The response time of a biosensor is a crucial metric in safety-critical applications such as medical diagnostics where an earlier diagnosis can markedly improve patient outcomes. However, the speed at which a biosensor reaches a final equilibrium state can be limited by poor mass transport and long molecular diffusion times that increase the time it takes target molecules to reach the active sensing region of a biosensor. While optimization of system and sensor design can promote molecules reaching the sensing element faster, a simpler and complementary approach for response time reduction that is widely applicable across all sensor platforms is to use time-series forecasting to predict the ultimate steady-state sensor response. In this work, we show that ensembles of long short-term memory (LSTM) networks can accurately predict equilibrium biosensor response from a small quantity of initial time-dependent biosensor measurements, allowing for significant reduction in response time by a mean and median factor of improvement of 18.6 and 5.1 respectively. The ensemble of models simultaneously estimates uncertainty, which is vital for ensuring confidence in the predictions and subsequent safety-related decisions that are made. This approach is demonstrated on real-time experimental data collected by exposing porous silicon biosensors to buffered protein solutions using a multi-channel fluidic cell that enables the automated measurement of 100 porous silicon biosensors in parallel. The dramatic improvement in sensor response time achieved using LSTM network ensembles and associated uncertainty quantification opens the door to trustworthy and faster responding biosensors, enabling more rapid medical diagnostics for faster clinical decision making that can lead to improved patient outcomes and healthcare access, as well as quicker identification of toxins in food and the environment.
This textbook gives a systematic treatment of statistical machine learning for astronomical research, based on Bayesian inference. It shows how modern data analysis techniques connect to classical statistical methods and follow from familiar statistical principles. The Bayesian perspective, used throughout, gives priority to uncertainty quantification and statistical rigor, which scientific inference in astronomy needs. The textbook progresses from probability theory and Bayesian inference through supervised learning, including linear regression with measurement uncertainties, logistic regression and classification. Unsupervised learning topics cover Principal Component Analysis and clustering methods. We then introduce sampling and Markov Chain Monte Carlo, followed by Gaussian Processes as probabilistic nonparametric methods and neural networks in the broader statistical context. Our theory-focused approach derives each method from first principles with complete mathematical development and then shows its astronomical applications. We put statistical insight first and explain why algorithms work, when they are appropriate, and how they connect to broader statistical principles. From a solid grounding in classical methods and their theory, the treatment leads to modern techniques such as neural networks. With this grounding, readers can apply the methods to astronomical research with proper attention to assumptions, limits and uncertainty propagation, as large astronomical surveys require. The 22 companion tutorials apply the methods to astronomical data in Jupyter notebooks.
We reinterpret and propose a framework for pricing path-dependent financial derivatives by estimating the full distribution of payoffs using Distributional Reinforcement Learning (DistRL). Unlike traditional methods that focus on expected option value, our approach models the entire conditional distribution of payoffs, allowing for risk-aware pricing, tail-risk estimation, and enhanced uncertainty quantification. We demonstrate the efficacy of this method on Asian options, using quantile-based value function approximators.
Quantitative remote sensing inversion aims to estimate continuous surface variables-such as biomass, vegetation indices, and evapotranspiration-from satellite observations, supporting applications in ecosystem monitoring, carbon accounting, and land management. With the evolution of remote sensing systems and artificial intelligence, traditional physics-based paradigms are giving way to data-driven and foundation model (FM)-based approaches. This paper systematically reviews the methodological evolution of inversion techniques, from physical models (e.g., PROSPECT, SCOPE, DART) to machine learning methods (e.g., deep learning, multimodal fusion), and further to foundation models (e.g., SatMAE, GFM, mmEarth). We compare the modeling assumptions, application scenarios, and limitations of each paradigm, with emphasis on recent FM advances in self-supervised pretraining, multi-modal integration, and cross-task adaptation. We also highlight persistent challenges in physical interpretability, domain generalization, limited supervision, and uncertainty quantification. Finally, we envision the development of next-generation foundation models for remote sensing inversion, emphasizing unified modeling capacity, cross-domain generalization, and physical interpretability.
