Predictive Uncertainty

Random forests (RFs) predict well but are opaque, whereas single decision trees are interpretable but unstable. Artificial representative trees (ARTs) were developed as interpretable surrogate models for RFs, but their use as standalone prediction models with uncertainty quantification has not been systematically investigated. We combine ARTs with leaf-wise Mondrian conformal predictive systems (CPS), enabling a single tree to provide continuous predictions, prediction intervals, and probabilities of exceeding arbitrary thresholds. We compared ARTs with CPS against decision trees with CPS and separate regression and probability trees across five simulation scenarios, 21 benchmark datasets, and a cross-sectional NHANES example data set. Repeated cross-validation assessed predictive performance, interpretability, and stability. ARTs with CPS yield compact, structurally stable trees with substantially more reproducible split-variable selection than decision trees across benchmark datasets and NHANES. Decision trees showed slightly better predictive performance and narrower prediction intervals, while coverage was broadly comparable. CPS-based trees generally achieved lower and less variable Brier scores than multi-model approaches. Combining ARTs with CPS therefore provides a single, interpretable, and stable model for continuous predictions and calibrated probabilities, balancing predictive performance with reproducibility and transparency in settings where stability and interpretability are essential.

Active subspace methods provide a framework for quantifying predictive uncertainty in high-dimensional models by identifying and performing inference along parameter directions that have the greatest influence on the model output. However, the construction of active subspaces requires storing many full-dimensional model gradients, which becomes prohibitive as model size increases. We address this limitation by proposing Compressed Active Subspaces (CAS), a scalable approach that first maps the model parameters to a compressed space using a structured isometric embedding and then constructs the active subspace within this reduced parameterization. Our approach substantially reduces the memory required for active subspace construction and enables Bayesian inference for large models where standard active subspace methods become impractical. We demonstrate the scalability of CAS on neural networks of increasing size while maintaining predictive performance and robust uncertainty estimates.

Earth system modeling is shifting from task-specific predictors toward foundation models with general spatiotemporal representation capabilities. Although these models can jointly encode dynamic Earth fields, external forcings, and static geographic context for multistep forecasting, accurate point predictions or statistically calibrated intervals alone are insufficient for high-impact applications such as extremeweather warning, flood control, renewable-energy dispatch, and emergency resource allocation. What matters in practice is whether predictive uncertainty can be translated into reliable decision risk under specific actions, loss functions, and risk preferences. We propose a decision-oriented uncertainty quantification framework for Earth system spatiotemporal foundation models. The framework produces predictive distributions of future states and uses a decision risk adapter to map forecast samples, decision context, and utility functions into action-conditional risks. A utility-aware calibration module further enforces reliability at the downstream decision-loss level rather than only at the forecast-value level. Calibrated risks are then used to select warning, dispatch, inspection, or resource-allocation actions. Compared with the strongest baseline, the proposed method reduces decision regret by 18.7%, lowers the missed-event rate from 14.2% to 9.1%, and improves expected utility by 11.6%, while maintaining 90.4% predictive coverage and reducing decision calibration error from 0.083 to 0.047. These results suggest that decision-oriented uncertainty quantification can improve the robustness and operational value of Earth system foundation models in risk-sensitive applications.

Uncertainty estimation for medical vision--language models (VLMs) using conformal prediction has gained increasing attention due to its distribution-free coverage guarantees. However, standard conformal prediction relies on exchangeability between calibration and test data and typically requires a sufficiently large calibration set to obtain reliable coverage. These assumptions are difficult to satisfy in few-shot transfer settings, where only a small labeled support set is available to adapt a pretrained VLM to a new medical task, while an unlabeled query set is used for evaluation. Supervised fine-tuning on the support set changes the model parameters and consequently shifts the nonconformity score distribution, breaking exchangeability between calibration and query samples and leading to unreliable coverage under distribution shift. Existing transductive conformal adaptation methods often preserve validity by avoiding supervised updates. While this helps maintain conformal assumptions, it underutilizes the scarce labeled support data and limits task adaptation, which is the primary objective in few-shot learning. In this setting, conformal prediction should serve as an uncertainty estimation layer that supports the adapted model, rather than preventing adaptation itself. To this end, we propose AlignCP, a framework that reconciles supervised few-shot adaptation with conformal uncertainty estimation under non-exchangeability. AlignCP learns a reweighted calibration distribution that reduces the score-level discrepancy between the labeled support set and the unlabeled query set. By aligning the one-dimensional nonconformity score distributions, AlignCP aims to close the coverage gap induced by adaptation without requiring query labels.

Although probabilistic statements are ubiquitous, foundational disagreements persist about their understanding, as exemplified by debates between Bayesians and frequentists; moreover, it is unclear when and why acting on them actually leads to desirable outcomes. Here, we argue that every probability is the output of a \emph{prediction method}, that is, it depends on both a particular way of constructing abstractions and a way of transforming them into predictions. Through this, we provide a unifying perspective on supposedly different kinds of probabilities and show that even supposedly objective ones are model-dependent. We demonstrate that when a finite calibration criterion is met, one can anticipate the distribution of utilities for a given policy and inform successful decision-making on finite sets of events. Based on the notion of prediction methods, inductive arguments, and the probability calculus, we explain the feasibility of the calibration criterion in many settings. Overall, we develop a coherent perspective on probabilities and their use, connecting key intuitions behind other interpretations along the way.

Counterfactual explanations formalize "what-if" scenarios by identifying modifications to an input instance that obtain a desired alternative prediction. Traditionally, whether generated via instance-specific optimization or amortized single pass models, these approaches rely on a single, deterministic point-estimate predictor. However, this ignores predictive uncertainty and hypothesis variability, leading to brittle explanations that frequently become invalid if the underlying model is retrained or updated. To address this fragility, we propose the Amortized Variational Counterfactual Generator (AVCG), a generalized optimization framework that formulates counterfactual generation as optimization over an arbitrary distribution of plausible predictive hypotheses rather than a single deterministic predictor. This formulation naturally accommodates Bayesian posteriors, Rashomon-restricted hypothesis spaces, and other uncertainty representations within a unified optimization framework. Evaluation across multiple benchmark datasets demonstrates that the AVCG framework produces counterfactual explanations that remain highly valid under predictive uncertainty and model changes, while maintaining competitive plausibility and single-pass runtime performance.

Prediction error is widely used to evaluate nuisance-function estimators in causal inference, but its relationship with causal estimator performance may differ across performance measures. We studied this question in a partially linear model using Monte Carlo simulations. We compared ordinary least squares (OLS), generalized additive models (GAMs), XGBoost, and Double Machine Learning with XGBoost (DML-XGBoost), evaluating nuisance-function prediction error, bias, RMSE, and 95% confidence interval coverage. We also examined a simple joint-error measure based on the absolute cross-product of estimation errors from the exposure and outcome nuisance functions. Across the simulated settings, XGBoost had the lowest RMSE among the non-oracle methods, while DML-XGBoost generally provided better confidence interval coverage. Prediction error did not consistently track causal bias across methods and settings, and the method with the best point-estimation performance did not necessarily have the best confidence interval coverage. The joint-error measure was only weakly associated with causal bias and did not provide a useful standalone measure of causal performance. These results suggest that prediction error is useful for assessing nuisance-function estimation, but it should not be treated as a direct measure of the quality of the resulting causal estimator.

Large-scale vision--language--action (VLA) policies have advanced generalist robot control, yet most remain stimulus-to-action black boxes: actions are exposed, but their explanatory state is not. They provide no native account of three explanatory signals: task progress, the next semantic transition, or local command reliability. Prior work shows that progress and event structure aid long-horizon control and that uncertainty supports monitoring; however, such capabilities are typically added or extracted only after action pretraining. The field therefore lacks a VLA foundation model whose explanatory state is jointly pretrained with control. Drawing on biological sensorimotor organization, in which outcome-sensitive, event-segmented, and probabilistic predictions structure behavior, we introduce LM-X. LM-X learns three directly supervised online signals: return-to-go (RTG) estimates visible progress and state quality; event-to-go (ETG) predicts the action sequence to the next semantic event; and heteroscedastic action-flow variance reports local command reliability. RTG conditions ETG and both condition action generation; uncertainty is estimated inside the action expert, making explanation part of control rather than a post-hoc description. We pretrain LM-X on more than 20,000 hours of heterogeneous real-robot trajectories, including over 1,000 hours of failed rollouts. A controlled gate favors joint over post-hoc training. LM-X achieves 74.1% success on 50 randomized-hard RoboTwin2.0 tasks and 73.5% on seven real-robot tasks, compared with 55.4% and 50.7% for GR00T N1.7. Its signals track progress and regression, anticipate event-scale motion, detect high-error actions, and provide advance failure warning. These results establish LM-X as an explainable VLA foundation model that couples transparent predictive state with stronger generalist control.

