Uncertainty Calibration

Modern artificial intelligence systems require calibrated uncertainty estimates that remain reliable in sequential and non-stationary environments. Online conformal prediction (OCP) addresses this challenge through adaptively updated prediction sets that provide deterministic long-run miscoverage guarantees. These guarantees, however, hinge on the assumption of perfect feedback about the coverage of past prediction sets. In practice, the observed miscoverage indicator may be corrupted by noise, communication failures, or adversarial manipulation, which can severely degrade OCP's calibration guarantees. In this paper, we study OCP under corrupted feedback. We first model feedback corruption as an arbitrary binary flip sequence, and analyze how feedback corruption affects and degrades the miscoverage performance of standard OCP. We then propose two robust schemes: robust OCP via filtering, which leverages the structural properties of the predicted threshold to filter corrupted feedback, and robust OCP via active compensation, which incorporates an active compensation mechanism to mitigate the effect of corrupted feedback. For both methods, we establish explicit miscoverage guarantees, which are further specialized for an independent stochastic flip model and for an arbitrary error model with memory bounds. Experiments on real-world datasets validate the proposed approach, showing markedly improved calibration and significantly smaller prediction sets compared with baseline OCP methods under corrupted feedback.

Effective medication management in Parkinson's Disease (PD) is challenging due to heterogeneous disease progression, variable patient response, and medication side effects. While AI models can forecast levodopa equivalent daily dose (LEDD) as a measure of medication needs, standard uncertainty quantification often fails to communicate the reliability of these predictions, treating high and low confidence clinical decisions identically. We introduce CASCADE (Calibrated Adaptive Scaling via Conformal And Distributional Estimation), a novel conformal prediction framework that propagates epistemic uncertainty from a screening classifier to adapt downstream predictions. Unlike standard conformal methods that rely on auxiliary residual regression, we leverage epistemic uncertainty from a primary classification task (identifying whether a medication change is needed) to dynamically scale the prediction intervals of a secondary regression task (predicting how much change). By mapping Venn-Abers multi-probabilistic uncertainty directly to non-conformity scores, our framework achieves continuous risk adaptation. We demonstrate that this cascade effect produces highly efficient intervals for confident patients (38.9% narrower than standard conformal baselines) while automatically expanding intervals to ensure robust coverage for uncertain cases, bridging the gap between discrete clinical decision-making and continuous dose forecasting in PD.

The maximum softmax probability (MSP) represents a default approach when evaluating uncertainty quantification for language model generation with structured output. Although cheap, it is often miscalibrated. Methods that probe the model's internal activations feed raw hidden states into opaque classifiers, reading activations as static snapshots and leaving implicit the layer-wise trajectory by which a representation is formed. Yet, similar endpoints can arise from very different paths, and how evidence accumulates, reinforces, or reverses across depth might reveal uncertainty that final probabilities obscure. We extract eleven scale-invariant geometric features, tracing the cumulative path of per-layer MLP updates, and feed them to a sparse linear probe. The probe outperforms MSP under selective abstention, with gains scaling with baseline miscalibration up to 21 AURC points. Because every feature has a closed-form geometric meaning, the probe's coefficients trace how and where along depth errors take shape -- which layers commit prematurely, which contradict the running state, where trajectories drift away from their endpoint.

Gaussian process (GP) predictive distributions are commonly used in Bayesian optimization (BO) to guide the selection of evaluation points for expensive objective functions. The choice of kernel and hyperparameters has a strong influence on the exploration--exploitation trade-off. For minimization, sampling criteria such as expected improvement (EI) depend on both the probability mass below the current best value and the shape of the predictive distribution in this region. This article studies goal-oriented calibration of GP predictive distributions below a low threshold $t$ in the noiseless setting, for standard GP models with hyperparameters selected by maximum likelihood. We consider two complementary forms of calibration below $t$ for inputs distributed according to a reference measure $μ$: occurrence calibration over the design space and thresholded $μ$-calibration on sublevel sets of the form $\{x\in\mathbb{X}, f(x)\le t\}$. We propose tcGP, a post-hoc method that combines these two forms of calibration for GP predictive distributions below $t$. With fixed GP hyperparameters, the exact EI sampling criterion based on tcGP generates a sequence of evaluation points that is dense in the design space. Experiments on standard benchmarks show improved lower-tail calibration and BO performance relative to standard GP models and globally calibrated GP models.

