizon problems and/or large datasets. To overcome these issues, we propose a new parameterization of the score functions and formulate an optimization program to compute the associated parameters. The optimal parameters directly lead to norm-ball regions that constitute minimal-average-radius conformal sets. We then provide a reformulation of the underlying optimization program to enable faster computation. We provide theoretical proofs on both the validity and efficiency of predictors constructed based on the proposed approach. Numerical results on various case studies demonstrate that our method outperforms state-of-the-art methods in terms of efficiency, with much lower computational requirements.

This paper presents a novel end-to-end framework for closed-form computation and visualization of critical point uncertainty in 2D uncertain scalar fields. Critical points are fundamental topological descriptors used in the visualization and analysis of scalar fields. The uncertainty inherent in data (e.g., observational and experimental data, approximations in simulations, and compression), however, creates uncertainty regarding critical point positions. Uncertainty in critical point positions, therefore, cannot be ignored, given their impact on downstream data analysis tasks. In this work, we study uncertainty in critical points as a function of uncertainty in data modeled with probability distributions. Although Monte Carlo (MC) sampling techniques have been used in prior studies to quantify critical point uncertainty, they are often expensive and are infrequently used in production-quality visualization software. We, therefore, propose a new end-to-end framework to address these challenges that comprises a threefold contribution. First, we derive the critical point uncertainty in closed form, which is more accurate and efficient than the conventional MC sampling methods. Specifically, we provide the closed-form and semianalytical (a mix of closed-form and MC methods) solutions for parametric (e.g., uniform, Epanechnikov) and nonparametric models (e.g., histograms) with finite support. Second, we accelerate critical point probability computations using a parallel implementation with the VTK-m library, which is platform portable. Finally, we demonstrate the integration of our implementation with the ParaView software system to demonstrate near-real-time results for real datasets.

Uncertainty quantification (UQ) is a critical aspect of artificial intelligence (AI) systems, particularly in high-risk domains such as healthcare, autonomous systems, and financial technology, where decision-making processes must account for uncertainty. This review explores the evolution of uncertainty quantification techniques in AI, distinguishing between aleatoric and epistemic uncertainties, and discusses the mathematical foundations and methods used to quantify these uncertainties. We provide an overview of advanced techniques, including probabilistic methods, ensemble learning, sampling-based approaches, and generative models, while also highlighting hybrid approaches that integrate domain-specific knowledge. Furthermore, we examine the diverse applications of UQ across various fields, emphasizing its impact on decision-making, predictive accuracy, and system robustness. The review also addresses key challenges such as scalability, efficiency, and integration with explainable AI, and outlines future directions for research in this rapidly developing area. Through this comprehensive survey, we aim to provide a deeper understanding of UQ's role in enhancing the reliability, safety, and trustworthiness of AI systems.

The Segment Anything Model (SAM) has revolutionized interactive segmentation through spatial prompting. While existing work primarily focuses on automating prompts in various settings, real-world annotation workflows involve iterative refinement where annotators observe model outputs and strategically place prompts to resolve ambiguities. Current pipelines typically rely on the annotator's visual assessment of the predicted mask quality. We postulate that a principled approach for automated interactive prompting is to use a model-derived criterion to identify the most informative region for the next prompt. In this work, we establish active prompting: a spatial active learning approach where locations within images constitute an unlabeled pool and prompts serve as queries to prioritize information-rich regions, increasing the utility of each interaction. We further present BALD-SAM: a principled framework adapting Bayesian Active Learning by Disagreement (BALD) to spatial prompt selection by quantifying epistemic uncertainty. To do so, we freeze the entire model and apply Bayesian uncertainty modeling only to a small learned prediction head, making intractable uncertainty estimation practical for large multi-million parameter foundation models. Across 16 datasets spanning natural, medical, underwater, and seismic domains, BALD-SAM demonstrates strong cross-domain performance, ranking first or second on 14 of 16 benchmarks. We validate these gains through a comprehensive ablation suite covering 3 SAM backbones and 35 Laplace posterior configurations, amounting to 38 distinct ablation settings. Beyond strong average performance, BALD-SAM surpasses human prompting and, in several categories, even oracle prompting, while consistently outperforming one-shot baselines in final segmentation quality, particularly on thin and structurally complex objects.

We study uncertainty quantification for partial differential equations subject to domain uncertainty. We parameterize the random domain using the model recently considered by Chernov and Le (2024) as well as Harbrecht, Schmidlin, and Schwab (2024) in which the input random field is assumed to belong to a Gevrey smoothness class. This approach has the advantage of being substantially more general than models which assume a particular parametric representation of the input random field such as a Karhunen-Loeve series expansion. We consider both the Poisson equation as well as the heat equation and design randomly shifted lattice quasi-Monte Carlo (QMC) cubature rules for the computation of the expected solution under domain uncertainty. We show that these QMC rules exhibit dimension-independent, essentially linear cubature convergence rates in this framework. In addition, we complete the error analysis by taking into account the approximation errors incurred by dimension truncation of the random input field and finite element discretization. Numerical experiments are presented to confirm the theoretical rates.

Accurate estimation of uncertainty in deep learning is critical for deploying models in high-stakes domains such as medical diagnosis and autonomous decision-making, where overconfident predictions can lead to harmful outcomes. In practice, understanding the reason behind a model's uncertainty and the type of uncertainty it represents can support risk-aware decisions, enhance user trust, and guide additional data collection. However, many existing methods only address a single type of uncertainty or require modifications and retraining of the base model, making them difficult to adopt in real-world systems. We introduce CUPID (Comprehensive Uncertainty Plug-in estImation moDel), a general-purpose module that jointly estimates aleatoric and epistemic uncertainty without modifying or retraining the base model. CUPID can be flexibly inserted into any layer of a pretrained network. It models aleatoric uncertainty through a learned Bayesian identity mapping and captures epistemic uncertainty by analyzing the model's internal responses to structured perturbations. We evaluate CUPID across a range of tasks, including classification, regression, and out-of-distribution detection. The results show that it consistently delivers competitive performance while offering layer-wise insights into the origins of uncertainty. By making uncertainty estimation modular, interpretable, and model-agnostic, CUPID supports more transparent and trustworthy AI. Related code and data are available at https://github.com/a-Fomalhaut-a/CUPID.

There are two major types of uncertainty one can model. Aleatoric uncertainty captures noise inherent in the observations. On the other hand, epistemic uncertainty accounts for uncertainty in the model -- uncertainty which can be explained away given enough data. Traditionally it has been difficult to model epistemic uncertainty in computer vision, but with new Bayesian deep learning tools this is now possible. We study the benefits of modeling epistemic vs. aleatoric uncertainty in Bayesian deep learning models for vision tasks. For this we present a Bayesian deep learning framework combining input-dependent aleatoric uncertainty together with epistemic uncertainty. We study models under the framework with per-pixel semantic segmentation and depth regression tasks. Further, our explicit uncertainty formulation leads to new loss functions for these tasks, which can be interpreted as learned attenuation. This makes the loss more robust to noisy data, also giving new state-of-the-art results on segmentation and depth regression benchmarks.

Deep learning tools have gained tremendous attention in applied machine learning. However such tools for regression and classification do not capture model uncertainty. In comparison, Bayesian models offer a mathematically grounded framework to reason about model uncertainty, but usually come with a prohibitive computational cost. In this paper we develop a new theoretical framework casting dropout training in deep neural networks (NNs) as approximate Bayesian inference in deep Gaussian processes. A direct result of this theory gives us tools to model uncertainty with dropout NNs -- extracting information from existing models that has been thrown away so far. This mitigates the problem of representing uncertainty in deep learning without sacrificing either computational complexity or test accuracy. We perform an extensive study of the properties of dropout's uncertainty. Various network architectures and non-linearities are assessed on tasks of regression and classification, using MNIST as an example. We show a considerable improvement in predictive log-likelihood and RMSE compared to existing state-of-the-art methods, and finish by using dropout's uncertainty in deep reinforcement learning.

Predicting not only the target but also an accurate measure of uncertainty is important for many machine learning applications and in particular safety-critical ones. In this work we study the calibration of uncertainty prediction for regression tasks which often arise in real-world systems. We show that the existing definition for calibration of a regression uncertainty [Kuleshov et al. 2018] has severe limitations in distinguishing informative from non-informative uncertainty predictions. We propose a new definition that escapes this caveat and an evaluation method using a simple histogram-based approach. Our method clusters examples with similar uncertainty prediction and compares the prediction with the empirical uncertainty on these examples. We also propose a simple, scaling-based calibration method that preforms as well as much more complex ones. We show results on both a synthetic, controlled problem and on the object detection bounding-box regression task using the COCO and KITTI datasets.

Foundation models for segmentation such as the Segment Anything Model (SAM) family exhibit strong zero-shot performance, but remain vulnerable in shifted or limited-knowledge domains. This work investigates whether uncertainty quantification can mitigate such challenges and enhance model generalisability in a domain-agnostic manner. To this end, we (1) curate UncertSAM, a benchmark comprising eight datasets designed to stress-test SAM under challenging segmentation conditions including shadows, transparency, and camouflage; (2) evaluate a suite of lightweight, post-hoc uncertainty estimation methods; and (3) assess a preliminary uncertainty-guided prediction refinement step. Among evaluated approaches, a last-layer Laplace approximation yields uncertainty estimates that correlate well with segmentation errors, indicating a meaningful signal. While refinement benefits are preliminary, our findings underscore the potential of incorporating uncertainty into segmentation models to support robust, domain-agnostic performance. Our benchmark and code are made publicly available.

Uncertainty estimation is essential for the safe clinical deployment of medical image segmentation systems, enabling the identification of unreliable predictions and supporting human oversight. While prior work has largely focused on pixel-level uncertainty, landmark-based segmentation offers inherent topological guarantees yet remains underexplored from an uncertainty perspective. In this work, we study uncertainty estimation for anatomical landmark-based segmentation on chest X-rays. Inspired by hybrid neural network architectures that combine standard image convolutional encoders with graph-based generative decoders, and leveraging their variational latent space, we derive two complementary measures: (i) latent uncertainty, captured directly from the learned distribution parameters, and (ii) predictive uncertainty, obtained by generating multiple stochastic output predictions from latent samples. Through controlled corruption experiments we show that both uncertainty measures increase with perturbation severity, reflecting both global and local degradation. We demonstrate that these uncertainty signals can identify unreliable predictions by comparing with manual ground-truth, and support out-of-distribution detection on the CheXmask dataset. More importantly, we release CheXmask-U (huggingface.co/datasets/mcosarinsky/CheXmask-U), a large scale dataset of 657,566 chest X-ray landmark segmentations with per-node uncertainty estimates, enabling researchers to account for spatial variations in segmentation quality when using these anatomical masks. Our findings establish uncertainty estimation as a promising direction to enhance robustness and safe deployment of landmark-based anatomical segmentation methods in chest X-ray. A fully working interactive demo of the method is available at huggingface.co/spaces/matiasky/CheXmask-U and the source code at github.com/mcosarinsky/CheXmask-U.

Automated segmentation of BUS images is important for precise lesion delineation and tumor characterization, but is challenged by inherent artifacts and dataset inconsistencies. In this work, we evaluate the use of a modified Residual Encoder U-Net for breast ultrasound segmentation, with a focus on uncertainty quantification. We identify and correct for data duplication in the BUSI dataset, and use a deduplicated subset for more reliable estimates of generalization performance. Epistemic uncertainty is quantified using Monte Carlo dropout, deep ensembles, and their combination. Models are benchmarked on both in-distribution and out-of-distribution datasets to demonstrate how they generalize to unseen cross-domain data. Our approach achieves state-of-the-art segmentation accuracy on the Breast-Lesion-USG dataset with in-distribution validation, and provides calibrated uncertainty estimates that effectively signal regions of low model confidence. Performance declines and increased uncertainty observed in out-of-distribution evaluation highlight the persistent challenge of domain shift in medical imaging, and the importance of integrated uncertainty modeling for trustworthy clinical deployment. Code available at: https://github.com/toufiqmusah/nn-uncertainty.git

We address the neural network robustness problem by adding Similarity (i.e., correctly predicted depth-matches into training)-awareness and Distance-to-training-distribution-awareness to the existing output Magnitude (i.e., decision-boundary)-awareness of the softmax function. The resulting SDM activation function provides strong signals of the relative epistemic (reducible) predictive uncertainty. We use this novel behavior to further address the complementary HCI problem of mapping the output to human-interpretable summary statistics over relevant partitions of a held-out calibration set. Estimates of prediction-conditional uncertainty are obtained via a parsimonious learned transform over the class-conditional empirical CDFs of the output of a final-layer SDM activation function. For decision-making and as an intrinsic model check, estimates of class-conditional accuracy are obtained by further partitioning the high-probability regions of this calibrated output into class-conditional, region-specific CDFs. The uncertainty estimates from SDM calibration are remarkably robust to test-time distribution shifts and out-of-distribution inputs; incorporate awareness of the effective sample size; provide estimates of uncertainty from the learning and data splitting processes; and are well-suited for selective classification and conditional branching for additional test-time compute based on the predictive uncertainty, as for selective LLM generation, routing, and composition over multiple models and retrieval. Finally, we construct SDM networks, LLMs with uncertainty-aware verification and interpretability-by-exemplar as intrinsic properties. We provide open-source software implementing these results.

Quantifying uncertainty is important for actionable predictions in real-world applications. A crucial part of predictive uncertainty quantification is the estimation of epistemic uncertainty, which is defined as an integral of the product between a divergence function and the posterior. Current methods such as Deep Ensembles or MC dropout underperform at estimating the epistemic uncertainty, since they primarily consider the posterior when sampling models. We suggest Quantification of Uncertainty with Adversarial Models (QUAM) to better estimate the epistemic uncertainty. QUAM identifies regions where the whole product under the integral is large, not just the posterior. Consequently, QUAM has lower approximation error of the epistemic uncertainty compared to previous methods. Models for which the product is large correspond to adversarial models (not adversarial examples!). Adversarial models have both a high posterior as well as a high divergence between their predictions and that of a reference model. Our experiments show that QUAM excels in capturing epistemic uncertainty for deep learning models and outperforms previous methods on challenging tasks in the vision domain.

