Uncertainty Calibration

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15 papers in the last four weeks, up 7% on the four weeks before. 0.1% of all new papers.

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Latest papers 179

Aug 10, 2026cs.CV

You Only Flow Once: Calibrated and Real-Time Radar Pose Estimation with Multi-Hypothesis Normalizing Flows

Sparse and noisy millimeter-wave radar point cloud observations often correspond to multiple plausible human poses, making deterministic pose estimation fundamentally ill-posed. Yet existing radar methods remain deterministic, collapsing this ambiguity into a single estimate. Diffusion-based alternatives can model multi-hypothesis distributions but require costly sequential denoising for each distribution sample and lack calibrated uncertainty. We propose Multi-Hypothesis Normalizing Flow Pose Generator (MH-NFPG), which models pose distributions from radar point clouds using a conditional normalizing flow. Specifically, we combine a spatiotemporal transformer backbone with a normalizing flow that transforms a Laplace base distribution into an expressive posterior, generated in parallel through a single forward pass. Leveraging this efficiency, we outperform diffusion-based alternatives in calibration across three radar benchmarks (MM-Fi, mmRadPose, mRI), improve pose accuracy on two, and match it on the third, while achieving over 20x faster inference for applications and reducing calibration error by up to 85%. We find that calibration degrades substantially for diffusion models, whereas our flow-based approach maintains reliable coverage, also in cross-environment settings. These results demonstrate normalizing flows as a practical alternative to diffusion models for real-time, uncertainty-aware radar pose estimation. Our code will be made publicly available.

Jonas Leo Mueller, Sebastian Hoefler, Dario Zanca +3

Department Artificial Intelligence in Biomedical Engineering (AIBE), Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany · Munich Center for Machine Learning (MCML), Munich, Germany · Chair of AI-supported Therapy Decisions, LMU München +1

Normalizing Flows Millimeter-Wave Radar mmWave Radar Sensing Human Pose Estimation Flow-Based Generative Modeling Uncertainty Calibration

Aug 10, 2026cs.RO

Particle-Based Conformal Prediction for Contact-Aware Uncertainty Calibration in Stratified Configuration Spaces

Reliable uncertainty representation is essential for deploying autonomous systems that interact with their environment, as robots must reason about how uncertainty arising from both stochasticity and model mismatch is impacted by contacts with obstacles (e.g., when navigating through a cluttered environment or inserting a part into an assembly). We propose Calibrated Particle-sets for Trans-dimensional Uncertainty Representation (CaPTURe), a geometry-aware, conformal prediction-based algorithm that generates probabilistically valid prediction regions of the unknown future system configuration using particle-based models of arbitrary fidelity. While calibrated uncertainty predictions are essential for safe and efficient planning, analytical or learned motion models are often inaccurate - due to limited data, simplifying assumptions, unmodeled effects, etc. - which can lead to unsafe executions or task failure. Additionally, when a robot contacts an obstacle, the distribution of its future configurations can become multimodal or disjoint, or lie along manifolds of lower intrinsic dimension than the space of possible robot configurations. Our method uses a calibration dataset of system transitions to locally calibrate motion uncertainty estimates, constructing regions guaranteed to contain the future robot configuration at a user-set probability. Our calibration procedure captures how motion uncertainty varies between contact-rich and contactless motions, leading to sufficient coverage in both cases. We evaluate our method on two simulated planning tasks: controlling a marble around a labyrinth and performing tight-tolerance peg-in-hole insertion with a manipulator. Compared to relevant baselines, CaPTURe achieves the user-specified coverage requirement both in and out of contact and achieves up to a 30% absolute improvement in task success rate over the best baseline.

Luís Marques, Kristian Popov, Dmitry Berenson

Department of Robotics, University of Michigan, Ann Arbor, MI, 48109, USA · Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, 48109, USA

Contact-Rich Robotic Manipulation Conformal Prediction Safe Motion Planning Robot Motion Planning Uncertainty Calibration

Aug 10, 2026cs.CL

When Confidence Fails: Overconfidence in LLMs under Uncertainty and Missing Clinical Information

Large Language Models (LLMs) have achieved strong performance in medical question answering and clinical reasoning tasks. However, their reliability under uncertainty remains poorly understood which raises critical concerns for deployment in high-stakes clinical settings. In such environments, incorrect predictions are inherently risky, but confident incorrect predictions can be particularly harmful as they may mislead clinical decision-making. In this paper, we conduct a systematic behavioral analysis of LLMs under clinical information uncertainty. We propose an evaluation framework based on the MedMCQA dataset consisting of two complementary uncertainty settings. First, we introduce linguistic uncertainty cues through prompt modifications to simulate ambiguous clinical contexts. Second, we construct an answer removal setting, wherein the correct option is deliberately excluded mandating the model to recognize insufficient information and abstain. We analyze both model accuracy and confidence behavior using multiple calibration metrics including calibration gap, Expected Calibration Error (ECE), and Unsafe Confident Error Rate (UCER) across 500 medical questions. Our results reveal a consistent failure mode, i.e., although accuracy degrades under increasing uncertainty, model confidence remains misaligned with accuracy. This leads to a substantial increase in unsafe confident errors, indicating that model confidence remains largely insensitive to clinically meaningful information loss. Furthermore, we observe significant variation across models in their ability to abstain when the correct answer is unavailable, with some models persistently producing high confidence hallucinated answers. These findings expose critical limitations in the epistemic reliability of current LLMs and highlight the need for uncertainty aware evaluation methods prior to their deployment in clinical workflows.

