Uncertainty Quantification

Accurate modeling of quantum many-body systems often requires computationally expensive simulations such as Density Matrix Renormalization Group (DMRG) or Quantum Monte Carlo (QMC) calculations. These methods, while precise, impose significant time and resource constraints, limiting their use in exhaustive parameter exploration. Moreover, these expensive simulations can contain variable errors over the large unknown parameter space, which needs to be quantified and propagated. Thus, predictive modelling is required to estimate the functional space accurately over scarcely sampled data with heteroskedastic noise, while preserving the physical relevance of the estimation. Therefore, we present a Physically Constrained Ensemble Gaussian Process (pc-EGP) framework designed to efficiently model complex and noisy quantum systems under physical consistency constraints. The proposed method first enforces physical constraints as a user controlled weighted penalty to the data-driven loss function of the Gaussian Process (GP) surrogates. Then an ensemble of such GP models is trained with variable noisy simulations via numerical quadrature method where these multiple GP(s) at different nodes is integrated as a quadrature weighted average. We first demonstrate the framework on synthetically generated data before applying to quantum systems. In the first case study, we leverage DMRG simulations of the Bose-Hubbard Model to predict the critical interaction parameter Uc governing the superfluid-to-Mott-insulator transition. In the second case study, we demonstrate our method on QMC simulations, of a quantum liquid confined inside a nanoporous silicate with the goal of optimizing a chemical environment to realize a one-dimensional superfluid. Compared to conventional GP, pc-EGP achieves a better balance of accuracy and physically meaningful predictions.

Dynamical systems reconstruction (DSR) aims to learn surrogate models that capture the dynamics underlying time-series data. Reliably deploying these surrogates requires uncertainty estimates consistent with the learned dynamics. We expose a dynamic-probabilistic consistency (DPC) gap: the pursuit of finite-horizon probabilistic objectives can degrade dynamics or decouple predictive uncertainty from the local tangent dynamics it ought to reflect. We isolate three mechanisms behind this gap: core collapse, noise masking, and blind uncertainty. Specifically, we show that open-loop Gaussian rollout objectives can penalize Jacobian-generated covariance growth in chaotic systems, encouraging optimization shortcuts that weaken physical expansion or decouple uncertainty from it. To mitigate this gap, we propose KAFFEE (Kalman-Aware Framework For Ergodic Emulation), a differentiable extended Kalman filter-based training framework that evaluates likelihood on local predictive residuals (innovations) while transporting covariance through learned local Jacobians. On stochastic hyperchaotic Lorenz-96, KAFFEE reduces the identified failure modes, improves reconstruction of dynamical invariants relative to open-loop objectives, and maintains competitive predictive scores. We further show that the DPC gap appears when probabilistically adapting a DSR foundation model across 13 chaotic systems, where KAFFEE enables in-context Bayesian filtering while largely preserving zero-shot dynamics.

This study presents a novel hybrid prognostic framework for uncertainty-aware Remaining Useful Life (RUL) estimation in turbofan engines using the NASA C-MAPSS dataset. The framework employs a state-aware strategy that bifurcates the engines operational lifespan into "healthy" and "degraded" regimes. An LSTM-based autoencoder, trained strictly on nominal data (RUL > 150 cycles), monitors reconstruction error to act as a robust state classifier. For the healthy regime, a Conditional Weibull Survival Analysis is used for Mean Residual Life estimation. For the degraded regime, a Probabilistic Neural Network with Monte Carlo Dropout captures both aleatoric and epistemic uncertainties. Rather than using rigid binary labels, a calibrated sigmoid function converts the autoencoders output into continuous state probabilities, dynamically weighting the final ensemble prediction. The primary strength of this framework is its generation of physically consistent uncertainty bands, yielding high-confidence predictions near end-of-life while accurately reflecting the inherent variance of early operation, providing a robust tool for risk-informed maintenance.

