Conformal Prediction

This paper studies multivariate conformal prediction (CP), a distribution-free uncertainty quantification framework with finite-sample coverage guarantees. The efficiency of multivariate prediction sets hinges critically on the residual geometry encoded by the nonconformity score, while existing minimum-volume methods rely on quantile thresholds that ignore tail residual severity and implicitly bind geometry learning to coverage level. We propose a tail-aware geometry learning framework for conformal ellipsoids that decouples tail sensitivity in geometry learning from the final coverage guarantee. Using a two-split design, we learn the metric matrix via volume minimization under a CVaR constraint on an estimation split, then apply standard conformal calibration on a held-out calibration split. The resulting problem is convex and admits a bounded-reweighting interpretation that prioritizes high-residual samples. Moreover, we theoretically characterize the trade-off between ellipsoidal volume and tail severity. Experimental results demonstrate the effectiveness of the proposed method.

We introduce Rolling Conformal Prediction (rolling-CP), a distribution-free predictive inference method for the setting of sequential model training. Specifically, given a data stream $(X_1,Y_1),(X_2,Y_2),\dots$, at each time $n$ the trained model may depend on the observed history $\{(X_i,Y_i)\}_{i<n}$. This setting arises naturally in modern sequential training, including one-pass training over massive datasets and continual fine-tuning or test-time adaptation of language models during deployment. Rolling-CP first calibrates each incoming observation against the current predictor and then rolls it into future training. In this way, we avoid the need for data splitting. Remarkably, although the models at times $n=1,2,\dots$ may have entirely different properties and accuracy levels, for exchangeable data it is nonetheless possible to establish a guarantee of marginal coverage, with a familiar universal factor-two guarantee (a worst case guarantee of $1-2α$ coverage, as compared to the target level $1-α$), without any assumptions of stability or any restrictions on the model training process. For i.i.d. data streams, we further prove high-probability training-conditional validity uniformly over time; under stability conditions, coverage guarantees sharpen towards $1-α$. Numerical experiments on sequential regression, multiclass SGD, and one-pass neural-network training further demonstrate the practical effectiveness of rolling-CP.

A classical question in statistics is which observable quantities to condition on when drawing inferences about unobservable targets. For conformal prediction in nonparametric uncertainty quantification, standard marginal validity offers limited resolution at the prediction values on which decisions are based, and fully conditional guarantees with respect to the covariates are provably unattainable. We address this gap by introducing a prediction-based conditioning framework that we refer to as Prediction-Interval-Conditional Prediction Intervals (PICPIs). Formally, a PICPI is an interval $I$ satisfying a self-consistency condition: $\mathbb{E} [Y \mid p(X) \in I] \in I,$ for predictive model $p$, contextual covariate $X$, and outcome $Y$. Thus, an interval simultaneously defines a stratum of prediction values and certifies that the mean outcome in that stratum lies in the same interval. This self-consistency condition yields data-adaptive strata without altering the original prediction. Such intervals can be constructed using practical algorithms. Under regularity of the prediction distribution, the constructed intervals cover all but an arbitrarily small fraction of prediction values and have widths that decrease at rate $n^{-1/3}$, up to logarithmic factors and the prediction error. Moreover, identifying these locally calibrated intervals can, in turn, inform downstream decision-making. We derive inference procedures for PICPIs in probabilistic prediction and multi-class classification, accompanied by theoretical guarantees. Empirical results are provided that compare PICPIs with existing interval-based baselines.

Uncertainty estimation for Vision-Language-Navigation (VLN) models is a critical task since it can help identify ambiguous and unreliable predictions, enabling agents to make safer navigation decisions. As one of the most advanced uncertainty estimation frameworks, conformal prediction (CP) offers a promising approach for uncertainty estimation in VLN. However, given that VLN agent requires a sequence of steps, standard calibration in conformal prediction fails to provide coverage guarantee it promises over a dependent, variable-length VLN episode. To this end, we propose Episode-Normalized Conformal Prediction (ENCP), which rescales a nonconformity score by the policy's residual confidence and calibrates one maximum score per episode. Under exchangeable calibration and test episodes, this construction covers the ground truth at every step with probability at least $1 - α$, while allowing dependence among steps within an episode. Across four VLN policies and three nonconformity scores on R2R and REVERIE dataset, ENCP meets all reported empirical step-coverage targets on the seen-to-unseen evaluation. These results demonstrate that ENCP can provide model-agnostic uncertainty estimates, which might be useful for determining when a VLN agent should defer to a more capable predictor, including human assistance.

Conformal counterfactual prediction constructs prediction sets with finite-sample coverage guarantees for counterfactual outcomes and individual treatment effects under the no-interference assumption. In this work, we relax this assumption by allowing each unit's potential outcomes to depend on other units' treatments and covariates. In this setting, propensity-score reweighting does not restore weighted exchangeability, and existing methods may fail to achieve valid coverage. To address this issue, we develop interference-adjusted weighted conformal prediction that accounts for interference by constructing an observable upper bound on the ideal and unobserved conformal $p$-value under the target intervention. The resulting prediction sets provide finite-sample marginal coverage guarantees for counterfactual outcomes and individual treatment effects in both transductive and inductive settings. We also derive a sharper construction when intervention-induced changes in nonconformity scores are bounded. Numerical experiments show that our methods preserve nominal coverage, whereas existing methods may not.

Conformal prediction provides distribution-free uncertainty quantification under exchangeability. However, this assumption is violated by label shift, where the marginal distribution of labels changes while the conditional distribution of inputs given labels remains stable. Under such shifts, standard conformal procedures no longer maintain their intended coverage behavior. Existing approaches address this via importance weighting. They pair the reweighting with residual-based nonconformity scores that ignore predictive uncertainty. The resulting intervals have uniform width. Bayesian conformal methods produce adaptive intervals by leveraging predictive distributions. They evaluate conformity under the source predictive, which is misaligned with the target domain under label shift. We propose the \emph{Label-Shift-Adjusted Bayesian Score} (LSA score), a nonconformity score derived from a posterior predictive tilting identity. This identity shows that the target predictive is an importance-weighted transformation of the source predictive. We use it to derive a direct correction to the Bayesian score. We evaluate the method on molecular property prediction under controlled label shift. The LSA score consistently yields shorter intervals than residual-based and source-based Bayesian scores. Coverage in the target domain remains comparable. Under stronger shift, all methods incur some coverage loss due to pseudo-label-based density-ratio estimation. The LSA score is defined for any source predictive with a tractable log-density. We instantiate it with Bayesian Ridge Regression, where the correction admits a closed form.

Neural operators such as the Fourier Neural Operator (FNO) achieve remarkable accuracy in approximating solutions to partial differential equations (PDEs). However, providing rigorous uncertainty estimates remains an open challenge. We propose Physics-Informed Conformal Prediction (PI-CP), a framework that embeds PDE residuals into the nonconformity score of split conformal prediction, producing prediction intervals that are (i) distribution-free with provable coverage guarantees, and (ii) spatially adaptive when the PDE residual correlates with prediction error -- tighter where physics is well-satisfied, wider where it is violated. Additionally, we prove that FNO's translation equivariance creates a fundamental approximation barrier for PDEs with Dirichlet boundary conditions, and show that coordinate channels resolve this with up to 63x error reduction. We validate PI-CP across six physics scenarios -- heat conduction (2D/3D), structural mechanics (2D/3D), Darcy flow, and Navier-Stokes -- demonstrating consistent 89-91% coverage for all four Conformal methods, while MC Dropout and Deep Ensembles are unstable (82-100%). FNO outperforms CNN and DeepONet by 10-12x.