Popular approaches for quantifying predictive uncertainty in deep neural networks often involve distributions over weights or multiple models, for instance via Markov Chain sampling, ensembling, or Monte Carlo dropout. These techniques usually incur overhead by having to train multiple model instances or do not produce very diverse predictions. This comprehensive and extensive survey aims to familiarize the reader with an alternative class of models based on the concept of Evidential Deep Learning: For unfamiliar data, they aim to admit "what they don't know", and fall back onto a prior belief. Furthermore, they allow uncertainty estimation in a single model and forward pass by parameterizing distributions over distributions. This survey recapitulates existing works, focusing on the implementation in a classification setting, before surveying the application of the same paradigm to regression. We also reflect on the strengths and weaknesses compared to other existing methods and provide the most fundamental derivations using a unified notation to aid future research.

Bayesian neural networks (BNN) and deep ensembles are principled approaches to estimate the predictive uncertainty of a deep learning model. However their practicality in real-time, industrial-scale applications are limited due to their heavy memory and inference cost. This motivates us to study principled approaches to high-quality uncertainty estimation that require only a single deep neural network (DNN). By formalizing the uncertainty quantification as a minimax learning problem, we first identify input distance awareness, i.e., the model's ability to quantify the distance of a testing example from the training data in the input space, as a necessary condition for a DNN to achieve high-quality (i.e., minimax optimal) uncertainty estimation. We then propose Spectral-normalized Neural Gaussian Process (SNGP), a simple method that improves the distance-awareness ability of modern DNNs, by adding a weight normalization step during training and replacing the output layer with a Gaussian process. On a suite of vision and language understanding tasks and on modern architectures (Wide-ResNet and BERT), SNGP is competitive with deep ensembles in prediction, calibration and out-of-domain detection, and outperforms the other single-model approaches.

Financial time-series forecasting is a high-stakes problem where regime shifts and shocks make point-accurate yet overconfident models dangerous. We propose Uncertainty-Gated Generative Modeling (UGGM), which treats uncertainty as an internal control signal that gates (i) representation via gated reparameterization, (ii) propagation via similarity and confidence routing, and (iii) generation via uncertainty-controlled predictive distributions, together with uncertainty-driven regularization and calibration to curb miscalibration. Instantiated on Weak Innovation AutoEncoder (WIAE-GPF), our UG-WIAE-GPF significantly improves risk-sensitive forecasting, delivering a 63.5\% MSE reduction on NYISO (0.3508 rightarrow 0.1281), with improved robustness under shock intervals (mSE: 0.2739 rightarrow 0.1748).

Vision-language models (VLMs), such as CLIP and SigLIP, have found remarkable success in classification, retrieval, and generative tasks. For this, VLMs deterministically map images and text descriptions to a joint latent space in which their similarity is assessed using the cosine similarity. However, a deterministic mapping of inputs fails to capture uncertainties over concepts arising from domain shifts when used in downstream tasks. In this work, we propose post-hoc uncertainty estimation in VLMs that does not require additional training. Our method leverages a Bayesian posterior approximation over the last layers in VLMs and analytically quantifies uncertainties over cosine similarities. We demonstrate its effectiveness for uncertainty quantification and support set selection in active learning. Compared to baselines, we obtain improved and well-calibrated predictive uncertainties, interpretable uncertainty estimates, and sample-efficient active learning. Our results show promise for safety-critical applications of large-scale models.

While Large Language Models (LLMs) demonstrate impressive capabilities, they still struggle with generating factually incorrect content (i.e., hallucinations). A promising approach to mitigate this issue is enabling models to express uncertainty when unsure. Previous research on uncertainty modeling has primarily focused on short-form QA, but realworld applications often require much longer responses. In this work, we introduce the task of Long-form Generation with Uncertainty(LoGU). We identify two key challenges: Uncertainty Suppression, where models hesitate to express uncertainty, and Uncertainty Misalignment, where models convey uncertainty inaccurately. To tackle these challenges, we propose a refinement-based data collection framework and a two-stage training pipeline. Our framework adopts a divide-and-conquer strategy, refining uncertainty based on atomic claims. The collected data are then used in training through supervised fine-tuning (SFT) and direct preference optimization (DPO) to enhance uncertainty expression. Extensive experiments on three long-form instruction following datasets show that our method significantly improves accuracy, reduces hallucinations, and maintains the comprehensiveness of responses.

We present Δ-UQ -- a novel, general-purpose uncertainty estimator using the concept of anchoring in predictive models. Anchoring works by first transforming the input into a tuple consisting of an anchor point drawn from a prior distribution, and a combination of the input sample with the anchor using a pretext encoding scheme. This encoding is such that the original input can be perfectly recovered from the tuple -- regardless of the choice of the anchor. Therefore, any predictive model should be able to predict the target response from the tuple alone (since it implicitly represents the input). Moreover, by varying the anchors for a fixed sample, we can estimate uncertainty in the prediction even using only a single predictive model. We find this uncertainty is deeply connected to improper sampling of the input data, and inherent noise, enabling us to estimate the total uncertainty in any system. With extensive empirical studies on a variety of use-cases, we demonstrate that Δ-UQ outperforms several competitive baselines. Specifically, we study model fitting, sequential model optimization, model based inversion in the regression setting and out of distribution detection, & calibration under distribution shifts for classification.

Most mobile robots for indoor use rely on 2D laser scanners for localization, mapping and navigation. These sensors, however, cannot detect transparent surfaces or measure the full occupancy of complex objects such as tables. Deep Neural Networks have recently been proposed to overcome this limitation by learning to estimate object occupancy. These estimates are nevertheless subject to uncertainty, making the evaluation of their confidence an important issue for these measures to be useful for autonomous navigation and mapping. In this work we approach the problem from two sides. First we discuss uncertainty estimation in deep models, proposing a solution based on a fully convolutional neural network. The proposed architecture is not restricted by the assumption that the uncertainty follows a Gaussian model, as in the case of many popular solutions for deep model uncertainty estimation, such as Monte-Carlo Dropout. We present results showing that uncertainty over obstacle distances is actually better modeled with a Laplace distribution. Then, we propose a novel approach to build maps based on Deep Neural Network uncertainty models. In particular, we present an algorithm to build a map that includes information over obstacle distance estimates while taking into account the level of uncertainty in each estimate. We show how the constructed map can be used to increase global navigation safety by planning trajectories which avoid areas of high uncertainty, enabling higher autonomy for mobile robots in indoor settings.

In recent years, the neural implicit surface has emerged as a powerful representation for multi-view surface reconstruction due to its simplicity and state-of-the-art performance. However, reconstructing smooth and detailed surfaces in indoor scenes from multi-view images presents unique challenges. Indoor scenes typically contain large texture-less regions, making the photometric loss unreliable for optimizing the implicit surface. Previous work utilizes monocular geometry priors to improve the reconstruction in indoor scenes. However, monocular priors often contain substantial errors in thin structure regions due to domain gaps and the inherent inconsistencies when derived independently from different views. This paper presents DebSDF to address these challenges, focusing on the utilization of uncertainty in monocular priors and the bias in SDF-based volume rendering. We propose an uncertainty modeling technique that associates larger uncertainties with larger errors in the monocular priors. High-uncertainty priors are then excluded from optimization to prevent bias. This uncertainty measure also informs an importance-guided ray sampling and adaptive smoothness regularization, enhancing the learning of fine structures. We further introduce a bias-aware signed distance function to density transformation that takes into account the curvature and the angle between the view direction and the SDF normals to reconstruct fine details better. Our approach has been validated through extensive experiments on several challenging datasets, demonstrating improved qualitative and quantitative results in reconstructing thin structures in indoor scenes, thereby outperforming previous work.

We develop analytical and particle-based methods for uncertainty propagation in random neural network models, where both the inputs and network parameters are allowed to be random. Building on the piecewise-linear structure of the Leaky ReLU activation function, we derive a local approximation of the neural network output with respect to perturbations in both its inputs and parameters. This approximation is exact for perturbations that preserve the network activation pattern, and it allows us to compute analytical expressions for the probability density function and characteristic function of the network output, together with closed-form approximations for its mean and covariance. We extend this uncertainty propagation framework to autonomous dynamical systems whose one-step evolution map is represented by a random neural network. Repeated application of this map defines an autoregressive model, for which we derive recursive equations to propagate uncertainty in both the state and network parameters over time. These equations explicitly account for the state-parameter cross-covariance that develops under successive iterations of the network. Numerical experiments on the Lorenz-63 system and the Kuramoto-Sivashinsky equation demonstrate accurate uncertainty propagation through the predictability horizon and the applicability of the proposed framework to high-dimensional dynamical systems.

We investigate composed image retrieval with text feedback. Users gradually look for the target of interest by moving from coarse to fine-grained feedback. However, existing methods merely focus on the latter, i.e., fine-grained search, by harnessing positive and negative pairs during training. This pair-based paradigm only considers the one-to-one distance between a pair of specific points, which is not aligned with the one-to-many coarse-grained retrieval process and compromises the recall rate. In an attempt to fill this gap, we introduce a unified learning approach to simultaneously modeling the coarse- and fine-grained retrieval by considering the multi-grained uncertainty. The key idea underpinning the proposed method is to integrate fine- and coarse-grained retrieval as matching data points with small and large fluctuations, respectively. Specifically, our method contains two modules: uncertainty modeling and uncertainty regularization. (1) The uncertainty modeling simulates the multi-grained queries by introducing identically distributed fluctuations in the feature space. (2) Based on the uncertainty modeling, we further introduce uncertainty regularization to adapt the matching objective according to the fluctuation range. Compared with existing methods, the proposed strategy explicitly prevents the model from pushing away potential candidates in the early stage, and thus improves the recall rate. On the three public datasets, i.e., FashionIQ, Fashion200k, and Shoes, the proposed method has achieved +4.03%, +3.38%, and +2.40% Recall@50 accuracy over a strong baseline, respectively.

High-quality calibrated uncertainty estimates are crucial for numerous real-world applications, especially for deep learning-based deployed ML systems. While Bayesian deep learning techniques allow uncertainty estimation, training them with large-scale datasets is an expensive process that does not always yield models competitive with non-Bayesian counterparts. Moreover, many of the high-performing deep learning models that are already trained and deployed are non-Bayesian in nature and do not provide uncertainty estimates. To address these issues, we propose BayesCap that learns a Bayesian identity mapping for the frozen model, allowing uncertainty estimation. BayesCap is a memory-efficient method that can be trained on a small fraction of the original dataset, enhancing pretrained non-Bayesian computer vision models by providing calibrated uncertainty estimates for the predictions without (i) hampering the performance of the model and (ii) the need for expensive retraining the model from scratch. The proposed method is agnostic to various architectures and tasks. We show the efficacy of our method on a wide variety of tasks with a diverse set of architectures, including image super-resolution, deblurring, inpainting, and crucial application such as medical image translation. Moreover, we apply the derived uncertainty estimates to detect out-of-distribution samples in critical scenarios like depth estimation in autonomous driving. Code is available at https://github.com/ExplainableML/BayesCap.

End-to-end (E2E) driving has become a cornerstone of both industry deployment and academic research, offering a single learnable pipeline that maps multi-sensor inputs to actions while avoiding hand-engineered modules. However, the reliability of such pipelines strongly depends on how well they handle uncertainty: sensors are noisy, semantics can be ambiguous, and interaction with other road users is inherently stochastic. Uncertainty also appears in multiple forms: classification vs. localization, and, crucially, in both static map elements and dynamic agents. Existing E2E approaches model only static-map uncertainty, leaving planning vulnerable to overconfident and unreliable inputs. We present UniUncer, the first lightweight, unified uncertainty framework that jointly estimates and uses uncertainty for both static and dynamic scene elements inside an E2E planner. Concretely: (1) we convert deterministic heads to probabilistic Laplace regressors that output per-vertex location and scale for vectorized static and dynamic entities; (2) we introduce an uncertainty-fusion module that encodes these parameters and injects them into object/map queries to form uncertainty-aware queries; and (3) we design an uncertainty-aware gate that adaptively modulates reliance on historical inputs (ego status or temporal perception queries) based on current uncertainty levels. The design adds minimal overhead and drops throughput by only sim0.5 FPS while remaining plug-and-play for common E2E backbones. On nuScenes (open-loop), UniUncer reduces average L2 trajectory error by 7\%. On NavsimV2 (pseudo closed-loop), it improves overall EPDMS by 10.8\% and notable stage two gains in challenging, interaction-heavy scenes. Ablations confirm that dynamic-agent uncertainty and the uncertainty-aware gate are both necessary.

Weather forecasting is fundamentally challenged by the chaotic nature of the atmosphere, necessitating probabilistic approaches to quantify uncertainty. While traditional ensemble prediction (EPS) addresses this through computationally intensive simulations, recent advances in Bayesian Deep Learning (BDL) offer a promising but often disconnected alternative. We bridge these paradigms through a unified hybrid Bayesian Deep Learning framework for ensemble weather forecasting that explicitly decomposes predictive uncertainty into epistemic and aleatoric components, learned via variational inference and a physics-informed stochastic perturbation scheme modeling flow-dependent atmospheric dynamics, respectively. We further establish a unified theoretical framework that rigorously connects BDL and EPS, providing formal theorems that decompose total predictive uncertainty into epistemic and aleatoric components under the hybrid BDL framework. We validate our framework on the large-scale 40-year ERA5 reanalysis dataset (1979-2019) with 0.25° spatial resolution. Experimental results show that our method not only improves forecast accuracy and yields better-calibrated uncertainty quantification but also achieves superior computational efficiency compared to state-of-the-art probabilistic diffusion models. We commit to making our code open-source upon acceptance of this paper.