Central to human-aligned AI is understanding the benefits of human-elicited labels over synthetic alternatives. While human soft-labels improve calibration by capturing uncertainty, prior studies conflate these benefits with the implicit correction of mislabeled data (mode shifts), obscuring true effects of soft-labels. We present a controlled audit of soft-label learning across MNIST and a synthetic variant, re-annotating subsets to extract human uncertainty. By decoupling soft-label supervision from underlying label mode shifts, we show that while human soft-labels do provide accuracy gains, their larger value lies in acting as a regularizer that improves model calibration on difficult samples and promotes stable convergence across training runs. Dataset cartography reveals models trained on human soft-labels mirror human uncertainty, whereas those trained on synthetic labels fail to align with humans. Broadly, this work provides a diagnostic testbed for human-AI uncertainty alignment.

Ensemble disagreement is widely used as a proxy for epistemic uncertainty in medical image segmentation. In practice, many studies form ensembles via K-fold cross-validation (CV), yet refer to them as ``deep ensembles'' (DE). Because CV members are trained on different data subsets, their disagreement mixes seed-driven variability with data-exposure effects, which can change how uncertainty should be interpreted. We audit recent segmentation uncertainty studies and find that terminology--implementation mismatches are common. We then compare a standard 5-fold CV ensemble to a 5-member DE (fixed training set, different random seeds) under otherwise identical configurations on three multi-rater segmentation datasets spanning three modalities. We evaluate uncertainty for calibration, failure detection, ambiguity modeling, and robustness under distribution shift. DE match segmentation accuracy while improving calibration and failure detection, whereas CV ensembles sometimes correlate more strongly with inter-rater variability on the studied datasets. Thus, ensemble construction should be chosen to match the research question: DE for reliability-oriented use (e.g., selective referral/failure detection) and CV ensembles as a proxy for ambiguity. We provide a lightweight nnU-Net modification enabling DE training within the default pipeline.

Uncertainty quantification (UQ) is critical for safety-critical domains like healthcare, yet it is rarely evaluated under realistic out-of-distribution (OOD) conditions. Here, we assessed predictive performance and uncertainty reliability for deep learning-based blood pressure (BP) estimation from photoplethysmography (PPG) signals under both in-distribution (ID) and OOD settings. Using an XResNet1D-50 trained on PulseDB and tested on four external datasets, we compared deep ensembles (DE) and Monte Carlo dropout (MCD) with Gaussian negative log-likelihood (GNLL) and mean squared error (MSE) losses, optionally followed by post-hoc recalibration via conformal prediction (CP), temperature scaling (TS), and isotonic regression (IR). The key findings of our study are as follows: (1) DE provides stronger predictive robustness under domain shift than MCD, an advantage that becomes clear primarily under external shift. (2) Recalibrated GNLL-based methods yield the best uncertainty calibration (e.g., GNLL+DE+CP for systolic blood pressure (SBP), GNLL+DE+TS for diastolic blood pressure (DBP)), while MSE-based uncertainty requires recalibration to become practically useful. (3) Across settings, CP and TS offer the most consistent gains, with IR remaining competitive in several cases. Overall, our results identify DE-based methods as most robust for predictive performance under domain shift, GNLL as strongest for native UQ, and recalibration as essential for making MSE-based uncertainty practical. These findings highlight the need to jointly assess predictive accuracy and calibration on external data for trustworthy cuffless BP estimation

A fundamental challenge in recommender systems is balancing reliability for Low-Active Users (LAUs) with diversity for High-Active Users (HAUs). The key to this balance lies in quantifying model uncertainty, which approximates the risk of prediction errors and reveals the limits of the model's current knowledge. On large-scale short-video and livestream platforms, model uncertainty can warn of low-quality recommendations that may lead to disengagement of LAUs and at the same time identify opportunities to diversify content recommendation for HAUs. To leverage this dichotomy, we introduce a unified, production-ready framework that calibrates uncertainty to drive differentiated strategies. Specifically, we implement a model-uncertainty-based risk-averse deboosting policy for LAUs to suppress unreliable recommendations, while employing a risk-seeking Upper Confidence Bound (UCB) strategy for HAUs to encourage exploration. Validated on a major livestream platform, our framework demonstrates significant improvements in retention (active hours) and satisfaction (quality watch time ratio) for LAUs as well as remarkable increases in interest diversity and category coverage for HAUs, proving the value of uncertainty-aware recommendation in industrial settings.