Conformal prediction converts point predictions into set-valued predictions with coverage guarantees under exchangeability between calibration and deployment data. We study conformal calibration transfer, where this requirement fails because labeled calibration is available only in a source space, while prediction sets are needed in a target space linked to the source through unlabeled paired observations (e.g., paired modalities or sensor changes). We propose Transported Conformal Calibration (TCC): we transport labeled source calibration into the target space using the paired data, and then correct residual post-transport mismatch using only unlabeled target inputs. We instantiate this correction with two complementary methods: TCC-KS, which uses a label-free uncertainty surrogate to detect mismatch and adjust calibration conservatively, and weighted-TCC, which reweights transported calibration toward the target domain for improved efficiency when weights are stable. We provide finite-sample target-domain coverage guarantees that adapt to an observable measure of mismatch. Across CIFAR-100-C, Tiny-ImageNet-C, and SEN12MS, we show reliable target-domain coverage transfer without labeled target calibration data, with label-free diagnostics that predict when correction is needed.

Cardiovascular screening models trained on national health surveys routinely report areas under the receiver operating characteristic curve (AUROC) near 0.89. We asked whether that accuracy reflects learning or target leakage, whether tabular foundation models change the answer, and whether the properties deployment requires survive joint examination. We benchmarked ten classifiers spanning linear, tree-ensemble, neural, glass-box, and tabular foundation classes for prevalent myocardial infarction in 442,067 respondents of the 2022 Behavioral Risk Factor Surveillance System across five feature tiers of decreasing leakage risk. Each was audited for discrimination, calibration, fairness at an explicit screening threshold, conformal coverage, explanation faithfulness, and inference cost, then applied -- models and thresholds frozen -- to 430,755 respondents of 2023. Removing two post-diagnostic features cost every model 0.049-0.051 AUROC, collapsing the field into a 0.0045-wide band. The glass-box explainable boosting machine was non-inferior to every alternative within a pre-specified 0.005 margin while scoring the cohort roughly 104 times faster than the strongest foundation model. One threshold detected 75.4% of women's infarctions against 89.0% of men's; editing the model's shape functions reduced the gap to 0.010. Marginal conformal prediction gave 0.86 coverage to men and 0.82 to adults over 60; Mondrian calibration repaired every stratum. Frozen models transported within 0.002 AUROC. Reported headroom in this literature is a property of the feature set, not the learner. Transparency cost nothing measurable and made fairness repair and uncertainty conditioning directly auditable. Evaluation practice, not model capacity, is the binding constraint.

Many applications in engineering, including wireless broadcasting, require designs that provide performance certificates at different target outage levels. This paper studies the problem of maximizing the weighted average of such certificates in the presence of uncertainty about the true system state. The problem is shown to be equivalent to an optimization over nested prediction sets, connecting to the literature on conformal prediction and extending prior art on single-level risk-averse decision making. Furthermore, we derive a dual formulation that decouples optimization across input values. Numerical experiments on a diversity-based wireless transmission system illustrate the cost of enforcing multi-level certificates with a single shared policy and trace the Pareto trade-off between multiple reliability levels.

In the last decade, kilometre-scale interferometric gravitational-wave detectors have observed hundreds of compact binary mergers, the majority of which are binary black holes. However, the data are noise-dominated, and multiple independent search algorithms (pipelines) are used to enhance sensitivity and improve robustness. Rather than the standard approach of selecting the most significant pipeline output, we combine the outputs from all pipelines using a conformal prediction-based framework to provide statistically rigorous confidence estimates for candidate events. While combining pipelines improves sensitivity and ranking robustness, it requires a principled statistical framework that remains valid as data properties evolve across observing runs. A key challenge is distribution shifts between simulated datasets used for training and calibration and the real, unlabelled, observations used for testing, which can invalidate coverage guarantees and bias confidence estimates. In this work, we address this challenge by incorporating likelihood-ratio reweighting into our conformal prediction framework to account for covariate shift. Using mock datasets containing simulated signals, we demonstrate that weighted conformal prediction restores well-calibrated coverage under covariate shift and increases the confidence of events near the detection threshold, recovering true signals that would otherwise be missed.

3D Gaussian Splatting (3DGS) renders novel views in real time, but an uncertainty heatmap does not certify that a rendered view meets a certain prediction coverage. We treat novel-view synthesis as structured regression and ask that, with probability at least $1-α$, RGB prediction boxes cover at least a $1-β$ fraction of pixels in a new view. We propose View-Structured Conformal Prediction (VSCP). It splits the pre-calibration scale into a spatial shape from the renderer and a transferable view-difficulty factor, which predicts the smallest view-wise multiplier that shape needs. A held-out quantile over views (View-CP) then gives finite-sample validity even when transferring to new scenes. The same factorization makes the analysis exact: a conformity score is the ratio of oracle to predicted view difficulty, and excess width separates into a test-side and a calibration-side term. Across 13 real scenes, pixel-pooled calibration reaches 89.9% marginal pixel coverage but only 61.4% view-event coverage at a 90% target, while View-CP reaches 91.7--92.0%. At matched coverage VSCP cuts width by 22.1% against a constant scale, and matches a ten-model ensemble's 21.0% reduction using only one model per scene and four rather than ten rasterization passes per query. VSCP also improves on the closest single-model baseline, the 3DGS-U field, by 4.7 points ($p=0.0225$). The view predictor transfers from bounded source families to all nine unbounded Mip-NeRF360 scenes. There the full scale beats the constant scale with 20.7% width saving on all nine scenes. It also keeps an 18.3% saving under a different densification backbone and runs at 216--280 FPS on an RTX4090.

Scenario optimization, conformal prediction, and related distribution-free certification methods use finite samples to construct decisions or prediction sets with violation-risk guarantees for fresh observations. In several classical settings, the conditional violation risk follows an exact beta law, whose tail has a beta-binomial representation and whose parameter is a support, calibration, or compression dimension. This paper identifies the deterministic boundary mechanism behind these formulas and derives the corresponding law when the observed boundary size is random. A decision rule is represented by an acceptance set for future observations, together with a boundary map selecting the sample points responsible for that set. The resulting pair is called a {\em proper projective boundary scheme} when held-out samples are accepted precisely if the full-sample boundary is retained, and accepted non-boundary samples can be deleted without changing that boundary. For every such scheme, the conditional law of the violation risk given the observed boundary size is determined by the boundary's cross-sample complexity profile. A stable profile yields the usual beta law, whereas a varying profile produces an exact profile correction. The framework covers scalar order-statistic calibration, support-reconstructive scenario programs, cascaded support-removal certificates, coordinatewise envelopes, and Pareto-frontier calibration with vector scores. It also yields conditional probabilistic certificates and a no-go result explaining why observed complexity alone is insufficient.

Post-hoc calibration corrects reported confidence, yet a multiclass calibrator can also change the associated top-1 prediction. Accuracy captures only the net effect of these changes on correctness, not how often predictions change; the Top-1 Prediction Change Rate (TPCR) instead measures this frequency. We propose Calibrator-Output Repair for Top-1 Decision Preservation (CORD), the first post-fit adapter to impose exact prediction preservation by repairing the full calibrated probability vector. From the original and calibrated outputs alone, CORD determines the mass assigned to the original top-1. The calibrated conditional distribution allocates the remaining mass over the other classes, yielding a repaired vector whose own argmax recovers the original prediction. On the calibration split, CORD coordinates the repaired masses to retain the calibrated outputs' mean mass on original predictions whenever attainable. The adapter alters neither the fitted calibrator nor its direct output, fits no additional supervised map, and requires no user- or validation-tuned hyperparameter. Across CIFAR-10/100 and ImageNet-1K, CORD attains zero TPCR by construction and lowers mean ECE, NLL, and Brier relative to the corresponding direct outputs in every dataset; paired gains persist under distribution shift and across calibration-set sizes. CORD thus removes the preservation constraint from calibrator fitting and assigns exact recovery of the original decision to subsequent output repair. Our code is available at https://github.com/labhai/CORD.