Solving chance-constrained optimal control problems for systems subject to non-stationary uncertainties is a significant challenge.Conventional robust model predictive control (MPC) often yields excessive conservatism by relying on static worst-case assumptions, while standard stochastic MPC methods struggle when underlying uncertainty distributions are unknown a priori.This article presents a Risk-Aware Adaptive Robust MPC (RAAR-MPC) framework,a hierarchical architecture that systematically orchestrates a novel synthesis of proactive, learning-based risk assessment and reactive risk regulation. The framework employs a medium-frequency risk assessment engine, which leverages Gaussian process regression and active learning, to construct a tight, data-driven characterization of the prediction error set from operational data.Concurrently, a low-timescale outer loop implements a self-correcting update law for an adaptive safety margin to precisely regulate the empirical risk and compensate for unmodeled dynamics.This dual-timescale adaptation enables the system to rigorously satisfy chance constraints with a user-defined probability, while minimizing the conservatism inherent in traditional approaches.We formally establish that the interplay between these adaptive components guarantees recursive feasibility and ensures the closed-loop system satisfies the chance constraints up to a user-defined risk level with high probability.Numerical experiments on a benchmark DC-DC converter under non-stationary parametric uncertainties demonstrate that our framework precisely achieves the target risk level, resulting in a significantly lower average cost compared to state-of-the-art robust and stochastic MPC strategies.
A methodology is developed, based on nonparametric Bayesian dictionary learning, for joint space-time wind field data extrapolation and estimation of related statistics by relying on limited/incomplete measurements. Specifically, utilizing sparse/incomplete measured data, a time-dependent optimization problem is formulated for determining the expansion coefficients of an associated low-dimensional representation of the stochastic wind field. Compared to an alternative, standard, compressive sampling treatment of the problem, the developed methodology exhibits the following advantages. First, the Bayesian formulation enables also the quantification of the uncertainty in the estimates. Second, the requirement in standard CS-based applications for an a priori selection of the expansion basis is circumvented. Instead, this is done herein in an adaptive manner based on the acquired data. Overall, the methodology exhibits enhanced extrapolation accuracy, even in cases of high-dimensional data of arbitrary form, and of relatively large extrapolation distances. Thus, it can be used, potentially, in a wide range of wind engineering applications where various constraints dictate the use of a limited number of sensors. The efficacy of the methodology is demonstrated by considering two case studies. The first relates to the extrapolation of simulated wind velocity records consistent with a prescribed joint wavenumber-frequency power spectral density in a three-dimensional domain (2D and time). The second pertains to the extrapolation of four-dimensional (3D and time) boundary layer wind tunnel experimental data that exhibit significant spatial variability and non-Gaussian characteristics.
Birds' Eye View (BEV) semantic segmentation is an indispensable perception task in end-to-end autonomous driving systems. Unsupervised and semi-supervised learning for BEV tasks, as pivotal for real-world applications, underperform due to the homogeneous distribution of the labeled data. In this work, we explore the potential of synthetic data from driving world models to enhance the diversity of labeled data for robustifying BEV segmentation. Yet, our preliminary findings reveal that generation noise in synthetic data compromises efficient BEV model learning. To fully harness the potential of synthetic data from world models, this paper proposes NRSeg, a noise-resilient learning framework for BEV semantic segmentation. Specifically, a Perspective-Geometry Consistency Metric (PGCM) is proposed to quantitatively evaluate the guidance capability of generated data for model learning. This metric originates from the alignment measure between the perspective road mask of generated data and the mask projected from the BEV labels. Moreover, a Bi-Distribution Parallel Prediction (BiDPP) is designed to enhance the inherent robustness of the model, where the learning process is constrained through parallel prediction of multinomial and Dirichlet distributions. The former efficiently predicts semantic probabilities, whereas the latter adopts evidential deep learning to realize uncertainty quantification. Furthermore, a Hierarchical Local Semantic Exclusion (HLSE) module is designed to address the non-mutual exclusivity inherent in BEV semantic segmentation tasks. Experimental results demonstrate that NRSeg achieves state-of-the-art performance, yielding the highest improvements in mIoU of 13.8% and 11.4% in unsupervised and semi-supervised BEV segmentation tasks, respectively. The source code will be made publicly available at https://github.com/lynn-yu/NRSeg.
Tensor Network (TN) Kernel Machines speed up model learning by representing parameters as low-rank TNs, reducing computation and memory use. However, most TN-based Kernel methods are deterministic and ignore parameter uncertainty. Further, they require manual tuning of model complexity hyperparameters like tensor rank and feature dimensions, often through trial-and-error or computationally costly methods like cross-validation. We propose Bayesian Tensor Network Kernel Machines, a fully probabilistic framework that uses sparsity-inducing hierarchical priors on TN factors to automatically infer model complexity. This enables automatic inference of tensor rank and feature dimensions, while also identifying the most relevant features for prediction, thereby enhancing model interpretability. All the model parameters and hyperparameters are treated as latent variables with corresponding priors. Given the Bayesian approach and latent variable dependencies, we apply a mean-field variational inference to approximate their posteriors. We show that applying a mean-field approximation to TN factors yields a Bayesian ALS algorithm with the same computational complexity as its deterministic counterpart, enabling uncertainty quantification at no extra computational cost. Experiments on synthetic and real-world datasets demonstrate the superior performance of our model in prediction accuracy, uncertainty quantification, interpretability, and scalability.