Maryam Tahermazandarani, Adnan Mahmood, Fahmida Islam +1

School of Computing, Macquarie University, Sydney, NSW 2109, Australia

Hallucination in Language Models LLM Uncertainty Estimation Clinical Language Model Evaluation Uncertainty Calibration LLM Abstention

Aug 9, 2026stat.ML

Physics-Informed Learning for Robust Acoustic Localization with Calibrated Uncertainty

Recent advances in Passive Acoustic Monitoring (PAM) offer an opportunity to obtain ecological spatial point-process data at unprecedented scale. However, realizing this opportunity necessitates the development of accurate and scalable localization methods. In real-world outdoor soundscapes, however, the assumptions underlying classical localization methods such as hyperbolic and score-based localization are routinely violated by multipath dominance, near-field effects, and complex propagation. Under these conditions, classical localization methods become brittle, with extreme errors possible even in small detection arrays. Rather than statistically replacing the underlying physics, we propose a method to refine it and increase robustness outside of ideal operating conditions: a learned model operating on physics-informed acoustic features corrects a fast hyperbolic solver where it produces implausible solutions, substantially reducing catastrophic worst-case errors while matching its median accuracy on field data. We further provide calibrated, geometry-aware uncertainty estimates suitable for propagation into downstream spatial models. Evaluating on distributed microphone arrays in real and simulated outdoor environments, we demonstrate that the proposed method yields robust, uncertainty-aware localization, providing a step toward scalable automated wildlife monitoring in complex acoustic environments.

Jennifer N. Kampe, Changwoo J. Lee, Xin Shen +5

Department of Biological and Environmental Science, University of Jyväskylä, Jyväskylä, Finland · Department of Statistical Science, Duke University, Durham, NC, USA · 3Digital Services, University of Jyväskylä, Jyväskylä, Finland +1

Physics-Informed ML Bioacoustics Passive Acoustic Monitoring Uncertainty Calibration

Aug 9, 2026cs.CV

CUPA-T2: Covariance-Aware Uncertainty Propagation and Alignment for T2 Mapping in Accelerated MRI

Quantitative T2* maps have strong potential for biomarker discovery but are limited by long scan times, rendering them impractical in clinical settings. Significant acceleration can be achieved through undersampling in k-space combined with learning-based reconstruction. However, reconstruction artifacts and noise can propagate into downstream T2* fitting, degrading its accuracy. We introduce CUPA-T2*, a framework that explicitly propagates voxel-wise inter-echo uncertainty from stochastic Monte Carlo dropout reconstructions to downstream T2* fitting via covariance-aware sampling. T2* fitting is performed with a heteroscedastic MLP and a correlation-based regularizer that encourages alignment between predicted variance and reconstruction uncertainty. Experiments on accelerated brain MRI data show tissue-dependent behavior: CUPA-T2* achieves competitive overall T2* fitting performance and improves white-matter performance at higher accelerations. Compared with a heteroscedastic baseline, the proposed framework substantially increases alignment between reconstruction uncertainty and predicted T2* variance, while also revealing a trade-off with calibration (ECE) and selective prediction performance (AURC). CUPA-T2* enables reconstruction uncertainty-aware T2* fitting and delivers voxel-wise uncertainty maps to support the interpretation of quantitative T2* estimates.

Gideon N. L. Rouwendaal, Natascha Niessen, Hannah Eichhorn +3

Institute of Machine Learning in Biomedical Imaging, Helmholtz Munich, Germany · School of Computation, Information & Technology, Technical University of Munich, Germany · GE HealthCare, Munich, Germany +6

Uncertainty Quantification Heteroscedastic Regression Uncertainty Propagation Uncertainty Calibration

Aug 6, 2026cs.CV

UQ-Loc: Uncertainty-Aware LiDAR Scene Coordinate Regression

LiDAR-based Scene Coordinate Regression (SCR) maps point clouds directly to 3D scene coordinates, enabling precise 6-DoF localisation without explicit map retrieval. However, existing methods produce deterministic predictions, discarding aleatoric uncertainty that could improve robustness and downstream decision-making. We present UQ-Loc, which extends the LightLoc architecture with an anisotropic Gaussian covariance head that predicts a full 3x3 positive-definite covariance matrix per voxel. Training uses a Negative Log-Likelihood (NLL) loss augmented with a kNN-based spatial smoothness regulariser, while inference employs a modified SC2-PCR solver with uncertainty-weighted seed scoring and a Mahalanobis-distance inlier test. We adopt Expected Calibration Error (ECE) as a principled metric for evaluating the quality of the predicted uncertainty. Experiments demonstrate that UQ-Loc achieves consistent improvement in 6-DoF localization accuracy while producing well-calibrated covariances.

Jacek Komorowski

Warsaw University of Technology, Warsaw, Poland

Uncertainty Quantification LiDAR Localization Uncertainty Calibration 3D Scene Understanding

Aug 4, 2026cs.LG

Unscented KalmanNet: Structure-Preserving Deep Learning with Calibrated Posterior Uncertainty under Incomplete Physics and Unknown Noise

Nonlinear state estimation requires sequentially fusing model-based predictions with noisy measurements. Under imperfect dynamics and unknown, time-varying noise statistics, this fusion can degrade in both accuracy and statistical consistency. Existing learning-aided filters largely treat accuracy and uncertainty estimation separately, limiting their ability to correct model-mismatch-induced bias while retaining an explicit, calibrated posterior covariance. This paper introduces Unscented KalmanNet (UKN), a model-based deep learning architecture that extends the Unscented Kalman Filter (UKF) with learned mechanisms for these two sources of filtering error while preserving explicit posterior covariance propagation. NoiseNet learns time-varying process and measurement covariances as bounded multiplicative corrections to baseline covariances, guaranteeing positive definiteness, while GainNet learns a bounded residual correction to the analytical UKF gain to compensate for model-mismatch-induced bias. A calibration-aware training objective couples state error with posterior covariance and innovation consistency terms through adaptive weighting, jointly optimizing accuracy and calibration. UKN is benchmarked against UKF, KalmanNet, and Bayesian KalmanNet on three synthetic systems and real-flight UZH-FPV data. It achieves the lowest state-estimation error in all four examples and reduces RMSE by 26.4-49.7% compared with UKF in the synthetic cases. Leave-one-sequence-out cross-validation over 11 flights shows 22.4% and 34.3% reductions in mean position and velocity RMSE, respectively. UKN also yields the lowest fold-to-fold variability, with dimension-normalized NEES and empirical coverage closest to nominal values among covariance-reporting filters. These results show that structured learned adaptation improves estimation accuracy while retaining calibrated uncertainty.

Minhyeok Ko, Abdollah Shafieezadeh

Department of Civil and Construction Engineering and Management, The University of Texas at Tyler, Tyler, TX, 75799, USA · Department of Civil, Environmental, and Geodetic Engineering, The Ohio State University, Columbus, OH, 43210, USA

Kalman Filtering Unscented Kalman Filtering Covariance Estimation Robot State Estimation Uncertainty Calibration

Aug 4, 2026cs.CL

DUD: Decoupled Update Dynamics for Reliable Uncertainty Quantification in Large Language Models

Accurate Uncertainty Quantification (UQ) is critical for reliable deployment of Large Language Models (LLMs), yet traditional probability-based metrics often fail to capture the model's true epistemic state. While recent mechanistic approaches leverage hidden state dynamics, they typically aggregate residual stream updates, conflating the distinct roles of parametric memory (Feed-Forward Networks) and contextual processing (Attention). We argue that this aggregation obscures fine-grained mechanistic conflicts, such as memory-context misalignment, that are fundamental indicators of uncertainty. To address this, we introduce \textbf{D}ecoupled \textbf{U}pdate \textbf{D}ynamics \textbf{(DUD)}, a framework that explicitly decouples FFN and Attention contributions via noise-induced causal interventions. By quantifying the independent restoration capabilities of each module, we construct a dual-stream dynamic profile that captures the model's internal fragility. Extensive experiments demonstrate that DUD significantly outperforms state-of-the-art baselines in both uncertainty estimation and calibration, while exhibiting superior cross-dataset generalization, validating decoupled dynamics as a robust proxy for model faithfulness.

Yixin Bu, Runze Xia, Guanyun Zou +3

College of Artificial Intelligence, Nanjing University of Aeronautics and Astronautics, China · 2The Key Laboratory of Brain-Machine Intelligence Technology, Ministry of Education, Nanjing, China · 3Researcher

LLM Uncertainty Estimation Uncertainty Calibration

Aug 3, 2026cs.CV

Deep Evidential Regression for Sparse Forest Height Estimation from Multimodal Satellite Imagery

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.

Laura Bader, Muhammad Ammar Ahmed, Xiao Xiang Zhu +1

LMU Munich, 80539 Munich, Germany · Munich Center for Machine Learning (MCML) · TU Munich, 80333 Munich

Remote Sensing Image Understanding Evidential Deep Learning Uncertainty Quantification Remote Sensing Uncertainty Calibration

Aug 3, 2026cs.CV

PNEC-Mamba: Prototype-Guided Positive-Negative Evidence Calibration for Hyperspectral Image Classification

In real-world hyperspectral scenes, pixel representations are often ambiguous due to factors such as spectral similarity, mixed pixels, and local context interference, which may simultaneously encode discriminative evidence and interfering information. Existing methods mainly focus on learning more powerful representations or modeling broader contexts, but rarely investigate whether the learned representations provide reliable evidence or introduce interference into classification decisions. To address this issue, we view hyperspectral image classification from the perspective of pixel-level evidence reliability modeling and propose PNEC-Mamba, a prototype-guided positive-negative evidence calibration framework. The framework progressively establishes semantic references, separates class-related evidence from interference, estimates pixel-level reliability, and performs selective calibration. First, a full-image state-space encoder extracts pixel representations, while dynamic class prototypes provide semantic references that evolve jointly with the feature space. Subsequently, positive and negative evidence is derived from pixel-prototype competition, explicitly separating discriminative cues that support classification from confusing signals associated with competing classes. Based on these evidence relationships, a multi-source uncertainty estimation strategy is introduced to assess pixel-level reliability, enabling stronger evidence calibration for uncertain regions. Finally, a full-resolution consistency refinement step is applied to recover local spatial details and improve boundary coherence in the final predictions. Extensive experiments on three benchmark datasets demonstrate that PNEC-Mamba achieves superior classification performance compared with state-of-the-art methods.

Mingzhen Xu, Can Xu, Di Wang +2

Prototype-Based Classification Hyperspectral Image Classification Uncertainty Calibration

Jul 30, 2026cs.CV

Uncertainty-Aware Deepfake Detection via Multi-View Structural Learning

Security-critical biometric and forensic applications require accurate predictions and reliable confidence estimates, particularly under distribution shift. This challenge is especially acute for deepfake detection, where foundation-model-based detectors often exhibit overconfident predictions on out-of-distribution manipulations, which limits their suitability for operational deployment. We propose an uncertainty-aware deepfake detection framework that identifies manipulations through inconsistencies across complementary evidence sources. The framework integrates three streams: a visual stream based on an adapted CLIP encoder, a semantic stream that models consistency among facial attributes through differentiable constraints, and a structural stream that captures class-dependent dependency patterns between semantic and forensic features. To effectively combine these signals, we introduce Inter-Branch Disagreement Calibration (IBDC), a disagreement-aware uncertainty modeling mechanism that links predictive uncertainty to conflicts among evidence streams. Extensive cross-dataset experiments using FaceForensics++ as the training source demonstrate that the proposed framework achieves state-of-the-art generalization across multiple out-of-distribution benchmarks while consistently improving calibration and selective prediction performance. These results show that combining complementary evidence with disagreement-aware uncertainty provides a robust foundation for trustworthy and well-calibrated deepfake detection under distribution shift.

Muhammad Umar Farooq, Kutub Uddin, Awais Khan +1

College of Innovation and Technology University of Michigan, Flint, MI 48502

Multi-View Learning OOD Generalization Deepfake Detection Uncertainty Calibration

Jul 29, 2026cs.CV

From Keypoints to Predictive Distributions: Post-Hoc Uncertainty for YOLO-Pose Models

YOLO-Pose models provide efficient keypoint localization, but do not quantify the associated spatial uncertainty. We introduce a lightweight post-hoc probabilistic extension that augments a trained YOLO-Pose model with calibrated bivariate predictive distributions over keypoint locations, centered at the model's original predictions. Concretely, we train additional probabilistic heads with an importance-weighted negative log-likelihood to predict an input-dependent

2\times2

dispersion matrix for each keypoint, followed by Gaussian calibration for broad downstream compatibility or Student-

t

calibration for distributional fidelity. Complementing this, we propose an evaluation protocol that combines a suite of distributional calibration diagnostics with average keypoint precision (AKP), a keypoint-level extension of the COCO AP protocol for assessing reliability rankings. Experiments on COCO show that the learned uncertainty estimates enable effective keypoint-level reliability ranking, Student-

t

calibration best captures the empirical residual distribution, and uncertainty-based pruning removes unreliable keypoints. A central application-level demonstration is vision-based aircraft landing, where calibrated covariances for runway keypoints support uncertainty-aware aircraft position estimation and downstream sensor fusion.

Alexej Klushyn, Juan Rivero Sesma, Florian Seligmann +3

Airbus Central Research & Technology · Airbus Commercial Aircraft · Karlsruhe Institute of Technology +2

Post-Hoc Calibration Uncertainty Quantification Uncertainty Calibration

Jul 27, 2026stat.ML

proxymate: Diagnosis and Adjustment of Proxy Estimates for Reliable Inference

Proxy outcomes (such as short-term behavioral signals, model predictions, or surrogate endpoints) are frequently used in place of primary outcomes that are too slow to mature, rare, or challenging to measure directly. But valid inference on a proxy does not guarantee valid inference on the primary estimate as proxy-based estimates can be systematically biased in ways that are difficult to predict, leading to improperly calibrated confidence intervals. We present proxymate, a framework and open-source Python package for proxy validation and adjustment. proxymate organizes into four levels: The Representativity Level (population validity), the Unit Level (measurement quality), the Estimate Level (decision validity), and the Domain Level (cross-domain transportability). Within each level, proxymate provides diagnostic checks, and targeted adjustment strategies that map specific failures to appropriate corrections. At Meta, proxymate has been adopted by many different use cases, spanning experimentation, prevalence estimation, and monitoring use cases, all facing different proxy challenges (limited human review time, long maturation window of outcomes, low detectability) and showcasing the modularity of the framework. Across all products, proxymate assessed and corrected millions of proxy, primary unit comparisons. It has facilitated launches across multiple work streams including enabling quick decision making on thousands of experiments.

Alexandra N. M. Darmon, Deeksha Sinha, Steve Wilkins-Reeves +1

Meta

Causal Inference Uncertainty Calibration

Jul 27, 2026cs.LG

FinAbstain: Uncertainty-Calibrated Multimodal RAG for Selective Financial Forecasting

Large language models (LLMs) can synthesize financial narratives but may express high confidence when evidence is sparse, stale, or contradictory. This failure is especially consequential in forecasting, where filings, news, prices, volume, and technical signals can disagree. We present FinAbstain, a research framework for uncertainty-calibrated multimodal retrieval-augmented generation (RAG) with selective prediction. A point-in-time retriever admits only information public at the forecast timestamp and supplies modality-specific evidence to fundamental, news, technical, risk, and verification agents. Their probabilistic assessments are aggregated with retrieval relevance, evidence contradiction, repeated-sample consistency, and historical calibration statistics. Temperature scaling, isotonic regression, conformal prediction, and a proposed hybrid uncertainty score are evaluated under a common chronological protocol. A controller predicts bullish, bearish, or neutral outcomes only when uncertainty is below a validated threshold; otherwise it abstains, requests evidence, reduces exposure, or routes the case to human review. The evaluation covers one- and five-day abnormal-return direction, twenty-day volatility intervals, and abstention decisions, using accuracy, calibration, risk--coverage, citation, trading, latency, and cost metrics. To make the design auditable before a full data collection is complete, we report explicitly labeled simulated results rather than empirical claims. These results illustrate the intended hypothesis: calibrated abstention may trade coverage for lower selective error and drawdown. The contribution is a time-safe architecture, a composite uncertainty formulation, and a reproducible evaluation blueprint for evidence-grounded selective financial forecasting.

Dorothy Torres, Wei Cheng, Henan Huang

School of Science, Technology, Engineering and Mathematics 263 Academy Street, Room S204 Jersey City, NJ 07306 · School of Electrical Engineering and Computer Science The Pennsylvania State University 207 Electrical Engineering West University Park, PA 16802, USA

Selective Prediction Financial Forecasting LLM-Based Time Series Forecasting Retrieval-Augmented Time Series Forecasting Uncertainty Calibration Finance

Jul 26, 2026cs.LG

Controllable Diversity in Normalization-Based Implicit Ensembles via Softmax-Temperature Modulation

Deep ensembles provide the most reliable uncertainty estimates in deep learning, but their cost grows linearly with the number of members. Implicit ensembles lower this cost by sharing a single backbone across members. Member diversity is a primary determinant of ensemble quality, yet no implicit ensemble can shape it during training; existing methods fix it at initialisation or build it into the architecture. We introduce

σ

N-Ens, a normalisation-based implicit ensemble that treats each member as a task in a multi-task architecture and modulates the shared backbone through sigmoid-bounded scalers. We also introduce a softmax-temperature regulariser, which shapes the equilibrium level of sharing between members and traces the accuracy-calibration frontier. Because only normalisation layers are replicated, the mechanism can wrap convolutional and transformer backbones alike, also allowing pretrained models to be adapted through a short fine-tune. We frame the epistemic uncertainty such an ensemble expresses as modulation uncertainty, and explain why its calibration holds under input corruption, and why its out-of-distribution detection is weaker. Our method is evaluated across ResNets and transformers on CIFAR-10/100, ImageNet and SST-2.

σ

N-Ens matches or outperforms deep ensembles at a fraction of their parameter cost, scales with ensemble size where partitioning methods collapse, and maintains calibration under distribution shift.

Mihai Suteu, Ovidiu Serban

Data Science Institute, Imperial College London

Ensemble Learning Ensemble Diversity Uncertainty Calibration Epistemic Uncertainty

Jul 23, 2026stat.ML

Priors learned from legacy reconstructions inherit undetectable overconfidence

Where truths are scarce (e.g., seismic and medical imaging), learned priors in ill-posed inverse problems are trained on archives of legacy reconstructions---i.e., an older method's outputs---and their reported uncertainty is taken as data-driven. We show that this prior is, in the population limit, exactly the regularizer that produced its archive of posterior samples, advanced one expectation--maximization step toward the truth. While the step improves the regularizer on the directions the measurements resolve, it leaves the regularizer's assumption on the operator's blind subspace unchanged. An archive of single-best reconstructions collapses the blind interval to zero width. Neither error is detectable in practice, as truths differing only on the blind subspace share the data law, and simulation-based calibration is neutral by construction. We identify from the operator alone which directions the measurements do not inform, and, given a handful of ground-truth models, build intervals there that contain the truth as often as they claim to. We validate these findings on a two-dimensional example with closed-form predictions and in controlled experiments on seismic-imaging and groundwater-flow operators, against priors trained on the truth.

Ali Siahkoohi, Sina Alemohammad

Department of Computer Science University of Central Florida Orlando, FL, USA · Department of Electrical and Computer Engineering The University of Texas at Austin Austin, TX, USA

Uncertainty Quantification Bayesian Inverse Problems Uncertainty Calibration

Jul 21, 2026cs.AI

Evaluating medical AI under missing information: same-provider judges and human raters change apparent safety

Readiness stress-testing of medical AI has focused on closed-ended and multimodal benchmarks. We extend it to open-ended clinical conversation under missing information, where safe behavior means recognizing absent information and qualifying, clarifying, or not over-committing - and where the evaluator becomes part of the measurement. We stress-test four models - three flagships (Claude Opus 4.8, GPT-5.5, Grok 4.3) and one mid-tier model (Gemini 3.5 Flash) - by deleting the latter half of the final user turn in HealthBench conversations, grading responses with a four-provider LLM-judge panel and a blinded clinician-anchored reference. Two evaluator-facing results are robust. First, judge choice materially changes apparent safety: inter-judge agreement is only moderate (Fleiss' kappa = 0.65), and after adjusting for each judge's general leniency (vote-level logistic regression), a positive same-provider association remains (exact permutation p = 0.04; GPT-5.5 ~ +0.10 on the probability scale) - large enough to change which model appears to over-commit least once its own-provider judge is excluded. Second, LLM judges are more permissive than clinicians on a blinded 50-item subsample: all four are significantly more lenient than the stricter independent clinician (crediting appropriate uncertainty on 66-84% of items vs 52%), and three of four than the author-influenced consensus (Grok directional only; judge-vs-consensus kappa = 0.20-0.43). On the author-audited clinical-underdetermined subset the permissiveness gap widened and the point-estimate model ordering held. A closed-ended MedQA anchor confirms accuracy is high and option-order effects are within a +/-5-point equivalence region for three of four models, so the safety gap is about calibration, not knowledge. We release the harness, prompts, per-item outputs, judge panel, perturbation audit, and human-annotation protocol.

Koyar Afrasyab

Kinvectum AB, Sweden

LLM-as-a-Judge Language Model Safety Evaluation Clinical Language Model Evaluation Uncertainty Calibration

Jul 20, 2026stat.ME

Calibrating Semantic Uncertainty from Observable Language-Model Probabilities

As generative artificial intelligence enters scientific and professional work, its uncertainty must be defined on the states that matter for inference and decision-making. Language models assign probabilities to words, whereas applications require uncertainty over meaningful states such as diagnoses, hypotheses or operational conditions. We introduce a \emph{semantic map}: a prespecified, testable bridge from probabilities over verbal responses to a posterior over declared finite states. The language distribution remains unrestricted; held-out calibration connects it to a reference posterior. We derive posterior-error bounds and conditions for existence, conditional uniqueness, presentation stability and stable inverse recovery. This distinction matters because language probabilities depend on prompt wording, while the target posterior should not change under information-equivalent rewording. Experiments use professional market text compiled from Federal Reserve economic and financial series, together with controlled simulations having exact posteriors. Across two fitted language models, language-derived probabilities outperform printed numerical confidence, recover held-out posteriors with valid uncertainty coverage, remain largely stable under paraphrase and respond appropriately to altered evidence. \textbf{Prompt engineering optimises a wording-dependent response; robust scientific use requires validated stability of application-relevant meaning.} The proposed map turns semantic uncertainty in generative systems into an identifiable and testable statistical measurement problem and, when its acceptance conditions hold, yields an auditable posterior estimate.

Matthew F. Dixon

Artificial Intelligence Finance Institute (AIFI), 69 Charlton St, New York, NY 10014, USA

Uncertainty Quantification Confidence Estimation in Language Models Language Model Calibration LLM Uncertainty Estimation Uncertainty Calibration

Jul 18, 2026stat.ML

Isotonic Conformal Prediction

A point prediction that is well calibrated on average can still be systematically biased conditional on its own value, undermining its use in downstream decision-making. We consider two objectives for reliable uncertainty quantification: self-calibration, requiring a point prediction to be unbiased conditional on its own value, and prediction-conditional validity, requiring a prediction interval to attain nominal coverage conditional on the prediction. Self-Calibrating Conformal Prediction (SC-CP) attains both objectives exactly in finite samples, but requires refitting its calibrator for every candidate outcome, which is computationally prohibitive for continuous outcomes. We propose Isotonic Conformal Prediction (ICP), a framework that decouples calibration from prediction-set construction by fitting a single isotonic recalibration map and constructing prediction intervals within strata of similar recalibrated predictions. Within this framework we develop two procedures. Split Isotonic Conformal Prediction (SICP) attains prediction-conditional validity in finite samples and self-calibration asymptotically, at the computational cost of split conformal prediction. Transductive Isotonic Conformal Prediction (TICP) attains both objectives exactly in finite samples through a per-test-point inner loop that avoids refitting the isotonic calibrator. On synthetic heteroscedastic regression problems and a real-world healthcare-utilization dataset, both procedures match the coverage of SC-CP at substantially lower computational cost.

Daniel Bensimon, Sean Xiang Yu, Eric D. Kolaczyk +1

Department of Mathematics and Statistics, McGill University; Mila · Eli Lilly and Company

Conformal Prediction Uncertainty Calibration Isotonic Regression Regression Prediction Intervals

Jul 15, 2026cs.CV

Posterior Variance Is a Constraint Map, Not an Error Map: Closed-Form Uncertainty for Radiative Gaussian Splatting in Sparse-View CT

Radiative Gaussian splatting reconstructs sparse-view CT fast and accurately, and recent work attaches per-Gaussian posteriors to yield per-voxel uncertainty maps. We ask what such a map actually measures: posterior variance is a data-constraint map, not an error map -- its alarms are trustworthy, its all-clears are not. Exploiting the strict linearity of X-ray rendering in the per-Gaussian densities, we derive a clamp-aware closed form that the unchanged rasterizer evaluates exactly in one forward pass, in volume and projection space: the infinite-sample limit of the sampling estimator of concurrent work, at ~8x lower cost. On the official 15-scene benchmark this uncertainty ranks true error on 14 of 15 scenes. Restricted to the object interior -- the tissue a clinician reads -- the ranking collapses (median Spearman 0.11, 0/15 pass), identically for a deep ensemble and for a strictly positive log-normal posterior: three constructions, two estimator families, no survivors. The mechanism is structural: about 90% of in-object error is bias that reproduces across retrainings, invisible to model disagreement; 73-81% of the full-volume correlation is carried by object/surround contrast; and an exactly solvable control puts the observed in-object ranking 4-5x below what a perfectly calibrated posterior with the same sigma-spread would score. The error scale, by contrast, is an engineering problem, and we solve it: reparameterizing the posterior contracts the cross-scene temperature spread from 19.3x to 2.6x, one scene-agnostic temperature transfers to unseen scenes (10/15 leave-one-scene-out), and the repaired scale tracks photon count at the Poisson-predicted -1/2 power. We distill evaluation practice that would have caught the illusion -- masked calibration, seed-wise bias decomposition, an exact-posterior reference -- and release all protocols, seeds and per-run evidence.

Chulin Zhao, Yiran Xu, Shu Liu

1Dundee International Institute, Central South University · 2Central South University

3D Gaussian Splatting Uncertainty Quantification Sparse-View CT Reconstruction Uncertainty Calibration

Jul 14, 2026eess.IV

Exact and Calibrated Diffusion Reconstruction for Digital Breast Tomosynthesis

Limited-angle digital breast tomosynthesis (DBT) reconstructs a volume from a few low-dose projections over a narrow arc. At a representative nine-view,

25^{\circ}

protocol more than 98% of image space is unmeasured, so a learned prior must supply structure in the missing wedge. Conditional diffusion priors achieve strong perceptual quality here but leave three clinical obstacles: inexact data consistency, unlocalized hallucination, and uncalibrated uncertainty. We enforce measurements exactly by replacing the per-step proximal update of a conditional diffusion sampler with exact Euclidean projection onto the data-consistent set, computed via an

m

-dimensional dual system with a one-time Gram matrix

AA^{\top}

factorization. This projection costs 4.5 ms per step (a

248\times

speedup) and drives the data residual to the double-precision floor (

2.4\times10^{-13}

). We prove it is the

ρ\to0

limit of the proximal step, provide a no-harm theorem, and show that exactly consistent sample ensembles have variance supported on null(

A

). Thus, the mean's entire error lies in the unmeasured subspace covered by the uncertainty map. On patient-derived breast phantoms, this improves fidelity at no depth-resolution cost. Conversely, a proximal step applied post-update degrades quality, isolating the consistency step's placement as decisive. Isotonic recalibration brings the ensemble spread to a calibrated error scale (expected calibration error

0.029\to0.008

; standardized error

4.7\to0.96

), ranking errors better than the pure prior. We also repair a 20.3% adjoint mismatch in a deployed projector via a materialized operator of record. This is the first data-consistent, uncertainty-calibrated learned reconstruction for limited-angle DBT. The solver naturally relaxes to discrepancy-ball and maximum-a-posteriori modes for noisy measurements.

Imade Bouftini

Image Reconstruction Sparse-View CT Reconstruction Computed Tomography Diffusion Posterior Sampling Uncertainty Calibration

Jul 14, 2026cs.LG

Learning-based Probabilistic Load Forecasting with Post-hoc and In-model Uncertainty

Smart-building load forecasters are often trained offline on dense, multivariate, high-frequency data, but deployment may provide only hourly, feature-limited inputs. Missing features must then be reconstructed, and their errors can propagate through the model. If this input uncertainty is not reflected, prediction intervals may become miscalibrated, affecting demand-response scheduling. Our work examines where uncertainty should be placed once inference inputs are reconstructed. We develop a unified one-day-ahead probabilistic forecasting framework that aligns temporal resolution, reconstructs the unavailable inputs, and derives causal features, and we compare a modular post-hoc residual-quantile scheme with an integrated in-model quantile-learning scheme. The comparison uses three mid-scale Deep Learning (DL) backbones: recurrent, hybrid recurrent, and attention-based Temporal Fusion Transformer (TFT) models, under identical inputs, forecasting horizon, preprocessing rules, and training budgets. Results show that uncertainty placement is backbone-dependent. Integrated quantile learning is most reliable with the TFT, yielding 2.2-3.6% MAPE and 28-83W RMSE on the labeled test window, while producing intervals about 5x narrower than the modular intervals at the closest-to-nominal coverage level. Diebold-Mariano tests support the TFT ranking and the mixed behavior of the recurrent backbones. A reconstruction-sensitivity test shows that reconstructed inputs increase the Quantile Score (QS) by 106% while interval width remains nearly unchanged, indicating that the model does not automatically absorb reconstruction-induced uncertainty. Robustness checks against non-DL baselines and seasonal hold-out weeks support this ranking. Our results expose the limits of post-hoc residual quantiles when inference depends on reconstructed inputs.