Accurate uncertainty estimation is essential for machine learning systems de- ployed in high-stakes domains such as medicine. Traditional approaches primarily rely on probability outputs from trained models (point predictions), which provide no formal guarantees on prediction coverage and often require additional calibra- tion techniques to improve reliability. In contrast, conformal prediction (region prediction) offers a principled alternative by generating prediction sets with finite- sample validity guarantees, ensuring that the ground truth is contained within the set at a specified confidence level. In this study, we explore the impact of pre-training approach, dataset scale and domain on both point and region-level uncertainty quantification, by studying domain-specific vision medical foundation models vs. general domain vision foundation models. We conduct a comprehensive evaluation across foundation models trained on retinal, histopathological, and Chest X-Rays data, applying various calibration techniques. Our results demonstrate that (1) pre-training on higher-quality domain-specific datasets along with self-supervised learning leads to better-calibrated point predictions than general domain pre-training, (2) stan- dard re-calibration methods alone cannot fully mitigate uncertainty discrepancies across models trained on different data sources, (3) domain-specific foundation model can lead to more efficient conformal prediction. These findings highlight the importance of careful model selection and the inte- gration of both point and region prediction to enhance the reliability and trust- worthiness of medical AI systems. Our work underscores the need for a holistic approach to uncertainty quantification in recent development of medical vision foundation model, ensuring robust and interpretable AI-driven decision-making.

Uncertainty quantification from finite data is central to machine learning, optimization, and automation systems, where decisions must remain reliable under limited samples and test-time distribution shift. Conformal prediction (CP) and distributionally robust optimization (DRO) offer two complementary approaches: CP constructs data-dependent prediction sets with distribution-free finite-sample validity under exchangeability, while DRO optimizes worst-case performance over an ambiguity set around an empirical distribution. We develop a unified probabilistic perspective on CP and DRO by viewing both as ways to turn finite calibration data into a data-dependent quantile estimator that a test score falls below with high probability. From this perspective, CP and DRO correct the empirical quantile along two coordinates of the same family of estimators: CP inflates the quantile level, whereas DRO shifts the quantile value through an ambiguity radius. Both methods provide the same calibration-conditional guarantee for the true distribution, requiring the target coverage to hold with high probability over the calibration sample. Their constructions differ, however: CP uses a closed-form, distribution-free level correction, while DRO uses a value-space correction whose certified radius depends on properties of the unknown distribution and additionally guarantees coverage uniformly over the ambiguity set. This distinction emerges in the tails of the score distribution. Because CP relies on sparse upper-tail order statistics of the calibration samples, its level inflation barely moves the estimator when those samples are dense near the target quantile but overshoots when they are sparse, whereas a well-chosen DRO radius corrects in value space and may avoid this overshoot.

This note studies the geometry of full conformal prediction (FullCP) regions generated by an empirical energy-form pairwise score. Candidate-score convexity alone does not guarantee connected FullCP regions, even for empirical averages of losses convex in the candidate argument. For the energy-form score, each leave-one-out training comparison reduces exactly to a pairwise-dissimilarity sublevel condition. Under symmetry, a constant diagonal, a diagonal lower bound, and attainment of the associated Fréchet-type objective, all comparison regions share a minimizer; if they are convex, every nontrivial exact conformal region is star-shaped about that point. For power distances $ρ_β(x,y)=\|x-y\|^β$, this geometry holds for $β\ge1$, while the conventional energy score is strictly proper for $0<β<2$. For $d=1,β=1$, every nontrivial empirical-CRPS FullCP region is a nonempty closed interval (possibly $\mathbb R$ when $m=1$). For $1<β<2$ and $m\ge2$, explicit data-checkable derivative bounds give Lipschitz control of the radial exits and exact conformal radial function. Combined with directional root search and classical Lipschitz extensions, they yield certified inner and outer radial envelopes of width at most $δ+2\widetilde Lh_{\mathcal U}$ and same-ray Hausdorff guarantees. An analytic two-dimensional example shows why preserving star-shaped but nonconvex geometry can matter. A staged two-dimensional study finds modest but systematic tightening of the generic certificate and frequent robust nonconvexity witnesses, with detected normalized radial departures typically small. The method is intended for low-dimensional multivariate outputs rather than high-dimensional scaling or runtime improvement.

In high-stakes applications, reliable confidence estimates are as important as the predictions themselves. Confidence calibration ensures that predicted probabilities reflect the likelihood of correctness, making it essential for safe deployment of deep learning models. However, existing methods typically assume access to clean validation data, which is often unrealistic due to label noise and domain shifts. This thesis develops methods for improving calibration under these conditions. First, we address calibration under label noise. Standard methods can produce misleading confidence estimates when labels are unreliable. We propose a framework that uses an estimated noise model to reconstruct noise-free confidence estimates by modeling the relationship between noisy and clean label distributions. We extend this approach to Conformal Prediction (CP), which provides set-valued predictions with guaranteed coverage. Our noise-aware CP method estimates clean conformity scores despite label noise, enabling reliable uncertainty quantification. Next, we study calibration in unsupervised domain adaptation, where a model trained on a labeled source domain is adapted to an unlabeled target domain. Since labeled target data are unavailable, we estimate target-domain accuracy from source performance and domain discrepancies, enabling calibration without target labels. We also consider privacy-preserving settings in which user labels and model outputs must remain protected. We propose a locally differentially private conformal prediction framework that provides valid uncertainty quantification while maintaining privacy guarantees and balancing privacy, computational feasibility, and prediction reliability. Our results bridge calibration theory and practical deployment in safety-critical applications, contributing to reliable, privacy-preserving, and noise-resilient neural network predictions.

Conformal prediction (CP) provides distribution-free prediction intervals for fixed forecasters, but its standard calibration procedure is often inefficient for time series data, where forecast errors are temporally dependent and change across time and operating conditions. Recent time series CP methods improve local calibration using recent, weighted, or localized residuals. Yet local calibration can remain indirect, since broad residual weighting or additional adaptation procedures may dilute the evidence most relevant to the current prediction. This motivates a simple retrieval and correction strategy that selects similar past residuals as local evidence and then corrects the coverage error left by retrieval. In this paper, we propose Retrieval--Corrected Conformal Prediction (RCCP), a retrieval-augmented calibration method for time series prediction intervals. RCCP builds an asymmetric interval from retrieved one-sided residuals and calibrates its normalized retrieval error with a scalar conformal correction. Thus, retrieval provides local residual evidence, while conformal correction determines the final scale needed for coverage. We provide a coverage-gap bound based on the stability of the normalized retrieval error distribution. Across standard benchmarks and backbone forecasters, RCCP attains the target coverage in every setting and achieves the lowest Winkler scores, with fewer severe misses. RCCP also achieves low calibration and inference overhead, showing that retrieval-corrected calibration is an effective and scalable approach to uncertainty quantification in time series forecasting. Code is available at https://github.com/jinsaaang/rccp.

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.

Interval prediction aims to achieve a target coverage level while producing intervals that are as short as possible. Many conformal regression pipelines first predict an uncertainty surrogate and then convert it into an interval through calibration or selection. This separation supports coverage calibration, but post hoc rules largely determine the final interval and do not fully use the learned output distribution. We observe that the resulting intervals have inherently hierarchical geometry: an interval can be recursively refined into nested subintervals, and binary trees naturally represent this structure. We formulate this hierarchy as next-interval prediction and propose PATH, which learns how probability mass flows from each interval to its next nested subintervals. PATH predicts a base leaf distribution and uses an autoregressive decoder to refine branch probabilities. Matching the distribution to the interval hierarchy aligns learning with extraction: PATH accumulates probability over adjacent output intervals and returns the shortest contiguous range reaching a selected mass. We compare PATH with 24 baselines for interval prediction on PATHBench, comprising 56 OpenML regression datasets. PATH substantially shortens the resulting intervals, achieving the lowest mean normalized length, 0.1473, while maintaining mean coverage of 0.9144. These results establish hierarchical output modeling as an effective approach for compact interval prediction on tabular data. Code is publicly available at https://github.com/pxcai/PATH.