Nonlinear conservation laws are at the heart of many of the most important dynamical systems in science and engineering. In practical applications, such systems are often subject to various sources of uncertainty, e.g. due to sparse or noisy measurements. Inferring physical quantities and fields of interest then becomes an ill-posed problem which both classical numerical methods and modern deep learning-based methods struggle to treat appropriately. Recent work has framed classical numerical methods as Bayesian inference under Gaussian process priors, resulting in a physics-aware treatment of uncertainties. Following this line of work, we develop a novel numerically conservative method for uncertainty-aware simulations of nonlinear conservation laws. We use recent sparse approximation techniques to scale up to large-scale forward and inverse problems. For forward simulation, we inherit the accuracy of classical solvers while providing structured uncertainty quantification. On inverse problems, we recover posteriors over nonparametric source fields in seconds -- outperforming neural baselines that take minutes to produce a less accurate point estimate.

Conditional generative models have recently achieved remarkable success in various applications. However, a suitable metric for evaluating the reliability of these models, which takes into account their inherent uncertainty, is still lacking. Existing metrics, which typically assess a single output, may fail to capture the variability or potential risks in generation. In this paper, we propose a novel evaluation metric called reliability score based on conformal prediction, which measures the worst-case performance within the prediction set at a pre-specified confidence level. However, computing this score is challenging due to the high-dimensional nature of the output space and the nonconvexity of both the metric function and the prediction set. To efficiently compute this score, we introduce Conformal ReLiability (CReL), a framework that can (i) construct the prediction set with desired coverage; and (ii) accurately optimize the reliability score within the constructed prediction set. We provide theoretical results on coverage and demonstrate empirically that our method produces more informative prediction sets than existing approaches. Experiments on synthetic data and the image-to-text and text-to-image tasks further demonstrate the interpretability of our new metric, and the validity and effectiveness of our computational framework. Source code can be found at https://ggc29.github.io/CReL/.

Epistemic uncertainty quantification (UQ) for deep neural networks (DNNs) is a requirement for safe adoption of AI in mission-critical settings. Several leading methods for UQ linearize DNNs to form Bayesian Generalized Linear Models (GLMs), where epistemic uncertainty is modeled via the predictive posterior distribution. Linearizing around the parameters of the final connected layer of a DNN is a commonly used approximation for reducing the computational burden of such GLMs, though it is often believed to come at the cost of degraded performance. In this work, we compare GLMs arising from full-network and last-layer linearization using both theoretical and empirical approaches. We first employ tools from random matrix theory to conduct a theoretical comparison; this analysis reveals no meaningful improvement in the UQ capabilities of full linearization. Coupled with a large-scale empirical evaluation across a range of modern machine learning tasks, we arrive at the following conclusion: a last-layer approximation yields comparable UQ performance while offering substantially improved computational efficiency.

Effective prognostics and health management of modern engines relies on accurate turbine gas temperature predictions and robust uncertainty quantification to ensure reliability and safety. This paper investigates five major approaches for constructing prediction intervals -- namely the Delta method, Bayesian Monte Carlo Dropout, Bootstrap method, Lower-Upper Bound Estimation, and Mean-Variance Estimation -- as a means of capturing the uncertainty in neural network predictions of turbine gas temperature. Each approach is implemented within a unified experimental framework that employs cross-validation for hyperparameter selection, repeated train-test splits for performance robustness, and multiple metrics to evaluate both the accuracy and tightness of the intervals. In particular, Coverage Probability, Normalized Mean Prediction Interval Width, and the Coverage Width-based Criterion are measured to comprehensively assess each method's reliability and sharpness. Experiments conducted on a representative turbine gas temperature dataset reveal distinct trade-offs among the five methods in terms of interval coverage, width, and stability. These findings provide a practical guide for selecting and tuning prediction interval methods in engine health management and prognostics, ensuring both interpretability and precision in real-world applications.