Brain-computer interfaces (BCIs) turn brain signals into functionally useful output, but they are not always accurate. A good Machine Learning classifier should be able to indicate how confident it is about a given classification, by giving a probability for its classification. Standard classifiers for Motor Imagery BCIs do give such probabilities, but research on uncertainty quantification has been limited to Deep Learning. We compare the uncertainty quantification ability of established BCI classifiers using Common Spatial Patterns (CSP-LDA) and Riemannian Geometry (MDRM) to specialized methods in Deep Learning (Deep Ensembles and Direct Uncertainty Quantification) as well as standard Convolutional Neural Networks (CNNs). We found that the overconfidence typically seen in Deep Learning is not a problem in CSP-LDA and MDRM. We found that MDRM is underconfident, which we solved by adding Temperature Scaling (MDRM-T). CSP-LDA and MDRM-T give the best uncertainty estimates, but Deep Ensembles and standard CNNs give the best classifications. We show that all models are able to separate between easy and difficult estimates, so that we can increase the accuracy of a Motor Imagery BCI by rejecting samples that are ambiguous.
Unlike classification, whose goal is to estimate the class of each data point in a dataset, prevalence estimation or quantification is a task that aims to estimate the distribution of classes in a dataset. The two main tasks in prevalence estimation are to adjust for bias, due to the prevalence in the training dataset, and to quantify the uncertainty in the estimate. The standard methods used to quantify uncertainty in prevalence estimates are bootstrapping and Bayesian quantification methods. It is not clear which approach is ideal in terms of precision (i.e. the width of confidence intervals) and coverage (i.e. the confidence intervals being well-calibrated). Here, we propose Precise Quantifier (PQ), a Bayesian quantifier that is more precise than existing quantifiers and with well-calibrated coverage. We discuss the theory behind PQ and present experiments based on simulated and real-world datasets. Through these experiments, we establish the factors which influence quantification precision: the discriminatory power of the underlying classifier; the size of the labeled dataset used to train the quantifier; and the size of the unlabeled dataset for which prevalence is estimated. Our analysis provides deep insights into uncertainty quantification for quantification learning.
6D pose confidence region estimation has emerged as a critical direction, aiming to perform uncertainty quantification for assessing the reliability of estimated poses. However, current sampling-based approach suffers from critical limitations that severely impede their practical deployment: 1) the sampling speed significantly decreases as the number of samples increases. 2) the derived confidence regions are often excessively large. To address these challenges, we propose a deterministic and efficient method for estimating pose confidence regions. Our approach uses inductive conformal prediction to calibrate the deterministically regressed Gaussian keypoint distributions into 2D keypoint confidence regions. We then leverage the implicit function theorem to propagate these keypoint confidence regions directly into 6D pose confidence regions. This method avoids the inefficiency and inflated region sizes associated with sampling and ensembling. It provides compact confidence regions that cover the ground-truth poses with a user-defined confidence level. Experimental results on the LineMOD Occlusion and SPEED datasets show that our method achieves higher pose estimation accuracy with reduced computational time. For the same coverage rate, our method yields significantly smaller confidence region volumes, reducing them by up to 99.9\% for rotations and 99.8\% for translations. The code will be available soon.
Hallucinations, defined as instances where Large Language Models (LLMs) generate false or misleading content, pose a significant challenge that impacts the safety and trust of downstream applications. We introduce UQLM, a Python package for LLM hallucination detection using state-of-the-art uncertainty quantification (UQ) techniques. This toolkit offers a suite of UQ-based scorers that compute response-level confidence scores ranging from 0 to 1. This library provides an off-the-shelf solution for UQ-based hallucination detection that can be easily integrated to enhance the reliability of LLM outputs.
