Uncertainty Quantification

Benchmarking the capabilities of AI systems, including Large Language Models (LLMs) and Vision Models, typically ignores the impact of uncertainty in the underlying ground truth answers from experts. This ambiguity is not just limited to human preferences, but is also consequential even in safety critical domains such as medicine where uncertainty is pervasive. In this paper, we introduce a probabilistic paradigm to theoretically explain how high certainty in ground truth answers is almost always necessary for even an expert to achieve high scores. In contrast, in datasets with high variation in ground truth answers, the expert performance drops substantially resulting in almost no difference between a random labeller and an expert. This characteristic also manifests when comparing models, where uncertainty obfuscates differences between poor and high performing models. Therefore, ignoring uncertainty in ground truth evaluation data can result in the misleading conclusion that a non-expert has similar performance to that of an expert. Given ground truth answer variability and commonly used F1 and accuracy metrics, we bring forth a simple and explainable probabilistic coin toss model to estimate expected accuracyand expected F1 and compare the estimated scores with those that experts (human or model) achieve across multiple domains -- including radiology and models such as GPT-5 and Gemini-3. The results lead to the recommendation that stratification by the probability of the ground truth answer becomes critical when expert performance is relatively low. Under stratified evaluation, performance comparison becomes more reliable in high certainty bins, mitigating the effect of the key confounding factor --- uncertainty.

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

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

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

Conformal regression provides finite-sample marginal coverage, but interval efficiency depends on how uncertainty is allocated across inputs. We propose Conformal Laplace-Aware Predictive Scaling (CLAPS), which combines a learned heteroscedastic scale with last-layer Laplace uncertainty. The learned variance also weights the training features used to construct the posterior, yielding a noise-weighted support correction in the learned representation. The correction contracts as weighted feature information increases and becomes most influential under localized support imbalance. Experiments show near-nominal marginal coverage and modest efficiency improvements over the matched aleatoric baseline on real regression benchmarks.

Conformal prediction (CP) constructs uncertainty sets for model outputs with finite-sample coverage guarantees. Yet ranking scores is straightforward only when they are scalar-valued, limiting CP to real-valued scores or ad-hoc one-dimensional reductions. Vector-valued scores arise naturally in multi-output regression and model aggregation, where each predictor in an ensemble provides its own score. Optimal transport (OT) defines vector ranks and center-outward multivariate quantile regions, though generally with asymptotic coverage guarantees. Applying a fixed transport map learned from calibration data to a new point introduces an uncontrolled approximation error. We restore finite-sample, distribution-free coverage by conformalizing vector-valued OT quantile regions. Each candidate's rank is defined by transporting the calibration scores augmented with that candidate's score, preserving the symmetry needed for validity. This appears to require a continuum of OT problems. However, we prove that the optimal assignment is piecewise constant across a fixed polyhedral partition of score space. This lets us characterize the entire prediction set in $O(n^3)$ time, matching the cost of a single assignment solve. It also addresses a limitation of prediction sets: they indicate which outcomes are plausible, but not their relative likelihood. In one dimension, conformal predictive distributions (CPDs) fill this gap by producing a predictive distribution with finite-sample calibration. Extending CPDs beyond one dimension remained an open problem. We construct, to our knowledge, the first multivariate CPDs with finite-sample calibration: a center-outward predictive distribution whose derived uncertainty regions have conformal coverage. We present both conservative and exact randomized versions; the latter generalizes the classical Dempster-Hill procedure.

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

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

We introduce Deep Jump Gaussian Processes (DJGP), a novel method for surrogate modeling of a piecewise continuous function on a high-dimensional domain. DJGP addresses the limitations of conventional Jump Gaussian Processes (JGP) in high-dimensional input spaces by integrating region-specific, locally linear projections with JGP modeling. These projections employ region-dependent matrices to capture local low-dimensional subspace structures, making them well suited to the inherently localized modeling behavior of JGPs, a variant of local Gaussian processes. To control model complexity, we place a Gaussian Process prior on the projection matrices, allowing them to evolve smoothly across the input space. The projected inputs are then modeled with a JGP to capture piecewise continuous relationships with the response. This yields a distinctive two-layer deep learning of GP/JGP. We further develop a scalable variational inference algorithm to jointly learn the projection matrices and JGP hyperparameters. Rigorous theoretical analysis and extensive empirical studies are provided to justify the proposed approach. In particular, we derive an oracle error bound for DJGP and decompose it into four distinct sources of error, which are then linked to practical implications. Experiments on synthetic and benchmark datasets demonstrate that DJGP achieves superior predictive accuracy and more reliable uncertainty quantification compared with existing methods.