We present Gaussian Splatting Anisotropic Visibility Field (GAVIS), a novel framework for uncertainty quantification and active mapping in 3DGS. Our key insight is that regions unseen from the training views yield unreliable predictions from the 3DGS. To address this, we introduce a principled and efficient method for quantifying the visibility field in 3DGS, defined as the anisotropic visibility of each particle with respect to the training views, and represented using spherical harmonics. The resulting visibility field is integrated into a Bayesian Network-based uncertainty-aware 3DGS rasterizer, enabling real-time (200 FPS) uncertainty quantification for synthesized views. Active mapping is further performed within a maximum information gain framework building on this formulation. Extensive experiments across diverse environments demonstrate that GAVIS consistently and significantly outperforms prior approaches in both accuracy and efficiency. Moreover, beyond standalone use, our method can be applied post-hoc to improve the performance of existing approaches.

Scaling data-driven energy forecasting to district level requires models that can be re-used across buildings with minimal target-domain data and honest uncertainty estimates. We present an uncertainty-aware transfer learning (TL) framework for cross-building energy forecasting based on the Temporal Fusion Transformer (TFT), evaluated on a newly released high-resolution real sub-meter dataset: an educational building at Aalborg University, Denmark (source) and the multi-typology NEST building at EMPA, Switzerland (target). We introduce the Transfer Robustness Index (TRI), an architecture-agnostic metric for quantifying generalization quality across domain gaps. A four-strategy layer-freezing ablation shows that Probe-Only fine-tuning, updating only 455 output-layer parameters out of 806K, achieves the best transfer quality (TRI = 3,097), outperforming full fine-tuning and suggesting that TFT encoders learn transferable temporal representations. Monte Carlo Dropout yields a prediction interval coverage probability of 93.2%, close to the nominal 95% target. A data-scarcity analysis further shows monotonic improvement with increasing target-domain data, providing practical guidance for district energy deployment.

Many applications require statistically valid inference across many related tasks, while using only a handful of high-quality labels per hypothesis. In AI evaluation, these tasks may correspond to model behaviors across prompts, subgroups, or hypotheses; in social science surveys, they may correspond to related questions, populations, or measurement conditions. Prediction-powered inference (PPI) uses abundant but inexpensive proxy measurements to improve inference from limited, ground-truth labels, but commonly used methods treat tasks independently and therefore fail to exploit shared structure across related tasks. This limitation is especially important in settings where only a small number of labels are available per task. To address this issue, we introduce a multi-task prediction-powered inference framework that uses labeled data from related tasks to improve power while preserving task-specific inference. Our methods exploit the shared structure in the proxy-ground-truth relationship through cross-task recalibration, while retaining within-task rectification and power tuning to construct accurate point estimates and confidence intervals. We prove that efficiency gains beyond power-tuned PPI are only possible when the proxy-ground-truth relationship contains nonlinear structure; affine cross-task recalibrations are asymptotically equivalent to using the original proxy. We complement our theoretical findings with experiments on synthetic and semi-synthetic datasets, as well as a case study auditing language models on election-related information during the 2024 U.S. presidential election. Using a large human-annotation study, we show that cross-task recalibration can substantially reduce confidence interval widths when labels are scarce.

Reduced-order models (ROMs) provide an efficient surrogate for complex multiscale systems, but their predictive accuracy is often compromised by truncation errors and the inadequate representation of interactions between resolved and unresolved scales. The missing effect of truncated (unresolved) scales on ROM (resolved) scales is often denoted as the closure problem. In this work, we formulate ROM closure modeling as a multi-fidelity (MF) learning problem and propose an uncertainty-aware MF framework based on conditional normalizing flow to enhance ROM predictive accuracy. The proposed approach learns a probabilistic mapping from low-fidelity (LF) ROM coefficients to high-fidelity (HF) coefficients, thereby improving predictive fidelity while quantifying the uncertainty associated with the learned closure. Two correction strategies are investigated: direct learning, in which HF coefficients are predicted directly from LF inputs, and residual learning, which learns the discrepancy between LF and HF coefficients and uses it to recover the corrected HF solution. The framework is demonstrated on a vortex merging problem governed by the two-dimensional Navier Stokes equations. Results show that both correction strategies improve ROM accuracy over uncorrected ROM, with residual learning achieving consistently better performance than direct learning. Moreover, the two proposed deep generative model-based strategies provide uncertainty quantification for the corrected ROM coefficients, which is critical for assessing prediction confidence and supporting the reliable use of ROMs in practical applications.