Clinical prediction models are often developed as if the outcome of interest were cleanly observed for every patient. This assumption fails when treatment decisions make the clinically relevant outcome permanently unobservable. As a case study of this problem, we consider post-cardiac-arrest neurological prognostication using a cohort of 2,497 patients, including 1,429 patients whose outcomes were rendered indeterminate by treatment decisions. These patients with indeterminate outcomes were reviewed by independent clinical experts, who provided their guesses of counterfactual outcomes about what would have happened to the patients. We refer to these patients as uncertain cases. We also have patients for whom we observe their clinically relevant outcomes; we refer to these patients as certain cases. We propose a framework for evaluating prediction models that explicitly splits the evaluation between certain and uncertain cases. Here, we cannot easily evaluate both types of cases in a uniform manner as the available target labels differ. We then propose a simple prediction model that uses target labels from both certain and uncertain cases in a manner that allows us to trade off between them. Across the proposed neural model and a collection of tabular baselines, models with similar certain-case AUROC can nevertheless differ substantially in both certain-case Brier score and their probability estimates for uncertain cases. Improving alignment with target labels of uncertain cases for our proposed model generally comes at the cost of worse accuracy on certain cases, highlighting an explicit tradeoff that standard evaluation conceals. These results show that when treatment decisions determine whether clinically meaningful outcomes remain observable, conventional evaluation metrics can miss important failure modes in the very patients for whom prognostic support matters most.

Modern stochastic predictors can model rich, multi-modal outcome distributions. However, this expressive power comes with challenges in ensuring robust predictions $-$ a critical requirement in safety-critical domains. Randomized smoothing is a leading technique for improving robustness, particularly against adversarial perturbations. Yet, in stochastic multi-modal regression settings, randomized smoothing often fails due to mode collapse, yielding averaged predictions that do not reflect the underlying distribution. To address this limitation, we propose clustered $α$-smoothing, a framework that (1) partitions noisy samples using an arbitrary clustering algorithm, (2) applies $α$-smoothing locally within each cluster, and (3) combines the resulting predictions into a mixture distribution. By interpreting the smoothing distribution as a mixture of $α$-smoothers, we derive a lower bound on the probability that the smoothed prediction lies within a union of compact regions corresponding to distinct modes. We empirically evaluate our framework on two benchmarks, demonstrating substantial improvements over state-of-the-art methods. In stochastic trajectory prediction on a driving simulator dataset, our approach achieves, on average, a $27\%$ lower Wasserstein distance to the ground-truth distribution compared to $α$-smoothing. In quadrotor control, where modes correspond to distinct feasible paths to a target, our method reduces the collision rate by $81\%$ relative to the state-of-the-art randomized smoothing.

Machine learning models should be robust, in the sense of remaining predictively consistent under permissible variations. A model's predictions should ideally remain unchanged when it is replaced by a functionally equivalent one, or when its inputs are subject to minor, admissible perturbations. If such changes alter a prediction significantly, then the prediction is "ambiguous" with respect to the model. Models should abstain from making such ambiguous predictions and/or should flag them for human inspection, especially in high-stakes decision-making scenarios. However, in practice, such ambiguity is not easy to identify once a model is deployed. Here, the Robust Ambiguity Detection (RAD) framework is advanced for quantifying predictive ambiguity using two complementary metrics: Model-Space Consistency and Feature-Space Consistency. These two scores, the RAD Score-Pair, visualised through the RAD Plot, provide an interpretable characterisation of the sources of ambiguity and the actions a user may consider in response. RAD is evaluated on synthetic datasets with systematically controlled overlap, as well as several real-world datasets where the level of ambiguity cannot be directly inspected. Finally, we demonstrate a downstream application of RAD where samples are ranked by their RAD Pareto-Rank and the most ambiguous are abstained from prediction, achieving performance comparable to existing rejection-based approaches.

Probabilistic long-term time-series forecasting commonly relies on trained models. Training-free conformal methods typically construct intervals around a pre-existing point forecaster and do not natively represent a complete predictive distribution; sequential variants additionally suffer from increasingly delayed feedback at long horizons. We propose KReF, a training-free retrieval framework that treats retrieved historical futures as a querylocal empirical predictive distribution. After robust preprocessing, KReF embeds each lookback using handcrafted statistics or frozen random Fourier features and retrieves similar historical lookback-future pairs. Their similarity weights directly define predictive masses, quantiles, CRPS, and a weighted-mean point forecast. KReF further uses the observed query lookback to construct a probability-integral-transform map and applies validation-selected expansion and shrinkage rates to adapt interval boundaries. Across six LTSF benchmarks and four horizons, KReF obtains the lowest CRPS in all 12 dataset-embedding settings and the lowest IS90 in 9 settings. Without gradient-based fitting, its point forecasts also match or surpass trained baselines on two of six datasets. An archive-oracle analysis further reveals substantial headroom under finer horizon- and channel-wise routing. These results establish retrieval as a useful and underexplored inductive bias for LTSF.

Predicting failures before they occur remains a major challenge in predictive maintenance, particularly when failures are rare, when equipment of the same family differ in sensor configurations, and when the goal is anticipation rather than diagnosis of an already observed fault. This paper proposes a common-to-target probabilistic model that learns shared failure-related patterns across a family of heterogeneous equipment and adapts parsimoniously to target equipment. The model explicitly accounts for sensor heterogeneity, operating context, and degradation dynamics to produce calibrated failureprobability estimates suitable for maintenance planning. Its performance is evaluated on a synthetic refrigerator dataset comprising 27 simulated refrigerators with varying sensor configurations, operating conditions, and failure types, providing a controlle

Probabilistic prediction heads in neural networks typically output either a Gaussian mixture or a single conformal region. Neither separates the distinct sources of uncertainty often present in real prediction tasks: a discrete choice among modes, bounded systematic drift within the chosen mode, and irreducible stochastic noise. We introduce the Hybrid Probabilistic Zonotope (HProbZ), an output head that represents these three sources as binary, bounded, and stochastic generators of a zonotope, and admits a closed-form likelihood by convolution. Sharing the bounded generator across prediction steps couples future predictions algebraically, so observing one step refines the predictive distribution at every remaining step in a single forward pass. We establish that the three generators are identifiable from the likelihood up to permutation, and that an HProbZ density is representationally distinct from any finite Gaussian mixture. The same shared structure provides analytic per-mode risk and distribution-free multi-modal conformal sets at inference time. Empirical analysis on representative prediction benchmarks supports the effectiveness of the design relative to same-encoder mixture baselines, while offering structural properties that mixture or convex-conformal predictors do not jointly provide.

Effective model-based reinforcement learning in stochastic environments requires planning that accounts for predictive uncertainty. Propagating full state distributions analytically offers a principled way to do this, but has traditionally required restrictive policy or reward structures to remain tractable. Consequently, modern deep reinforcement learning has largely retreated to either stochastic sampling, which introduces significant target variance, or deterministic point estimates that ignore predictive covariance entirely. We investigate whether distribution-aware planning is possible without these constraints. Using a quadratic action-value parameterization, we first reduce the Bellman backup to an expectation over the state-value function alone; the key idea is then a compatibility principle between the predictive transition distribution and the value function class, under which this expectation is analytic in the distribution's moments. We instantiate this principle with a Gaussian transition model paired with a radial-basis value function, yielding a closed-form backup that propagates both predictive mean and covariance. Empirically, our approach reduces target variance and yields well-calibrated predictive uncertainty under stochastic observations in continuous control, providing a principled framework for planning with learned distribution models.