Deep Neural Networks have become the dominant solution for Autonomous Driving perception, but their opacity conflicts with emerging Trustworthy AI guidelines and complicates safety assurance, debugging, and human oversight. While theoretical frameworks for safe and Explainable AI (XAI) exist, concrete implementations of Trustworthy AI for 3D scene understanding remain scarce. We address this gap by proposing a Trustworthy AI perception module that is remarkably robust, integrates faithful explainability, and calibrated uncertainty estimates. Building on a transformer-based detector, we derive explanation from the attention mechanism at inference time and validate their faithfulness using perturbation-based consistency tests. We further integrate an uncertainty estimation and calibration module, and apply robustness-enhancing training methods. Experiments show faithful saliency behavior, improved robustness, and well-calibrated uncertainty estimates. Finally, we deploy these Trustworthy AI elements in a prototype vehicle and provide an XAI Interface that visualizes documentation artifacts, model uncertainty state, and saliency maps, demonstrating the feasibility of trustworthy perception monitoring in real time. Supplementary materials are available at https://tillbeemelmanns.github.io/trustworthy_ai/ .

Rapid and accurate wildfire smoke severity assessment from satellite images is essential for emergency response, air quality modeling, and human health risk management. Existing deep learning approaches treat smoke detection as a binary task, producing point estimates without any measure of prediction confidence. We propose a probabilistic framework to categorize a satellite patch into Light, Moderate, and Heavy severity classes and to provide decomposed epistemic and aleatoric uncertainty in a single forward pass. Our architecture uses the backbone of a pre-trained EfficientNet-B3 and a CBAM module with an evidential deep learning head that predicts Dirichlet concentration parameters, directly estimating vacuity (epistemic) and dissonance (aleatoric) without Monte Carlo sampling. Evaluated on 16,298 real satellite patches derived from the Wildfire Detection dataset, our model achieves 93.8% weighted test accuracy (91.1% unweighted) with ECE=0.0274. Selective prediction retaining the most certain 50% of patches achieves 96.7% accuracy. As image quality degrades, uncertainty increases monotonically, and vacuity is a practical scan quality measure. The Moderate class represents transitional smoke conditions that exhibit the highest epistemic uncertainty (mean vacuity = 0.187), confirming the model correctly identifies ambiguous smoke boundary regions. CBAM spatial attention maps localize to structurally distinctive scene regions, and t-SNE demonstrates the clear cluster separation of Light and Heavy smoke.

In this study, we explore in depth a few under-studied topics at the intersection of uncertainty estimation and segmentation. Prior work has shown that the quality of uncertainty estimates can be very sensitive to a range of variables. As one of the main uses of uncertainty estimation is to help identify and deal with prediction errors in practical scenarios, any factors that affect this must be clearly identified. For example, do more challenging domains or different datasets and architectures result in worse performance when using uncertainty estimates? Can prior frames in a video sequence in fact provide useful uncertainty estimates comparable to other approaches? Is it possible to combine uncertainty estimation approaches, taking advantage of sample diversity, to get better estimates? Finally, when might it make sense to use an ensemble-based uncertainty estimate over a deterministic network? We address these questions by creating a framework for and executing a large scale study across many variables such as datasets, backbones, and downstream tasks, for both semantic and panoptic segmentation. We find that a) the more challenging task of panoptic segmentation usually results in worse performance while high performance variance between datasets and backbones indicates that generalization is not guaranteed, b) time series samples can be useful for specific configurations, but in many cases are not worth the cost, c) sample diversity shows the most promise in the downstream task of calibration, but otherwise fails to beat simpler alternatives, d) a deterministic approach is adequate for some downstream tasks, but ensembles allow for significant improvements if the right conditions can be achieved in deployment.