Black-box machine learning models are now routinely used in high-risk settings, like medical diagnostics, which demand uncertainty quantification to avoid consequential model failures. Conformal prediction is a user-friendly paradigm for creating statistically rigorous uncertainty sets/intervals for the predictions of such models. Critically, the sets are valid in a distribution-free sense: they possess explicit, non-asymptotic guarantees even without distributional assumptions or model assumptions. One can use conformal prediction with any pre-trained model, such as a neural network, to produce sets that are guaranteed to contain the ground truth with a user-specified probability, such as 90%. It is easy-to-understand, easy-to-use, and general, applying naturally to problems arising in the fields of computer vision, natural language processing, deep reinforcement learning, and so on. This hands-on introduction is aimed to provide the reader a working understanding of conformal prediction and related distribution-free uncertainty quantification techniques with one self-contained document. We lead the reader through practical theory for and examples of conformal prediction and describe its extensions to complex machine learning tasks involving structured outputs, distribution shift, time-series, outliers, models that abstain, and more. Throughout, there are many explanatory illustrations, examples, and code samples in Python. With each code sample comes a Jupyter notebook implementing the method on a real-data example; the notebooks can be accessed and easily run using our codebase.

We present a scheme to obtain an inexpensive and reliable estimate of the uncertainty associated with the predictions of a machine-learning model of atomic and molecular properties. The scheme is based on resampling, with multiple models being generated based on sub-sampling of the same training data. The accuracy of the uncertainty prediction can be benchmarked by maximum likelihood estimation, which can also be used to correct for correlations between resampled models, and to improve the performance of the uncertainty estimation by a cross-validation procedure. In the case of sparse Gaussian Process Regression models, this resampled estimator can be evaluated at negligible cost. We demonstrate the reliability of these estimates for the prediction of molecular energetics, and for the estimation of nuclear chemical shieldings in molecular crystals. Extension to estimate the uncertainty in energy differences, forces, or other correlated predictions is straightforward. This method can be easily applied to other machine learning schemes, and will be beneficial to make data-driven predictions more reliable, and to facilitate training-set optimization and active-learning strategies.

The analysis of parametric and non-parametric uncertainties of very large dynamical systems requires the construction of a stochastic model of said system. Linear approaches relying on random matrix theory and principal componant analysis can be used when systems undergo low-frequency vibrations. In the case of fast dynamics and wave propagation, we investigate a random generator of boundary conditions for fast submodels by using machine learning. We show that the use of non-linear techniques in machine learning and data-driven methods is highly relevant. Physics-informed neural networks is a possible choice for a data-driven method to replace linear modal analysis. An architecture that support a random component is necessary for the construction of the stochastic model of the physical system for non-parametric uncertainties, since the goal is to learn the underlying probabilistic distribution of uncertainty in the data. Generative Adversarial Networks (GANs) are suited for such applications, where the Wasserstein-GAN with gradient penalty variant offers improved convergence results for our problem. The objective of our approach is to train a GAN on data from a finite element method code (Fenics) so as to extract stochastic boundary conditions for faster finite element predictions on a submodel. The submodel and the training data have both the same geometrical support. It is a zone of interest for uncertainty quantification and relevant to engineering purposes. In the exploitation phase, the framework can be viewed as a randomized and parametrized simulation generator on the submodel, which can be used as a Monte Carlo estimator.

End-to-end weather forecasting systems produce skillful global gridded and station forecasts directly from raw Earth observations, replacing the numerical weather prediction pipeline, including data assimilation, at a fraction of its cost. These systems are deterministic and issue no uncertainty. Here we render the Aardvark Weather model probabilistic by attaching one stochastic mechanism to each component: learned, input-dependent noise at the observation encoder, capturing aleatoric uncertainty inherited from the observing system, and Monte Carlo dropout in the processor, capturing epistemic uncertainty in the learned dynamics. The resulting nested ensemble attributes forecast spread to the two sources through a law-of-total-variance decomposition, cross-checked by withholding observation streams. Probabilistic finetuning significantly improves the mean forecast, by 4.2% on average across variables and lead times. The ensemble is calibrated against ERA5 through the medium range (spread-skill ratio 0.98), keeps station RMSE within 2.4% of the deterministic model while beating it in CRPS at every lead time, and trails the operational ECMWF ensemble. The encoder branch behaves as observation-driven uncertainty. Component-attributed uncertainty makes end-to-end forecasts more transparent, a step toward observation-driven digital twins of the atmosphere.

Human motion forecasters are increasingly accurate and fast, but reliable deployment requires uncertainty estimates that are structured, calibrated, and efficient. Bayesian and ensemble-based uncertainty estimates often require repeated stochastic inference [15, 26], while conformal calibration alone does not provide an epistemic signal or preserve trajectory covariance structure [14, 50]. We introduce SPARC (Single-Pass Adaptive Risk Calibration), a Bayesian-conformal uncertainty layer for motion forecasting. A deterministic MLP backbone predicts the future mean, and a conjugate Bayesian last layer converts time-domain feature leverage into an analytic horizon-wise epistemic scale κ_t(x). This scale inflates a graph-temporal Gaussian covariance without changing its correlation structure, and split conformal calibration produces 95% marginal prediction tubes with finite-sample validity under exchangeability. The key interface is the structured factorization κ_t(x)Σ_{str,t}(x), which injects feature-space epistemic uncertainty into trajectory densities without Monte Carlo sampling. Across nine dataset-protocol blocks and deterministic, multimodal, and calibration baselines, SPARC ranks first on NLL and on the combined MPJPE+NLL criterion while retaining competitive point accuracy and efficient calibrated tubes. Ranking windows by κ separates high-error cases, making the scale usable as a lightweight risk monitor.

Calibrated uncertainty estimates in machine learning are crucial to many fields such as autonomous vehicles, medicine, and weather and climate forecasting. While there is extensive literature on uncertainty calibration for classification, the classification findings do not always translate to regression. As a result, modern models for predicting uncertainty in regression settings typically produce uncalibrated and overconfident estimates. To address these gaps, we present a calibration method for regression settings that does not assume a particular uncertainty distribution over the error: Calibrating Regression Uncertainty Distributions Empirically (CRUDE). CRUDE makes the weaker assumption that error distributions have a constant arbitrary shape across the output space, shifted by predicted mean and scaled by predicted standard deviation. We detail a theoretical connection between CRUDE and conformal inference. Across an extensive set of regression tasks, CRUDE demonstrates consistently sharper, better calibrated, and more accurate uncertainty estimates than state-of-the-art techniques.

Multimodal semantic understanding often has to deal with uncertainty, which means the obtained messages tend to refer to multiple targets. Such uncertainty is problematic for our interpretation, including inter- and intra-modal uncertainty. Little effort has studied the modeling of this uncertainty, particularly in pre-training on unlabeled datasets and fine-tuning in task-specific downstream datasets. In this paper, we project the representations of all modalities as probabilistic distributions via a Probability Distribution Encoder (PDE) by utilizing sequence-level interactions. Compared to the existing deterministic methods, such uncertainty modeling can convey richer multimodal semantic information and more complex relationships. Furthermore, we integrate uncertainty modeling with popular pre-training frameworks and propose suitable pre-training tasks: Distribution-based Vision-Language Contrastive learning (D-VLC), Distribution-based Masked Language Modeling (D-MLM), and Distribution-based Image-Text Matching (D-ITM). The fine-tuned models are applied to challenging downstream tasks, including image-text retrieval, visual question answering, visual reasoning, and visual entailment, and achieve state-of-the-art results.

Diffusion-based image super-resolution methods have demonstrated significant advantages over GAN-based approaches, particularly in terms of perceptual quality. Building upon a lengthy Markov chain, diffusion-based methods possess remarkable modeling capacity, enabling them to achieve outstanding performance in real-world scenarios. Unlike previous methods that focus on modifying the noise schedule or sampling process to enhance performance, our approach emphasizes the improved utilization of LR information. We find that different regions of the LR image can be viewed as corresponding to different timesteps in a diffusion process, where flat areas are closer to the target HR distribution but edge and texture regions are farther away. In these flat areas, applying a slight noise is more advantageous for the reconstruction. We associate this characteristic with uncertainty and propose to apply uncertainty estimate to guide region-specific noise level control, a technique we refer to as Uncertainty-guided Noise Weighting. Pixels with lower uncertainty (i.e., flat regions) receive reduced noise to preserve more LR information, therefore improving performance. Furthermore, we modify the network architecture of previous methods to develop our Uncertainty-guided Perturbation Super-Resolution (UPSR) model. Extensive experimental results demonstrate that, despite reduced model size and training overhead, the proposed UWSR method outperforms current state-of-the-art methods across various datasets, both quantitatively and qualitatively.

This study addresses the challenging problem of active view selection and uncertainty quantification within the domain of Radiance Fields. Neural Radiance Fields (NeRF) have greatly advanced image rendering and reconstruction, but the limited availability of 2D images poses uncertainties stemming from occlusions, depth ambiguities, and imaging errors. Efficiently selecting informative views becomes crucial, and quantifying NeRF model uncertainty presents intricate challenges. Existing approaches either depend on model architecture or are based on assumptions regarding density distributions that are not generally applicable. By leveraging Fisher Information, we efficiently quantify observed information within Radiance Fields without ground truth data. This can be used for the next best view selection and pixel-wise uncertainty quantification. Our method overcomes existing limitations on model architecture and effectiveness, achieving state-of-the-art results in both view selection and uncertainty quantification, demonstrating its potential to advance the field of Radiance Fields. Our method with the 3D Gaussian Splatting backend could perform view selections at 70 fps.

When using large language models (LLMs) in high-stakes applications, we need to know when we can trust their predictions. Some works argue that prompting high-performance LLMs is sufficient to produce calibrated uncertainties, while others introduce sampling methods that can be prohibitively expensive. In this work, we first argue that prompting on its own is insufficient to achieve good calibration and then show that fine-tuning on a small dataset of correct and incorrect answers can create an uncertainty estimate with good generalization and small computational overhead. We show that a thousand graded examples are sufficient to outperform baseline methods and that training through the features of a model is necessary for good performance and tractable for large open-source models when using LoRA. We also investigate the mechanisms that enable reliable LLM uncertainty estimation, finding that many models can be used as general-purpose uncertainty estimators, applicable not just to their own uncertainties but also the uncertainty of other models. Lastly, we show that uncertainty estimates inform human use of LLMs in human-AI collaborative settings through a user study.

Predictions made by deep learning models are prone to data perturbations, adversarial attacks, and out-of-distribution inputs. To build a trusted AI system, it is therefore critical to accurately quantify the prediction uncertainties. While current efforts focus on improving uncertainty quantification accuracy and efficiency, there is a need to identify uncertainty sources and take actions to mitigate their effects on predictions. Therefore, we propose to develop explainable and actionable Bayesian deep learning methods to not only perform accurate uncertainty quantification but also explain the uncertainties, identify their sources, and propose strategies to mitigate the uncertainty impacts. Specifically, we introduce a gradient-based uncertainty attribution method to identify the most problematic regions of the input that contribute to the prediction uncertainty. Compared to existing methods, the proposed UA-Backprop has competitive accuracy, relaxed assumptions, and high efficiency. Moreover, we propose an uncertainty mitigation strategy that leverages the attribution results as attention to further improve the model performance. Both qualitative and quantitative evaluations are conducted to demonstrate the effectiveness of our proposed methods.

Probabilistic circuits (PCs) are models that allow exact and tractable probabilistic inference. In contrast to neural networks, they are often assumed to be well-calibrated and robust to out-of-distribution (OOD) data. In this paper, we show that PCs are in fact not robust to OOD data, i.e., they don't know what they don't know. We then show how this challenge can be overcome by model uncertainty quantification. To this end, we propose tractable dropout inference (TDI), an inference procedure to estimate uncertainty by deriving an analytical solution to Monte Carlo dropout (MCD) through variance propagation. Unlike MCD in neural networks, which comes at the cost of multiple network evaluations, TDI provides tractable sampling-free uncertainty estimates in a single forward pass. TDI improves the robustness of PCs to distribution shift and OOD data, demonstrated through a series of experiments evaluating the classification confidence and uncertainty estimates on real-world data.

Researchers in explainable artificial intelligence have developed numerous methods for helping users understand the predictions of complex supervised learning models. By contrast, explaining the uncertainty of model outputs has received relatively little attention. We adapt the popular Shapley value framework to explain various types of predictive uncertainty, quantifying each feature's contribution to the conditional entropy of individual model outputs. We consider games with modified characteristic functions and find deep connections between the resulting Shapley values and fundamental quantities from information theory and conditional independence testing. We outline inference procedures for finite sample error rate control with provable guarantees, and implement efficient algorithms that perform well in a range of experiments on real and simulated data. Our method has applications to covariate shift detection, active learning, feature selection, and active feature-value acquisition.

Reliable probability estimation is of crucial importance in many real-world applications where there is inherent (aleatoric) uncertainty. Probability-estimation models are trained on observed outcomes (e.g. whether it has rained or not, or whether a patient has died or not), because the ground-truth probabilities of the events of interest are typically unknown. The problem is therefore analogous to binary classification, with the difference that the objective is to estimate probabilities rather than predicting the specific outcome. This work investigates probability estimation from high-dimensional data using deep neural networks. There exist several methods to improve the probabilities generated by these models but they mostly focus on model (epistemic) uncertainty. For problems with inherent uncertainty, it is challenging to evaluate performance without access to ground-truth probabilities. To address this, we build a synthetic dataset to study and compare different computable metrics. We evaluate existing methods on the synthetic data as well as on three real-world probability estimation tasks, all of which involve inherent uncertainty: precipitation forecasting from radar images, predicting cancer patient survival from histopathology images, and predicting car crashes from dashcam videos. We also give a theoretical analysis of a model for high-dimensional probability estimation which reproduces several of the phenomena evinced in our experiments. Finally, we propose a new method for probability estimation using neural networks, which modifies the training process to promote output probabilities that are consistent with empirical probabilities computed from the data. The method outperforms existing approaches on most metrics on the simulated as well as real-world data.