Predictive processing theories portray the brain as a hierarchical prediction engine that minimizes prediction error, yet they lack operational definitions for the structure of a "prediction," the standardized response to a prediction error, and the mechanism that maintains consistency across successive updates. Bayesian cognitive science attempts to subsume all uncertainty under probabilistic belief updating, but it presupposes a closed hypothesis space and provides no generative account of how the objects over which probabilities are distributed become discrete, identifiable referents in the first place. This paper introduces Predictive Set Theory (PST), a formal generative framework that reconstructs cognitive architecture from first principles. PST anchors cognition in a minimal set of operations---a sensor formalized as an identity function, set-theoretic state refresh, and three fundamental forms of reference chains (reference, counter-reference, and semi-reference)---and rigorously derives core cognitive functions including state sequences, demand, comparison, efficiency, and finite-horizon probabilistic planning. Rather than modeling neural mechanisms, PST constitutes a design specification for any system that must maintain internal consistency while acting under incomplete information and irreversible risk. The framework offers novel resolutions to classical problems such as Russell's paradox, the cognitive status of Gödelian incompleteness, the grounding of negative feedback, and the comprehension of film editing. The primary purpose of this paper is to establish, through the public academic record, the originality and completeness of the Predictive Set Theory framework.

Accurate estimation of forest height from satellite imagery is essential for applications such as carbon accounting, biodiversity monitoring, and ecosystem management. While recent deep learning approaches provide accurate predictions, they typically do not quantify predictive uncertainty. This limitation is particularly relevant in geospatial settings characterized by sparse supervision and geographic distribution shift. In this work, we investigate Deep Evidential Regression (DER) for forest height estimation on the TreeUQ benchmark, a large-scale dataset designed for the joint estimation of tree count and average tree height at 10 m resolution, based on Sentinel-1/-2 data as well as tree inventory data over the federal state of Bavaria. To account for the extreme label sparsity of the tree inventory data, we introduce a masked evidential loss for dense geospatial prediction. Using a U-Net architecture with multimodal Sentinel-1 and Sentinel-2 inputs, the proposed approach jointly predicts tree height and associated uncertainty estimates in a single forward pass. Experimental results show that DER achieves predictive performance comparable to a deterministic U-Net while additionally providing well-calibrated uncertainty estimates. These findings demonstrate the potential of evidential learning as an efficient framework for uncertainty-aware forest structure estimation from Earth observation data.

In this paper, we propose and study a robust variant of the smart predict-then-optimize approach that accounts for prediction shifts due to disturbance in the covariate feature space. While traditional integrated-learning-and-optimization models assume that side information is perfectly revealed, empirical data-driven features are frequently corrupted or noisy at the time of decision-making, leading to fragile operational policies. To bridge this gap, we integrate principles of robust optimization directly into the predictive-prescriptive pipeline via a smart predict-then-robustly optimize loss and establish a computationally tractable convex surrogate, designed to hedge against worst-case feature perturbations. On the theoretical front, we formalize the structural validity of this surrogate by proving its approximation error probability decays exponentially according to a sub-Gaussian concentration profile. Furthermore, we establish that under mild assumptions, the surrogate is Fisher consistent with high probability. We also prove necessary conditions under which our framework outperforms standard smart predict-then-optimize and maintain its superiority even when the standard method is equipped with regularized upstream predictions. Numerical experiments validate that our robust framework consistently yields significant performance improvements over standard methods, both in out-of-sample terms and in training stability.

In this work, we incorporate first principle physics into the construction of data-driven methods by considering a model that accounts for the different sources of energy losses during vehicle operations. Our results show that Bayesian linear regression based on this physics-aware model can improve the reliability of the expected energy consumption, as compared with standard linear regression. Further, it is shown that more complex machine learning models such as neural networks and gradient boosted regression trees, based on the same physical model, can further improve the accuracy in energy forecasting and significantly outperform standard versions of the same machine learning models. In addition to point predictions of the energy consumption, we develop a framework for estimating the corresponding uncertainty in the form of predicted standard deviation. Our results show that all of the models learn to estimate the uncertainty reasonably well.

A trained molecular property model can be refined at test time by correcting each prediction with the measured labels of the most similar training molecules, a retraining-free procedure we call neighbor fusion; evidential neural networks make it principled by using their aleatoric and epistemic uncertainty to parameterize a Bayesian update. Our main contribution, PG-EVIKAL, learns a property-distance metric to re-rank structurally similar neighbors by their property relevance before fusion, building on EVIKAL (scalar Kalman filter) and GP-EVIKAL (Gaussian process variant handling correlated neighbors). Evaluated on 16 molecular datasets, PG-EVIKAL reduces RMSE relative to the evidential model baseline on 14 of them, with a median reduction of 19.4%, and improves calibration; in sequential-assay scenarios it further incorporates newly measured molecules, refining predictions as they arrive without retraining. This work demonstrates that evidential uncertainty decomposition is not merely a calibration objective but an actionable inference resource that enables test-time refinement of molecular property predictions.

Reliable forecasting of several interrelated environmental variables - such as regional precipitation and temperature, or other correlated geophysical fields - across many locations calls for accurate predictions accompanied by trustworthy statements of their uncertainty. Modern deep-learning models forecast such variables accurately but usually report no uncertainty, and forcing them to output uncertainty through maximum likelihood tends to degrade their accuracy, especially when the variables are strongly correlated. Motivated by this tension, we develop TSCoNet, a two-stage convolutional-recurrent model coupled with a Gaussian copula that jointly forecasts multiple variables over space and time while quantifying predictive uncertainty. The method first learns accurate mean forecasts and then, holding the mean fixed, refines a shared representation to estimate the predictive variance, yielding calibrated prediction intervals after a standard recalibration, so that uncertainty is added without sacrificing point accuracy. We study the approach on simulated non-stationary spatial fields on the sphere and on a real dataset of monthly precipitation and temperature for fifty cities over 2000-2020. The model matches the accuracy of a strong deterministic forecaster while supplying calibrated prediction intervals that the deterministic model cannot, giving a single tool that provides both accurate point forecasts and reliable uncertainty for multivariate spatio-temporal data.

Predict-then-optimize systems usually compress uncertainty into a point forecast and then solve a downstream optimization problem as if the forecast were reliable. Distributionally robust optimization (DRO) offers protection against misspecification, but the ambiguity set is often centered at historical samples and uses a fixed radius. We propose \emph{learned predictive ambiguity sets} (LPAS): a deep contextual model outputs a finite nominal scenario distribution, a state-dependent Wasserstein radius, and optionally an anisotropic ground metric. These outputs define a contextual ambiguity set that feeds a DRO decision layer. The radius is trained by a combination of conditional quantile calibration, size regularization, and downstream decision loss, so that robustness is adaptive rather than globally fixed. We derive the finite dual form used by the decision layer, present a staged training algorithm, and evaluate the method on distributionally robust portfolio optimization with 20 S&P 500 constituents from 2018--2026. The proposed method substantially improves over equal-weight, predict-then-optimize, and historical Wasserstein DRO baselines, achieving 26.28% annualized return, Sharpe ratio 1.30, final wealth 1.61, and lower tail loss than a deep fixed-radius DRO baseline while using a smaller average radius. The results show that learned ambiguity radii can recover most of the performance of strong fixed-radius DRO while reducing unnecessary conservatism and improving regime adaptivity.

Hypergraph neural networks have shown powerful capability in modeling higher-order relations, yet their predictive uncertainty remains underexplored. Unlike pairwise graphs, uncertainty in hypergraphs arises not only from noisy attributes and ambiguous labels, but also from variations in node-hyperedge incidence structures and complex higher-order dependencies. Existing approaches mainly estimate uncertainty from final predictions or rely on computationally expensive ensembles and Bayesian inference, limiting their ability to capture uncertainty evolution during representation learning. In this paper, we propose Hypergraph Neural Stochastic Diffusion(HyperNSD), a stochastic differential equation framework for uncertainty estimation on hypergraphs. HyperNSD models hypergraph representations as stochastic processes evolving over node-hyperedge incidence structures. A learnable drift function captures deterministic higher-order diffusion dynamics, while a learnable stochastic forcing function characterizes structural ambiguity and representation noise. Predictive uncertainty is directly quantified through the variability of stochastic representation trajectories, providing an intrinsic uncertainty measure beyond post-hoc confidence scores. We formulate HyperNSD with neural drift and diffusion networks, enabling joint learning of prediction and uncertainty propagation. Theoretical analyses establish well posedness, perturbation stability,permutation equivariance, and numerical convergence of the proposed stochastic dynamics. Experiments on multiple hypergraph benchmarks demonstrate that HyperNSD achieves reliable uncertainty estimation for out-of-distribution and misclassification detection while preserving competitive prediction accuracy. These results provide a principled stochastic-dynamical framework for trustworthy higher-order representation learning.