Conformal prediction (CP) and its extension, conformal risk control (CRC), are established frameworks for quantifying uncertainty in supervised machine learning through formal guarantees. However, recent breakthroughs in artificial intelligence (AI) have been driven by unsupervised generative models, such as large language models (LLMs) and image generators, which are not directly compatible with CP or CRC. In this work we introduce conformal generation (Conf-Gen), a general framework adapting CRC to generative tasks while relaxing its theoretical assumptions. Conf-Gen unifies and generalizes previous attempts to apply CP to LLMs, and extends conformal methodology to entirely new domains. We demonstrate the flexibility of Conf-Gen through some novel applications, including obtaining conformal guarantees on: image generators producing non-memorized images, conversational AI systems having asked enough clarifying questions, and the output of AI agents being correct.

Generative modeling provides a powerful framework for learning data distributions. These models initially relied on probabilistic methods such as Gaussian Processes (GP) for uncertainty-aware predictions and shifted towards larger trainable models to learn more complex distributions. In this work, we introduce Random Process (RP) Flow, a Flow Matching-based framework that represents the vector field as a neural implicit function. Unlike modern generative methods, our setting involves a single observed field, from which only sparse measurements are available. RP Flow uses Random Fourier Features to learn an implicit signal representation that can be queried at any arbitrary location from a limited set of observations, while encoding uncertainty through ensemble sampling. We propose constructing a Bayesian posterior by GP regression in the source space to generate high-quality samples. Our empirical results demonstrate that this framework generates realistic samples along with calibrated uncertainty estimates, even under challenging conditions such as high frequency, high sparsity, or high dimensionality. These findings position RP Flow as a milestone towards generative models for reconstruction tasks where data is scarce and uncertainty must remain traceable.

Recent Tabular Foundation Models (TFMs) have demonstrated state-of-the-art predictive performance, often surpassing Gradient-Boosted Decision Trees (GBDTs). However, the trustworthiness of these models, particularly their uncertainty quantification, has been largely overlooked. We investigate this gap through an extensive study comparing TFMs, GBDTs, and classical baselines on the 112 datasets of the TALENT benchmark. Our results reveal a performance-uncertainty trade-off: although TFMs achieve the highest predictive performance, measured by AUC, they exhibit lower conditional coverage under conformal prediction, measured by SSCS, compared to GBDTs. Complementary experiments on synthetic datasets further characterize the regimes in which this effect intensifies. We conclude that while TFMs advance predictive frontiers, achieving well-calibrated uncertainty remains a major open challenge for their reliable adoption. Code is available at: https://github.com/jose-melo/high-performance-low-reliability

In safety-critical robotics applications, guaranteed and practical uncertainty quantification (UQ) in perception is vital. Many existing works either offer no formal containment guarantee, rely on restrictive modeling assumptions, or focus only on pose estimation rather than a complete SLAM pipeline. This paper presents provably guaranteed UQ algorithms for 3D-3D landmark-based SLAM. The algorithms consist of three basic UQ modules: forward UQ for mapping, backward UQ for pose tracking, and pose compound. Each module produces a certified uncertainty set; when the input uncertainty bounds are deterministic, the output sets inherit deterministic guarantees, i.e., they provably contain the true poses and landmarks. Specifically, we use polytopes to represent uncertainty sets, enabling tractable computations and a unified treatment of pose uncertainty. To enhance algorithms' practical usability, we incorporate conformal prediction to calibrate measurement uncertainty from data with prescribed probability. Simulations and experiments demonstrate that the proposed algorithms provide both strong theoretical guarantees and practical usability. The code is open-sourced at https://github.com/LIAS-CUHKSZ/Polytopic-SLAM-Uncertainty-Quantification.

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.