Fetal ultrasound standard-plane classification underpins reliable prenatal biometry and anomaly screening, yet real-world deployment is limited by domain shift, image noise, and poor calibration of predicted probabilities. This paper presents a practical framework for uncertainty-calibrated explainable AI in fetal plane classification. We synthesize uncertainty estimation methods (Monte Carlo dropout, deep ensembles, evidential learning, and conformal prediction) with post-hoc and uncertainty-aware explanations (Grad-CAM variants, LIME-style local surrogates, and uncertainty-weighted multi-resolution activation maps), and we map these components to a clinician-facing workflow. Using FETAL_PLANES_DB as a reference benchmark, we define a reporting protocol that couples accuracy with calibration and selective prediction, including expected calibration error, Brier score, coverage-risk curves, and structured error analysis with explanations. We also discuss integration points for quality control and human-in-the-loop review, where uncertainty flags trigger re-acquisition or expert confirmation. The goal is a reproducible, clinically aligned blueprint for building fetal ultrasound classifiers whose confidence and explanations remain trustworthy under noisy acquisition conditions.

Foundation models have become a dominant paradigm in machine learning, achieving remarkable performance across diverse tasks through large-scale pretraining. However, they often yield overconfident, uncalibrated predictions. The standard approach to quantifying epistemic uncertainty are ensembles of multiple independently trained models. But their computational cost scales linearly with ensemble size, making them impractical for large foundation models. We propose Singular Value Ensemble (SVE), a parameter-efficient implicit ensembling method. SVE builds on a simple, but powerful core assumption: namely, that the singular vectors of the weight matrices correspond to meaningful directions in the representation space. If the singular vectors are indeed meaningful (orthogonal) "knowledge directions", then a model ensemble can be obtained by modulating only how strongly each direction contributes to the output. Rather than learning new parameters for each ensemble member, we freeze the singular vectors and only train per-member singular values that rescale the contribution of each direction in that shared knowledge basis. Ensemble diversity emerges naturally during joint training as stochastic initialization and random batch sampling cause different members to converge to different combinations of the same underlying knowledge. SVE performs comparable to an explicit ensemble, while increasing the parameter count of the base model by <1%, making principled uncertainty estimation accessible in resource-constrained settings. We validate SVE on NLP and vision tasks with various different backbones and show that it improves calibration while maintaining predictive accuracy.

Reliable estimation of predictive uncertainty is crucial for machine learning applications, particularly in high-stakes scenarios where hedging against risks is essential. Despite its significance, there is no universal agreement on how to best quantify predictive uncertainty. In this work, we revisit core concepts to propose a framework for information-theoretic measures of predictive uncertainty. Our proposed framework categorizes predictive uncertainty measures according to two factors: (I) The predicting model (II) The approximation of the true predictive distribution. Examining all possible combinations of these two factors, we derive a set of predictive uncertainty measures that includes both known and newly introduced ones. We extensively evaluate these measures across a broad set of tasks, identifying conditions under which certain measures excel. Our findings show the importance of aligning the choice of uncertainty measure with the predicting model on in-distribution (ID) data, the limitations of epistemic uncertainty measures for out-of-distribution (OOD) data, and that the disentanglement between measures varies substantially between ID and OOD data. Together, these insights provide a more comprehensive understanding of predictive uncertainty measures, revealing their implicit assumptions and relationships.

This paper proposes a two-time scale neurodynamic duplex approach to solve distributionally robust geometric joint chance-constrained optimization problems. The probability distributions of the row vectors are not known in advance and belong to a certain distributional uncertainty set. In our paper, we study three uncertainty sets for the unknown distributions. The neurodynamic duplex is designed based on three projection equations. The main contribution of our work is to propose a neural network-based method to solve distributionally robust joint chance-constrained optimization problems that converges in probability to the global optimum without the use of standard state-of-the-art solving methods. We show that neural networks can be used to solve multiple instances of a problem. In the numerical experiments, we apply the proposed approach to solve a problem of shape optimisation and a telecommunication problem.

To provide rigorous uncertainty quantification for online learning models, we develop a framework for constructing uncertainty sets that provably control risk -- such as coverage of confidence intervals, false negative rate, or F1 score -- in the online setting. This extends conformal prediction to apply to a larger class of online learning problems. Our method guarantees risk control at any user-specified level even when the underlying data distribution shifts drastically, even adversarially, over time in an unknown fashion. The technique we propose is highly flexible as it can be applied with any base online learning algorithm (e.g., a deep neural network trained online), requiring minimal implementation effort and essentially zero additional computational cost. We further extend our approach to control multiple risks simultaneously, so the prediction sets we generate are valid for all given risks. To demonstrate the utility of our method, we conduct experiments on real-world tabular time-series data sets showing that the proposed method rigorously controls various natural risks. Furthermore, we show how to construct valid intervals for an online image-depth estimation problem that previous sequential calibration schemes cannot handle.

In this paper, we focus on the task of conditional image generation, where an image is synthesized according to user instructions. The critical challenge underpinning this task is ensuring both the fidelity of the generated images and their semantic alignment with the provided conditions. To tackle this issue, previous studies have employed supervised perceptual losses derived from pre-trained models, i.e., reward models, to enforce alignment between the condition and the generated result. However, we observe one inherent shortcoming: considering the diversity of synthesized images, the reward model usually provides inaccurate feedback when encountering newly generated data, which can undermine the training process. To address this limitation, we propose an uncertainty-aware reward modeling, called Ctrl-U, including uncertainty estimation and uncertainty-aware regularization, designed to reduce the adverse effects of imprecise feedback from the reward model. Given the inherent cognitive uncertainty within reward models, even images generated under identical conditions often result in a relatively large discrepancy in reward loss. Inspired by the observation, we explicitly leverage such prediction variance as an uncertainty indicator. Based on the uncertainty estimation, we regularize the model training by adaptively rectifying the reward. In particular, rewards with lower uncertainty receive higher loss weights, while those with higher uncertainty are given reduced weights to allow for larger variability. The proposed uncertainty regularization facilitates reward fine-tuning through consistency construction. Extensive experiments validate the effectiveness of our methodology in improving the controllability and generation quality, as well as its scalability across diverse conditional scenarios. Code will soon be available at https://grenoble-zhang.github.io/Ctrl-U-Page/.

The 100 MW cryogenic liquid oxygen/hydrogen multi-injector combustor BKD operated by the DLR Institute of Space Propulsion is a research platform that allows the study of thermoacoustic instabilities under realistic conditions, representative of small upper stage rocket engines. We use data from BKD experimental campaigns in which the static chamber pressure and fuel-oxidizer ratio are varied such that the first tangential mode of the combustor is excited under some conditions. We train an autoregressive Bayesian neural network model to forecast the amplitude of the dynamic pressure time series, inputting multiple sensor measurements (injector pressure/ temperature measurements, static chamber pressure, high-frequency dynamic pressure measurements, high-frequency OH* chemiluminescence measurements) and future flow rate control signals. The Bayesian nature of our algorithms allows us to work with a dataset whose size is restricted by the expense of each experimental run, without making overconfident extrapolations. We find that the networks are able to accurately forecast the evolution of the pressure amplitude and anticipate instability events on unseen experimental runs 500 milliseconds in advance. We compare the predictive accuracy of multiple models using different combinations of sensor inputs. We find that the high-frequency dynamic pressure signal is particularly informative. We also use the technique of integrated gradients to interpret the influence of different sensor inputs on the model prediction. The negative log-likelihood of data points in the test dataset indicates that predictive uncertainties are well-characterized by our Bayesian model and simulating a sensor failure event results as expected in a dramatic increase in the epistemic component of the uncertainty.

We study the problem of mapping parameter-space variance of a 3D Morphable Model (3DMM) to per-vertex variance. Traditionally, per-vertex variance has been used to visually interpret the learnt parameter uncertainty. However, mapping parameter-space variance to per-vertex using sampling or error propagation is computationally expensive. We address this limitation with Closed-form Local Uncertainty Estimation (CLUE), which transforms the parameter-space distribution of 3DMMs to per-vertex through a closed-form approximation. CLUE emerges from applying the law of total variance to the 3DMM formulation and is modular, computationally inexpensive, and accurate. Subsequently, we study integrating CLUE for in-the-wild human mesh recovery training and inference. We note that CLUE uses the geometry of the mesh as an inductive bias when predicting the variance, leading to improved accuracy. Through extensive qualitative and quantitative experiments, we show that when used in training, CLUE (1) outpeforms probabilistic and deterministic baselines on metrics evaluating joints, mesh and uncertainty, and (2) has the highest uncertainty-to-error spearman correlation, with applications including data mining and interpretable uncertainty. Finally, we share SAM 3D Body (CLUE) which augments the regressor with high resolution uncertainty maps. We make our code available at https://github.com/vita-epfl/CLUE

We propose the Distance-informed Neural Process (DNP), a novel variant of Neural Processes that improves uncertainty estimation by combining global and distance-aware local latent structures. Standard Neural Processes (NPs) often rely on a global latent variable and struggle with uncertainty calibration and capturing local data dependencies. DNP addresses these limitations by introducing a global latent variable to model task-level variations and a local latent variable to capture input similarity within a distance-preserving latent space. This is achieved through bi-Lipschitz regularization, which bounds distortions in input relationships and encourages the preservation of relative distances in the latent space. This modeling approach allows DNP to produce better-calibrated uncertainty estimates and more effectively distinguish in- from out-of-distribution data. Empirical results demonstrate that DNP achieves strong predictive performance and improved uncertainty calibration across regression and classification tasks.

Deep neural networks (NNs) are powerful black box predictors that have recently achieved impressive performance on a wide spectrum of tasks. Quantifying predictive uncertainty in NNs is a challenging and yet unsolved problem. Bayesian NNs, which learn a distribution over weights, are currently the state-of-the-art for estimating predictive uncertainty; however these require significant modifications to the training procedure and are computationally expensive compared to standard (non-Bayesian) NNs. We propose an alternative to Bayesian NNs that is simple to implement, readily parallelizable, requires very little hyperparameter tuning, and yields high quality predictive uncertainty estimates. Through a series of experiments on classification and regression benchmarks, we demonstrate that our method produces well-calibrated uncertainty estimates which are as good or better than approximate Bayesian NNs. To assess robustness to dataset shift, we evaluate the predictive uncertainty on test examples from known and unknown distributions, and show that our method is able to express higher uncertainty on out-of-distribution examples. We demonstrate the scalability of our method by evaluating predictive uncertainty estimates on ImageNet.

Accurate remaining useful life (RUL) prediction hinges on the quality of health indicators (HIs), yet existing methods often fail to disentangle complex degradation mechanisms in multi-sensor systems or quantify uncertainty in HI reliability. This paper introduces a novel framework for HI construction, advancing three key contributions. First, we adapt Reconstruction along Projected Pathways (RaPP) as a health indicator (HI) for RUL prediction for the first time, showing that it outperforms traditional reconstruction error metrics. Second, we show that augmenting RaPP-derived HIs with aleatoric and epistemic uncertainty quantification (UQ) via Monte Carlo dropout and probabilistic latent spaces- significantly improves RUL-prediction robustness. Third, and most critically, we propose indicator groups, a paradigm that isolates sensor subsets to model system-specific degradations, giving rise to our novel method, I-GLIDE which enables interpretable, mechanism-specific diagnostics. Evaluated on data sourced from aerospace and manufacturing systems, our approach achieves marked improvements in accuracy and generalizability compared to state-of-the-art HI methods while providing actionable insights into system failure pathways. This work bridges the gap between anomaly detection and prognostics, offering a principled framework for uncertainty-aware degradation modeling in complex systems.

Ensemble disagreement is widely used as a proxy for epistemic uncertainty in medical image segmentation. In practice, many studies form ensembles via K-fold cross-validation (CV), yet refer to them as ``deep ensembles'' (DE). Because CV members are trained on different data subsets, their disagreement mixes seed-driven variability with data-exposure effects, which can change how uncertainty should be interpreted. We audit recent segmentation uncertainty studies and find that terminology--implementation mismatches are common. We then compare a standard 5-fold CV ensemble to a 5-member DE (fixed training set, different random seeds) under otherwise identical configurations on three multi-rater segmentation datasets spanning three modalities. We evaluate uncertainty for calibration, failure detection, ambiguity modeling, and robustness under distribution shift. DE match segmentation accuracy while improving calibration and failure detection, whereas CV ensembles sometimes correlate more strongly with inter-rater variability on the studied datasets. Thus, ensemble construction should be chosen to match the research question: DE for reliability-oriented use (e.g., selective referral/failure detection) and CV ensembles as a proxy for ambiguity. We provide a lightweight nnU-Net modification enabling DE training within the default pipeline.