Uncertainty quantification is essential when deploying machine learning models in safety-critical applications. Online conformal prediction (OCP) provides theoretically principled uncertainty quantification for arbitrary black-box classifiers and non-i.i.d. data streams by constructing prediction sets that are guaranteed to contain the true label at a user-specified frequency. OCP usually updates prediction sets using feedback from previously deployed predictions. We instead study an OCP setting beyond feedback: on each round, the learner can either output a prediction set or query the correct label, but not both. Thus, no deployed prediction is ever evaluated directly. We reduce this problem to a partial monitoring game in which prediction actions return no observation and a separate query action reveals the label. The reward function is constructed in a way that encourages the learner to output small prediction sets while ensuring that the correct label is covered with a sufficiently high probability. To solve this game, we develop OCP with queries (OCPQ) by adapting the label efficient forecaster of Cesa-Bianchi, Lugosi, and Stoltz (2004) to our setting. For any black box classifier and any (non-i.i.d.) oblivious data stream of length $T$, OCPQ has $O(T^{2/3})$ expected regret and expected coverage at least $β-O(T^{-1/3})$ for a user-defined $β$, while querying only an expected $T^{-1/3}$ fraction of rounds. This provides coverage comparable to bandit-based OCP methods while requiring no feedback from deployed prediction sets. Experiments on real-world datasets further demonstrate the effectiveness of our approach.

Conformal prediction endows arbitrary black-box predictors with finite-sample, distribution-free marginal coverage, yet marginal validity can hide severe covariate-specific miscalibration, while exact distribution-free conditional coverage is finite-sample unattainable. Randomly localized conformal prediction (RLCP) mitigates this gap by calibrating near the test point while preserving marginal coverage. Existing theory, however, lacks finite-sample guarantees for the realized localized set that jointly control conditional validity and oracle efficiency. We provide such guarantees. For any fixed score, under Hölder regularity of the conditional score CDF and standard density and kernel assumptions, we prove high-probability bounds, uniform over a realized localization neighbourhood, for the conditional-coverage gap and the length error relative to the oracle. The bounds decompose into an $O(h^β)$ localization bias and a calibration term decreasing with calibration size, clarifying the bandwidth bias-variance tradeoff and when RLCP tracks the oracle. We also analyze data-split learned scores: when the score targets a pivotal score, as in conformalized quantile regression, uniform local guarantees decompose into fixed-score calibration and uniform score-estimation errors, showing that improved learning sharpens localized guarantees.

In this paper, we consider a class of multiblock nonconvex nonsmooth optimization problems, which covers many applications such as the analysis of pre-earthquake anomalies and machine learning. To solve this class of problems, we propose the inertial block proximal linearized method with two-phase adaptive momentum (IBPL$^+$-TP). Compared to the current methods, our method possesses three main advantages: (1) it introduces a two-phase adaptive momentum strategy to effectively update the extrapolation parameters, (2) it allows using two different extrapolation points to accelerate the convergence, (3) it allows the extrapolation parameters of these two extrapolation points to be independent of and unconstrained by all other parameters. While maintaining the above advantages, we prove that our method ensures the monotonic convergence of the objective function of this class of problems, and we also prove that the sequence generated by our method globally converges to a critical point, as well as establish the convergence rate of our method. To demonstrate the effectiveness of our method, we apply it to solve two nonconvex and nonsmooth machine learning problems, namely sparse nonnegative matrix factorization with $\ell_0$-constraints and sparse nonnegative CP decomposition with $\ell_0$-constraints. The numerical experimental results on solving these problems show that our method outperforms several state-of-the-art methods.

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

Translating natural language instructions into machine-interpretable formal specifications enables robots and autonomous systems to plan, reason, and formally verify their behavior. However, existing translation models typically generate a specification for every input, even when the result is unreliable or fails to capture the user's intent, creating risks in safety-critical applications. Inspired by selective conformal prediction, we propose a selective translation framework that not only generates formal specifications but also determines when they can be trusted. Reliability is scored by two complementary black-box signals, the fidelity of the specification back-translated into natural language and the dispersion of repeated translations under exact semantic equivalence, which fail on different errors and jointly separate incorrect translations more sharply than either alone. Conformal risk control calibrates this score into a decision that accepts a specification or abstains, with a distribution-free bound on the rate at which incorrect specifications are accepted for execution, and a conformal anomaly detector on instruction embeddings screens out-of-distribution inputs before any translation is attempted. The proposed framework is general across formal specification languages, with experiments on Signal Temporal Logic (STL), Linear Temporal Logic (LTL), and geometric Spatio-Temporal Logic (SpaTiaL) demonstrating improved translation reliability, robustness under the evaluated cross-tier shifts, and effective uncertainty-aware abstention. This work establishes a foundation for trustworthy natural language interfaces by enabling AI systems to recognize when generated specifications may not be reliable.

Predicting fatigue failure in steel components experimentally is costly because it requires testing across multiple compositions and processing conditions. This has spurred research on data-driven prediction models. Studies using the NIMS MatNavi steel fatigue dataset often report high point-prediction accuracy but rely on aggregate error metrics, leaving uncertainty about the reliability of individual predictions and whether accuracy is consistent across the fatigue-strength spectrum. This paper is the first to apply conformal prediction to steel fatigue strength, comparing five interval-construction methods across 50 independent data splits and distinguishing marginal coverage from coverage within specific sub-regions of the predicted property. A gradient-boosting point model achieves an R^2 of 0.976 +/- 0.009 and a mean absolute error of 18.3 +/- 2.3 MPa. Split-conformal prediction provides valid marginal coverage (0.918) but drops to 0.755 in the highest-strength quartile, where design margins are most critical, a pattern also observed with a Gaussian process baseline. A cross-fitted, normalized conformal method restores near-uniform coverage across all quartiles (0.869-0.938) without a significant increase in interval width, by scaling the interval based on a cross-fitted estimate of local prediction difficulty rather than using a single global width. Diagnostic analysis traces the residual gap in the highest-strength quartile to elevated residual variance (2.7x the pooled Q1-Q3 level) rather than a systematic bias, situating the shortfall against a proven distribution-free limit on exact conditional coverage. Marginal coverage claims for ML-based fatigue-strength predictions can conceal systematic unreliability precisely where engineering decisions are most risky; therefore, conditional coverage should be routinely assessed alongside marginal coverage.

Adaptive conformal prediction can recover clinically important heartbeat classes missed by a point classifier, but delayed feedback makes its decisions sensitive to event order. We introduce ConformalShift, a bounded event-reordering attack that suppresses the ventricular class for rescued events without modifying ECG waveforms, labels, classifier scores, or the event multiset. ConformalShift searches for feasible permutations of authentic preceding events that lower the ventricular threshold before a selected target is evaluated. On disjoint MIT--BIH confirmation records, the attack suppressed 66.7% of eligible targets for Extra Trees and 60.0% for HistGradientBoosting, compared with random-schedule rates of 4.4% and 12.0%, respectively. Transferred configurations also outperformed random scheduling on INCART, while reducing the displacement budget weakened the attack on both datasets. These results show that adaptive monitors in healthcare can be compromised through the timing of authentic information, even when waveforms, labels, classifier outputs, and event contents remain unchanged.