Knowledge distillation is a powerful method for model compression, enabling the efficient deployment of complex deep learning models (teachers), including large language models. However, its underlying statistical mechanisms remain unclear, and uncertainty evaluation is often overlooked, especially in real-world scenarios requiring diverse teacher expertise. To address these challenges, we introduce \textit{Multi-Teacher Bayesian Knowledge Distillation} (MT-BKD), where a distilled student model learns from multiple teachers within the Bayesian framework. Our approach leverages Bayesian inference to capture inherent uncertainty in the distillation process. We introduce a teacher-informed prior, integrating external knowledge from teacher models and task-specific training data, offering better generalization, robustness, and scalability. Additionally, an entropy-based weighting mechanism adaptively adjusts each teacher's influence, allowing the student to combine multiple sources of expertise effectively. MT-BKD enhances the interpretability of the student model's learning process, improves predictive accuracy, and provides uncertainty quantification. We validate MT-BKD on both synthetic and real-world tasks, including protein subcellular location prediction and image classification. Our experiments show improved performance and robust uncertainty quantification, highlighting the strengths of our MT-BKD framework.

RGB-D direct visual odometry (VO) benefits from metric depth measurements but often degrades in the presence of dynamic objects, occlusions, illumination changes, and unreliable depth, which violate the photometric and geometric consistency assumptions of direct alignment. We propose Con-DSO, a consistency-aware RGB-D direct sparse odometry framework that addresses these challenges through a unified learned uncertainty model. A dual-branch consistency network is trained on adjacent RGB-D frame pairs using flow-guided photometric errors and projective depth-consistency errors to predict pixel-level photometric and geometric uncertainty. The predicted uncertainty is first converted into pairwise quality to guide support-pixel selection and is then fused across adjacent frame pairs to form a host-side quality prior for keyframe-based tracking. To account for the different roles of photometric and depth information in direct RGB-D optimization, the quality prior is incorporated through a decoupled photometric-geometric weighting scheme, with the geometric weight applied only to the translational component of pose estimation. Experiments on five public RGB-D benchmarks demonstrate consistent improvements over direct RGB-D odometry baselines, achieving more than 20% reduction in absolute trajectory error on ICL-NUIM and approximately 50% to 80% reductions on RGB-D Scenes V2, TUM/BONN, and OpenLORIS. These results demonstrate that learned consistency-aware uncertainty can substantially improve the robustness of RGB-D direct visual odometry in challenging environments.

As Low Earth Orbit has grown more crowded with space debris, the need for reliable and efficient debris removal solutions becomes more urgent. An active tether-net system with maneuverable units is one of the promising solutions to this problem, whose success is dependent on the robustness of the net maneuver and closing decisions. These in turn are impacted by the uncertainties attributed to i) noisy observation of the target debris state (e.g., sensing errors), and ii) imperfect simulations of the complex net dynamics and net/debris interaction behavior, over which the decision system is trained. This paper focuses on the first of these two uncertainty sources, and presents a pipeline to propagate and quantify the resulting uncertainty in the debris capture performance expressed in terms of Capture Quality Index (CQI). This quantification is uniquely performed for both an active tether-net using a fixed baseline control and one using a trained neuro-control policy to guide the net maneuver during the deployment phase. Two different uncertainty quantification (UQ) techniques, namely Sobol's variance-based sensitivity analysis and perturbation-based method are exploited. A high-fidelity simulator and a lower-fidelity surrogate-based environment are used to demonstrate trade-offs between prediction accuracy versus ease of resolving uncertainties.

Neural cellular automata (NCA) provide a lightweight alternative to encoder-decoder segmentation networks. However, it can be difficult to decide when a prediction should be trusted. Here, we study uncertainty estimation for NCA-based medical image segmentation without modifying the underlying architecture or retraining the model. Our approach is motivated by viewing the NCA as a dynamical system where convergent attractors correspond to confident predictions. Concretely, we propose resilience, a simple measure that leverages the intrinsic iterative structure of NCAs by probing the stability of the final prediction under small perturbations of the automaton state. Predictions that return to the same solution are deemed confident, while those that change substantially are flagged as uncertain. We evaluate uncertainty by its ability to predict segmentation quality using selective prediction metrics ($Δ$Dice@90 and AURC) and ranking metrics (AUROC and AUPRC). Across multiple medical segmentation benchmarks, resilience identifies failure cases more reliably than baselines, improving trust and safety in NCA-based models.