We introduce an efficient Bayesian deep ensemble method for predictive regression designed to enhance interpretability while maintaining competitive predictive performance and computational efficiency. Our method combines the statistical rigor of Bayesian inference with the scalability of deep ensembles, providing calibrated uncertainty estimates that enable its use not only for standalone prediction but also as a component within broader learning systems. To achieve these goals, our work relies on three key design components: (i) low-dimensional ensemble representation: predictions are expressed as a combination of a small number of trained neural predictors, enabling scalable inference whose cost depends on ensemble size rather than dataset size; (ii) closed-form Bayesian aggregation: ensemble predictions are combined using Bayesian linear regression, yielding interpretable posterior weights and calibrated uncertainty without approximate inference; and (iii) Independent ensemble training: multiple neural networks are trained separately, producing diverse predictive representations that improve robustness and uncertainty calibration. Empirical results on standard regression benchmarks demonstrate that the proposed approach achieves competitive predictive performance while maintaining reliable uncertainty estimates across settings.

3D Gaussian splatting (3DGS) is a strong representation for real-time novel-view synthesis, but its standard training pipeline relies on point estimates and hand-tuned heuristics, providing no native uncertainty or principled complexity control. This is most limiting under sparse views or fixed acquisition budgets, where a model must identify weakly supported geometry and select informative views. We introduce a rendering-aware Bayesian 3DGS framework that tracks Gaussian geometry with a Normal-Inverse-Wishart posterior over means and covariances using renderer-derived surrogate summaries. An optional Dirichlet-process extension adds a probabilistic component-usage signal, and the training schedule makes the closed-form versus approximate inference boundary explicit. Re-rendering posterior geometry samples yields native predictive uncertainty for interval calibration and active view selection. In a fixed-budget 16-to-32 active-view task, native NIW acquisition improves PSNR by +0.453 dB and LPIPS by -0.0146 over a scoring-only 3-member standard-ensemble baseline, winning 29/39 scene-seed pairs and 10/13 scene means; it also improves over PPU-style (+0.355 dB) and NIW-proxy (+0.401 dB) acquisition. NIW native intervals reduce 95% coverage error by about 17x relative to a shared proxy (0.046 vs. 0.796) and are about 10x closer to nominal coverage than a 3-member deep ensemble (0.047 vs. 0.454) at roughly one-third the training cost. As a reconstruction compatibility check, paired NIW-vs-standard analysis over 39 scene-seed runs yields +0.030 dB PSNR with 1.6% additional training time. These results position Bayesian 3DGS as a practical probabilistic scene representation for decision-facing tasks such as active view selection.

Uncertainty quantification of hydrological predictions is necessary to inform operational decisions. Recent generative machine-learning methods have advanced probabilistic streamflow prediction, but have remained confined to lumped models that predict a basin outlet directly. At the same time, deterministic LSTM runoff models are increasingly applied at grid or catchment scale and routed through river networks to produce spatially continuous, physically consistent discharge fields. This technical note argues that moving probabilistic prediction from lumped to distributed models introduces a specific new requirement: the joint distribution of upstream runoff generation must be sampled jointly. In lumped inference, the model predicts the outlet distribution directly and can modulate spread from basin attributes. In distributed inference, downstream discharge is obtained by routing many upstream runoff predictions, so independent local sampling averages uncertainty away. Using Japan as a case study, we train two probabilistic basin-scale runoff LSTMs and route their runoff through a Hayami routing scheme. Randomly matching upstream ensemble members produces severely under-dispersed downstream ensembles, whereas a simple quantile matching strategy restores much of the spread of the direct basin-scale reference. The shift from lumped to distributed probabilistic hydrology therefore requires explicit attention to the spatial joint structure of runoff uncertainty.

Safe motion planning in dynamic environments requires reasoning about the uncertainty in predicted obstacle motion without sacrificing real-time performance. Existing conformal approaches conformalize a scalar score that aggregates per-obstacle prediction errors, losing spatial coherence and scaling poorly with scene density. We instead conformalize the entire predicted distance field at once. This functional conformal prediction (FCP) framework yields a distribution-free, field-level lower bound, from which safety follows uniformly: any trajectory satisfying the resulting constraint is certified safe, independent of how the control space is sampled. The key enabler is that the residual distance field is empirically low-rank and approximately time-invariant, which makes the bound decomposable in coefficient space. An envelope is fitted offline via functional PCA and a Gaussian-mixture inductive conformal procedure, then refined online by a lightweight adaptive functional conformal (AFCP) update on a low-dimensional vector. This keeps the per-step cost largely insensitive to obstacle count and retains long-run field coverage under distribution shift. We embed the envelope as a tightened safety constraint in a sampling-based model predictive controller, FCP-MPC. On the ETH--UCY pedestrian benchmarks and a dense 3D quadrotor task with up to 280 dynamic obstacles, FCP-MPC attains a favorable balance of safety, feasibility, and efficiency, reaching goals where pointwise and egocentric conformal baselines become too conservative or too expensive, while keeping per-step computation far below online uncertainty-reasoning baselines.

We investigate a forecasting framework based on a simple discrete-time dynamic model with coefficients varying in time. The parameters of the model are recovered within a deep learning framework, which makes it possible to retain a transparent parametric structure while simultaneously accounting for complex and nonstationary patterns in the observed phenomenon. Our analysis covers two specifications of the noise process. Besides the standard Gaussian setting, we also consider Laplace-distributed noise, which can offer a more adequate description in the presence of heavier tails and sharper local fluctuations. For both cases, we formulate the predictive scheme of the model and analyze the associated uncertainty quantification, including the construction of prediction intervals. The results illustrate that a relatively simple model, when combined with time-dependent parameter estimation, can serve as a mathematically tractable and practically flexible tool for forecasting complex dynamics under different noise assumptions. The general model is stated for TVAR($p$), while the prediction-interval formulas and the numerical experiments are developed for the TVAR(1) case.

Probabilistic downscaling is the task of modeling the conditional distribution of high-resolution fields given coarse inputs, and is a central challenge to atmospheric science, climate modeling, and other multiscale physical systems. A widely used paradigm decomposes the problem into a deterministic mean predictor followed by a stochastic residual generator. While effective in idealized settings, this mean--residual approach frequently produces biased and under-dispersive ensembles in real-world applications. Is this merely generic predictive uncertainty miscalibration? We show that the root cause is more fundamental: residual target misspecification, the residual distribution induced during training differs systematically from the one required at test time due to downscaling bias. To close this gap, we introduce ReMatch (Residual Distribution Matching). ReMatch aligns the training residual distribution toward the test-time regime via optimal transport in a low-dimensional PCA space. This preserves the statistical benefits of the mean--residual framework while reducing the train--test mismatch in the residual targets seen by the stochastic generator. On a controlled synthetic benchmark with varying bias levels and a real-world HRRR--ERA5 wind field downscaling task, ReMatch substantially reduces under-dispersion, improves calibration (SSR and CRPS), and outperforms strong baselines, including the standard mean--residual model and its variants, as well as state-of-the-art super-resolution models. Our code is available at https://github.com/sdean-group/ReMatch.git.

Forecasting visual fields (VFs) is critical for personalized monitoring and treatment planning in glaucoma. This is inherently uncertain due to heterogeneous disease progression and measurement variability, yet most existing methods produce single deterministic predictions that fail to represent this uncertainty. We formulate VF forecasting as a probabilistic prediction problem and the use of conditioned denoising diffusion models to generate distributions of plausible future VFs from longitudinal observations with irregular follow-up intervals. Experiments on two independent VF cohorts show that diffusion-based predictions produce well-calibrated distributions for clinically relevant VF measures. When reduced to a standard point-estimate, the proposed approach achieves state-of-the-art accuracy compared to clinical baselines and prior learning-based methods. Our results highlight the advantages of distributional modeling for VF forecasting and support a shift from point-estimate prediction toward uncertainty-aware, clinically interpretable risk assessment in glaucoma.