Recently, deep generative models have been used for posterior inference in inverse problems, including high-stakes applications in medical imaging and scientific discovery, where the uncertainty of a prediction can matter as much as the prediction itself. However, posterior uncertainty is difficult to interpret because it can mix ambiguity inherent to the forward operator with uncertainty propagated through inference. We introduce a structural decomposition of posterior uncertainty that isolates intrinsic ambiguity. A cascade formulation makes this ambiguity accessible for calibration analysis, enabling qualitative diagnostics and simulation-based calibration tests that reveal failure modes that remain hidden when models are selected by reconstruction quality alone. We first validate the approach on a Gaussian example with analytical posterior structure, then illustrate the decomposition on accelerated magnetic resonance imaging (MRI), and finally apply the calibration diagnostics to electroencephalography (EEG) source imaging.

This work studies lung tumor growth prediction from sparse and irregular longitudinal computed tomography (CT) observations with measurement variability. A Bayesian physics-informed neural network is developed by combining Gompertz growth dynamics with low-dimensional Bayesian inference in the log-volume domain. The framework employs a two-stage inference strategy combining maximum a posteriori (MAP) estimation and Hamiltonian Monte Carlo (HMC) sampling to estimate posterior predictive distributions and uncertainty intervals. The method was evaluated on longitudinal data from the National Lung Screening Trial (30 patients). Results show that the model captures heterogeneous tumor growth patterns while maintaining reasonable prediction accuracy under limited observations. Compared with deterministic modeling approaches, the proposed approach additionally provides calibrated uncertainty estimates. The inferred posterior parameter correlations were consistent with expected biological growth behavior. The proposed framework achieved a cohort-level log-space RMSE of approximately 0.20 together with well-calibrated 95% credible interval coverage across 30 patients. These findings suggest that Bayesian physics-informed modeling may be useful for uncertainty-aware tumor growth assessment when only limited longitudinal follow-up scans are available.

Calibration is commonly evaluated by comparing model confidence with its empirical correctness, implicitly treating reliability as a function of the confidence score alone. However, this view can hide substantial structure: models may be systematically overconfident on some kinds of inputs and underconfident on others, causing global reliability diagnostics to obscure localised calibration failures. To address this, we formulate the problem of discovering hidden miscalibration regimes without assuming access to predefined data slices. We define the corresponding miscalibration field and propose a diagnostic framework for estimating it. Our approach learns a calibration-aware representation of the input space and estimates signed local miscalibration by kernel smoothing in the learned geometry. Across four real-world LLM benchmarks and twelve LLMs, we find that input-dependent calibration heterogeneity is prevalent. We further show that the discovered fields are actionable: they support local confidence correction and reduce calibration error in systematically miscalibrated regions where confidence-based methods such as isotonic regression and temperature scaling are less effective.

Conformal prediction provides a principled framework for uncertainty quantification with finite-sample coverage guarantees. While recent work has extended conformal prediction to online and sequential settings, existing methods typically focus on a single coverage level and do not ensure consistency across multiple confidence levels. In many real-world applications, such as weather forecasting, macroeconomic prediction, and risk management, different users operate under heterogeneous risk tolerances and require calibrated uncertainty estimates across a range of coverage levels. In such settings, it is desirable to produce prediction sets corresponding to different coverage levels that are nested and valid simultaneously. In this paper, we propose two novel online conformal prediction methods that output \emph{nested prediction sets} across a range of coverage levels, enabling simultaneous uncertainty quantification across the entire risk spectrum. Beyond interpretability, jointly estimating multiple coverage levels is known to improve statistical efficiency in classical quantile regression by enforcing non-crossing constraints and sharing information across quantiles. Our approaches leverage an online optimization perspective with small regret that translates to quantile estimation error control while enforcing nestedness of prediction sets. Empirical results on synthetic and real-world datasets, including applications in forecasting tasks with heterogeneous risk requirements, demonstrate that our method achieves stable coverage across all levels, strictly nested prediction sets, and improved efficiency compared to existing online conformal baselines.

Approximate Bayesian inference typically revolves around computing the posterior parameter distribution. In practice, however, the main object of interest is often a model's predictions rather than its parameters. In this work, we propose to bypass the parameter posterior and focus directly on approximating the posterior predictive distribution. We achieve this by drawing inspiration from self-training within self-supervised and semi-supervised learning. Essentially, we quantify a Bayesian model's predictive uncertainty by refitting on self-predicted data. The idea is strikingly simple: If a model assigns high likelihood to self-predicted data, these predictions are of low uncertainty, and vice versa. This yields a deterministic, sampling-free approximation of the posterior predictive. The modular structure of our Self-Supervised Laplace Approximation (SSLA) further allows us to plug in different prior specifications, enabling classical Bayesian sensitivity (w.r.t. prior choice) analysis. In order to bypass expensive refitting, we further introduce an approximate version of SSLA, called ASSLA. We study (A)SSLA both theoretically and empirically in regression models ranging from Bayesian linear models to Bayesian neural networks. Across a wide array of regression tasks with simulated and real-world datasets, our methods outperform classical Laplace approximations in predictive calibration while remaining computationally efficient.

Probabilistic partial least squares (PPLS) is a central likelihood-based model for two-view learning when one needs both interpretable latent factors and calibrated uncertainty. Building on the identifiable parameterization of Bouhaddani et al.\ (2018), existing fitting pipelines still face two practical bottlenecks: noise--signal coupling under joint EM/ECM updates and nontrivial handling of orthogonality constraints. Following the fixed-noise scalar-likelihood protocol, we develop an end-to-end framework that combines noise pre-estimation, constrained likelihood optimization, and prediction calibration in one pipeline. We estimate the observation noise from the low-eigenvalue noise subspace and enforce orthogonality through exact Stiefel-manifold optimization. The noise-subspace estimator attains a signal-strength-independent leading finite-sample rate and matches a minimax lower bound, whereas a full-spectrum noise estimator carries a deterministic bias under the same model. We further extend the framework to sub-Gaussian settings via optional Gaussianization and provide closed-form standard errors through a block-structured Fisher analysis. Across synthetic high-noise settings and two multi-omics benchmarks (TCGA-BRCA and PBMC CITE-seq), the method achieves near-nominal coverage without post-hoc recalibration, reaches Ridge-level point accuracy on TCGA-BRCA at rank $r=3$, matches or exceeds PO2PLS on cross-view prediction while providing native calibrated uncertainty, and improves stability of parameter recovery.

System identification (SysID) is critical for modeling dynamical systems from experimental data, yet traditional approaches often fail to capture nonlinear behaviors. While deep learning offers powerful tools for modeling such dynamics, incorporating uncertainty quantification is essential to ensure reliable predictions. This paper presents a systematic framework for constructing and training interval Neural Networks (INNs) for uncertainty-aware SysID. By extending crisp neural networks into interval counterparts, we develop Interval LSTM and NODE models that propagate uncertainty through interval arithmetic without probabilistic assumptions. This design allows them to represent uncertainty and produce prediction intervals. For training, we propose two strategies: Cascade INN (C-INN), a two-stage approach converting a trained crisp NN into an INN, and Joint INN (J-INN), a one-stage framework jointly optimizing prediction accuracy and interval precision. Both strategies employ uncertainty-aware loss functions and parameterization tricks to ensure reliable learning. Comprehensive experiments on multiple SysID datasets demonstrate the effectiveness of both approaches and benchmark their performance against well-established uncertainty-aware baselines: C-INN achieves superior point prediction accuracy, whereas J-INN yields more accurate and better-calibrated prediction intervals. Furthermore, to reveal how uncertainty is represented across model parameters, the concept of channel-wise elasticity is introduced, which is used to identify distinct patterns across the two training strategies. The results of this study demonstrate that the proposed framework effectively integrates deep learning with uncertainty-aware modeling.

Bayesian optimization (BO) is a widely used framework for optimizing expensive black-box functions, commonly based on Gaussian process (GP) surrogate models. Its effectiveness relies on uncertainty quantification that is both sharp (informative) and well-calibrated along the BO trajectory. In practice, GP kernel hyperparameters are unknown and are refit online from sequentially collected (non-i.i.d.) data, which can yield miscalibrated or overly conservative uncertainty and lies outside the fixed-kernel assumptions of standard BO regret theory. We propose Online Sharp-Calibrated Bayesian Optimization (OSCBO), a BO algorithm that adaptively balances GP sharpness and calibration by casting hyperparameter selection as a constrained online-learning problem. We also show that OSCBO preserves sublinear regret bounds by leveraging the theoretical guarantees of the underlying online learning algorithm. Empirically, OSCBO performs competitively across synthetic and real-world benchmarks, ranking among the strongest methods in final simple regret while maintaining robust cumulative-regret behavior.