Sarah Al-Shareeda, Gulcihan Ozdemir, Heung Seok Jeon

aComputer System and Technology Department, Faculty of Computer Science and Information Technology, University of Malaya, Malaysia · bInformatics Institute, Istanbul Technical University, Turkey · cComputer Engineering Department Konkuk University South Korea

Quantile Regression Time Series Forecasting Probabilistic Forecasting Electricity Load Forecasting Uncertainty Calibration Probabilistic Time Series Forecasting

Jul 13, 2026q-bio.NC

Imputation-free transformer learning enables robust Alzheimer's disease prediction and calibrated uncertainty quantification across heterogeneous clinical cohorts

Accurate diagnostic classification and disease-severity prediction for Alzheimer's disease are hampered by the incompleteness and heterogeneity of real-world clinical data. Left unaddressed, these barriers prevent reliable disease modelling and hinder effective clinical evaluation. Conventional imputation strategies introduce systematic bias, distort inter-feature relationships, and yield overconfident predictions, limitations especially consequential in diagnostic settings. Here, we propose NITROGEN, an imputation-free transformer that jointly models within-patient feature dependencies and between-patient relational structure through masked and intersample attention, enabling robust multimodal learning directly from partially observed records. We trained NITROGEN on ADNI (N=7858 scans), and evaluated it on two independent cohorts: OASIS-3 (N=2675 scans) and AIBL (N=1286 scans). Across cohorts and diagnostic and cognitive score prediction tasks, NITROGEN showed robust calibration and uncertainty quantification advantages over tree-based ensemble methods, while maintaining competitive discriminative performance. Cross-cohort and cross-method analyses identified cortical thickness in the temporal pole, age, and APOE genotype as important, though not individually sufficient, features for AD classification. We further introduced a modality-aware uncertainty adjustment that augments predictive uncertainty proportionally to the importance of absent modalities, enabling calibrated confidence when diagnostic information is unavailable. Together, our results show that imputation-free attention learning preserved meaningful discrimination under cohort shift, revealing expected degradation on more distributionally different cohorts, and demonstrate that evaluating models along calibration, interpretability, and cross-cohort reliability, not accuracy alone, is essential for clinical deployment.

Christelle Schneuwly Diaz, Narmina Baghirova, Duy-Thanh Vu +4

Platform of Bioinformatics, Lausanne University Hospital. · Faculty of Biology and Medicine, University of Lausanne. · 3Leenaards Memory Centre, Lausanne University Hospital. +1

Missing-Modality Learning Learning with Missing Data Clinical Outcome Prediction Alzheimer's Disease Classification Neuroimaging Alzheimer's Disease Progression Prediction+1

Jul 13, 2026cs.LG

Bet on Features: Anytime-Valid and Feature-Aware Auditing of Conditional Quantile Forecasters

Black-box conditional quantile forecasts are widely used for sequential decisions under asymmetric costs, such as inventory planning in supply chain management. Once deployed, such forecasters must be monitored continuously as data streams drift and regimes change; this invalidates standard, fixed-horizon backtests for calibration. Further, existing backtests do not take into account that the notion of calibration is, in fact, information-dependent: forecasts can look calibrated to an auditor with coarse information while being miscalibrated to an auditor with richer information. We develop a distribution-free and game-theoretic testing framework for continuously auditing black-box conditional quantile forecasters with non-i.i.d. losses, such that the resulting evidence process is powerful against predictably chosen alternatives specified by the features available to the auditor. We first formalize notions of conditional quantile calibration when different sets of features are available to the auditor, establishing that the coarseness of the auditor's information set determines the hardness of the testing problem. We then identify the sets of alternatives for which the auditor can achieve power, and focusing on contextual bets linear in the features, we derive finite-time detection guarantees for such alternatives, all without an i.i.d. assumption. The resulting evidence processes are interpretable at the feature level, as they quantify fine-grained, "feature-aware" evidence for miscalibration. We empirically validate these methods on simulated and real data, finding that a popular time series forecaster (Chronos-2) is highly miscalibrated w.r.t. multiple relevant features.

Ivane Antonov, Sohom Mukherjee, Richard Pibernik +1

Julius-Maximilians-Universität Würzburg · INSEAD

Quantile Regression Sequential Hypothesis Testing Uncertainty Calibration Black-Box Model Auditing

Jul 12, 2026cs.LG

When does distribution shift break graph neural networks calibration?

Graph neural networks (GNNs) are increasingly deployed in real-world applications where distribution shift is un-avoidable. However, how such shifts affect model calibration, defined as the agreement between predictive confidence and actual accuracy, remains poorly understood, and existing graph calibration methods typically rely on labeled validation data from the deployment distribution. In this work, I present the first closed-form theoretical characterization of GNN calibration under distribution shift. I show that calibration is governed by a single scalar quantity that explicitly depends on structural changes between the source and target graphs, as well as feature quality. This characterization precisely identifies when a model becomes over-confident, under-confident, or remains calibrated, and directly yields the optimal temperature scaling strategy. I further extend the analysis to graph convolutional networks with symmetric normalization, multi-class classification, and covariate shift, and derive a theoretical upper bound on the expected calibration error. My analysis also reveals that, under homogeneous distribution shift, a single global temperature is theoretically optimal, providing a principled explanation for why more complex node-wise recalibration methods offer no additional benefit. Building on these theoretical insights, I propose STAC, a source-free, label-free calibration method. Experiments on synthetic benchmarks demonstrate substantial calibration improvements, while evaluations on five real-world graph datasets show that reliable calibration without target labels remains challenging despite the strong predictive power of the theory.

Abderaouf Bahi

Computer Science and Applied Mathematics Laboratory (LIMA), Faculty of Science and Technology, Chadli Bendjedid University, El Tarf 36000, Algeria

Graph Neural Networks Model Calibration Uncertainty Calibration

Jul 11, 2026stat.ML

TSCoNet: A Two-Stage Copula CNN-LSTM for Uncertainty-Aware Spatio-Temporal Forecasting

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.