Conformal prediction (CP) is a distribution-free framework for uncertainty quantification that has recently been adapted to large language models (LLMs), providing prediction sets with finite-sample coverage guarantees under exchangeability. Yet for LLMs, nonconformity scores are often induced by an inference pipeline, not just a fixed model, making them depend not only on the data distribution but also on configurable factors such as the prompt template, decoding parameters, and deployment setting. Since such configurations are routinely modified in practice but rarely treated as a source of shift, their impact on CP validity remains poorly understood. We call this \emph{configuration shift} and study it systematically along three axes: prompt template, decoding temperature, and weight quantization. In a broad empirical study spanning $9$ LLMs, $4$ datasets, and $4$ nonconformity scores, we find that configuration shift consistently erodes CP validity, often driving empirical coverage below the target. By contrast, efficiency is largely preserved: valid prediction sets remain close in size to the i.i.d. baseline. We derive coverage lower bounds that attribute this loss to a discrepancy between calibration and test score distributions, and use their finite-sample plug-in versions as empirical diagnostics of shift severity. We further show that these findings lead to practical mitigations: bound-inspired recalibration is effective with limited test examples, while fragility-aware calibration ensembling recovers much of the lost coverage without test data.

High-stakes decision systems in credit scoring, fraud detection, healthcare, and industrial safety require reliable uncertainty quantification under severe class imbalance and asymmetric error costs. Standard marginal conformal prediction (CP) provides valid overall coverage guarantees; however, we show that it severely under-covers rare, costly minority classes, with minority-class coverage dropping to as low as 0.5% on certain datasets. To characterize and address this limitation, we conduct a comprehensive benchmark comparing marginal CP, class-conditional (Mondrian) CP, and cost-controlled abstention mechanisms across 15 real-world imbalanced tabular datasets, 7 classification models, 3 probability calibration techniques, and 10 random seeds, resulting in 3,150 experimental runs. Our results show that Mondrian CP restores valid minority-class coverage, achieving an average minority-coverage improvement of 61.7 percentage points over marginal CP (p < 1e-80). Furthermore, combining Mondrian CP with cost-controlled abstention significantly reduces expected decision cost compared with standard decision boundaries, confidence-based rejectors, and risk-controlled rejectors under realistic human review budgets. We further quantify dataset-specific break-even thresholds at which deferring ambiguous instances to human experts becomes cost-effective. These findings provide practical guidance for deploying distribution-free, cost-aware uncertainty quantification in high-stakes decision support systems.

Adaptive conformal inference (ACI) of Gibbs and Cand{è}s and its variants are the standard approach to online conformal prediction under distribution shift, but they suffer from three fundamental limitations. First, their guarantees control only the \emph{signed} long-run coverage error: persistent miscoverage in one direction can be masked by compensating errors later, so a method can satisfy the theoretical guarantee while being badly wrong for extended periods. Second, existing guarantees say nothing about prediction-set size, so validity can be achieved trivially at the cost of unduly wide prediction sets. Third, the efficiency guarantees that do exist compare against a \emph{fixed} predictor chosen in hindsight, a benchmark that becomes increasingly less meaningful once the data-generating distribution shifts, since the very notion of an optimal threshold then changes over time. We consider a unified online learning framework that simultaneously controls absolute, non-cancelling coverage violation and prediction-set efficiency against a dynamically evolving benchmark for three important models. In the fully adversarial setting, exploiting the fact that the standard ACI update is exactly projected online gradient descent on the pinball loss, we derive simultaneous coverage and efficiency guarantees for arbitrary monotone Lipschitz efficiency objectives, with no distributional or {\it convexity} assumptions. In the stochastic setting with full-score feedback, we propose a sliding-window quantile tracker and establish a matching minimax lower bound showing our algorithm is rate-optimal. In the covariate-dependent stochastic setting, we develop a partitioned ACI algorithm that tracks a function-valued oracle threshold, and derive simultaneous coverage and efficiency guarantees.

Conformal prediction (CP) provides distribution-free uncertainty quantification, and its extension to graphs is an active research direction. Diffused Adaptive Prediction Sets (DAPS) is a widely used graph-aware diffusion baseline, propagating Adaptive Prediction Sets (APS) non-conformity scores along edges with a uniform coefficient $λ$. We identify a fundamental shortcoming of this design: the uniform low-pass diffusion presupposes graph homophily and proves detrimental on heterophilic graphs, enlarging the mean prediction-set size by up to 10.6% relative to plain APS. To mitigate this, we propose HeAD-CP, a family of node-wise diffusion variants whose coefficients are determined by a label-free local-homophily estimate derived from the GNN softmax. Three variants, namely signed-$γ$, edge-compatibility, and a DAPS-baseline-with-correction, are most effective at extreme heterophily, intermediate heterophily, and moderate-to-high homophily, respectively, and all preserve the marginal coverage guarantee. On ten benchmarks, the HeAD-CP family stays at or below plain APS on every dataset, while DAPS exceeds APS on six. The post-hoc oracle over the family improves over DAPS on 8/10 datasets at $p<0.01$ (paired Wilcoxon), with the largest gains on heterophilic graphs (10.3% on Texas); on the two homophilic datasets where DAPS still wins (CiteSeer, PubMed), it retains a marginal advantage of at most 0.002, statistically insignificant on CiteSeer ($p=0.23$). Designing a calibrated label-free selector that approaches this oracle is the main outstanding empirical question.

Vine copulas provide a flexible framework for modeling complex multivariate distributions through a hierarchical decomposition into bivariate pair-copulas. Fitting a D-vine requires selecting a copula family and parameter configuration for each pair-copula from a set of candidates encoding different dependence patterns. As the number of variables and candidate families increases, the number of possible configurations grows combinatorially. Existing fitting procedures address this challenge through sequential greedy decisions, committing to a single locally optimal family at each step and potentially discarding configurations that would yield a better global fit. To overcome this limitation, we propose a novel estimation framework that combines gradient-based maximum likelihood estimation, enabled by our fully differentiable implementation, with a beam-search strategy that maintains multiple competing D-vine configurations throughout the fitting process. This allows a broader exploration of the configuration space while remaining computationally tractable. Building on the fitted D-vine, we introduce a localized anomaly detection framework that exploits the hierarchical decomposition to produce both global anomaly scores and edge-level explanations. Statistical guarantees are provided through Mondrian conformal prediction, while the pair-copula structure enables the localization of anomalies to specific variable relationships. We evaluate the proposed framework on both benchmark and real-world datasets, demonstrating its effectiveness for interpretable anomaly detection with uncertainty quantification.

We propose ABF-T-GLCP, a model-agnostic framework for forecasting and uncertainty quantification in nonstationary multivariate time series. The central idea is to learn an adaptive predictive state representation for point forecasting and reuse it for conformal calibration. The forecasting module combines horizon-specific temporal experts through a learned gate and refines predictions using sparse predictive transfer across related series. The uncertainty module, Gate-Localized Conformal Prediction (GLCP), uses the learned gate state, together with temporal recency, to select locally relevant calibration residuals, thereby coupling uncertainty calibration to the predictive regimes used by the forecasting model. This shared representation allows point forecasts and prediction intervals to adapt consistently under evolving temporal dynamics while retaining the model-agnostic nature of conformal prediction and yielding approximate local coverage under mild stability conditions. Experiments on a large-scale high-frequency commodity forecasting benchmark show consistent gains in point forecasting accuracy and substantially narrower prediction intervals with empirical coverage close to the nominal level. Additional results indicate that the framework extends beyond the motivating financial application.