While Deep Neural Networks (DNNs) achieve remarkable performance, their tendency to produce overconfident predictions. Evidential Deep Learning (EDL) mitigates this by formulating predictions as a Dirichlet distribution over class probabilities to explicitly quantify epistemic uncertainty. However, we found that the conventional EDL suffers from two fundamental limitations: a Kullback-Leibler (KL) penalty that only suppresses the evidence of negative classes, producing excessively high evidence therefore decreasing the model's ability to quantify uncertainty, and an absence in theoretical guarantee of setting Dirichlet parameter $α=e+1$. In this paper, we propose a mathematically principled framework, Variational Inference Evidential Deep Learning (VI-EDL). By reformulating evidential learning through the lens of variational inference, we derive an Evidence Lower Bound (ELBO), which prevents the evidence from growing excessively. Theoretically, we rigorously establish a generalization bound and reveal how the predicted uncertainty, feature and network complexity affect this bound, and why setting $\boldsymbolα = \mathbf{e} + \mathbf{1}$ can minimize it. Extensive experiments on standard visual and medical datasets demonstrate that VI-EDL achieves state-of-the-art performance, showing excellent performance in out-of-distribution detection, noise detection and autonomous driving scenario. The code is available in https://github.com/seutjw/VI-EDL.

Reliable uncertainty quantification in continuous-time (CT) representation learning remains nascent, particularly within CT attention literature. We introduce the Neuronal Stochastic Attention Circuit (NSAC), a novel biologically-inspired CT attention architecture that reformulates attention logit computation as the solution of an Ornstein-Uhlenbeck stochastic differential equation modulated by input-dependent, nonlinear interlinked gates derived from repurposed C. elegans Neuronal Circuit Policies (NCPs) wiring mechanism. It induces a Gaussian distribution over logits that propagates principled stochasticity through a logistic-normal distribution over attention weights to yield probabilistic output. A two-term objective function combining Gaussian negative log-likelihood with an epistemic-separation regularizer enforces higher predictive variance under distributional shifts and enables joint quantification of aleatoric and epistemic uncertainty. Theoretically, we provide: (i) state stability bounds; (ii) closed-form guarantees; and (iii) frozen-coefficient error approximation. Empirically, we implement NSAC in a diverse set of learning tasks including: (i) irregular CT function approximation; (ii) multivariate regression; (iii) long-range forecasting; (iv) Industry 4.0; and (v) lane-keeping of autonomous vehicles. We observe that NSAC remains competitive against several baselines in terms of accuracy and produces informative uncertainty estimates while being interpretable at the neuronal cell level.

Stochastic gradient descent (SGD) is a foundational algorithm for large-scale statistical learning and stochastic optimization. However, statistical inference based on SGD iterates remains challenging when stochastic gradients have infinite variance, as the relevant limiting distributions depend on unknown nuisance parameters. In this paper, we develop an efficient, model-agnostic methodology for constructing confidence regions from SGD trajectories that applies in both finite- and infinite-variance regimes. The procedure is based on a joint weak convergence result for the Polyak-Ruppert averaged estimator and an empirical second-moment normalizer constructed from stochastic gradients along the SGD trajectory. This joint limit yields a self-normalized statistic in which the leading tail-dependent scaling terms cancel. We then use a subsampling calibration scheme to estimate the relevant critical values, avoiding explicit estimation of tail indices, slowly varying functions, or stable-law parameters. The resulting confidence regions are straightforward to implement and are asymptotically valid under both the finite- and infinite-second-moment regimes. Simulation studies show reliable coverage in various settings, supporting the proposed method as a practical tool for uncertainty quantification in stochastic optimization.