We study the problem of single-image 3D object reconstruction. Recent works have diverged into two directions: regression-based modeling and generative modeling. Regression methods efficiently infer visible surfaces, but struggle with occluded regions. Generative methods handle uncertain regions better by modeling distributions, but are computationally expensive and the generation is often misaligned with visible surfaces. In this paper, we present SPAR3D, a novel two-stage approach aiming to take the best of both directions. The first stage of SPAR3D generates sparse 3D point clouds using a lightweight point diffusion model, which has a fast sampling speed. The second stage uses both the sampled point cloud and the input image to create highly detailed meshes. Our two-stage design enables probabilistic modeling of the ill-posed single-image 3D task while maintaining high computational efficiency and great output fidelity. Using point clouds as an intermediate representation further allows for interactive user edits. Evaluated on diverse datasets, SPAR3D demonstrates superior performance over previous state-of-the-art methods, at an inference speed of 0.7 seconds. Project page with code and model: https://spar3d.github.io

In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in machine learning, notably in the setting of classification, have been proposed on the basis of second-order probability distributions, i.e., predictions in the form of distributions on probability distributions. A completely conclusive solution has not yet been found, however, as shown by recent criticisms of commonly used uncertainty measures associated with second-order distributions, identifying undesirable theoretical properties of these measures. In light of these criticisms, we propose a set of formal criteria that meaningful uncertainty measures for predictive uncertainty based on second-order distributions should obey. Moreover, we provide a general framework for developing uncertainty measures to account for these criteria, and offer an instantiation based on the Wasserstein distance, for which we prove that all criteria are satisfied.

The recent performance leap of Large Language Models (LLMs) opens up new opportunities across numerous industrial applications and domains. However, erroneous generations, such as false predictions, misinformation, and hallucination made by LLMs, have also raised severe concerns for the trustworthiness of LLMs', especially in safety-, security- and reliability-sensitive scenarios, potentially hindering real-world adoptions. While uncertainty estimation has shown its potential for interpreting the prediction risks made by general machine learning (ML) models, little is known about whether and to what extent it can help explore an LLM's capabilities and counteract its undesired behavior. To bridge the gap, in this paper, we initiate an exploratory study on the risk assessment of LLMs from the lens of uncertainty. In particular, we experiment with twelve uncertainty estimation methods and four LLMs on four prominent natural language processing (NLP) tasks to investigate to what extent uncertainty estimation techniques could help characterize the prediction risks of LLMs. Our findings validate the effectiveness of uncertainty estimation for revealing LLMs' uncertain/non-factual predictions. In addition to general NLP tasks, we extensively conduct experiments with four LLMs for code generation on two datasets. We find that uncertainty estimation can potentially uncover buggy programs generated by LLMs. Insights from our study shed light on future design and development for reliable LLMs, facilitating further research toward enhancing the trustworthiness of LLMs.

Transformers have become a standard architecture in machine learning, demonstrating strong in-context learning (ICL) abilities that allow them to learn from the prompt at inference time. However, uncertainty quantification for ICL remains an open challenge, particularly in noisy regression tasks. This paper investigates whether ICL can be leveraged for distribution-free uncertainty estimation, proposing a method based on conformal prediction to construct prediction intervals with guaranteed coverage. While traditional conformal methods are computationally expensive due to repeated model fitting, we exploit ICL to efficiently generate confidence intervals in a single forward pass. Our empirical analysis compares this approach against ridge regression-based conformal methods, showing that conformal prediction with in-context learning (CP with ICL) achieves robust and scalable uncertainty estimates. Additionally, we evaluate its performance under distribution shifts and establish scaling laws to guide model training. These findings bridge ICL and conformal prediction, providing a theoretically grounded and new framework for uncertainty quantification in transformer-based models.

Uncertainty quantification is essential when dealing with ill-conditioned inverse problems due to the inherent nonuniqueness of the solution. Bayesian approaches allow us to determine how likely an estimation of the unknown parameters is via formulating the posterior distribution. Unfortunately, it is often not possible to formulate a prior distribution that precisely encodes our prior knowledge about the unknown. Furthermore, adherence to handcrafted priors may greatly bias the outcome of the Bayesian analysis. To address this issue, we propose to use the functional form of a randomly initialized convolutional neural network as an implicit structured prior, which is shown to promote natural images and excludes images with unnatural noise. In order to incorporate the model uncertainty into the final estimate, we sample the posterior distribution using stochastic gradient Langevin dynamics and perform Bayesian model averaging on the obtained samples. Our synthetic numerical experiment verifies that deep priors combined with Bayesian model averaging are able to partially circumvent imaging artifacts and reduce the risk of overfitting in the presence of extreme noise. Finally, we present pointwise variance of the estimates as a measure of uncertainty, which coincides with regions that are more difficult to image.

We propose a novel method to capture data points near decision boundary in neural network that are often referred to a specific type of uncertainty. In our approach, we sought to perform uncertainty estimation based on the idea of adversarial attack method. In this paper, uncertainty estimates are derived from the input perturbations, unlike previous studies that provide perturbations on the model's parameters as in Bayesian approach. We are able to produce uncertainty with couple of perturbations on the inputs. Interestingly, we apply the proposed method to datasets derived from blockchain. We compare the performance of model uncertainty with the most recent uncertainty methods. We show that the proposed method has revealed a significant outperformance over other methods and provided less risk to capture model uncertainty in machine learning.

Epistemic Uncertainty is a measure of the lack of knowledge of a learner which diminishes with more evidence. While existing work focuses on using the variance of the Bayesian posterior due to parameter uncertainty as a measure of epistemic uncertainty, we argue that this does not capture the part of lack of knowledge induced by model misspecification. We discuss how the excess risk, which is the gap between the generalization error of a predictor and the Bayes predictor, is a sound measure of epistemic uncertainty which captures the effect of model misspecification. We thus propose a principled framework for directly estimating the excess risk by learning a secondary predictor for the generalization error and subtracting an estimate of aleatoric uncertainty, i.e., intrinsic unpredictability. We discuss the merits of this novel measure of epistemic uncertainty, and highlight how it differs from variance-based measures of epistemic uncertainty and addresses its major pitfall. Our framework, Direct Epistemic Uncertainty Prediction (DEUP) is particularly interesting in interactive learning environments, where the learner is allowed to acquire novel examples in each round. Through a wide set of experiments, we illustrate how existing methods in sequential model optimization can be improved with epistemic uncertainty estimates from DEUP, and how DEUP can be used to drive exploration in reinforcement learning. We also evaluate the quality of uncertainty estimates from DEUP for probabilistic image classification and predicting synergies of drug combinations.

Monitoring machine learning models once they are deployed is challenging. It is even more challenging to decide when to retrain models in real-case scenarios when labeled data is beyond reach, and monitoring performance metrics becomes unfeasible. In this work, we use non-parametric bootstrapped uncertainty estimates and SHAP values to provide explainable uncertainty estimation as a technique that aims to monitor the deterioration of machine learning models in deployment environments, as well as determine the source of model deterioration when target labels are not available. Classical methods are purely aimed at detecting distribution shift, which can lead to false positives in the sense that the model has not deteriorated despite a shift in the data distribution. To estimate model uncertainty we construct prediction intervals using a novel bootstrap method, which improves upon the work of Kumar & Srivastava (2012). We show that both our model deterioration detection system as well as our uncertainty estimation method achieve better performance than the current state-of-the-art. Finally, we use explainable AI techniques to gain an understanding of the drivers of model deterioration. We release an open source Python package, doubt, which implements our proposed methods, as well as the code used to reproduce our experiments.

Despite the remarkable progress in generative modelling, current diffusion models lack a quantitative approach to assess image quality. To address this limitation, we propose to estimate the pixel-wise aleatoric uncertainty during the sampling phase of diffusion models and utilise the uncertainty to improve the sample generation quality. The uncertainty is computed as the variance of the denoising scores with a perturbation scheme that is specifically designed for diffusion models. We then show that the aleatoric uncertainty estimates are related to the second-order derivative of the diffusion noise distribution. We evaluate our uncertainty estimation algorithm and the uncertainty-guided sampling on the ImageNet and CIFAR-10 datasets. In our comparisons with the related work, we demonstrate promising results in filtering out low quality samples. Furthermore, we show that our guided approach leads to better sample generation in terms of FID scores.

Applying a machine learning model for decision-making in the real world requires to distinguish what the model knows from what it does not. A critical factor in assessing the knowledge of a model is to quantify its predictive uncertainty. Predictive uncertainty is commonly measured by the entropy of the Bayesian model average (BMA) predictive distribution. Yet, the properness of this current measure of predictive uncertainty was recently questioned. We provide new insights regarding those limitations. Our analyses show that the current measure erroneously assumes that the BMA predictive distribution is equivalent to the predictive distribution of the true model that generated the dataset. Consequently, we introduce a theoretically grounded measure to overcome these limitations. We experimentally verify the benefits of our introduced measure of predictive uncertainty. We find that our introduced measure behaves more reasonably in controlled synthetic tasks. Moreover, our evaluations on ImageNet demonstrate that our introduced measure is advantageous in real-world applications utilizing predictive uncertainty.

Reward models (RM) play a critical role in aligning generations of large language models (LLM) to human expectations. However, prevailing RMs fail to capture the stochasticity within human preferences and cannot effectively evaluate the reliability of reward predictions. To address these issues, we propose Uncertain-aware RM (URM) and Uncertain-aware RM Ensemble (URME) to incorporate and manage uncertainty in reward modeling. URM can model the distribution of disentangled attributes within human preferences, while URME quantifies uncertainty through discrepancies in the ensemble, thereby identifying potential lack of knowledge during reward evaluation. Experiment results indicate that the proposed URM achieves state-of-the-art performance compared to models with the same size, demonstrating the effectiveness of modeling uncertainty within human preferences. Furthermore, empirical results show that through uncertainty quantification, URM and URME can identify unreliable predictions to improve the quality of reward evaluations.

Large language models (LLMs) are increasingly used in critical domains, where their susceptibility to hallucinations makes reliable and interpretable uncertainty quantification essential. We introduce the concept of mechanistic uncertainty, which characterizes uncertainty through the stability of the internal mechanisms supporting a model's response. We propose the causal robustness margin, defined as the minimum internal intervention required to deviate the model's output, and operationalize it using Jacobian Lens (Jlens) concept steering vectors. Across three open-source models and five question-answering datasets, we show incorrect and uncertain responses exhibit smaller margins, indicating greater sensitivity to internal interventions, and the causal robustness margin complements conventional output-based uncertainty methods. Finally, in a controlled knowledge injection experiment, correctly learned facts show increased margins after training, suggesting that causal robustness provides a mechanistically grounded signal of uncertainty.

Although large language models (LLMs) are capable of performing various tasks, they still suffer from producing plausible but incorrect responses. To improve the reliability of LLMs, recent research has focused on uncertainty quantification to predict whether a response is correct or not. However, most uncertainty quantification methods have been evaluated on questions requiring a single clear answer, ignoring the existence of data uncertainty that arises from irreducible randomness. Instead, these methods only consider model uncertainty, which arises from a lack of knowledge. In this paper, we investigate previous uncertainty quantification methods under the presence of data uncertainty. Our contributions are two-fold: 1) proposing a new Multi-Answer Question Answering dataset, MAQA, consisting of world knowledge, mathematical reasoning, and commonsense reasoning tasks to evaluate uncertainty quantification regarding data uncertainty, and 2) assessing 5 uncertainty quantification methods of diverse white- and black-box LLMs. Our findings show that entropy and consistency-based methods estimate the model uncertainty well even under data uncertainty, while other methods for white- and black-box LLMs struggle depending on the tasks. Additionally, methods designed for white-box LLMs suffer from overconfidence in reasoning tasks compared to simple knowledge queries. We believe our observations will pave the way for future work on uncertainty quantification in realistic setting.

GUI grounding, which localizes interface elements from screenshots given natural language queries, remains challenging for small icons and dense layouts. Test-time zoom-in methods improve localization by cropping and re-running inference at higher resolution, but apply cropping uniformly across all instances with fixed crop sizes, ignoring whether the model is actually uncertain on each case. We propose UI-Zoomer, a training-free adaptive zoom-in framework that treats both the trigger and scale of zoom-in as a prediction uncertainty quantification problem. A confidence-aware gate fuses spatial consensus among stochastic candidates with token-level generation confidence to selectively trigger zoom-in only when localization is uncertain. When triggered, an uncertainty-driven crop sizing module decomposes prediction variance into inter-sample positional spread and intra-sample box extent, deriving a per-instance crop radius via the law of total variance. Extensive experiments on ScreenSpot-Pro, UI-Vision, and ScreenSpot-v2 demonstrate consistent improvements over strong baselines across multiple model architectures, achieving gains of up to +13.4\%, +10.3\%, and +4.2\% respectively, with no additional training required.

Large Language Models (LLMs) have recently been successfully applied to regression tasks -- such as time series forecasting and tabular prediction -- by leveraging their in-context learning abilities. However, their autoregressive decoding process may be ill-suited to continuous-valued outputs, where obtaining predictive distributions over numerical targets requires repeated sampling, leading to high computational cost and inference time. In this work, we investigate whether distributional properties of LLM predictions can be recovered without explicit autoregressive generation. To this end, we study a set of regression probes trained to predict statistical functionals (e.g., mean, median, quantiles) of the LLM's numerical output distribution directly from its internal representations. Our results suggest that LLM embeddings carry informative signals about summary statistics of their predictive distributions, including the numerical uncertainty. This investigation opens up new questions about how LLMs internally encode uncertainty in numerical tasks, and about the feasibility of lightweight alternatives to sampling-based approaches for uncertainty-aware numerical predictions.

Deep Learning-based image super-resolution (SR) has been gaining traction with the aid of Generative Adversarial Networks. Models like SRGAN and ESRGAN are constantly ranked between the best image SR tools. However, they lack principled ways for estimating predictive uncertainty. In the present work, we enhance these models using Monte Carlo-Dropout and Deep Ensemble, allowing the computation of predictive uncertainty. When coupled with a prediction, uncertainty estimates can provide more information to the model users, highlighting pixels where the SR output might be uncertain, hence potentially inaccurate, if these estimates were to be reliable. Our findings suggest that these uncertainty estimates are decently calibrated and can hence fulfill this goal, while providing no performance drop with respect to the corresponding models without uncertainty estimation.