Conformal prediction certifies that a classifier's prediction sets cover the truth, and that certificate is marginal. Many recognition benchmarks build distribution shift into evaluation, placing disjoint conditions in the training and test splits. Under that shift the certificate stays reassuring while per class coverage fails silently: on a real cross subject skeleton benchmark marginal coverage holds near ninety percent while the worst class is covered about seventy percent and ten of sixty classes fall below eighty percent. This class specific undercoverage stays hidden behind a single reassuring marginal number. Once the shift acts jointly on covariates and labels, the target class conditional score law is unidentified, so no label free method is at once per class valid and efficient uniformly over target laws consistent with the observed source joint distribution and target covariate marginal. The per class labels needed to recover every class threshold to a given tolerance grow as the inverse square of that tolerance and the logarithm of the class count, with matching bounds for classwise threshold procedures. Pseudo labels do not shortcut it: the best prediction powered estimator gains at most a small constant factor where coverage collapses. Across three real shifts and an image corruption benchmark, source label calibration recovers much of the gap while marginal coverage holds, and stops once it breaks.

Large language models frequently violate fundamental scientific principles when generating technical content, undermining their reliability in scientific applications. We introduce Scientific Feasibility Control SFC, a graph-structured conformal prediction framework that provides statistical guarantees for scientific reasoning validity through progressive absolute-coherent-factuality validation. Our approach decomposes scientific reasoning into atomic absolute-coherent-factuality units requiring both individual correctness against physical laws and logical substantiation from preceding context, addressing the cascade effect where early scientific errors contaminate subsequent reasoning steps. Unlike independence-based methods that treat claims in isolation, SFC models logical dependencies as approximate deducibility graphs and operates through real-time validation with dynamic branching when scientific violations are detected, the system branches to alternative generation paths using verified context as foundation. We demonstrate SFC across established scientific reasoning benchmarks including PhyX multimodal physics, MATH, ScienceQA, and ARC Challenge, achieving 50.1 percent accuracy on PhyX physics reasoning, substantially outperforming recent reasoning models including DeepSeek-R1 49.8 percent and GPT-4 45.8 percent while providing 91.7 percent scientific validity with formal conformal coverage guarantees at alpha equals 0.10 confidence level and reducing scientific law violations by 73 percent across multiple model architectures.

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.

Quantum state tomography is sample-starved, and the states one prepares live on a narrow, learnable manifold. A $k{=}0$ prior-only control shows that on concentrated families a prior estimate is already near-optimal, so ``high fidelity at few measurements'' can be family memorization rather than tomography; genuine measurement-efficiency needs a model that conditions on the measurements and demonstrably uses them. On a shared matrix-product-state (MPS) core parameterization we study two routes. ApproachA learns a generative prior over MPS cores with measurement-guided posterior inference (gold-standard-validated, but whose few-measurement accuracy the control shows is largely the prior). ApproachB, our main proposal, is a \emph{fixed-protocol amortized} MPS estimator trained once with a gauge-invariant fidelity loss; we deliberately do not rest it on a permutation-invariant set encoder (a plain MLP matches it). The decisive lever is the measurement design: motivated by the fact that local reduced density matrices determine a $χ$-MPS, conditioning on an \emph{informative local} Pauli set rather than random strings turns a modest, memorization-prone estimator into a high-fidelity one ($\approx\!0.95$, up to $+0.59$ over prior-only, decisively passing a shuffled-measurement control). A dropout ensemble, conformally recalibrated, gives $\approx\!90\%$-coverage intervals -- including for observables never measured, where a shot-based interval does not exist. Quality holds as the system grows (fidelity $0.90$ at $n{=}10$, gain \emph{growing} in $n$; $0.88$ at bond dimension $χ{=}4$), the parameterization is polynomial (native contraction to $20$ qubits), and we close the loop on IBM hardware ($5$ states at $0.97$ from hardware-measured Paulis).

We introduce a new conformal prediction method that constructs calibrated prediction sets over collections of spatial events, such as tropical cyclone genesis and earthquake locations. Forecasting natural hazards has become increasingly important, due to their significant economic impact, and quantifying the uncertainty of predictions is critical for accurate risk assessment. Our approach works by representing spatial point clouds as empirical measures so that we can score them using (sliced) Wasserstein distance, then constraining the resulting distribution-valued prediction set to be supported only near the training data manifold. We derive a coverage lower bound for the intersected sets and show that, in practice, this gap can be made small through a simple data-adaptive selection criterion. Because the resulting set is not analytically tractable, we introduce a modified flow-based sampling procedure, which allows us to represent and apply these prediction sets in practice as ensembles. Numerical experiments on synthetic data, tropical cyclone genesis, and earthquake occurrences show that our method achieves near-nominal coverage, with significantly lower energy distance and manifold distance than highest predictive density region (HDR) baselines along with generative model baselines.

We revisit one-zero and two-zero textures of the neutrino mass matrix under current experimental and cosmological constraints. We identify the phenomenologically viable texture structures using the latest results on neutrino oscillation parameters, the cosmological bound on the sum of neutrino masses, the kinematic bound on the effective electron-neutrino mass, and limits from neutrinoless double-beta decay. For two-zero textures, several structures are still allowed if only the CMB bound on the neutrino mass sum is imposed. Among them, the $B$-series textures show a characteristic prediction for the Dirac CP phase, with $δ_{\rm CP}$ lying around $π/2$ and $3π/2$, and are within the reach of future neutrinoless double-beta decay searches. When the stronger CMB+BAO constraint is included, however, only the $A$-series textures remain viable. Therefore, we also analyze one-zero textures by using machine learning techniques, particularly flow matching. It turns out that some of the texture structures are already excluded by current data, while the allowed ones give distinct predictions for $\sum_i m_i$, $m_{ν_e}^{\rm eff}$, $\langle m_{ee}\rangle$, and $δ_{\rm CP}$. We further discuss how the one-zero texture structures can arise from non-invertible selection rules.

Radiomic features derived from medical images and segmentation masks are used to support decision making in clinical imaging pipelines. In practice, these features are often computed from predicted masks, but segmentation models can be overconfident or poorly calibrated, making derived measurements appear more reliable than they are. Conformal prediction (CP) provides distribution-free prediction intervals with finite-sample marginal coverage guarantees, but black-box intervals for segmentation-derived radiomics can be inefficient because they ignore test-time information about image appearance, mask geometry, and segmentation uncertainty. We propose ConRad, a conformal framework for scalar radiomic targets that uses covariates derived from the predicted mask, input image, predicted radiomics, and boundary uncertainty to construct adaptive intervals while maintaining coverage. Across five 2D medical imaging datasets and 171 retained radiomic targets, we show that ConRad improves feature-level efficiency compared to baselines while maintaining near-nominal empirical coverage. Ablation results further indicate that segmentation boundary uncertainty features are the largest contributors to interval efficiency.

Finance, sensing, and demand streams violate the exchangeability that IID conformal prediction and the IID bootstrap assume, and existing libraries implement either a general resampling engine or conformal calibration without the other. tsbootstrap provides block, residual, sieve, and wild resampling, classical bootstrap confidence intervals, and adaptive conformal calibrators (EnbPI, ACI, NexCP, AgACI) through a single typed API in which a specification object selects each method. In a controlled coverage study the IID bootstrap undercovers sharply under dependence; dependence-aware methods reduce the coverage deficit, the sieve nearest to nominal under short-memory linear dependence. On the shared fixed-statistic path a compiled backend runs several times faster than arch, and a streaming reduce avoids materializing the $O(Bn)$ replicate tensor, limiting peak extra memory to $O(B)$ for the statistic array. The software is MIT licensed (v0.6.1).