We introduce Fuzzy PyTorch, a framework for rapid evaluation of numerical variability in deep learning (DL) models. As DL is increasingly applied to diverse tasks, understanding variability from floating-point arithmetic is essential to ensure robust and reliable performance. Tools assessing such variability must be scalable, efficient, and integrate seamlessly with existing frameworks while minimizing code modifications. Fuzzy PyTorch enables this by integrating stochastic arithmetic into PyTorch through Probabilistic Rounding with Instruction Set Management, a novel library interfacing with Verificarlo, a numerical analysis compiler. The library offers stochastic rounding mode and a novel mode; up-down rounding. Comparative evaluations show Fuzzy PyTorch maintains model performance and achieves runtime reductions of 5x to 60x versus Verrou, a state-of-the-art tool. We further demonstrate scalability by running models from 1 to 341 million parameters, confirming applicability across small and large DL architectures. Overall, Fuzzy PyTorch provides an efficient, scalable, and practical solution for assessing numerical variability in deep learning, enabling researchers and practitioners to quantify and manage floating-point uncertainty without compromising performance or computational efficiency.

Recent advances in uncertainty quantification increasingly emphasise the distinction between aleatory and epistemic uncertainty in machine learning, motivating the need for more unified frameworks. However, despite much progress in producing reliable predictions, existing methods often lack rigorous guarantees when generalising beyond the training domain. We propose a conformalised imprecise inference framework for robust extrapolation, which is model-agnostic and augments predictive models with imprecision and distance awareness. The proposed approach yields imprecise predictions (probability boxes) that remain valid under distributional shift, maintaining coverage while adaptively expanding uncertainty in extrapolation regimes. Experiments on synthetic and benchmark datasets demonstrate improved robustness and reliable coverage compared to standard probabilistic approaches, particularly under limited data.

Single-pass uncertainty quantification (UQ) methods for classification represent uncertainty by predicting a tractable distribution over the class probability vector. While existing approaches primarily focus on enhancing the expressiveness of this distribution, they often provide limited insight into how predictive uncertainty is structured and aggregated, resulting in weak interpretability. We introduce the courtroom analogy, which conceptualizes uncertainty-aware classification as a structured debate among class-specific advocates. Each advocate forms a probabilistic opinion, and a final verdict is reached by aggregating these opinions using input-dependent plausibility weights. In this framework, each advocate's opinion is modeled as a Dirichlet distribution whose concentration parameter is decomposed into shared evidence and class-specific advocacy. This yields a structured mixture of Dirichlet distributions with semantically interpretable parameters. To instantiate this formulation, we propose Mixture of Dirichlet EXperts (MoDEX), a single-pass neural architecture that predicts the courtroom parameters, enabling efficient and expressive UQ while explicitly modeling uncertainty aggregation. We demonstrate that MoDEX enjoys strong theoretical properties and achieves state-of-the-art UQ performance across diverse benchmarks, yielding interpretable uncertainty estimates with meaningful semantics.

Evidential Deep Learning (EDL) has emerged as an efficient, sampling-free strategy for uncertainty estimation. A series of EDL variants have been proposed to address specific limitations of the original framework, achieving notable success. However, the underlying theoretical structure of EDL and the relationships among these variants have received limited systematic investigation. In this work, we establish a principled theoretical foundation for EDL by interpreting it within a generalized Bayesian framework that includes prior specification, posterior update, and training objective. We further characterize evidential uncertainty from a Bayesian distributional uncertainty viewpoint, established via asymptotic analysis. Building on this perspective, we further propose Generalized Evidential Deep Learning (GEDL), a unified and extensible framework that explicitly disentangles the roles of individual components and systematically relates GEDL to existing variants. Extensive experiments demonstrate that GEDL yields comparable results on classification, uncertainty estimation and OOD detections, with theoretical grounding.