Large Language and Vision-Language Models (LLMs/VLMs) are increasingly used in safety-critical applications, yet their opaque decision-making complicates risk assessment and reliability. Uncertainty quantification (UQ) helps assess prediction confidence and enables abstention when uncertainty is high. Conformal prediction (CP), a leading UQ method, provides statistical guarantees but relies on static thresholds, which fail to adapt to task complexity and evolving data distributions, leading to suboptimal trade-offs in accuracy, coverage, and informativeness. To address this, we propose learnable conformal abstention, integrating reinforcement learning (RL) with CP to optimize abstention thresholds dynamically. By treating CP thresholds as adaptive actions, our approach balances multiple objectives, minimizing prediction set size while maintaining reliable coverage. Extensive evaluations across diverse LLM/VLM benchmarks show our method outperforms Least Ambiguous Classifiers (LAC) and Adaptive Prediction Sets (APS), improving accuracy by up to 3.2%, boosting AUROC for hallucination detection by 22.19%, enhancing uncertainty-guided selective generation (AUARC) by 21.17%, and reducing calibration error by 70%-85%. These improvements hold across multiple models and datasets while consistently meeting the 90% coverage target, establishing our approach as a more effective and flexible solution for reliable decision-making in safety-critical applications. The code is available at: {https://github.com/sinatayebati/vlm-uncertainty}.

We propose a new uncertainty estimator for gradient-free optimisation of black-box simulators using deep generative surrogate models. Optimisation of these simulators is especially challenging for stochastic simulators and higher dimensions. To address these issues, we utilise a deep generative surrogate approach to model the black box response for the entire parameter space. We then leverage this knowledge to estimate the proposed uncertainty based on the Wasserstein distance - the Wasserstein uncertainty. This approach is employed in a posterior agnostic gradient-free optimisation algorithm that minimises regret over the entire parameter space. A series of tests were conducted to demonstrate that our method is more robust to the shape of both the black box function and the stochastic response of the black box than state-of-the-art methods, such as efficient global optimisation with a deep Gaussian process surrogate.

A set of novel approaches for estimating epistemic uncertainty in deep neural networks with a single forward pass has recently emerged as a valid alternative to Bayesian Neural Networks. On the premise of informative representations, these deterministic uncertainty methods (DUMs) achieve strong performance on detecting out-of-distribution (OOD) data while adding negligible computational costs at inference time. However, it remains unclear whether DUMs are well calibrated and can seamlessly scale to real-world applications - both prerequisites for their practical deployment. To this end, we first provide a taxonomy of DUMs, and evaluate their calibration under continuous distributional shifts. Then, we extend them to semantic segmentation. We find that, while DUMs scale to realistic vision tasks and perform well on OOD detection, the practicality of current methods is undermined by poor calibration under distributional shifts.

Offline reinforcement learning enables agents to leverage large pre-collected datasets of environment transitions to learn control policies, circumventing the need for potentially expensive or unsafe online data collection. Significant progress has been made recently in offline model-based reinforcement learning, approaches which leverage a learned dynamics model. This typically involves constructing a probabilistic model, and using the model uncertainty to penalize rewards where there is insufficient data, solving for a pessimistic MDP that lower bounds the true MDP. Existing methods, however, exhibit a breakdown between theory and practice, whereby pessimistic return ought to be bounded by the total variation distance of the model from the true dynamics, but is instead implemented through a penalty based on estimated model uncertainty. This has spawned a variety of uncertainty heuristics, with little to no comparison between differing approaches. In this paper, we compare these heuristics, and design novel protocols to investigate their interaction with other hyperparameters, such as the number of models, or imaginary rollout horizon. Using these insights, we show that selecting these key hyperparameters using Bayesian Optimization produces superior configurations that are vastly different to those currently used in existing hand-tuned state-of-the-art methods, and result in drastically stronger performance.

Biomedical image segmentation is critical for accurate identification and analysis of anatomical structures in medical imaging, particularly in cardiac MRI. Manual segmentation is labor-intensive, time-consuming, and prone to errors, highlighting the need for automated methods. However, current machine learning approaches face challenges like overfitting and data demands. To tackle these issues, we propose a new UU-Mamba model, integrating the U-Mamba model with the Sharpness-Aware Minimization (SAM) optimizer and an uncertainty-aware loss function. SAM enhances generalization by locating flat minima in the loss landscape, thus reducing overfitting. The uncertainty-aware loss combines region-based, distribution-based, and pixel-based loss designs to improve segmentation accuracy and robustness. Evaluation of our method is performed on the ACDC cardiac dataset, outperforming state-of-the-art models including TransUNet, Swin-Unet, nnUNet, and nnFormer. Our approach achieves Dice Similarity Coefficient (DSC) and Mean Squared Error (MSE) scores, demonstrating its effectiveness in cardiac MRI segmentation.

Representation learning has significantly driven the field to develop pretrained models that can act as a valuable starting point when transferring to new datasets. With the rising demand for reliable machine learning and uncertainty quantification, there is a need for pretrained models that not only provide embeddings but also transferable uncertainty estimates. To guide the development of such models, we propose the Uncertainty-aware Representation Learning (URL) benchmark. Besides the transferability of the representations, it also measures the zero-shot transferability of the uncertainty estimate using a novel metric. We apply URL to evaluate eleven uncertainty quantifiers that are pretrained on ImageNet and transferred to eight downstream datasets. We find that approaches that focus on the uncertainty of the representation itself or estimate the prediction risk directly outperform those that are based on the probabilities of upstream classes. Yet, achieving transferable uncertainty quantification remains an open challenge. Our findings indicate that it is not necessarily in conflict with traditional representation learning goals. Code is provided under https://github.com/mkirchhof/url .

We develop a method to generate predictive regions that cover a multivariate response variable with a user-specified probability. Our work is composed of two components. First, we use a deep generative model to learn a representation of the response that has a unimodal distribution. Existing multiple-output quantile regression approaches are effective in such cases, so we apply them on the learned representation, and then transform the solution to the original space of the response. This process results in a flexible and informative region that can have an arbitrary shape, a property that existing methods lack. Second, we propose an extension of conformal prediction to the multivariate response setting that modifies any method to return sets with a pre-specified coverage level. The desired coverage is theoretically guaranteed in the finite-sample case for any distribution. Experiments conducted on both real and synthetic data show that our method constructs regions that are significantly smaller compared to existing techniques.

Uncertainty quantification (UQ) is vital for trustworthy deep learning, yet existing methods are either computationally intensive, such as Bayesian or ensemble methods, or provide only partial, task-specific estimates, such as single-forward-pass techniques. In this paper, we propose a post-hoc single-forward-pass framework that jointly captures aleatoric and epistemic uncertainty without modifying or retraining pretrained models. Our method applies Split-Point Analysis (SPA) to decompose predictive residuals into upper and lower subsets, computing Mean Absolute Residuals (MARs) on each side. We prove that, under ideal conditions, the total MAR equals the harmonic mean of subset MARs; deviations define a novel Self-consistency Discrepancy Score (SDS) for fine-grained epistemic estimation across regression and classification. For regression, side-specific quantile regression yields prediction intervals with improved empirical coverage, which are further calibrated via SDS. For classification, when calibration data are available, we apply SPA-based calibration identities to adjust the softmax outputs and then compute predictive entropy on these calibrated probabilities. Extensive experiments on diverse regression and classification benchmarks demonstrate that our framework matches or exceeds several state-of-the-art UQ methods while incurring minimal overhead. Our source code is available at https://github.com/zzz0527/SPC-UQ.

The main objective of Prognostics and Health Management is to estimate the Remaining Useful Lifetime (RUL), namely, the time that a system or a piece of equipment is still in working order before starting to function incorrectly. In recent years, numerous machine learning algorithms have been proposed for RUL estimation, mainly focusing on providing more accurate RUL predictions. However, there are many sources of uncertainty in the problem, such as inherent randomness of systems failure, lack of knowledge regarding their future states, and inaccuracy of the underlying predictive models, making it infeasible to predict the RULs precisely. Hence, it is of utmost importance to quantify the uncertainty alongside the RUL predictions. In this work, we investigate the conformal prediction (CP) framework that represents uncertainty by predicting sets of possible values for the target variable (intervals in the case of RUL) instead of making point predictions. Under very mild technical assumptions, CP formally guarantees that the actual value (true RUL) is covered by the predicted set with a degree of certainty that can be prespecified. We study three CP algorithms to conformalize any single-point RUL predictor and turn it into a valid interval predictor. Finally, we conformalize two single-point RUL predictors, deep convolutional neural networks and gradient boosting, and illustrate their performance on the Commercial Modular Aero-Propulsion System Simulation (C-MAPSS) data sets.

Large Language Models (LLMs) that can express interpretable and calibrated uncertainty are crucial in high-stakes domains. While methods to compute uncertainty post-hoc exist, they are often sampling-based and therefore computationally expensive or lack calibration. We propose a three-stage pipeline to post-train LLMs to efficiently infer calibrated uncertainty estimates for their responses. First, we compute fine-grained entropy-based uncertainty scores on the training data, capturing the distributional variability of model outputs in embedding space. Second, these scores are calibrated via Platt scaling, producing reliable and human-interpretable uncertainty signals. Finally, the target LLM is post-trained via reinforcement learning to align its policy with these calibrated signals through a verifiable reward function. Unlike post-hoc uncertainty estimation methods, our approach provides interpretable and computationally efficient uncertainty estimates at test time. Experiments show that models trained with our pipeline achieve better calibration than baselines and generalize to unseen tasks without further processing, suggesting that they learn a robust uncertainty reasoning behavior.

Deep learning has recently revealed the existence of scaling laws, demonstrating that model performance follows predictable trends based on dataset and model sizes. Inspired by these findings and fascinating phenomena emerging in the over-parameterized regime, we examine a parallel direction: do similar scaling laws govern predictive uncertainties in deep learning? In identifiable parametric models, such scaling laws can be derived in a straightforward manner by treating model parameters in a Bayesian way. In this case, for example, we obtain O(1/N) contraction rates for epistemic uncertainty with respect to the number of data N. However, in over-parameterized models, these guarantees do not hold, leading to largely unexplored behaviors. In this work, we empirically show the existence of scaling laws associated with various measures of predictive uncertainty with respect to dataset and model sizes. Through experiments on vision and language tasks, we observe such scaling laws for in- and out-of-distribution predictive uncertainty estimated through popular approximate Bayesian inference and ensemble methods. Besides the elegance of scaling laws and the practical utility of extrapolating uncertainties to larger data or models, this work provides strong evidence to dispel recurring skepticism against Bayesian approaches: "In many applications of deep learning we have so much data available: what do we need Bayes for?". Our findings show that "so much data" is typically not enough to make epistemic uncertainty negligible.

Although Neural Radiance Fields (NeRFs) have markedly improved novel view synthesis, accurate uncertainty quantification in their image predictions remains an open problem. The prevailing methods for estimating uncertainty, including the state-of-the-art Density-aware NeRF Ensembles (DANE) [29], quantify uncertainty without calibration. This frequently leads to over- or under-confidence in image predictions, which can undermine their real-world applications. In this paper, we propose a method which, for the first time, achieves calibrated uncertainties for NeRFs. To accomplish this, we overcome a significant challenge in adapting existing calibration techniques to NeRFs: a need to hold out ground truth images from the target scene, reducing the number of images left to train the NeRF. This issue is particularly problematic in sparse-view settings, where we can operate with as few as three images. To address this, we introduce the concept of a meta-calibrator that performs uncertainty calibration for NeRFs with a single forward pass without the need for holding out any images from the target scene. Our meta-calibrator is a neural network that takes as input the NeRF images and uncalibrated uncertainty maps and outputs a scene-specific calibration curve that corrects the NeRF's uncalibrated uncertainties. We show that the meta-calibrator can generalize on unseen scenes and achieves well-calibrated and state-of-the-art uncertainty for NeRFs, significantly beating DANE and other approaches. This opens opportunities to improve applications that rely on accurate NeRF uncertainty estimates such as next-best view planning and potentially more trustworthy image reconstruction for medical diagnosis. The code is available at https://niki-amini-naieni.github.io/instantcalibration.github.io/.

We develop a data-driven deep neural operator framework to approximate multiple output states for a diesel engine and generate real-time predictions with reasonable accuracy. As emission norms become more stringent, the need for fast and accurate models that enable analysis of system behavior have become an essential requirement for system development. The fast transient processes involved in the operation of a combustion engine make it difficult to develop accurate physics-based models for such systems. As an alternative to physics based models, we develop an operator-based regression model (DeepONet) to learn the relevant output states for a mean-value gas flow engine model using the engine operating conditions as input variables. We have adopted a mean-value model as a benchmark for comparison, simulated using Simulink. The developed approach necessitates using the initial conditions of the output states to predict the accurate sequence over the temporal domain. To this end, a sequence-to-sequence approach is embedded into the proposed framework. The accuracy of the model is evaluated by comparing the prediction output to ground truth generated from Simulink model. The maximum mathcal L_2 relative error observed was approximately 6.5%. The sensitivity of the DeepONet model is evaluated under simulated noise conditions and the model shows relatively low sensitivity to noise. The uncertainty in model prediction is further assessed by using a mean ensemble approach. The worst-case error at the (mu + 2sigma) boundary was found to be 12%. The proposed framework provides the ability to predict output states in real-time and enables data-driven learning of complex input-output operator mapping. As a result, this model can be applied during initial development stages, where accurate models may not be available.

Pixel-based language models aim to solve the vocabulary bottleneck problem in language modeling, but the challenge of uncertainty quantification remains open. The novelty of this work consists of analysing uncertainty and confidence in pixel-based language models across 18 languages and 7 scripts, all part of 3 semantically challenging tasks. This is achieved through several methods such as Monte Carlo Dropout, Transformer Attention, and Ensemble Learning. The results suggest that pixel-based models underestimate uncertainty when reconstructing patches. The uncertainty is also influenced by the script, with Latin languages displaying lower uncertainty. The findings on ensemble learning show better performance when applying hyperparameter tuning during the named entity recognition and question-answering tasks across 16 languages.