Conformal prediction is being adopted in drug discovery to put an honest number on model reliability: pick an error rate alpha, and the method returns prediction sets containing the true label with probability at least 1 - alpha. We show this guarantee can be dangerous on imbalanced datasets. Across four datasets, standard (marginal) conformal prediction hits its global 90% coverage target while leaving the minority class badly exposed: realized minority coverage falls to 64.8% on blood-brain-barrier penetration and to 4.2% on clinical-trial toxicity, where the rare class is nearly abandoned. The failure is not tied to one model: a random forest, a graph network, and a frozen chemical language model all reproduce it (p < 0.001 in every case), with severity tracking baseline calibration on rare labels rather than architecture. A conservation identity explains the effect: the minority's shortfall equals the majority's surplus amplified by the imbalance ratio, predicting the measured gap to within one point and ordering severity across datasets. The failure survives realistic scaffold splits and a second conformal score, while aggregate accuracy and overall coverage stay reassuringly high, which is exactly why it is easy to miss. Class-conditional (Mondrian) conformal prediction closes the gap on every dataset, restoring minority coverage to target for a modest increase in prediction-set size. We localize the failures to generic molecular scaffolds - plain benzene and pyridine cores occurring in both classes - propose a one-number diagnostic, and show with a cost model that abstaining on affected compounds flips a screening campaign from net-negative to net-positive utility. Our contribution is demonstrating on real chemistry how severe and invisible this known conformal-theory gap becomes under imbalance, and laying out a practical protocol restoring per-class reliability.

Bayes-assisted conformal prediction combines the strengths of Bayesian modelling with exact, distribution-free frequentist coverage guarantees. Although conformal validity is preserved even when the Bayesian working model (BWM) is misspecified, the size of the resulting prediction sets can degrade substantially when the prior is poorly aligned with the observed data. We address this limitation by introducing RoBAS (Robust Bayes-Assisted Shrinkage): a Bayes-assisted framework for constructing robust nonconformity scores, with two instantiations: one induced by a heavy-tailed BWM, and a closed-form empirical Bayes shrinkage score. The resulting scores adapt to the quality of the working information encoded in the prior: when this information is reliable, they exploit it to produce efficient prediction sets; when it is weak or inaccurate, they revert to the Distance-To-Average (DTA) score, a robust non-informative baseline. We evaluate the proposed scores on tabular and image regression tasks where the training distribution may differ from the calibration and test distributions, while the calibration and test data themselves remain exchangeable. We find that they are competitive with widely used scores in the absence of such shift, while substantially reducing interval widths in shifted settings.

Predicting thermal volatility in high-performance EV powertrains is difficult as internal temperatures are rarely observable outside the lab, and models calibrated on lab drive cycles fail when deployed against real-world loads. We study this lab-to-track transfer problem using conformal prediction, offering distribution-free uncertainty bounds. We implement Ensemble Batch Prediction Intervals (EnbPI; Xu & Xie, 2021), a leave-one-out bootstrap-ensemble conformal method for autocorrelated time series, and calibrate it on real CALCE lithium-ion cycler data (A123 SP20 cells, FUDS profile). We evaluate it under a genuine, measured covariate shift: a second real CALCE test condition (US06 Highway Driving Schedule at 45°C). The unweighted EnbPI bound, achieving its nominal 95% coverage in-distribution (measured: 95.00%), degrades to 70.13% empirical coverage under this real shift. We introduce a weighted EnbPI procedure combining EnbPI's ensemble residuals with density-ratio weighting (Tibshirani et al., 2019), estimating the density ratio via a probabilistic domain classifier. This recovers coverage to 72.42%, a modest, honestly-reported improvement, not a complete fix. We additionally apply the calibrated model to real 2023 Formula 1 telemetry (Monza and Silverstone, driver VER) as an unsupervised out-of-distribution diagnostic. Because no internal thermal channel exists in public trackside telemetry, we report only unsupervised flag rates (65.6% at Monza, 58.0% at Silverstone, well above the 5% in-distribution base rate) and note inconsistent associations between flags and braking/DRS zones. We conclude that conformal domain adaptation is a promising but only partially solved tool for this problem, detailing exactly where it falls short.

Predictions are increasingly used to guide high-stakes decisions, from treatment selection to policy making. To ensure reliability with imperfect predictions, uncertainty quantification methods such as conformal prediction build prediction sets with coverage guarantees. However, statistical validity alone does not immediately determine the decisions to take, nor the optimality thereof. This gap is especially delicate in counterfactual settings where the outcome that materializes depends on the action taken, so uncertainty cannot be specified independently of the decision rule. We develop a decision-theoretic framework for uncertainty-informed counterfactual decisions. We identify a novel notion of \emph{policy-coupled coverage} -- namely, coverage of the realized outcome under the action induced by the prediction sets themselves -- as the optimal and lossless interface between uncertainty and action. It plays three roles. First, it justifies acting via a natural max-min rule as minimax-optimal under distributional ambiguity. Second, optimizing prediction sets under policy-coupled coverage is equivalent both to a stronger universal-coverage formulation and to the direct risk-averse optimization over policies and utility certificates; this equivalence yields the explicit form of the population-optimal prediction sets. Third, it admits a two-stage procedure, Policy-Coupled Risk-Averse Conformal Prediction (PC-RACP), that approximates these optimal sets with rigorous finite-sample coverage. Simulations and a real email-marketing experiment confirm that PC-RACP delivers higher utility than existing approaches while maintaining valid coverage, and that ignoring the counterfactual structure of the decision problem is suboptimal for both validity and utility.

Conventional traction control architectures intervene only after the adhesion limit of a tire has already been breached. This paper investigates whether Rolling Split Conformal Prediction , monitoring the volatility of non-conformity residuals from a per-driver Random Forest model of expected slip behavior , can serve as a statistically grounded pre-incident warning signal, ahead of gross traction loss. Unlike an earlier internal draft of this work, the evaluation reported here corrects a confound in the slip proxy (vehicle speed is included as an explicit model feature, not left implicit in the target's denominator), uses every racing lap for each driver rather than only the fastest lap, and is scored against real, timestamped incident labels extracted from FIA Race Control Messages and track-limits lap deletions rather than narrated post-hoc. The result is negative: across 19 drivers and 55,563 test-phase telemetry samples, the rolling-volatility detector achieves a mean precision of essentially 0.0 and mean recall of 0.0 against 14 ground-truth incidents, while flagging on average 15.3% of all samples as anomalous , too high a false-alarm rate for any early-warning use. A static 95th-percentile threshold baseline performs no better in any way that would justify the added complexity of the conformal-volatility formulation. Residual autocorrelation diagnostics show the split-conformal exchangeability assumption is violated for every driver (Ljung-Box p < 0.001, n = 19/19), which is one plausible driver of the high false-alarm rate. We report this as a methodologically rigorous negative finding, diagnose its likely causes, and outline what a genuinely predictive version of this approach would require.

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

While conformal prediction provides a general framework for uncertainty quantification in predictive inference, its application is often limited by computational cost. Recent methods, including Jackknife+ and Jackknife-minmax, achieve faster computation by trading a slight loss of efficiency relative to full conformal prediction, but still requires computing leave-one-out refits for all observations. In this paper, we further accelerate conformal prediction by incorporating approximate leave-one-out (ALO) estimators, and establish asymptotic coverage and efficiency. While our proof draws on methods developed for analyzing the consistency of ALO cross-validation risk estimators in high-dimensional statistics, it requires adaptations to handle conformal prediction, where leave-$i$-out residuals are needed for predictions at $x_{n+1}$ rather than just at the training covariate $x_i$. Simulation results validate our theoretical findings, showing that the ALO-based methods achieve coverage and efficiency comparable to the exact methods, while significantly reducing the runtime.