We introduce a user defined probabilistic polygonal representation for plane curves. Given a curve, we select vertices on the curve and connect consecutive vertices by line segments to obtain a polygonal approximation. Each segment is equipped with a user defined uncertainty parameter in the normal direction. This yields a collection of thin probabilistic geometric primitives that retain the geometrz of the underlying curve while extending it beyond the idealized deterministic one dimensional formulation. For each segment, we define a Random Variable that is uniform distributed in the tangent direction of the segment and Gaussian distributed in the normal direction of the segment. By matching the first and the second central moments, this construction induces a Gaussian component whose mean lies at the segment midpoint and whose covariance encodes both tangential and normal uncertainty. Combining the segment wise components with appropriate weights yields a Gaussian Mixture Model (GMM) representation of the user defined probabilistic polygonal representation of the plane curve. The proposed framework provides an analytically tractable probabilistic model that preserves local geometry, and uncertainty in the normal direction. It applies to smooth, closed, open, non regular, and self intersecting plane curves, allows adaptive discretization and varying uncertainty in the normal direction, and as a result supports uncertainty aware geometric modeling. Experiments on a collection of canonical plane curves show that the resulting GMM capture local tangent, local normal, and local arc length; resulting in the global shape of the underlying curves to be truthfully captured as well. The representation is particularly relevant for applications in uncertainty aware CAD and digital twins, probabilistic obstacle modeling in robotics, and probabilistic trajectory planning.

Inferring network-wide traffic states from sparse observations with high accuracy and trustworthy uncertainty quantification is essential for intelligent transportation systems, yet it remains challenging due to the underdetermined nature of the problem, multifaceted disturbances in sensing networks, and the inherent conflicts among multiple inference sub-tasks when modeled jointly. We propose the Task-Aware Attentive Neural Process (TA-ANP), a unified probabilistic framework for resilient and trustworthy global traffic state inference (GTSI) by fusing floating car data (FCD) with sparse fixed-detector measurements. By casting GTSI as a stochastic process, TA-ANP leverages the meta-learning properties of neural processes to adapt rapidly to changes in sensing configurations without retraining. A task-aware multi-query attention module with distinct spatiotemporal inductive biases is introduced to jointly handle three GTSI sub-tasks, while mitigating cross-task interference. For uncertainty quantification, we combine neural processes with Monte Carlo Dropout to capture both aleatoric and epistemic uncertainty. To support metropolis-scale evaluation, we construct the Metropolitan Multi-Source Traffic Dataset (MMTD), integrating fixed-loop sensor measurements, FCD statistics, and OpenStreetMap road-network data over an urban network of 2,371 road segments. Experiments on MMTD show that TA-ANP achieves state-of-the-art performance across all sub-tasks under deterministic and probabilistic metrics. The resulting well-calibrated uncertainties enable more efficient fixed-sensor placement with fewer sensor deployments. Under a Damage-Repair-Addition sensing lifecycle, TA-ANP demonstrates superior resilience in terms of disturbance absorption, performance recovery, and adaptability to unseen sensing configurations.

Annotator disagreement in emotion classification reflects ambiguity intrinsic to emotion concepts and is essential for predictor-quality assessment in subjective NLP. Yet no prior work integrates soft-label learning with Bayesian deep learning to evaluate uncertainty along axes including annotator-distribution fidelity. We train a linear head on a frozen RoBERTa via cyclical stochastic gradient Markov chain Monte Carlo (cSG-MCMC), targeting the empirical annotator distribution with a soft-label objective under a five-axis evaluation. On the 28-emotion GoEmotions benchmark, the proposed method outperforms Monte Carlo Dropout and Deep Ensemble simultaneously on three axes -- Jensen-Shannon divergence (JSD) to the annotator distribution, Spearman correlation between per-emotion aleatoric uncertainty and disagreement, and selective-prediction Area Under the Risk-Coverage Curve (AURC) and Area Under the ROC Curve (AUROC) -- showing independent axes are jointly attainable from one posterior. Post-hoc temperature scaling exhibits a bidirectional effect, establishing hard-label calibration and annotator-JSD as independent dimensions and motivating joint reporting as an honest protocol.