Large language models for subjectivity analysis are typically trained with aggregated labels, which compress variations in human judgment into a single supervision signal. This paradigm overlooks the intrinsic uncertainty of low-agreement samples and often induces overconfident predictions, undermining reliability and generalization in complex subjective settings. In this work, we advocate uncertainty-aware subjectivity analysis, where models are expected to make predictions while expressing uncertainty that reflects human disagreement. To operationalize this perspective, we propose a two-phase Disagreement Perception and Uncertainty Alignment (DPUA) framework. Specifically, DPUA jointly models label prediction, rationale generation, and uncertainty expression under an uncertainty-aware setting. In the disagreement perception phase, adaptive decoupled learning enhances the model's sensitivity to disagreement-related cues while preserving task performance. In the uncertainty alignment phase, GRPO-based reward optimization further improves uncertainty-aware reasoning and aligns the model's confidence expression with the human disagreement distribution. Experiments on three subjectivity analysis tasks show that DPUA preserves task performance while better aligning model uncertainty with human disagreement, mitigating overconfidence on boundary samples, and improving out-of-distribution generalization.

Multicalibration requires predicted scores to agree with label probabilities across rich families of subgroups and score-dependent tests, but existing methods require clean input-label pairs for evaluation and post-processing. This assumption fails in weakly supervised learning (WSL) regimes -- including positive-unlabeled, unlabeled-unlabeled, and positive-confidence learning -- where clean labels are costly or unavailable even though reliable uncertainty estimates may be crucial. We address this gap by developing estimators of multicalibration error and post-hoc correction methods for WSL settings in which clean input-label pairs are unavailable. We propose a unified framework for estimating and correcting multicalibration under weak supervision by combining contamination-matrix risk rewrites with witness-based calibration constraints, yielding corrected multicalibration moments with finite-sample guarantees. We further propose weak-label multicalibration boost (WLMC), a generic post-hoc recalibration algorithm under weak supervision. Finally, we conduct experiments across multiple weak-supervision settings to evaluate multicalibration behavior and offer empirical insight into uncertainty estimation under weak supervision.

We introduce the Free Energy Manifold (FEM), a score-trained conditional energy model specialized for inference in hybrid Bayesian networks with discrete and continuous variables. FEM represents each conditional factor as an energy landscape over learned discrete-parent embeddings and continuous observations, enabling posterior evaluation, generative sampling, and compositional inference across multiple continuous leaves by energy addition under conditional independence. A central finding is the mode-bridge artifact: standard conditional energy models can create low-energy ridges between separated modes of the same class, producing overconfident posteriors at off-data interior points. We analyze this failure and propose valley regularization, an off-data calibration term that restores near-uniform posteriors in such regions while preserving in-data fit. Across synthetic multimodal hybrid-BN benchmarks, FEM substantially reduces KL divergence relative to classical baselines and a vanilla conditional EBM, including large gains at mode-bridge midpoint queries and in multi-leaf evidence composition. We also evaluate high-cardinality discrete-parent settings and a UCI Breast Cancer sanity check, showing that FEM is most useful when multimodal or compositional Bayesian-network inference is required, while discriminative classifiers remain preferable for closed-world classification tasks.

Medicine is rife with high-stakes uncertainty. Doctors routinely make clinical judgments and decisions that juggle many fundamental unknowns, like predictions about what might be causing a patients' symptoms or decisions about what treatment to try next. Despite increasing interest in developing AI systems that aid or even replace doctors in clinical settings, current systems struggle with calibrated reasoning under uncertainty, and are often deeply opaque about their reasoning. We propose a framework for AI systems that can make practically useful but formally transparent clinical predictions under uncertainty. Given a clinical situation, our framework (MedMSA) uses language models to retrieve relevant prior knowledge, but constructs a formal probabilistic model to support calibrated and verifiable inferences under uncertainty. We show how an initial proof-of-concept of this framework can be used for differential diagnosis, producing an uncertainty-weighted list of potential diagnoses that could explain a patients' symptoms, and discuss future applications and directions for applying this framework more generally for safe clinical collaborations.