Temporal link prediction (TLP) is typically evaluated by predictive performance on unseen edges, but this criterion can conflate predictive accuracy with recovery of the underlying causal mechanism. In stochastic models, Fisher information governs the Cramér--Rao (CR) bound on parameter estimation error: higher Fisher information permits more accurate parameter recovery. We show that, under comonotonicity conditions between Fisher information and entropy, binary logistic models exhibit an estimation--prediction tradeoff: regimes with higher Fisher information, and hence smaller CR bounds, also have higher irreducible predictive entropy. To study this tradeoff in TLP, we introduce a probabilistic causal generator for temporal graphs with transient edges and known ground-truth causal structure, and validate the phenomenon empirically.

Longitudinal modelling of Alzheimer's disease progression is clinically useful only if it can describe not just the most likely next diagnosis, but how a patient may evolve over time and how reliable that forecast is. Most deep learning approaches reduce this problem to single-step classification, treating cognitively normal, mild cognitive impairment, and dementia as flat categories while providing limited insight into how uncertainty accumulates across future visits. We propose a probabilistic framework that combines ordinal diagnosis prediction, multi-horizon trajectory generation, and decomposed uncertainty estimation. A Temporal Fusion Transformer encoder is adapted with a CORAL ordinal output layer, asymmetric loss weighting, and converter oversampling to respect disease-stage ordering and improve sensitivity to MCI-to-dementia transitions. Conditioned on the learned patient-context representation, an autoregressive Mixture Density Network generates five-year probabilistic trajectories for diagnosis state, CDR Sum of Boxes, MMSE orientation, and hippocampal volume. On ADNI, the model outperforms linear, recurrent, and transformer baselines for next-visit diagnosis prediction, with the strongest gains on MCI-versus-dementia discrimination. Generated trajectories achieve near-nominal 90% credible interval coverage, widening uncertainty across the forecast horizon, and biomarker dynamics consistent with expected Alzheimer's disease progression. We further separate aleatoric from epistemic uncertainty using analytic mixture variance and a five-member bootstrap ensemble, which provides the strongest encoder diversity and output-level epistemic signal. Epistemic uncertainty is higher for rare progression archetypes, MCI and dementia patients, and under external evaluation on OASIS-3, where it increases alongside prediction error.

Plugging predictions of unknown parameters into downstream optimization problems, often referred to as the ``predict-then-optimize'' paradigm, has long been a standard approach in decision-making under uncertainty. However, improved predictive accuracy does not, in general, translate into improved decision quality. This disconnect has motivated growing interest in decision-focused learning (DFL) within the operations research community. This tutorial reviews recent developments in DFL and highlights key methodological insights, with a particular focus on stochastic linear programming as the downstream decision-making problem. We discuss why several widely used tools in traditional statistical learning are not directly suited to decision-focused settings and must be rethought, including (i) data collection strategies driven purely by predictive uncertainty and (ii) distributional distance measures such as the Wasserstein distance. We summarize properties of DFL that distinguish it from conventional predictive modeling and provide insights into the development of new decision-focused tools.

Soil properties relevant to agricultural and environmental applications are conventionally measured using elaborate laboratory methods involving physical and chemical processing. While highly accurate, these conventional methods are costly and time-consuming. In contrast, optical spectroscopy paired with machine learning enables rapid and cost-effective predictions of multiple soil properties. However, spectroscopic modelling is often considered unreliable, as the predictive accuracy varies between soil properties and individual samples. To balance this trade-off between cost and reliability, we introduce reject-to-remeasure: an AI-based measurement framework that combines probabilistic modelling with uncertainty-guided rejection. In this framework, soil samples are first analysed using spectroscopy, after which predictions are rejected if their predictive uncertainty exceeds predefined quality constraints. Rejected samples are subsequently remeasured using conventional laboratory procedures. On a regional visible-near-infrared spectral soil library from Québec, we demonstrate that reject-to-remeasure with modern foundation models (TabPFNv2.5 and TabICLv2) can facilitate the integration of optical spectroscopy into routine laboratory workflows while meeting user-defined accuracy requirements and reducing measurement costs.

Long-horizon spacecraft trajectory forecasting suffers from error accumulation due to the absence of corrective observations in the forecast regime, making reliable uncertainty estimation crucial for safety-critical decision-making such as space domain awareness and conjunction assessment. While high-fidelity physics-based orbit propagators provide accurate deterministic forecasts, they typically lack calibrated uncertainty estimates over long horizons. We introduce a Predictor--Corrector framework in which a physics-based continuous-time $\textit{deterministic}$ forecaster is augmented with a learned continuous-time $\textit{probabilistic}$ Corrector that models forecast errors. The proposed Corrector can be wrapped around an existing deterministic propagator to improve forecast accuracy while producing sharp and calibrated full-covariance uncertainty estimates. The Corrector is based on Latent Neural Controlled Differential Equations (Latent NCDEs) and models the probabilistic temporal evolution of forecast errors in continuous time, naturally supporting irregular sampling and missing features. We further introduce a loss function that promotes calibration and sharpness in long-horizon uncertainty propagation. We evaluate the proposed framework on long-horizon spacecraft trajectory forecasting using real-world data from NASA's Crustal Dynamics Data Information System (CDDIS), wrapping the Corrector around NASA's General Mission Analysis Tool (GMAT). Across forecast horizons of 2--4 days without observations and six rolling test windows, the proposed approach consistently improves accuracy and uncertainty calibration compared to deterministic baselines and Latent ODE-based correctors, demonstrating the effectiveness of the continuous-time probabilistic Corrector for trajectory forecasting.

Two dominant approaches have emerged for generating probabilistic forecasts of physical systems: generative models, such as diffusion or flow matching; and ensembles of deterministic models with stochasticity injected, trained using the continuous ranked probability score (CRPS) loss. While both approaches have demonstrated strong predictive accuracy, the reliability of their uncertainties has not been systematically assessed. We address this gap by developing a framework to evaluate both approaches across diverse 2D spatiotemporal physical systems, under matched model size and computational budget. We assess the reliability of probabilistic emulation by inspecting the empirical coverage of predictive intervals, while also considering accuracy and computational efficiency metrics. CRPS-trained ensembles typically achieve more reliable uncertainties on both single-step prediction and autoregressive rollouts, demonstrating better coverage than the standard alternative of training generative models in a latent space. Moreover, the CRPS approach offers significantly faster inference. When generative models are trained in ambient rather than a compressed latent space, which is often infeasible for high-dimensional problems, they exhibit comparable coverage to CRPS-trained ensembles, though with substantially larger inference latency. In contrast, when CRPS-trained ensembles are trained in latent space they do not show a marked degradation in coverage with respect to ambient space. Both generative models and CRPS-trained ensembles demonstrate good predictive accuracy. To facilitate future research and application, we release AutoCast, a modular framework implementing both generative models and CRPS-trained ensembles, alongside AutoSim, a flexible dataset generation package for rapid prototyping.

The standard taxonomy of predictive uncertainty defines epistemic uncertainty as the part removable by collecting more data, while the standard measure identifies it with a mutual-information term. We prove the definition and the measure are extensionally inconsistent. On an explicit construction, the measure assigns all uncertainty to the epistemic class, yet no quantity of training data reduces it. Reducibility is instead a property of the pair (uncertainty, acquisition class), and the dichotomy resolves into three parts: aleatoric, sample-reducible epistemic, and mechanism-reducible epistemic uncertainty. An exact identity for the value of an observation shows that in-distribution data never reduces mechanism-irreducible uncertainty and generically increases it. Ensemble disagreement, the deployed epistemic estimate, tracks the training procedure rather than the epistemic term. It collapses to zero beneath a positive truth under consistent training, and equals hyperparameter-scaled initialization noise under interpolation. A finite-sample falsification test and seed-swept experiments confirm the theory.

Statistical post-processing has proven to be an effective tool in improving ensemble forecast of different weather variables. Case studies show that post-processing can remedy the typically underdispersive and potentially biased behaviour of the ensemble while optimizing a proper scoring rule expressing the forecast skill. The price of these positive effects is generally a deterioration in sharpness; the width of the central prediction intervals and the uncertainty of the predictions are increasing, especially for shorter lead times. This work aims to reduce the extent of the latter phenomenon for neural network-based parametric post-processing methods by extending the network's loss function with a penalty term. We demonstrate the effect of the proposed technique for 2m temperature ensemble forecasts of the European Centre for Medium-Range Weather Forecasts downloaded from the EUPPBench benchmark dataset and verified against synoptic observations. Here, the predictive distribution is Gaussian, and we use the continuous ranked probability score (CRPS) as loss function. The case studies confirm a substantial relative decrease ($8.2\%-12.5\%$) in the width of the nominal central prediction interval compared to the width of the predictive distribution computed without the penalty term, while there is no deterioration in the mean CRPS of probabilistic forecasts and in the RMSE of the predictive mean.

Returned products in circular factories re-enter production with heterogeneous degradation states, usage histories, and remaining capability. Reuse cannot be decided from the current inspection alone, because future function fulfillment and component integrity may evolve differently under the next service scenario. Existing PHM approaches support degradation prediction, but often target fixed operating conditions or isolated component benchmarks, while material-fatigue assessment is rarely linked to system-level functional prognosis. This paper addresses this gap for an angle grinder by combining uncertainty-aware functional prediction with component-level fatigue assessment in an instance-specific reliability workflow. The proposed framework combines the current tool state with recent force--torque usage windows. A convolutional encoder extracts loading patterns from spindle forces and shaft torque, and an LSTM backbone predicts nine functional variables as Gaussian mean and variance estimates. In parallel, the same loading history is translated into output-shaft fatigue information through finite-element-supported stress reconstruction, S--N/Miner damage evaluation with Haibach extension, and Paris-law crack-growth analysis. A streaming replay algorithm consolidates both branches into functional, material, and system reliability trajectories. Held-out tests show mean $2\%$-tolerance accuracy of 0.9652 across nine outputs. Thermal variables are predicted near-perfectly, while drive motor current and load speed remain the most demanding dynamic outputs, with $R^2$ values of 0.9750 and 0.9924. Torque history is especially important for these variables, and the conventional LSTM outperforms GRU and xLSTM in the short-history setting. Reliability calibration is most informative for drive motor current, where predicted and observed exceedance probabilities ...

Multi-step time series forecasting (MSF) is commonly evaluated using point-wise error metrics such as mean squared error (MSE), implicitly treating the conditional mean as a sufficient target. We show that this can be misleading under conditional uncertainty, where the conditional expectation becomes unrepresentative of typical realized values at longer horizons. We formalize this effect through a conditional uncertainty gap and prove that whenever this gap is nonzero, no deterministic predictor can simultaneously minimize MSE and match the marginal distribution of realized futures. This establishes a fundamental, model-agnostic trade-off between point accuracy and marginal realism in MSF evaluation. Using controlled stochastic dynamical systems and nine real-world forecasting benchmarks, we empirically characterize the resulting accuracy--realism frontier and \textbf{quantify the practical cost of MSE-only model selection}. As conditional uncertainty increases with forecast horizon, the attainable set expands into a pronounced Pareto front, separating MSE-optimal but under-dispersed predictors from methods that trade accuracy for realistic marginal variability. \textbf{Across benchmarks, we find that small relaxations in MSE ($\boldsymbol{\le 5\%}$) frequently unlock disproportionate gains in marginal realism, with median improvements of $\mathbf{17.3\%}$ and gains exceeding $\mathbf{30\%}$ in some datasets.} We further show that common forecasting strategies systematically occupy different regions of this frontier: direct multi-output predictors concentrate near the accuracy-optimal extreme, while recursive strategies and sample-based inference favors marginal realism. Together, these results expose a structural failure mode of MSE-based evaluation in long-horizon forecasting and recast strategy and inference selection as navigation of an unavoidable accuracy--realism trade-off.

Modeling temporal evolution is important to analyzing and reasoning about scientific phenomena, yet most machine learning methods provide deterministic forward predictions that overlook multiple plausible outcomes and rarely support backward reasoning, limiting their usefulness in practical scientific workflows. We present a framework that integrates diffusion-based generative modeling with interactive visual analytics for scientific exploration. We introduce DiffUNet^2, a conditional diffusion model that enables bidirectional, any-to-any generation across time and captures distributions of plausible system evolutions. Built upon the model, our interactive system supports branching timeline exploration, user-guided state editing, and probability-space navigation, enabling scientists to actively explore alternative hypotheses rather than passively observe predictions. We evaluate the model on 5 datasets across different scientific domains to validate its predictive accuracy and probability-space ensemble quality. In collaboration with domain experts, we demonstrate the effectiveness of our approach in supporting practical scientific temporal data analysis workflows. By integrating modeling and visual interaction, our approach enables scientists to interactively explore system dynamics, transforming generative models into tools for hypothesis-driven scientific analysis.

Complex, dynamic, and interactive driving environments pose significant challenges for autonomous driving, primarily due to the pervasive uncertainty of surrounding traffic. A fundamental bottleneck in current systems is the disconnect between highly expressive uncertainty modeling and interpretable, safe motion planning. In this paper, we propose a novel sample-conditioned differentiable planning framework that bridges this gap by explicitly incorporating diffusion-generated future trajectories into the optimization process. Rather than compressing predictions into a single deterministic future or relying on black-box end-to-end architectures, our approach leverages a conditional diffusion model to generate a diverse set of plausible future scenarios. Crucially, these samples are directly fed into a differentiable planner, which explicitly mitigates predictive uncertainty via an empirical Conditional Value-at-Risk (CVaR) tail-risk constraint. This allows the planner to optimize a physically interpretable trajectory that is robust to rare yet safety-critical interactions. Furthermore, we introduce a directed graph representation for scene context that yields substantial improvements in both predictive effectiveness and computational efficiency. Validated through extensive open-loop and closed-loop evaluations on the Waymo Open Motion and Argoverse 2 datasets, our framework significantly outperforms state-of-the-art baselines in safety, efficiency, and ride comfort.

Probabilistic time series forecasting has attracted increasing attention in financial applications due to the need to quantify risk and uncertainty in future observations. We propose ProbRes, a post-hoc probabilistic calibration method that explicitly learns and incorporates volatility dynamics into probabilistic forecasting, enabling effective handling of heteroskedastic data. During training, ProbRes employs two architecture-agnostic modules to separately model the conditional mean and conditional volatility. At the inference stage, it generates predictive distributions by resampling normalized residuals. ProbRes is applicable to both univariate and multivariate time series and remains robust under a wide range of error distributions, including non-Gaussian innovations with conditional heteroskedasticity. Theoretical results demonstrate ProbRes's validity and experiments on both synthetic and real-world datasets show that ProbRes accurately captures predictive distributions and produces well-calibrated prediction intervals.

Energy consumption prediction is essential for efficient grid management, demand-side optimization, and sustainable energy planning. Although advanced machine learning methods have been employed for better prediction performance, existing works have two key limitations: (1) they usually formulate this task as a purely time-series prediction problem without explicitly modeling the spatial dependencies among different regions, and (2) they fail to provide reliable predictions with uncertainty estimates under abnormal situations such as extreme weather events. To advance existing research, we propose EnergyMamba, an uncertainty-aware spatiotemporal learning framework for accurate and reliable energy consumption prediction, which comprises two key components: (i) a novel Graph-Enhanced Selective State Space Model (GE-Mamba) that injects spatial context learned from the grid topology into the temporal dynamics, enabling coupled spatiotemporal modeling, and (ii) an Adaptive Sequential Conformalized Quantile Regression (AS-CQR) module, which includes locally adaptive normalization and an online feedback mechanism to dynamically calibrate prediction intervals under potential distribution shifts. We evaluate EnergyMamba on four large-scale real-world datasets from Florida, New York, and California. Results show EnergyMamba achieves around 5% improvement in prediction accuracy and 6% improvement in uncertainty quantification over 15 state-of-the-art baselines.