As large language models (LLMs) are increasingly integrated into high-stakes decision-making, the ability to reliably quantify uncertainty has become a critical requirement for safety and trust. However, current uncertainty quantification methods primarily operate at the output level, often failing to distinguish whether uncertainty arises from the model's lack of knowledge or from ambiguity in the user's input. While input-centric uncertainty quantification has recently emerged as a promising direction, it remains relatively underexplored and typically relies on coarse, input-level information. Consequently, users are provided with scalar uncertainty scores that offer little actionable guidance on which parts of the input should be clarified to improve reliability. To address this limitation, we propose Shapley-based input uncertainty Quantification (ShaQ), a framework for span-level attribution of input-induced uncertainty. Our approach models ambiguous spans in the input as players in a cooperative game and quantifies their contributions using Shapley values, defined via the weighted average of marginal reductions in conditional entropy obtained by clarifying each span coalition. Unlike existing input-level approaches, our formulation captures complex interactions among spans and provides a principled decomposition in which individual attributions sum exactly to the total input-induced uncertainty. We evaluate ShaQ on the AmbigQA and AmbiEnt benchmarks, where it achieves state-of-the-art performance in ambiguity detection. We further demonstrate its utility on MediTOD, showing that ShaQ can localize under-specified clinical utterances and facilitate human-AI collaboration in high-stakes settings. Overall, ShaQ improves uncertainty estimation and provides actionable insights for targeted input clarification.

Conventional uncertainty-aware temporal difference (TD) learning often assumes a zero-mean Gaussian distribution for TD errors, leading to inaccurate error representations and compromised uncertainty estimation. We introduce a novel framework for generalized Gaussian error modeling in deep reinforcement learning to enhance the flexibility of error distribution modeling by incorporating additional higher-order moment, particularly kurtosis, thereby improving the estimation and mitigation of data-dependent aleatoric uncertainty. We examine the influence of the shape parameter of the generalized Gaussian distribution (GGD) on aleatoric uncertainty and provide a closed-form expression that demonstrates an inverse relationship between uncertainty and the shape parameter. Additionally, we propose a theoretically grounded weighting scheme to address epistemic uncertainty by fully leveraging the GGD. We refine batch inverse variance weighting with bias reduction and kurtosis considerations, enhancing robustness. Experiments with policy gradient algorithms demonstrate significant performance gains.

Generative video models demonstrate impressive text-to-video capabilities, spurring widespread adoption in many real-world applications. However, like large language models (LLMs), video generation models tend to hallucinate, producing plausible videos even when they are factually wrong. Although uncertainty quantification (UQ) of LLMs has been extensively studied in prior work, no UQ method for video models exists, raising critical safety concerns. To our knowledge, this paper represents the first work towards quantifying the uncertainty of video models. We present a framework for uncertainty quantification of generative video models, consisting of: (i) a metric for evaluating the calibration of video models based on robust rank correlation estimation with no stringent modeling assumptions; (ii) a black-box UQ method for video models (termed S-QUBED), which leverages latent modeling to rigorously decompose predictive uncertainty into its aleatoric and epistemic components; and (iii) a UQ dataset to facilitate benchmarking calibration in video models. By conditioning the generation task in the latent space, we disentangle uncertainty arising due to vague task specifications from that arising from lack of knowledge. Through extensive experiments on benchmark video datasets, we demonstrate that S-QUBED computes calibrated total uncertainty estimates that are negatively correlated with the task accuracy and effectively computes the aleatoric and epistemic constituents.

Large language models (LLMs) have transformed natural language processing, but their reliable deployment requires effective uncertainty quantification (UQ). Existing UQ methods are often heuristic and lack a probabilistic foundation. This paper begins by providing a theoretical justification for the role of perturbations in UQ for LLMs. We then introduce a dual random walk perspective, modeling input-output pairs as two Markov chains with transition probabilities defined by semantic similarity. Building on this, we propose a fully probabilistic framework based on an inverse model, which quantifies uncertainty by evaluating the diversity of the input space conditioned on a given output through systematic perturbations. Within this framework, we define a new uncertainty measure, Inv-Entropy. A key strength of our framework is its flexibility: it supports various definitions of uncertainty measures, embeddings, perturbation strategies, and similarity metrics. We also propose GAAP, a perturbation algorithm based on genetic algorithms, which enhances the diversity of sampled inputs. In addition, we introduce a new evaluation metric, Temperature Sensitivity of Uncertainty (TSU), which directly assesses uncertainty without relying on correctness as a proxy. Extensive experiments demonstrate that Inv-Entropy outperforms existing semantic UQ methods. The code to reproduce the results can be found at https://github.com/UMDataScienceLab/Uncertainty-Quantification-for-LLMs.

Accurate histopathologic interpretation is key for clinical decision-making; however, current deep learning models for digital pathology are often overconfident and poorly calibrated in out-of-distribution (OOD) settings, which limit trust and clinical adoption. Safety-critical medical imaging workflows benefit from intrinsic uncertainty-aware properties that can accurately reject OOD input. We implement the Spectral-normalized Neural Gaussian Process (SNGP), a set of lightweight modifications that apply spectral normalization and replace the final dense layer with a Gaussian process layer to improve single-model uncertainty estimation and OOD detection. We evaluate SNGP vs. deterministic and MonteCarlo dropout on six datasets across three biomedical classification tasks: white blood cells, amyloid plaques, and colorectal histopathology. SNGP has comparable in-distribution performance while significantly improving uncertainty estimation and OOD detection. Thus, SNGP or related models offer a useful framework for uncertainty-aware classification in digital pathology, supporting safe deployment and building trust with pathologists.

Solving complex tasks usually requires LLMs to generate long multi-step reasoning chains. Previous work has shown that verifying the correctness of individual reasoning steps can further improve the performance and efficiency of LLMs on such tasks and enhance solution interpretability. However, existing verification approaches, such as Process Reward Models (PRMs), are either computationally expensive, limited to specific domains, or require large-scale human or model-generated annotations. Thus, we propose a lightweight alternative for step-level reasoning verification based on data-driven uncertainty scores. We train transformer-based uncertainty quantification heads (UHeads) that use the internal states of a frozen LLM to estimate the uncertainty of its reasoning steps during generation. The approach is fully automatic: target labels are generated either by another larger LLM (e.g., DeepSeek R1) or in a self-supervised manner by the original model itself. UHeads are both effective and lightweight, containing less than 10M parameters. Across multiple domains, including mathematics, planning, and general knowledge question answering, they match or even surpass the performance of PRMs that are up to 810x larger. Our findings suggest that the internal states of LLMs encode their uncertainty and can serve as reliable signals for reasoning verification, offering a promising direction toward scalable and generalizable introspective LLMs.

Multimodal medical imaging fuses complementary anatomical and functional information, yet modalities frequently disagree in pathologically heterogeneous regions. Current segmentation models handle this in one of two inadequate ways: deterministic fusion that averages away disagreement, or post-hoc uncertainty estimation decoupled from the fusion process that produces it. Both obscure the clinically critical question: why is this prediction unreliable? We present EnTrust, a framework that treats inter-modal conflict as the primary source of predictive uncertainty. Our EnFuse module decomposes multimodal features into three disentangled components: shared anatomical consensus (F_c), modality-specific cues (F_{u,m}), and spatially localized conflict signals (F_{cf}), with independence enforced via a cross-covariance objective. This structured decomposition conditions SegDiff, a diffusion-based generative segmentation model whose sampled hypotheses diverge specifically in regions of modal disagreement. TrustMap then translates this hypothesis divergence into calibrated, pixel-wise uncertainty using ensemble entropy, conflict-guided perturbation probing, and a learned calibration head, enabling clinicians to understand not only where predictions are uncertain, but why. Across four benchmarks spanning brain, cardiac, lesion, and oncology domains, EnTrust achieves state-of-the-art segmentation accuracy while reducing calibration error by 40% compared to the strongest baseline. Notably, it outperforms 5x deep ensembles using a single model at roughly half the memory footprint. Code and checkpoints are available at https://github.com/GenMI-Lab/EnTrust.git.

Artificial intelligence (AI)-based data-driven weather forecasting models have experienced rapid progress over the last years. Recent studies, with models trained on reanalysis data, achieve impressive results and demonstrate substantial improvements over state-of-the-art physics-based numerical weather prediction models across a range of variables and evaluation metrics. Beyond improved predictions, the main advantages of data-driven weather models are their substantially lower computational costs and the faster generation of forecasts, once a model has been trained. However, most efforts in data-driven weather forecasting have been limited to deterministic, point-valued predictions, making it impossible to quantify forecast uncertainties, which is crucial in research and for optimal decision making in applications. Our overarching aim is to systematically study and compare uncertainty quantification methods to generate probabilistic weather forecasts from a state-of-the-art deterministic data-driven weather model, Pangu-Weather. Specifically, we compare approaches for quantifying forecast uncertainty based on generating ensemble forecasts via perturbations to the initial conditions, with the use of statistical and machine learning methods for post-hoc uncertainty quantification. In a case study on medium-range forecasts of selected weather variables over Europe, the probabilistic forecasts obtained by using the Pangu-Weather model in concert with uncertainty quantification methods show promising results and provide improvements over ensemble forecasts from the physics-based ensemble weather model of the European Centre for Medium-Range Weather Forecasts for lead times of up to 5 days.

Counterfactual explanations are attracting significant attention due to the flourishing applications of machine learning models in consequential domains. A counterfactual plan consists of multiple possibilities to modify a given instance so that the model's prediction will be altered. As the predictive model can be updated subject to the future arrival of new data, a counterfactual plan may become ineffective or infeasible with respect to the future values of the model parameters. In this work, we study the counterfactual plans under model uncertainty, in which the distribution of the model parameters is partially prescribed using only the first- and second-moment information. First, we propose an uncertainty quantification tool to compute the lower and upper bounds of the probability of validity for any given counterfactual plan. We then provide corrective methods to adjust the counterfactual plan to improve the validity measure. The numerical experiments validate our bounds and demonstrate that our correction increases the robustness of the counterfactual plans in different real-world datasets.

Recent advancements in Large Language Models (LLMs) have enabled sophisticated reasoning and content generation, yet their inherent stochasticity poses significant challenges for ensuring predictive credibility. While traditional uncertainty taxonomy paradigms, such as the dichotomy of aleatoric and epistemic uncertainties, provide conceptual foundations, they often fail to capture the multi-component and multi-stage nature of LLM generation and struggle to evaluate the effectiveness of various Uncertainty Quantification (UQ) methods. In this paper, we propose a granular uncertainty taxonomy that systematically attributes LLM uncertainty into input-level, parameter-level, token-level, and decoding-process sources. Correspondingly, we categorize existing UQ methods into Bayesian, ensemble, consensus-based, and single-pass approaches. Furthermore, we introduce a comprehensive evaluation framework covering diverse generation settings and metrics. We empirically evaluate 21 typical UQ methods across three prominent LLM families, including Qwen3, Llama 3.2, and DeepSeek-V3, on benchmarks such as TriviaQA, GSM8K, and HumanEval. Our experimental results demonstrate that (i) the effectiveness of UQ methods is sensitive to task types and generation settings; (ii) consensus-based methods, typed Deg and EigV, consistently outperform other UQ approaches; and (iii) larger model scales correlate with lower uncertainty estimates, suggesting an empirical scaling law for LLM uncertainty. This work bridges the gap between theoretical origins and practical deployment, providing a versatile diagnostic tool for systematically quantifying uncertainty in LLM applications.

A crucial requirement for reliable deployment of deep learning models for safety-critical applications is the ability to identify out-of-distribution (OOD) data points, samples which differ from the training data and on which a model might underperform. Previous work has attempted to tackle this problem using uncertainty estimation techniques. However, there is empirical evidence that a large family of these techniques do not detect OOD reliably in classification tasks. This paper gives a theoretical explanation for said experimental findings and illustrates it on synthetic data. We prove that such techniques are not able to reliably identify OOD samples in a classification setting, since their level of confidence is generalized to unseen areas of the feature space. This result stems from the interplay between the representation of ReLU networks as piece-wise affine transformations, the saturating nature of activation functions like softmax, and the most widely-used uncertainty metrics.

Physics-Informed Neural Networks (PINNs) are a novel computational approach for solving partial differential equations (PDEs) with noisy and sparse initial and boundary data. Although, efficient quantification of epistemic and aleatoric uncertainties in big multi-scale problems remains challenging. We propose \PINN a novel method of computing global uncertainty in PDEs using a Bayesian framework, by combining local Bayesian Physics-Informed Neural Networks (BPINN) with domain decomposition. The solution continuity across subdomains is obtained by imposing the flux continuity across the interface of neighboring subdomains. To demonstrate the effectiveness of PINN, we conduct a series of computational experiments on PDEs in 1D and 2D spatial domains. Although we have adopted conservative PINNs (cPINNs), the method can be seamlessly extended to other domain decomposition techniques. The results infer that the proposed method recovers the global uncertainty by computing the local uncertainty exactly more efficiently as the uncertainty in each subdomain can be computed concurrently. The robustness of \$PINN is verified by adding uncorrelated random noise to the training data up to 15% and testing for different domain sizes.