Second-order calibration error quantifies how closely a higher-order predictor's epistemic-uncertainty estimate matches the conditional variance of the label probability on its level sets. We characterize the minimax rate of estimating the second-order calibration error for binary classification in the regime where a small perturbation is applied to the classifier outputs. Our procedure is simple: add independent bandwidth-$h$ sech noise to the score coordinates, then regress $Y^{(1)}$ and $Y^{(1)}Y^{(2)}$ on the perturbed score using low-degree polynomials. Crucially, the sech perturbation makes the calibration functions analytic in a suitable strip. The resulting estimator has error $O_h(\log^{3/2}n/\sqrt n)$, with explicit constants. In the same setting, a matching $Ω(1/\sqrt{n})$ lower bound establishes minimax optimality up to logarithmic factors. As a corollary, we give a finite-sample guarantee for perturbed second-order Platt scaling, yielding a post-hoc procedure that recalibrates both the mean prediction and the epistemic-variance estimate of the perturbed higher-order predictor. Along the way, we give an explicit two-moment formulation of second-order calibration error and relate it quantitatively to the bucketed formulation of Ahdritz et al. [2025]. Our experiments confirm the predicted rate and the quality of the recalibrated uncertainties.

Conformal Prediction (CP) is a distribution-free method for constructing prediction sets with marginal finite-sample coverage guarantees, making it a suitable framework for reliable uncertainty quantification in safety-critical object detection. However, object detection introduces structured multi-output predictions, complicating the application of classical CP theory developed for single outputs. In addition, standard, unscaled CP produces fixed-width prediction intervals across inputs, leading to unnecessary width for low-uncertainty predictions. While scaled CP addresses this by adapting the interval width to an input-dependent uncertainty estimate, prior work has neither systematically compared unscaled and scaled CP for multi-class object detection, nor integrated CP with a complementary uncertainty quantification method in this setting. We fill this gap by: (i) applying CP coordinate-wise to bounding box corners with a Bonferroni correction for box-level guarantees; (ii) scaling the resulting intervals using per-prediction aleatoric uncertainty estimates derived from a probabilistic object detector trained with loss attenuation, evaluated in uncalibrated and two calibrated variants; (iii) extending to a two-step pipeline that constructs prediction sets for the class using RAPS and conditions the conformalized bounding boxes on the predicted class set. Across three autonomous driving datasets (KITTI, BDD, CODA), including a cross-domain setting under distribution shift, scaled CP consistently improves interval sharpness over unscaled CP, achieving up to 19% higher IoU and 39% lower interval scores, without sacrificing coverage. Class-wise calibration further improves coverage for both variants with a negligible effect on sharpness. Together, these improvements yield more actionable uncertainty estimates for real-time, real-world object detection.

Inference-time scaling methods rely on Process Reward Models (PRMs), which are often poorly calibrated and overestimate success probabilities. We propose, to our knowledge, the first use of conditional optimal transport for calibrating PRMs, modifying conditional OT (CondOT) map learning \cite{bunne2022supervised} to estimate a monotonic conditional quantile function over success probabilities estimated by the PRM, conditioned on PRM hidden states. This yields structurally valid quantile estimates and enables efficient extraction of confidence bounds at arbitrary levels, which we integrate into the instance-adaptive scaling (IAS) framework of \cite{park2025know}. We evaluate on mathematical reasoning benchmarks spanning moderate-difficulty problems (MATH-500) and harder out-of-distribution problems (AIME). For PRMs with reliable ranking signals, our method substantially improves calibration over both uncalibrated PRMs and quantile regression. On downstream Best-of-N IAS performance, our method generally improves over uncalibrated PRMs. These results establish conditional optimal transport as another principled and practical approach to PRM calibration, offering structural guarantees and flexible uncertainty estimation.

In many scientific domains, including experimentation, researchers rely on measurements of proxy outcomes to achieve faster and more frequent reads, especially when the primary outcome of interest is challenging to measure directly. While proxies offer a more readily accessible observation for inference, the ultimate goal is to draw statistical inferences about the primary outcome parameter and proxy data are typically imperfect in some ways. To correct for these imperfections, current statistical inference methods often depend on strict identifying assumptions (such as surrogacy, covariate/label shift, or missingness assumptions). These assumptions can be difficult to validate and may be violated by various additional sources of distribution shift, potentially leading to biased parameter estimates and miscalibrated uncertainty quantification. We introduce an estimate-level framework, inspired by domain adaptation techniques, to empirically calibrate proxy-based inference. This framework models the proxy-primary metric discrepancy as a random effect at the parameter level, estimating its distribution from aggregated historical observations across past domains (e.g., experiments, time periods, or distinct segments). This method avoids the requirement for retaining individual-level response data. Additionally, this adjustment can be layered on top of existing proxy-correction methods (such as prediction-powered inference or importance weighting) to account for additional biases not addressed by those corrections. To manage uncertainty when the number of historical domains is limited, we provide both a method-of-moments estimator and a domain bootstrap procedure. We further validate this approach using publicly available datasets and real-world experiments.