Quantum error correction (QEC) is essential for scalable quantum computing, yet decoding errors via conventional algorithms result in limited accuracy (i.e., suppression of logical errors) and high overheads, both of which can be alleviated by inference-based decoders. To date, such machine-learning (ML) decoders lack two key properties crucial for practical fault tolerance: reliable uncertainty quantification and robust generalization to previously unseen QEC codes. To address this gap, we propose a Quantum Bayesian graph Attention decoder (QuBA) that enables expressive error-pattern recognition alongside calibrated uncertainty estimates. Building on QuBA, we further develop a multi-phase training framework with enhanced cross-domain robustness enabling decoding beyond the training set called Sequential Aggregate Generalization under Uncertainty (SAGU). Experiments on bivariate bicycle (BB) codes and their coprime variants demonstrate that (i)QuBA consistently outperforms the classical baseline belief propagation (BP), achieving up to a two orders of magnitude reduction in logical error rate (LER) under confident-decision bounds on the coprime BB code $[[154,6,16]]$; (ii) SAGU achieves decoding performance comparable to or even outperforming QuBA's domain-specific training approach.

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

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

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

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

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

Accurate uncertainty quantification is critical for reliable predictive modeling. Existing methods typically address either aleatoric uncertainty due to measurement noise or epistemic uncertainty resulting from limited data, but not both in a balanced manner. We propose CLEAR, a calibration method with two distinct parameters, $γ_1$ and $γ_2$, to combine the two uncertainty components and improve the conditional coverage of predictive intervals for regression tasks. CLEAR is compatible with any pair of aleatoric and epistemic estimators; we show how it can be used with (i) quantile regression for aleatoric uncertainty and (ii) ensembles drawn from the Predictability-Computability-Stability (PCS) framework for epistemic uncertainty. Across 17 diverse real-world datasets, CLEAR achieves an average improvement of 28.3% and 17.5% in the interval width compared to the two individually calibrated baselines while maintaining nominal coverage. Similar improvements are observed when applying CLEAR to Deep Ensembles (epistemic) and Simultaneous Quantile Regression (aleatoric). The benefits are especially evident in scenarios dominated by high aleatoric or epistemic uncertainty. Project page: https://unco3892.github.io/clear/

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

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

Physics-informed neural networks (PINNs) have proven an effective tool for solving differential equations, in particular when considering the inverse problem. When inferring solutions and parameters of the differential equation from data, uncertainty estimates are preferable to point estimates, as they give an idea about the accuracy of the solution. In this work, we consider the inverse problem for a selection of basic differential equations and employ repulsive ensembles of PINNs (RE-PINN) to obtain such estimates. The repulsion is implemented by adding a particular repulsive term to the loss function, which has the property that the ensemble predictions correspond to the true Bayesian posterior in the limit of infinite ensemble members. Where available, we compare the ensemble predictions to Monte Carlo baselines. Whereas the standard ensemble tends to collapse to maximum-a-posteriori solutions, the repulsive ensemble produces significantly more accurate uncertainty estimates and exhibits higher sample diversity. The experimental results demonstrate that RE-PINN can accurately infer the Bayesian posterior in low-dimensional problems, establishing repulsive ensembles as a promising framework for uncertainty quantification in PINNs.

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

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

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

Echocardiography plays a fundamental role in the extraction of clinical parameters (e.g. left ventricular volume and ejection fraction) required to determine the presence and severity of heart-related conditions. When deploying automated techniques for computing these parameters, uncertainty estimation is highly valuable for assessing their utility. Since clinical parameters are usually derived from segmentation maps, there is no clear path for converting pixel-wise uncertainty values into uncertainty estimates in the downstream clinical metric calculation. In this work, we propose a novel uncertainty estimation method based on contouring rather than segmentation. Our method explicitly predicts contour location uncertainty from which contour samples can be drawn. Finally, the sampled contours can be used to propagate uncertainty to clinical metrics. Our proposed method not only provides accurate uncertainty estimations for the task of contouring but also for the downstream clinical metrics on one public and two private cardiac ultrasound datasets containing a total of 1264 patients. Code is available at: https://github.com/ThierryJudge/contouring-uncertainty.