As Large Language Models (LLMs) are integrated into safety-critical applications involving sequential decision-making in the real world, it is essential to know when to trust LLM decisions. Existing LLM Uncertainty Quantification (UQ) methods are primarily designed for single-turn question-answering formats, resulting in multi-step decision-making scenarios, e.g., LLM agentic system, being underexplored. In this paper, we introduce a principled, information-theoretic framework that decomposes LLM sequential decision uncertainty into two parts: (i) internal uncertainty intrinsic to the current decision, which is focused on existing UQ methods, and (ii) extrinsic uncertainty, a Mutual-Information (MI) quantity describing how much uncertainty should be inherited from preceding decisions. We then propose UProp, an efficient and effective extrinsic uncertainty estimator that converts the direct estimation of MI to the estimation of Pointwise Mutual Information (PMI) over multiple Trajectory-Dependent Decision Processes (TDPs). UProp is evaluated over extensive multi-step decision-making benchmarks, e.g., AgentBench and HotpotQA, with state-of-the-art LLMs, e.g., GPT-4.1 and DeepSeek-V3. Experimental results demonstrate that UProp significantly outperforms existing single-turn UQ baselines equipped with thoughtful aggregation strategies. Moreover, we provide a comprehensive analysis of UProp, including sampling efficiency, potential applications, and intermediate uncertainty propagation, to demonstrate its effectiveness. Codes will be available at https://github.com/jinhaoduan/UProp.
Despite their widespread use, large language models (LLMs) are known to hallucinate incorrect information and be poorly calibrated. This makes the uncertainty quantification of these models of critical importance, especially in high-stakes domains, such as autonomy and healthcare. Prior work has made Bayesian deep learning-based approaches to this problem more tractable by performing inference over the low-rank adaptation (LoRA) parameters of a fine-tuned model. While effective, these approaches struggle to scale to larger LLMs due to requiring further additional parameters compared to LoRA. In this work we present $\textbf{Scala}$ble $\textbf{B}$ayesian $\textbf{L}$ow-Rank Adaptation via Stochastic Variational Subspace Inference (ScalaBL). We perform Bayesian inference in an $r$-dimensional subspace, for LoRA rank $r$. By repurposing the LoRA parameters as projection matrices, we are able to map samples from this subspace into the full weight space of the LLM. This allows us to learn all the parameters of our approach using stochastic variational inference. Despite the low dimensionality of our subspace, we are able to achieve competitive performance with state-of-the-art approaches while only requiring ${\sim}1000$ additional parameters. Furthermore, it allows us to scale up to the largest Bayesian LLM to date, with four times as a many base parameters as prior work.
Forecasting geopolitical conflict from data sources like the Global Database of Events, Language, and Tone (GDELT) is a critical challenge for national security. The inherent sparsity, burstiness, and overdispersion of such data cause standard deep learning models, including the Temporal Fusion Transformer (TFT), to produce unreliable long-horizon predictions. We introduce STFT-VNNGP, a hybrid architecture that won the 2023 Algorithms for Threat Detection (ATD) competition by overcoming these limitations. Designed to bridge this gap, our model employs a two-stage process: first, a TFT captures complex temporal dynamics to generate multi-quantile forecasts. These quantiles then serve as informed inputs for a Variational Nearest Neighbor Gaussian Process (VNNGP), which performs principled spatiotemporal smoothing and uncertainty quantification. In a case study forecasting conflict dynamics in the Middle East and the U.S., STFT-VNNGP consistently outperforms a standalone TFT, showing a superior ability to predict the timing and magnitude of bursty event periods, particularly at long-range horizons. This work offers a robust framework for generating more reliable and actionable intelligence from challenging event data, with all code and workflows made publicly available to ensure reproducibility.
Parameter estimation is a foundational step in statistical modeling, enabling us to extract knowledge from data and apply it effectively. Bayesian estimation of parameters incorporates prior beliefs with observed data to infer distribution parameters probabilistically and robustly. Moreover, it provides full posterior distributions, allowing uncertainty quantification and regularization, especially useful in small or truncated samples. Utilizing the left-truncated log-logistic (LTLL) distribution is particularly well-suited for modeling time-to-event data where observations are subject to a known lower bound such as precipitation data and cancer survival times. In this paper, we propose a Bayesian approach for estimating the parameters of the LTLL distribution with a fixed truncation point \( x_L > 0 \). Given a random variable \( X \sim LL(\alpha, \beta; x_L) \), where \( \alpha > 0 \) is the scale parameter and \( \beta > 0 \) is the shape parameter, the likelihood function is derived based on a truncated sample \( X_1, X_2, \dots, X_N \) with \( X_i > x_L \). We assume independent prior distributions for the parameters, and the posterior inference is conducted via Markov Chain Monte Carlo sampling, specifically using the Metropolis-Hastings algorithm to obtain posterior estimates \( \hat{\alpha} \) and \( \hat{\beta} \). Through simulation studies and real-world applications, we demonstrate that Bayesian estimation provides more stable and reliable parameter estimates, particularly when the likelihood surface is irregular due to left truncation. The results highlight the advantages of Bayesian inference outperform the estimation of parameter uncertainty in truncated distributions for time to event data analysis.