We introduce a probabilistic approach for dating historical manuscript pages from visual features alone. Instead of aggregating centuries into classes as is standard in the previous literature, we pose dating as an evidential deep regression problem over a continuous year axis, allowing our neural network to output a full predictive distribution with decomposed aleatoric and epistemic uncertainty in a single forward pass. Our architecture combines an EfficientNet-B2 backbone with a Normal-Inverse-Gamma (NIG) output head trained with a joint negative-log-likelihood and evidence-regularization objective. On the DIVA-HisDB benchmark (150 pages, 3 medieval codices, 151,936 patches), our model scores a test MAE of 5.4 years, well below the 50-year century-label supervision granularity, with 93% of patches within 5 years and 97% within 10 years. Our approach achieves \textbf{PICP=92.6%}, the best calibration among all compared methods, in a single forward pass, outperforming MC Dropout (PICP=88.2%, 50 passes) and Deep Ensembles (PICP=79.7%, 5 models) at $5\times$ lower inference cost. Uncertainty decomposition shows aleatoric uncertainty is a strong predictor of dating error (Spearman $ρ=0.729$), and a selective prediction about the most certain 20% of patches can provide \textbf{0.5 years MAE}. We show that predicted uncertainty increases as image degradation worsens, spatial decomposition maps explain which script regions cause aleatoric uncertainty, and page-level aggregation reduces MAE to 4.5 years with $ρ=0.905$ between uncertainty and page-level error.

Global navigation satellite system (GNSS) positioning is widely used for urban navigation, but the covariance reported by the GNSS solver is often unreliable in urban canyons. Existing differentiable factor graph optimization (DFGO) methods learn measurement weighting through the solver, but they still use position-only objectives. As a result, the position estimate may improve while the reported covariance remains too small, too large, or incorrectly oriented. We propose CredibleDFGO (CDFGO), a differentiable GNSS factor graph framework that makes covariance credibility an explicit training target. A Weighting Generation Network (WGN) predicts per-satellite reliability weights, and a differentiable Gauss-Newton solver maps these weights to a position estimate and a Hessian-derived posterior covariance. We use proper scoring rules to supervise the East-North predictive distribution end to end. We study negative log-likelihood (NLL), the energy score (ES), and their combination. Results on three UrbanNav test scenes show consistent gains in covariance credibility. Positioning accuracy also improves on the medium-urban and harsh-urban scenes; on the deep-urban scene, both the mean horizontal error and the 95th-percentile error improve. On the harsh-urban Mong Kok (MK) scene, CDFGO-Combined reduces the mean horizontal error from 13.77 m to 11.68 m, reduces NLL from 40.63 to 6.59, and reduces ES from 12.31 to 9.05 relative to DFGO (MAE). Case studies link the MK improvement to better axis-wise consistency, more credible local covariance ellipses, and satellite-level reweighting.

Quantifying uncertainty in neural network predictions is essential for high-stakes domains such as autonomous driving, healthcare, and manufacturing. While existing approaches often depend on costly sampling or restrictive distributional assumptions, we propose Hyperspherical Confidence Mapping (HCM), a simple yet principled framework for sampling-free and distribution-free uncertainty estimation. HCM decomposes outputs into a magnitude and a normalized direction vector constrained to lie on the unit hypersphere, enabling a novel interpretation of uncertainty as the degree of violation of this geometric constraint. This yields deterministic and interpretable estimates applicable to both regression and classification. Experiments across diverse benchmarks and real-world industrial tasks demonstrate that HCM matches or surpasses ensemble and evidential approaches, with far lower inference cost and stronger confidence-error alignment. Our results highlight the power of geometric structure in uncertainty estimation and position HCM as a versatile alternative to conventional techniques.