Engine Health Management (EHM) depends on reliable forecasting of Remaining Useful Life (RUL) and on tracking thermal indicators such as turbine gas temperature (TGT). In practice, real-world fleet data are heterogeneous and non-stationary, and point predictions alone are insufficient for risk-aware maintenance decisions. This paper presents a multi-task scientific machine learning framework for turbine prognostics that jointly predicts turbine gas temperature untrimmed (TGTU), Delta Turbine Gas Temperature (DTGT), and RUL, with quantified uncertainty in the form of prediction intervals whose empirical coverage is evaluated. A shared sequence encoder (convolutional front-end with residual bidirectional LSTM layers and attention pooling) feeds task-specific heads, including mean--variance estimation for probabilistic regression and, optionally, a survival head for threshold-based event modeling. The framework is designed to be tunable via a small set of practitioner-facing parameters (e.g., DTGT thresholding rules and RUL target construction) so that deployment can align with in-house policies and proprietary criteria. The predictive performance of the proposed framework is evaluated using both point and interval metrics, including mean absolute error (MAE), prediction interval coverage probability (PICP), mean prediction interval width (MPIW), and the coverage--width criterion (CWC). Results are reported both in aggregate and stratified by flight phase and maintenance segment to highlight operational-context effects and to support uncertainty-aware monitoring.

Chance-constrained Model Predictive Path Integral (MPPI) control is increasingly adopted for navigation in dynamic environments to explicitly bound collision risk. However, these probabilistic guarantees implicitly assume that upstream uncertainties from localization and perception are well-calibrated. In practice, estimators are often miscalibrated, inducing characteristic closed-loop failure modes: overconfidence leads to systematic safety violations, while underconfidence triggers overly conservative freezing or probability dilution. To address this critical gap, our primary contribution is a rigorous evaluation methodology applying proper scoring rules to assess the statistical validity of predicted collision risks during closed-loop execution. Concurrently, Dual-Uncertainty Chance-Constrained Tube MPPI (DUCCT-MPPI) is proposed as a real-time, risk-aware planning architecture. DUCCT-MPPI jointly integrates localization uncertainty via a one-tube Unscented Transform (UT) approximation and dynamic obstacle prediction uncertainty via Monte Carlo aggregation. Through extensive physics-based simulations, the framework demonstrates robust failure-mitigation, seamlessly transitioning to safe, conservative maneuvering without succumbing to functional deadlocks in highly cluttered environments. In highly cluttered environments, DUCCT-MPPI achieves superior robustness, outperforming established Monte Carlo MPPI baselines by nearly 28% in navigation success rate, while simultaneously recording the lowest travel times and minimizing induced social forces. Ultimately, these findings establish that reliable probabilistic safety in autonomous navigation dictates not only expressive risk models but statistically valid uncertainty estimates throughout the entire autonomy stack.

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.

Decision trees assign identical confidence to instances near and far from each split threshold. We introduce ternary decision trees, which augment each split node with an uncertainty zone of half-width delta. A decision-theoretic framework characterises the optimal zone width delta* as the solution to a node-local cost-minimisation problem; four formal properties are established: accuracy decomposition, a sufficiency condition for decided accuracy improvement, an exact efficiency characterisation (eta = Dec-Acc minus Acc_u, the accuracy gap between decided and boundary-uncertain predictions), and asymptotic consistency of the margin method. Instances within the zone receive predictions by weighted blending of both child subtrees and are flagged as boundary-uncertain. We propose and evaluate five delta-estimation methods: quality-plateau (plateau width of the split criterion curve), class-overlap (empirical class-distribution overlap), gain-ratio (split quality relative to split entropy), node-bootstrap (threshold variance under node-level resampling), and margin (SVM-inspired distance to the nearest cross-class training example). All methods reuse statistics already computed during standard CART split finding, requiring no external noise specification. Evaluated across 71 of the 72 OpenML-CC18 datasets with 5-fold cross-validation, all five methods with probabilistic routing significantly outperform standard CART on decided accuracy (Wilcoxon signed-rank, p < 0.001). The margin method achieves the best efficiency (0.104 accuracy gain per unit flagging rate), wins on 42 of 72 datasets, and requires zero hyperparameters. Analysis on Breiman synthetic benchmarks confirms margin is self-calibrating on clean data. On mammography, node-bootstrap achieves +0.71% decided accuracy by flagging 10.8% of cases as boundary-uncertain.

Accurate wind power forecasting requires reliable uncertainty quantification, yet most existing methods report a single predictive uncertainty that conflates epistemic and aleatoric sources. This paper applies the law of total variance to the joint setting of heteroscedastic neural network regression and Bayesian posterior approximation, deriving an explicit decomposition of total uncertainty (TU) into aleatoric (AU) and epistemic (EU) components. The resulting estimators are compatible with standard posterior-approximation methods and with $β$-NLL training to regulate the mean--variance learning trade-off. A wind power--specific evaluation framework is proposed to validate disentanglement without access to ground-truth uncertainty labels, comprising three modules: controlled synthetic experiments to verify responses to heteroscedastic noise and distribution shift; data-property--driven validation on a real-world wind turbine SCADA dataset; and dataset-size scaling experiments to examine the predicted asymptotic behavior of EU. Across synthetic and real-world experiments, the decomposed AU and EU components respond in theoretically consistent directions to noise structure, distributional shift, and training-scale variation, supporting the theoretical consistency and operational utility of the proposed decomposition and evaluation protocol.

In high-stakes automated decision-making, access to predictive uncertainty is essential for enabling users -- human or downstream systems -- to accept or reject predictions based on application-specific cost trade-offs. Such uncertainty-augmented (UA) systems -- i.e., systems that output both predictions and uncertainty scores -- are currently being assessed in the literature in a variety of ways, using separate metrics to evaluate the predictions and the uncertainty scores, setting a cost function with a fixed rejection cost or integrating over a coverage-risk curve. We argue that these evaluation approaches are inadequate for assessing overall performance of the UA system for decision making under uncertainty and propose a novel family of metrics, ECUAS$_n$, formulated as proper scoring rules for the task of interest. The parameter $n$ controls the trade-off between the cost of incorrect predictions and imperfect uncertainties depending on the needs of the use-case. We demonstrate the advantages of the ECUAS$_n$ metrics both theoretically and empirically, through experiments on diverse classification and generation datasets, including a manually annotated subset of TriviaQA.

Bayesian conformal optimisation methods often use the same held-out data both to search for efficient prediction sets and to certify coverage or risk. This coupling is natural for high-probability risk-control guarantees, but it is not necessary when the target is standard finite-sample marginal conformal coverage. We propose Decoupled Conformal Optimisation (DCO), a train-tune-calibrate design principle that uses an independent tuning split for efficiency-oriented structural selection and a fresh calibration split for the final conformal quantile. Conditional on the tuned structure, standard split-conformal exchangeability yields finite-sample marginal coverage for any candidate class, without a confidence parameter or multiple-testing correction. DCO therefore targets a different finite-sample guarantee from PAC-style methods: marginal conformal coverage rather than high-probability risk control. Under consistency assumptions on the coupled risk bound, the two approaches nevertheless converge to the same population threshold. Across classification and regression benchmarks, including ImageNet-A, CIFAR-100, Diabetes, California Housing, and Concrete, DCO tracks the nominal coverage level closely while often reducing average prediction-set size or interval width relative to PAC-style calibration. On ImageNet-A, for example, the average set size decreases from $26.52$ to $25.26$ and the 95th-percentile set size from $58.95$ to $53.73$; on Diabetes, the average interval width decreases from $2.098$ to $1.914$.

Robotic systems often use predictive uncertainty to decide whether to act autonomously or defer to a fallback policy. In threshold-gated autonomy, uncertainty matters mainly through its ability to rank likely errors. Standard metrics such as expected calibration error and AUROC do not directly test whether uncertainty changes act/defer decisions. We therefore evaluate uncertainty using Spearman rank correlation, paired bootstrap equivalence testing, and act/defer agreement. Across three temporal activity-recognition benchmarks, we find a dataset-dependent competence regime below which uncertainty provides a weak and unstable error ranking. Above this regime, softmax heuristics, MC Dropout, and ensembles produce similar gating behavior, while threshold choice has a much larger effect on execution outcomes. A multi-seed embodied simulation shows the same pattern for collision rate and cost once realized autonomy is matched. Under temporal covariate shift, ranking quality remains stable, but fine grained semantic OOD detection remains near chance. These results suggest that simple uncertainty proxies can suffice for selective gating once the base model is competent, but not for semantic novelty detection.

Modeling uncertainty in heavy-tailed time series remains a critical challenge for deep probabilistic forecasting models, which often struggle to capture abrupt, extreme events. While Lévy stable distributions offer a natural framework for modeling such non-Gaussian behaviors, the intractability of their probability density functions severely limits conventional likelihood-based inference. To address this, we introduce DeepLévy, a neural framework that learns mixtures of Lévy stable distributions by minimizing the discrepancy between empirical and parametric characteristic functions. DeepLévy incorporates a mixture mechanism that adaptively learns context-dependent weights and parameters over multiple Lévy components, enabling flexible multi-horizon uncertainty modeling. Evaluations on both real and synthetic datasets demonstrate that DeepLévy outperforms state-of-the-art deep probabilistic forecasting approaches in tail risk metrics, especially under extreme volatility.

Automated mental health prediction using textual data has shown promising results with deep learning and large language models. However, deploying these models in high-stakes real-world settings remains challenging, as existing approaches largely rely on semantic representations and often produce overconfident predictions under ambiguous, noisy, or shifted data. Moreover, most methods lack reliable uncertainty estimation, undermining trust in risk-sensitive mental health applications. To address these limitations, we formulate the task as a multi-view learning problem that integrates semantic information from encoder-only models with higher-level reasoning information from decoder-only models, where reasoning-aware representations and uncertainty modeling are obtained in a trustworthy manner. To ensure reliable fusion, we adopt an evidential learning framework based on Subjective Logic to explicitly model uncertainty and introduce an evidential fusion strategy that balances complementary views while discounting unreliable evidence. Benchmarking on three real-world datasets, Dreaddit, SDCNL, and DepSeverity, reports accuracies of 0.835, 0.731, and 0.751, respectively, demonstrating its potential for reliable mental health prediction. Additional experiments on robustness to noise and case studies for interpretability confirm that our proposed framework not only improves predictive performance but also provides trustworthy uncertainty estimates and human-understandable reasoning signals, making it suitable for risk-sensitive applications in mental health assessment.

Financial market forecasting is inherently uncertain, yet most deep learning approaches rely on point predictions that provide only single-value estimates without quantifying uncertainty. Such predictions are insufficient for risk-aware decision-making, as they fail to capture the range of possible outcomes and the associated confidence of forecasts.The problem can be solved using prediction intervals, which allow obtaining an upper and lower bound for the prediction, thus enabling uncertainty representation in the model. Yet, the current methods tend to disregard relationships between assets or cannot simultaneously ensure good calibration and sharpness of the resulting intervals in dynamically changing market regimes. In our work, we propose a spatio-temporal graph-based approach with a bi-level chaotic fusion technique to solve this problem. Our model uses separate nonlinear transformation functions to estimate the interval center and width. Additionally, a volatility-aware gating mechanism is used to make predictions dependent on the regime in which the market operates. Temporal dependencies are considered by embedding graph structures and sequentially modeling them. Training is conducted according to a Lower-Upper Bound Estimation (LUBE) objective. Our experimental results show significant improvements compared to existing baselines (LSTM, GRU, GCN, HGNN) when applied to data from 2016 to 2026 with 43 leading companies in eight sectors of the NSE. It provides the lowest Winkler score (0.0778), tightest prediction intervals (PIAW = 0.1407), and highest coverage (PICP = 96.6%), with all differences statistically significant (p < 0.001) according to the Diebold-Mariano test.

Deep neural networks achieve impressive results across diverse applications, yet their overconfidence on unseen inputs necessitates reliable epistemic uncertainty modeling. Existing methods for uncertainty modeling face a fundamental dilemma: Bayesian approaches provide principled estimates but remain computationally prohibitive, while efficient second-order predictors lack rigorous connections between their specific objectives and epistemic uncertainty quantification. To resolve this dilemma, we introduce Dirichlet-approximated possibilistic posterior predictions (DAPPr), a principled framework grounded in possibility theory. We define a possibilistic posterior over parameters, project it to the prediction space via supremum operators, and approximate the projected posterior using learnable Dirichlet possibility functions. This projection-and-approximation strategy yields a simple training objective with closed-form solutions. Despite its simplicity, extensive experiments across diverse benchmarks show that DAPPr achieves competitive or superior uncertainty quantification performance over state-of-the-art second-order predictors while maintaining both principled derivation and computational efficiency. Code is available at https://github.com/MaxwellYaoNi/DAPPr.

Background: Timely, uncertainty-aware forecasting from irregular electronic health records (EHR) can support critical-care decisions, yet most approaches either impute to a grid or sacrifice interpretability. We introduce StructGP, a continuous-time multi-task Gaussian process that couples process convolutions with differentiable structure learning to uncover a sparse, ordered directed acyclic graph (DAG) of inter-variable dependencies while preserving principled uncertainty. We further propose LP-StructGP, which augments StructGP with latent pathways-shared, temporally shifted trajectories inferred via subject-specific coupling filters and a softmax gating mechanism-to capture cross-patient progression patterns. Both models are trained under sparsity and acyclicity constraints (augmented Lagrangian, Adam) using scalable low-rank updates. Results: In simulations, the approach reliably recovers ground-truth graphs (Structural Hamming Distance approaching 0 as cohorts grow) and pathway assignments (high Adjusted Rand Index). On a MIMIC-IV septic shock cohort (n=1,008; norepinephrine, creatinine, mean arterial pressure), StructGP improves short-horizon (6 h) forecasting over independent-task baselines (average RMSE 0.68 [95%CI: 0.63--0.74] vs. 0.88 [0.83-0.94]) and, with 15 additional inputs, markedly outperforms unstructured kernels (0.63 [0.58-0.69] vs. 3.02 [2.85-3.18]) with superior calibration (coverage 0.96 vs. 0.84). On the PhysioNet Challenge (12k patients, 41 variables), StructGP attains competitive accuracy (MAE 3.72e-2) relative to a state-of-the-art graph neural model while maintaining calibrated uncertainty. Conclusion: These results show that structured process convolutions with latent pathways deliver interpretable, scalable, and well-calibrated forecasting for irregular clinical time series.

Machine Learning is becoming more prevalent in science and engineering, but many approaches do not provide meaningful uncertainty estimates and predictions may also violate known physical knowledge. We propose a Bayesian framework to embed linear relationships across inputs and outputs into the learning process, whilst characterizing full predictive uncertainty over both the model parameters and the domain knowledge. We evaluated our method on learning the single particle battery model subject to voltage and energy balances, showing its ability to provide reduced credible intervals and constraint violations compared to standard Bayesian neural networks based on variational inference.

Transformer-based scientific foundation models are increasingly deployed in high-stakes settings, but current architectures give deterministic outputs and provide limited support for calibrated predictive uncertainty. We propose Stochastic Attention, a sample average lightweight inference-time modification that randomizes attention by replacing softmax weights with normalized multinomial samples controlled by a single concentration parameter, and produces predictive ensembles without retraining. To set this parameter, we introduce a calibration objective that matches the stochastic attention output with the target, yielding an efficient univariate post-hoc tuning problem. We evaluate this mechanism on scientific foundation models for weather and time-series forecasting, as well as several regression tasks. Across benchmarks against uncertainty-aware baselines, we find that Sample Average Stochastic Attention achieves the strongest native calibration and the sharpest prediction intervals at comparable calibration, with adaptation costs nearly three orders of magnitude lower than the next-best baseline.

This paper proposes a multi-step probabilistic forecasting framework using a single neural-network based model to generate simultaneous point and interval forecasts. Our approach ensures non-crossing prediction intervals (PIs) through a model structure design that strictly satisfy a target coverage probability (PICP) while maximizing sharpness. Unlike existing methods that rely on manual weight tuning for scalarized loss functions, we treat point and PI forecasting as a multi-objective optimization problem, utilizing multi-gradient descent to adaptively select optimal weights. Key innovations include a new PI loss function based on an extended log-barrier with an adaptive hyperparameter to guarantee the coverage, a hybrid architecture featuring a shared temporal model with horizon-specific submodels, and a training strategy. The proposed loss is scale-independent and universally applicable; combined with our training algorithm, the framework eliminates trial-and-error hyperparameter tuning for balancing multiple objectives. Validated by an intra-day solar irradiance forecasting application, results demonstrate that our proposed loss consistently outperforms those in current literature by achieving target coverage with the narrowest PI widths. Furthermore, when compared against LSTM encoder-decoder and Transformer architectures--including those augmented with Chronos foundation models--our method remains highly competitive and can be seamlessly adapted to any deep learning structure.