Large language models are informing decisions with ever-higher stakes. As the consequences of their errors grow, a central question becomes harder to ignore: how much can we trust an individual answer? Yet recognizing when to defer remains difficult because language models can present incorrect conclusions with fluent explanations and an authoritative tone. Uncertainty quantification seeks to address this disconnect by estimating the reliability of individual predictions. However, many existing methods require repeated generations or separately trained components, and their scalar estimates do not reveal where uncertainty arises or how it evolves during reasoning. Recent work has also shown that generation length can be strongly associated with uncertainty estimates and correctness, raising the question of how much of an estimator's predictive power comes from uncertainty-specific information rather than output length alone. Mechanistic interpretability offers a way to address these limitations by connecting human-interpretable concepts to intermediate model states. Building on this capability, we introduce the U-Space, a low-dimensional subspace that makes a model's evolving uncertainty measurable and interpretable. We identify semantic anchors for doubt and certainty, map their unembedding directions back into the residual space, and combine their contrasts into an orthogonal basis. The U-Lens projects each token state onto these basis vectors, yielding an interpretable token-level uncertainty map that can be inspected directly or aggregated into a scalar uncertainty score. Our approach requires no correctness labels, repeated generations, or training. Across reasoning benchmarks, its confidence score outperforms established baselines under both standard and length-controlled evaluation and transfers more reliably than supervised estimators. Code: https://github.com/s2labres/U-Space.

We present a state-of-the-art model for fine-grained probability estimation of propositions conditioned on context. Recent advances in large language models (LLMs) have significantly enhanced their reasoning capabilities, particularly on well-defined tasks with complete information. However, LLMs continue to struggle with making accurate and well-calibrated probabilistic predictions under uncertainty or partial information. While incorporating uncertainty into model predictions often boosts performance, obtaining reliable estimates of that uncertainty remains understudied. In particular, LLM probability estimates tend to be coarse and biased towards more frequent numbers. Through a combination of human and synthetic data creation and assessment, scaling to larger models, and better supervision, we propose a set of strong and precise probability estimation models. We conduct systematic evaluations across tasks that rely on conditional probability estimation and show that our approach consistently outperforms existing fine-tuned and prompting-based methods by a large margin.

The Helmholtz equation governs time-harmonic wave propagation, and in dissipative media a complex modulus renders its squared wavenumber κ^2 complex. Inferring such fields from sparse, noisy data calls for solvers that also quantify their own uncertainty. Physics-informed Gaussian-process (GP) regression supplies this by returning a posterior over the solution, yet operator-conditioned formulations have been developed almost exclusively for real-valued fields. We extend operator-informed GP regression to complex-valued Helmholtz problems by realifying the complex operator into an equivalent coupled real block, which enables inference with standard real-valued GP conditioning. The construction admits a family of priors, from a proper diagonal prior to coregionalized and multiscale variants, and conditions on PDE residuals and boundary traces. On benchmark problems in one to three dimensions, the solver is competitive with finite-difference and neural-network baselines at a far smaller interior-constraint budget. Unlike those deterministic baselines, it returns a posterior over the complex wavefield rather than a point estimate. Applied to in vivo brain magnetic resonance elastography, a proper multiscale prior reconstructs the shear curl field to a correlation of 0.77 with measurement, above a 0.75 target. The gain arises from the multiscale kernel rather than from real--imaginary coupling. We further identify a low-frequency accuracy ceiling set by model mismatch and a posterior uncertainty that is not yet calibrated. Calibrated uncertainty therefore emerges as the central next step for probabilistic wavefield inference in dissipative media.

Deep Neural Networks (DNNs) have demonstrated remarkable performance across various domains, including computer vision and natural language processing. However, they often struggle to accurately quantify the uncertainty of their predictions, limiting their broader adoption in critical real-world applications. Uncertainty Quantification (UQ) for Deep Learning seeks to address this challenge by providing methods to improve the reliability of uncertainty estimates. Although numerous techniques have been proposed, a unified tool offering a seamless workflow to evaluate and integrate these methods remains lacking. To bridge this gap, we introduce Torch-Uncertainty, a PyTorch and Lightning-based framework designed to streamline DNN training and evaluation with UQ techniques and metrics. In this paper, we outline the foundational principles of our library and present comprehensive experimental results that benchmark a diverse set of UQ methods across classification, segmentation, and regression tasks. Our library is available at https://github.com/ENSTA-U2IS-AI/Torch-Uncertainty

Trainable evaluation metrics for machine translation (MT) exhibit strong correlation with human judgements, but they are often hard to interpret and might produce unreliable scores under noisy or out-of-domain data. Recent work has attempted to mitigate this with simple uncertainty quantification techniques (Monte Carlo dropout and deep ensembles), however these techniques (as we show) are limited in several ways -- for example, they are unable to distinguish between different kinds of uncertainty, and they are time and memory consuming. In this paper, we propose more powerful and efficient uncertainty predictors for MT evaluation, and we assess their ability to target different sources of aleatoric and epistemic uncertainty. To this end, we develop and compare training objectives for the COMET metric to enhance it with an uncertainty prediction output, including heteroscedastic regression, divergence minimization, and direct uncertainty prediction. Our experiments show improved results on uncertainty prediction for the WMT metrics task datasets, with a substantial reduction in computational costs. Moreover, they demonstrate the ability of these predictors to address specific uncertainty causes in MT evaluation, such as low quality references and out-of-domain data.

Large language models have demonstrated remarkable capabilities in complex mathematical reasoning tasks, but they inevitably generate errors throughout multi-step solutions. Process-level Reward Models (PRMs) have shown great promise by providing supervision and evaluation at each intermediate step, thereby effectively improving the models' reasoning abilities. However, training effective PRMs requires high-quality process reward data, yet existing methods for constructing such data are often labour-intensive or inefficient. In this paper, we propose an uncertainty-driven framework for automated process reward data construction, encompassing both data generation and annotation processes for PRMs. Additionally, we identify the limitations of both majority vote and PRMs, and introduce two generic uncertainty-aware output aggregation methods: Hybrid Majority Reward Vote and Weighted Reward Frequency Vote, which combine the strengths of majority vote with PRMs. Extensive experiments on ProcessBench, MATH, and GSMPlus show the effectiveness and efficiency of the proposed PRM data construction framework, and demonstrate that the two output aggregation methods further improve the mathematical reasoning abilities across diverse PRMs. The code and data will be publicly available at https://github.com/Jiuzhouh/UnPRM.

Diffusion models are a class of probabilistic generative models that have been widely used as a prior for image processing tasks like text conditional generation and inpainting. We demonstrate that these models can be adapted to make predictions and provide uncertainty quantification for chaotic dynamical systems. In these applications, diffusion models can implicitly represent knowledge about outliers and extreme events; however, querying that knowledge through conditional sampling or measuring probabilities is surprisingly difficult. Existing methods for conditional sampling at inference time seek mainly to enforce the constraints, which is insufficient to match the statistics of the distribution or compute the probability of the chosen events. To achieve these ends, optimally one would use the conditional score function, but its computation is typically intractable. In this work, we develop a probabilistic approximation scheme for the conditional score function which provably converges to the true distribution as the noise level decreases. With this scheme we are able to sample conditionally on nonlinear userdefined events at inference time, and matches data statistics even when sampling from the tails of the distribution.

Uncertainty in LiDAR measurements, stemming from factors such as range sensing, is crucial for LIO (LiDAR-Inertial Odometry) systems as it affects the accurate weighting in the loss function. While recent LIO systems address uncertainty related to range sensing, the impact of incident angle on uncertainty is often overlooked by the community. Moreover, the existing uncertainty propagation methods suffer from computational inefficiency. This paper proposes a comprehensive point uncertainty model that accounts for both the uncertainties from LiDAR measurements and surface characteristics, along with an efficient local uncertainty analytical method for LiDAR-based state estimation problem. We employ a projection operator that separates the uncertainty into the ray direction and its orthogonal plane. Then, we derive incremental Jacobian matrices of eigenvalues and eigenvectors w.r.t. points, which enables a fast approximation of uncertainty propagation. This approach eliminates the requirement for redundant traversal of points, significantly reducing the time complexity of uncertainty propagation from O (n) to O (1) when a new point is added. Simulations and experiments on public datasets are conducted to validate the accuracy and efficiency of our formulations. The proposed methods have been integrated into a LIO system, which is available at https://github.com/tiev-tongji/LOG-LIO2.

Contrastively trained encoders have recently been proven to invert the data-generating process: they encode each input, e.g., an image, into the true latent vector that generated the image (Zimmermann et al., 2021). However, real-world observations often have inherent ambiguities. For instance, images may be blurred or only show a 2D view of a 3D object, so multiple latents could have generated them. This makes the true posterior for the latent vector probabilistic with heteroscedastic uncertainty. In this setup, we extend the common InfoNCE objective and encoders to predict latent distributions instead of points. We prove that these distributions recover the correct posteriors of the data-generating process, including its level of aleatoric uncertainty, up to a rotation of the latent space. In addition to providing calibrated uncertainty estimates, these posteriors allow the computation of credible intervals in image retrieval. They comprise images with the same latent as a given query, subject to its uncertainty. Code is available at https://github.com/mkirchhof/Probabilistic_Contrastive_Learning

As large language models continue to be widely developed, robust uncertainty quantification techniques will become crucial for their safe deployment in high-stakes scenarios. In this work, we explore how conformal prediction can be used to provide uncertainty quantification in language models for the specific task of multiple-choice question-answering. We find that the uncertainty estimates from conformal prediction are tightly correlated with prediction accuracy. This observation can be useful for downstream applications such as selective classification and filtering out low-quality predictions. We also investigate the exchangeability assumption required by conformal prediction to out-of-subject questions, which may be a more realistic scenario for many practical applications. Our work contributes towards more trustworthy and reliable usage of large language models in safety-critical situations, where robust guarantees of error rate are required.

We consider the problem of performing Bayesian inference in probabilistic models where observations are accompanied by uncertainty, referred to as "uncertain evidence." We explore how to interpret uncertain evidence, and by extension the importance of proper interpretation as it pertains to inference about latent variables. We consider a recently-proposed method "distributional evidence" as well as revisit two older methods: Jeffrey's rule and virtual evidence. We devise guidelines on how to account for uncertain evidence and we provide new insights, particularly regarding consistency. To showcase the impact of different interpretations of the same uncertain evidence, we carry out experiments in which one interpretation is defined as "correct." We then compare inference results from each different interpretation illustrating the importance of careful consideration of uncertain evidence.

Scaling laws are used to plan multi-million-dollar training runs, but fitting those laws can itself cost millions. In modern large-scale workflows, assembling a sufficiently informative set of pilot experiments is already a major budget-allocation problem rather than a routine preprocessing step. We formulate scaling-law fitting as budget-aware sequential experimental design: given a finite pool of runnable experiments with heterogeneous costs, choose which runs to execute so as to maximize extrapolation accuracy in a high-cost target region. We then propose an uncertainty-aware method for sequentially allocating experimental budget toward the runs most useful for target-region extrapolation. Across a diverse benchmark of scaling-law tasks, our method consistently outperforms classical design-based baselines, and often approaches the performance of fitting on the full experimental set while using only about 10% of the total training budget. Our code is available at https://github.com/PlanarG/active-sl.

Quantitative tools are increasingly appealing for decision support in healthcare, driven by the growing capabilities of advanced AI systems. However, understanding the predictive uncertainties surrounding a tool's output is crucial for decision-makers to ensure reliable and transparent decisions. In this paper, we present a case study on pulmonary nodule detection for lung cancer screening, enhancing an advanced detection model with an uncertainty quantification technique called conformal risk control (CRC). We demonstrate that prediction sets with conformal guarantees are attractive measures of predictive uncertainty in the safety-critical healthcare domain, allowing end-users to achieve arbitrary validity by trading off false positives and providing formal statistical guarantees on model performance. Among ground-truth nodules annotated by at least three radiologists, our model achieves a sensitivity that is competitive with that generally achieved by individual radiologists, with a slight increase in false positives. Furthermore, we illustrate the risks of using off-the-shelve prediction models when faced with ontological uncertainty, such as when radiologists disagree on what constitutes the ground truth on pulmonary nodules.

In this paper, we tackle the problem of active robotic 3D reconstruction of an object. In particular, we study how a mobile robot with an arm-held camera can select a favorable number of views to recover an object's 3D shape efficiently. Contrary to the existing solution to this problem, we leverage the popular neural radiance fields-based object representation, which has recently shown impressive results for various computer vision tasks. However, it is not straightforward to directly reason about an object's explicit 3D geometric details using such a representation, making the next-best-view selection problem for dense 3D reconstruction challenging. This paper introduces a ray-based volumetric uncertainty estimator, which computes the entropy of the weight distribution of the color samples along each ray of the object's implicit neural representation. We show that it is possible to infer the uncertainty of the underlying 3D geometry given a novel view with the proposed estimator. We then present a next-best-view selection policy guided by the ray-based volumetric uncertainty in neural radiance fields-based representations. Encouraging experimental results on synthetic and real-world data suggest that the approach presented in this paper can enable a new research direction of using an implicit 3D object representation for the next-best-view problem in robot vision applications, distinguishing our approach from the existing approaches that rely on explicit 3D geometric modeling.

Identifying how much a model {p}_{theta}(Y|X) knows about the stochastic real-world process p(Y|X) it was trained on is important to ensure it avoids producing incorrect or "hallucinated" answers or taking unsafe actions. But this is difficult for generative models because probabilistic predictions do not distinguish between per-response noise (aleatoric uncertainty) and lack of knowledge about the process (epistemic uncertainty), and existing epistemic uncertainty quantification techniques tend to be overconfident when the model underfits. We propose a general strategy for teaching a model to both approximate p(Y|X) and also estimate the remaining gaps between {p}_{theta}(Y|X) and p(Y|X): train it to predict pairs of independent responses drawn from the true conditional distribution, allow it to "cheat" by observing one response while predicting the other, then measure how much it cheats. Remarkably, we prove that being good at cheating (i.e. cheating whenever it improves your prediction) is equivalent to being second-order calibrated, a principled extension of ordinary calibration that allows us to construct provably-correct frequentist confidence intervals for p(Y|X) and detect incorrect responses with high probability. We demonstrate empirically that our approach accurately estimates how much models don't know across ambiguous image classification, (synthetic) language modeling, and partially-observable navigation tasks, outperforming existing techniques.