Accurate energy demand forecasting is essential for the reliable operation and planning of modern sustainable energy systems. Spatial-temporal graph neural networks (STGNNs) have recently achieved strong performance in point forecasting by jointly modeling temporal dynamics and relational dependencies across interconnected energy nodes. However, in real-world energy systems, accurate point forecasts alone are insufficient, as operators also require reliable uncertainty estimates to support risk-aware decision-making, grid stability, and operational planning under uncertainty. Conformal prediction provides a principled and model-agnostic framework for uncertainty quantification with statistical coverage guarantees, making it particularly attractive for safety-critical energy applications. However, existing conformal prediction approaches often fail to fully capture the complex spatial-temporal structure of energy systems. To address these limitations, we propose STOIC (Spatial-Temporal Graph Conformal Prediction with In-Context Learning), a novel framework that integrates graph-based forecasting with the zero-shot calibration capabilities of tabular foundation models. STOIC first generates point forecasts using an STGNN and subsequently reformulates spatial-temporal residuals into a tabular representation suitable for in-context learning. Leveraging a tabular foundation model, STOIC calibrates prediction intervals without task-specific retraining, effectively capturing both sequential and relational dependencies. We evaluate STOIC on five diverse benchmarks, including synthetic simulations as well as real-world electricity and district heating networks. Across all datasets, STOIC consistently outperforms existing conformal prediction baselines, delivering more reliable and robust uncertainty estimates for complex graph-structured energy time series.

Conformal prediction and its variants, including the split conformal prediction, provide a distribution-free framework for uncertainty quantification by constructing prediction intervals or sets with finite-sample coverage guarantees. The statistical efficiency of these intervals depends critically on how the data are split into training and calibration samples. Despite its practical importance, a principled characterization of the training-calibration split that minimizes prediction interval length while maintaining coverage has remained largely unresolved. In this paper, we develop a theoretical framework for optimal data splitting in split conformal prediction. We first analyze the problem in a general setting and derive analytical characterizations of the length-optimal split ratio under both symmetric and asymmetric regimes. We then show how the general results specialize to several commonly used regression settings, including linear regression, nonparametric regression, and neural networks, thereby demonstrating the scope of the framework. We also describe a data-based method for selecting the optimal proportion. Our analysis clarifies how model-related features govern the optimal allocation of samples between training and calibration and provides principled guidance for constructing shorter prediction intervals. Experiments on both synthetic and real-world datasets demonstrate the applicability of the proposed methodology across a variety of practical scenarios.

Conformal predictions have attracted significant attention in the field of uncertainty quantification, mainly because of their strong marginal coverage guarantees. Full conditional guarantee is not an attainable goal, a well known fact in conformal predictions literature. As a result, several approaches have tried to approximate this behavior by adapting the conformal sets of test-time samples according to their similarity to calibration examples. Although the latter has gained traction and shown impressive performances for regression problems, its application to image classification remains under-explored. We conduct an extensive benchmarking on natural image classification tasks with vision-language models (VLMs), using our open source implementation of a recent localized conformal prediction algorithm. We show that straightforward usage of the cosine similarity between test-time and calibration visual features, an intuitive choice for VLMs, is not sufficient to improve over the non-local baselines. In response, we propose a simple non-linear transformation of the cosine similarities, which conserves marginal coverage guarantees and achieves statistically significant mean set sizes reduction. Code is available at https://github.com/cfuchs2023/lcp-vlm/.

Conformal prediction provides distribution-free reliability guarantees for vision systems, but these guarantees depend on how prediction errors are measured in the output space. Many vision tasks produce outputs on curved spaces (e.g. gaze directions on the sphere or 3D head rotations), yet intermediate prediction heads, residuals, uncertainty estimates, or conformal scores are often defined in flat coordinate charts such as yaw-pitch or Euler angles. We show that this scoring choice introduces systematic geometric distortion near coordinate singularities (large pitch angles on the sphere and poses approaching gimbal lock in 3D rotations). Across four datasets (ETH-XGaze, Gaze360, BIWI, AFLW2000-3D), slice-conditional coverage at a nominal 90% target drops by 30-50 percentage points in these regions, falling to 38.9% on ETH-XGaze and 42.0% on Gaze360 at gaze pitch above 70 degrees, and to 57.5% on BIWI and 55.2% on AFLW2000-3D at head pose pitch above 60 degrees near gimbal lock, despite marginal coverage remaining near 90%. We prove that this is structural. Scalar thresholding changes the size of chart-coordinate prediction sets but leaves their distorted axis ratios unchanged. To diagnose this hidden failure mode, we show that a simple geometric quantity, the Riemannian volume density, strongly correlates with where coverage collapse occurs. Finally, we show that coordinate-free geodesic scoring removes this distortion. It requires no retraining and adds negligible computational cost.

Conformal prediction guarantees marginal coverage, but a pooled calibration quantile can hide systematic undercoverage across heterogeneous regions of the feature space. We introduce Self-Organized Conformal Prediction (SOCP), a calibration scheme that discovers input-space groups with an unsupervised Self-Organizing Map (SOM) trained without calibration labels. At prediction time, the query's best-matching unit (BMU) draws a calibration buffer from one cell, a fixed grid neighborhood, or a prototype-based enlargement. When fixed neighborhoods are too sparse, Regime 3 adds cells by prototype distance, using a global budget selected from training-cell occupancies and the planned calibration size before any calibration score is observed. The predictor and nonconformity score remain unchanged. Cell-only retrieval has exact cell-conditional validity, and each fixed union of cells has exact retrieved-set validity. Interpreting a neighborhood threshold at its central cell incurs an explicit Kolmogorov-Smirnov (KS) bias term. Across ten regression and classification benchmarks, SOCP reduces the weighted coverage gap relative to pooled split conformal prediction on nine datasets. The mean relative change is $-14.3\%$, at a mean output-size change of $+3.0\%$. Under fixed-neighborhood retrieval, SO composition lowers the ten-seed mean WCovGap in $43$ of the $50$ dataset-score comparisons, while SO-SCP lowers it on average over paired seeds for every dataset at all three tested external partition granularities. These results provide a concise route to group-local calibration without supervised partitions or predictor retraining with a diagnostic toolkit, while keeping the cost and limits of locality explicit.

Prediction sets should have high coverage to be useful, but some coverage notions are more practically relevant than others. In the classification setting, class-conditional coverage requires that the prediction set (i.e., the set of candidate labels for a new test point) must achieve the target accuracy level within each class, which may be challenging to satisfy when many classes are rare and have few calibration points. At the other extreme, marginal coverage requires only that coverage holds on average over the distribution of all classes, which can lead to low-probability labels being essentially ignored. To find a middle ground, recent work has introduced macro-coverage, defined as the unweighted average of class-conditional coverages. Macro-coverage offers a compromise between marginal coverage and class-conditional coverage that is particularly appropriate for long-tailed settings. In this work, we show that label-weighted conformal prediction can be used to produce prediction sets with a finite-sample macro-coverage guarantee, and more generally a guarantee on a family of generalized macro-coverage objectives that aggregate coverage at the level of arbitrary class groupings and take a weighted average. We further characterize the form of the smallest prediction sets satisfying a given generalized macro-coverage objective and propose a corresponding conformal score function. We validate our theoretical results on two large-scale image classification datasets.

Quantifying the evolution of uncertainty is critical to both probabilistic forecasting and data assimilation in numerical weather prediction. In this study, we investigate the applicability of conformal prediction (CP), a recent machine learning (ML) method, to quantify uncertainty in a controlled, idealized setting. We use the one dimensional modified shallow water model, designed to mimic the convective process. CP provides a set of possible outcomes with a chosen confidence level. Here, we compare and evaluate the average empirical coverage, the average interval length, miss low, miss high and average interval score loss (AISL) for three variants of CP, namely a) Standard CP, b) