Uncertainty Quantification

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.

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.

We propose a data-driven approach to quantify the uncertainty of models constructed by kernel methods. Our approach minimizes the needed distributional assumptions, hence, instead of working with, for example, Gaussian processes or exponential families, it only requires knowledge about some mild regularity of the measurement noise, such as it is being symmetric or exchangeable. We show, by building on recent results from finite-sample system identification, that by perturbing the residuals in the gradient of the objective function, information can be extracted about the amount of uncertainty our model has. Particularly, we provide an algorithm to build exact, non-asymptotically guaranteed, distribution-free confidence regions for ideal, noise-free representations of the function we try to estimate. For the typical convex quadratic problems and symmetric noises, the regions are star convex centered around a given nominal estimate, and have efficient ellipsoidal outer approximations. Finally, we illustrate the ideas on typical kernel methods, such as LS-SVC, KRR, $\varepsilon$-SVR and kernelized LASSO.

Inference for factor models is often hampered by the lack of tractable and accurate variance estimates, which can materially distort downstream analyses. In practice, uncertainty in the residual covariance matrix is frequently either ignored or addressed through computationally intensive resampling methods that tend to be unstable. This paper develops a unified analytical framework for inference in exploratory factor analysis under several widely used extraction rules, including least-squares, principal-factor, iterative principal-component, alpha, and image factoring. By treating these estimators as implicitly defined functions of the sample covariance matrix, we derive closed-form Jacobians that translate perturbations in the covariance matrix into changes in the resulting factor solutions. Combined with the delta method and consistent estimators of the sample covariance matrix, the proposed approach yields standard errors that are straightforward to compute and remain valid under non-Gaussianity, heteroskedasticity, and serial or cross-sectional dependence. Simulation evidence confirms that the analytical standard errors accurately capture finite-sample variability while avoiding both the instability of bootstrap procedures and the restrictive assumptions underlying Fisher information-based inference. An application to a factor-augmented structural vector autoregressive (SVAR) model further demonstrates how accounting for this source of uncertainty can substantially affect impulse-response inference. Taken together, the results provide a practical and general tool for propagating estimation uncertainty in settings where factor extraction serves as an intermediate step.

In this paper, we describe a representation for spatial information, called the stochastic map, and associated procedures for building it, reading information from it, and revising it incrementally as new information is obtained. The map contains the estimates of relationships among objects in the map, and their uncertainties, given all the available information. The procedures provide a general solution to the problem of estimating uncertain relative spatial relationships. The estimates are probabilistic in nature, an advance over the previous, very conservative, worst-case approaches to the problem. Finally, the procedures are developed in the context of state-estimation and filtering theory, which provides a solid basis for numerous extensions.

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.

Advanced Air Mobility (AAM) operations are planned to utilize strategic flight planning services that predict temporal uncertainties to validate flight plans against hazards such as weather cells, restricted airspaces, and CNS disruption areas. This paper presents a Kalman Filter-based uncertainty propagation method that models Flight Management System (FMS) correction behavior through a sigmoid-blended measurement noise covariance. The sigmoid formulation generalizes existing discrete FMS activation thresholds into a continuous, tunable function that smoothly transitions the filter's measurement noise based on progress toward each waypoint. When the measurement noise is high, due to an inverse relationship, the Kalman gain is small and thus uncertainty grows; as the aircraft nears a waypoint, measurement noise decreases as a function of progress, the Kalman gain increases, and state covariance contracts which models the FMS progressively correcting toward the planned trajectory. The approach is computationally efficient (up to two orders of magnitude faster than Monte Carlo methods), scales with control inputs, and is parametrically tunable for different classes of aircraft. The measurement noise covariance is calibrated using real Automatic Dependent Surveillance-Broadcast (ADS-B) data from commercial Instrument Flight Rules (IFR) flights serving as surrogates for future AAM operations, achieving coverage probability conservative relative to theoretical Gaussian predictions at the 1-sigma confidence level on a hold out verification dataset (N = 36). Parameter sensitivity analysis across multiple flight routes demonstrates robust behavior, and comparative evaluation against Monte Carlo and Linear Propagation methods contextualizes the method's computational and accuracy trade-offs.

Trusted multi-view classification typically relies on a view-wise evidential fusion process: each view independently produces class evidence and uncertainty, and the final prediction is obtained by aggregating these independent opinions. While this design is modular and uncertainty-aware, it implicitly assumes that evidence from different views is numerically comparable. In practice, however, this assumption is fragile. Different views often differ in feature space, noise level, and semantic granularity, while independently trained branches are optimized only for prediction correctness, without any constraint enforcing cross-view consistency in evidence strength. As a result, the uncertainty used for fusion can be dominated by branch-specific scale bias rather than true sample-level reliability. To address this issue, we propose Trusted Multi-view learning with Unified Routing (TMUR), which decouples view-specific evidence extraction from fusion arbitration. TMUR uses view-private experts and one collaborative expert, and employs a unified router that observes the global multi-view context to generate sample-level expert weights. Soft load-balancing and diversity regularization further encourage balanced expert utilization and more discriminative expert specialization. We also provide theoretical analysis showing why independent evidential supervision does not identify a common cross-view evidence scale. Extensive experiments on 14 datasets and comparisons with 15 recent baselines demonstrate that TMUR consistently improves both classification performance and reliability. Code is available at https://github.com/YilinZhang107/TMUR.

An LLM judge evaluates outputs at scale. Experts should label only where it is least sure. Its natural escalation signal conflates two uncertainties: aleatoric, real disagreement in the expert pool, which labels cannot reduce, and epistemic, the judge's ignorance, which labels do reduce. A small Bayesian model separates them: a regression on labels already collected learns how far to trust a black-box judge's prediction. Both components follow as simple formulas, with no sampling or further judge calls. The components isolate on a real LLM judge against exactly known truth, and stated confidence is no guide to its actual error. On real human disagreement (ChaosNLI) the epistemic ranking removes 83% more error than total uncertainty for the same expert labels, though simply escalating the least-labelled items does as well there. We demonstrate we can estimate where a judge is ignorant rather than where experts genuinely disagree, and propose using this to direct expert labelling. Code and data are available at https://github.com/composo-ai/judge-uncertainty-decomposition.

Practical uncertainty quantification (UQ) for large language models must decide, from a single generation, whether a specific answer should be trusted. Existing methods either sample multiple generations, read only output-token probabilities, or reduce the model's internal computation to a single hidden state. We introduce ActMap, a white-box representation that compresses the generation-time hidden-state trajectory (every layer, every generated token) into a fixed $12\times32\times128$ tensor of temporal-statistic channels that preserves structure across transformer depth and pooled hidden coordinates. The map is captured during the generation pass with no measurable overhead, has a fixed shape across model depths and hidden sizes, and occupies 96 KiB: a compact artifact that can be retained for audit-relevant generations and probed directly, with occlusion analysis localizing the classifier's signal to mid-depth regions of the map. A lightweight classifier, instantiated as a compact Vision Transformer, reads an estimated correctness probability from each map in a fraction of a millisecond; capacity-matched MLPs perform comparably, indicating the representation itself carries the result. Trained and evaluated in-domain on short-answer QA, direct-answer math, and summarization factuality with three instruction-tuned 7-8B models, ActMap consistently outperforms sampling, token-probability, attention, and embedding baselines, and matches ACT-ViT, a detector trained on dense activation tensors $67\times$ larger, at essentially the same mean AUROC with lower calibration error on ten of twelve pairs. The resulting score supports abstention, routing, and selective verification from a single generation, making it a practical primitive for scalable oversight of deployed models.