Modern industrial systems are often subject to multiple failure modes, and their conditions are monitored by multiple sensors, generating multiple time-series signals. Additionally, time-to-failure data are commonly available. Accurately predicting a system's remaining useful life (RUL) requires effectively leveraging multi-sensor time-series data alongside multi-mode failure event data. In most existing models, failure modes and RUL prediction are performed independently, ignoring the inherent relationship between these two tasks. Some models integrate multiple failure modes and event prediction using black-box machine learning approaches, which lack statistical rigor and cannot characterize the inherent uncertainty in the model and data. This paper introduces a unified approach to jointly model the multi-sensor time-series data and failure time concerning multiple failure modes. This proposed model integrate a Cox proportional hazards model, a Convolved Multi-output Gaussian Process, and multinomial failure mode distributions in a hierarchical Bayesian framework with corresponding priors, enabling accurate prediction with robust uncertainty quantification. Posterior distributions are effectively obtained by Variational Bayes, and prediction is performed with Monte Carlo sampling. The advantages of the proposed model is validated through extensive numerical and case studies with jet-engine dataset.
Reliable uncertainty quantification is crucial for the trustworthiness of machine learning applications. Inductive Conformal Prediction (ICP) offers a distribution-free framework for generating prediction sets or intervals with user-specified confidence. However, standard ICP guarantees are marginal and typically require a fresh calibration set for each new prediction to maintain their validity. This paper addresses this practical limitation by demonstrating how e-conformal prediction, in conjunction with Hoeffding's inequality, can enable the repeated use of a single calibration set with a high probability of preserving the desired coverage. Through a case study on the CIFAR-10 dataset, we train a deep neural network and utilise a calibration set to estimate a Hoeffding correction. This correction allows us to apply a modified Markov's inequality, leading to the construction of prediction sets with quantifiable confidence. Our results illustrate the feasibility of maintaining provable performance in conformal prediction while enhancing its practicality by reducing the need for repeated calibration. The code for this work is publicly available.
We propose a novel statistical inference framework for streaming principal component analysis (PCA) using Oja's algorithm, enabling the construction of confidence intervals for individual entries of the estimated eigenvector. Most existing works on streaming PCA focus on providing sharp sin-squared error guarantees. Recently, there has been some interest in uncertainty quantification for the sin-squared error. However, uncertainty quantification or sharp error guarantees for entries of the estimated eigenvector in the streaming setting remains largely unexplored. We derive a sharp Bernstein-type concentration bound for elements of the estimated vector matching the optimal error rate up to logarithmic factors. We also establish a Central Limit Theorem for a suitably centered and scaled subset of the entries. To efficiently estimate the coordinate-wise variance, we introduce a provably consistent subsampling algorithm that leverages the median-of-means approach, empirically achieving similar accuracy to multiplier bootstrap methods while being significantly more computationally efficient. Numerical experiments demonstrate its effectiveness in providing reliable uncertainty estimates with a fraction of the computational cost of existing methods.
Conventional benchmarks for crop type classification from optical satellite time series typically assume access to labeled data from the same year and rely on fixed calendar-day sampling. This limits generalization across seasons, where crop phenology shifts due to interannual climate variability, and precludes real-time application when current-year labels are unavailable. Furthermore, uncertainty quantification is often neglected, making such approaches unreliable for crop monitoring applications. Inspired by ecophysiological principles of plant growth, we propose a simple, model-agnostic sampling strategy that leverages growing degree days (GDD), based on daily average temperature, to replace calendar time with thermal time. By uniformly subsampling time series in this biologically meaningful domain, the method emphasizes phenologically active growth stages while reducing temporal redundancy and noise. We evaluate the method on a multi-year Sentinel-2 dataset spanning all of Switzerland, training on one growing season and testing on other seasons. Compared to state-of-the-art baselines, our method delivers substantial gains in classification accuracy and, critically, produces more calibrated uncertainty estimates. Notably, our method excels in low-data regimes and enables significantly more accurate early-season classification. With only 10 percent of the training data, our method surpasses the state-of-the-art baseline in both predictive accuracy and uncertainty estimation, and by the end of June, it achieves performance similar to a baseline trained on the full season. These results demonstrate that leveraging temperature data not only improves predictive performance across seasons but also enhances the robustness and trustworthiness of crop-type mapping in real-world applications.