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

Modern artificial intelligence is supported by machine learning models (e.g., foundation models) that are pretrained on a massive data corpus and then adapted to solve a variety of downstream tasks. To summarize performance across multiple tasks, evaluation metrics are often aggregated into a summary metric, e.g., average accuracy across 10 question-answering tasks. When aggregating evaluation metrics, it is useful to incorporate uncertainty in the aggregate metric in order to gain a more realistic understanding of model performance. Our objective in this work is to demonstrate how statistical methodology can be used for quantifying uncertainty in metrics that have been aggregated across multiple tasks. The methods we emphasize are bootstrapping, Bayesian hierarchical (i.e., multilevel) modeling, and the visualization of task weightings that consider standard errors. These techniques reveal insights such as the dominance of a specific model for certain types of tasks despite an overall poor performance. We use a popular ML benchmark, the Visual Task Adaptation Benchmark (VTAB), to demonstrate the usefulness of our approaches.

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

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

Statistical Taylor expansion is a rigorous extension of conventional Taylor expansion that replaces each precise input variable with a random variable of known distribution and sample count, then computes the mean, deviation, and a bounding reliability of every result. By tracking the propagation of input uncertainties through all intermediate steps, it renders the final result path-independent, with precise quantification of the tracking quality. This path-independence sets it fundamentally apart from conventional numerical approaches, which are path-dependent. This study presents an implementation called variance arithmetic and demonstrates its performance across diverse mathematical applications. This study also reveals the potentially substantial impact of numerical errors in library functions, the defect of applying input uncertainties as weights in conventional regression, and the modeling error of the discrete Fourier transformation. The concept of statistical algebra is also introduced.

Purpose: Uncertainty Quantification (UQ) has gained traction in an attempt to improve the interpretability and robustness of machine learning predictions. Specifically (medical) biosignals such as electroencephalography (EEG), electrocardiography (ECG), electrooculography (EOG), and electromyography (EMG) could benefit from good UQ, since these suffer from a poor signal-to-noise ratio, and good human interpretability is pivotal for medical applications. To determine how uncertainty estimation can be used for biosignal tasks, we investigate current methods, use cases, applications, evaluations, and uncertainty measures. Methods: In this paper, we systematically review the state of the art of applying Uncertainty Quantification to Machine Learning tasks in the biosignal domain. All works from Web of Science, Scopus, IEEE XPlore and PsycINFO that discuss uncertainty in Machine Learning on one of the aforementioned biosignals is included. Results: We present various methods, shortcomings, uncertainty measures and theoretical frameworks that currently exist in this application domain based on the 53 reviewed papers and related literature. We address misconceptions in the field, provide recommendations for future work, and discuss gaps in the literature in relation to diagnostic implementations as well as control for prostheses or brain-computer interfaces. Conclusion: Overall it can be concluded that promising UQ methods are available, but that research is needed on how people and systems may interact with an uncertainty-model in a (clinical) environment.

Embedding models, often obtained via self-supervised learning, extract general-purpose representations from data. Quantifying the reliability of these representations is crucial, as many downstream models rely on them as input for their own tasks. To this end, we introduce a formal definition of representation reliability: the representation for a given test point is considered to be reliable if the downstream models built on top of that representation can, on average, consistently generate accurate predictions for that test point across various downstream tasks. However, accessing the downstream data to quantify the representation reliability is often limited or restricted for various reasons. We propose training-free methods for estimating the representation reliability without access to the downstream data. Our method is based on the concept of neighborhood consistency (NC) across distinct pre-trained representation spaces. The key insight is to find shared neighboring points as anchors to align these representation spaces before comparing them. We provide theoretical justifications for NC and develop two practical approaches: (1) directly computing NC when multiple pre-trained models are available, and (2) a perturbation-based NC (PNC), which creates synthetic ensembles from a single model through isotropic Gaussian noise, avoiding the computational cost of training deep ensembles. We further propose PNC-spread tuning, which systematically determines the perturbation magnitude by maximizing the spread of the PNC scores on a reference set. We demonstrate through comprehensive numerical experiments that our methods effectively capture the representation reliability with a high degree of correlation, achieving robust and favorable performance compared with baseline methods.