Jongwook Kim, Jong-Min Kim

Statistics Discipline, Division of Science and Mathematics, University of Minnesota-Morris

Multivariate Time Series Forecasting Copula Models Spatiotemporal Forecasting Uncertainty Calibration Probabilistic Time Series Forecasting Regression Prediction Intervals

Jul 9, 2026eess.IV

Calibrated Hybrid CNN-Transformer for Retinal OCT Classification

Deep models for retinal optical coherence tomography (OCT) classification report high accuracy but rarely report whether their confidence can be trusted -- a gap that matters when a wrong-but-confident reading delays sight-saving treatment. We pair a hybrid convolutional-Transformer encoder with a gradient-boosting (XGBoost) classification head and a three-part clinical safety layer: confidence calibration, out-of-distribution (OOD) rejection, and per-prediction uncertainty flagging. On four-class OCT (84,495 scans) the model reaches 95.4% accuracy while cutting calibration error twelve-fold (expected calibration error, ECE = 0.0024), so the confidence it reports tracks its true accuracy. To our knowledge this is the first OCT classifier to validate all three safety mechanisms jointly, with public weights and reproducible multi-seed evaluation.

Animesh Kumar

School of Computing, Newcastle University, UK

Uncertainty Quantification Optical Coherence Tomography Uncertainty Calibration

Jul 7, 2026cs.LG

Heckman-Corrected Epistemic Uncertainty: Selection on Unobservables Defeats Importance Weighting

Training data for machine learning is routinely collected by a selection process the model never sees: loans are observed only when granted, outcomes only when a test was ordered. The standard fixes -- importance weighting, covariate-shift correction, MAR imputation -- assume selection is ignorable given observables. Econometrics solved the harder case in 1979: Heckman's two-equation model jointly fits a probit selection equation and an outcome equation linked through correlated errors, and the inverse-Mills-ratio term corrects for selection on unobservables, where importance weighting is structurally helpless. We instantiate this for deep epistemic uncertainty: a deep outcome network, a linear selection head, and a joint bivariate-normal likelihood over all units, ensembled for predictive variance. In a controlled generator where sampling probability depends on an unobservable correlated (rho up to 0.9) with the outcome noise, deep ensembles, MC dropout, and GP baselines are overconfident exactly where data was avoided: coverage of nominal-90% intervals falls to 64.4% at rho=0.9, and importance weighting with oracle propensities does not fix it (43.1%) -- reweighting corrects the covariate distribution, not the conditional bias E[y|x,selected] != E[y|x]. The Heckman correction restores coverage (88.9%) when the selection equation has an instrument -- a variable affecting selection but not the outcome -- and degrades measurably without one (40.3%); we chart this honesty curve rather than hide it. On real tabular data with induced MNAR selection, the corrected intervals are the best-calibrated (lowest region-ECE) non-oracle method in selected-against regions; baselines matching its raw coverage do so only by over-widening everywhere. Our estimators reproduce classic Stata output to seven digits. We state which identification regime a practitioner is in, and release the code.

Gunner Levi Howe

Independent Researcher

Selection Bias Uncertainty Calibration Regression Prediction Intervals Epistemic Uncertainty

Jul 6, 2026astro-ph.IM

Interpretable Human-Label-Free Deep Learning for Real-Bogus Classification with Uncertainty Quantification

Time-domain surveys generate many transient candidates, making Real-Bogus classification a critical step in automated discovery pipelines. Reliable labels are costly, while community labels can be noisy and survey-dependent. We aim to develop a Real-Bogus classification framework that can be trained without human-labeled data using injected transients and bogus-dominated survey data, remains robust under strong class contamination, and provides calibrated uncertainty quantification. We combine simulated transient injections with a contaminated survey class and train a dual-network model using asymmetric co-teaching for classes with different label-noise levels. We evaluate performance on a benchmark subset and analyze the learned representation with latent-space visualization tools. For uncertainty quantification (UQ), we compare MC dropout and deep ensembles and propose a low-cost hybrid strategy that exploits the dual-network setting to improve calibration. We extend the evaluation to the light-curve domain to assess recovery of light-curve classes. The method achieves strong Real-Bogus performance on the labeled subset and remains stable under severe class contamination. It recovers transient light-curve classes with high fidelity, while single-source identification is limited by ambiguity in light-curve-derived labels. Our hybrid UQ approach achieves competitive calibration relative to more expensive ensemble baselines. Latent-space analyses indicate that uncertainty aligns with the decision boundary and reveal subclasses within the bogus population. Our results show that injection-driven, weakly supervised training can enable scalable and consistent Real-Bogus classification without human-labeled training data while providing calibrated uncertainties. The method is suited for transfer to forthcoming surveys by re-running the injection-based training pipeline.

Raphaël Bonnet-Guerrini, Bruno Sanchez, Dominique Fouchez +5

Università degli Studi di Milano, Department of Computer Science, Milan 20133, Italy · Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France · Université Côte d’Azur, INRIA, CNRS, Laboratoire J.A.Dieudonné, Maasai team, Nice 06000, France +2

Weakly Supervised Learning Learning with Noisy Labels Uncertainty Calibration

Jul 6, 2026cs.SD

Towards Robust Uncertainty-Aware Speaker Modeling

Speaker embeddings aggregate frame-level acoustic features into compact representations for speaker recognition. Recent uncertainty-aware speaker modeling approaches further characterize the reliability of speaker embeddings by estimating their associated uncertainty. However, existing methods often suffer from inaccurate uncertainty estimation and uncertainty miscalibration under domain shifts. To address these challenges, we propose a robust uncertainty modeling framework from both estimation and adaptation perspectives. Specifically, we introduce an Inter- and Intra-Speaker-Aware Uncertainty Softmax that incorporates both inter-speaker separability and intra-speaker variability into uncertainty learning, enabling uncertainty estimates to better capture the reliability of speaker embeddings. Furthermore, we propose an Uncertainty-Calibrated Domain Adaptation (UCDA) framework to mitigate uncertainty miscalibration caused by domain mismatch. Extensive experiments on both in-domain and cross-domain benchmarks demonstrate that the proposed approach consistently improves uncertainty reliability and speaker recognition robustness.

Junjie Li, Yang Xiao, Kong Aik Lee

Department of Electrical and Electronic Engineering, The Hong Kong Polytechnic University, Hong Kong SAR · The University of Melbourne, Australia

Speaker Verification Uncertainty Quantification Domain Adaptation Uncertainty Calibration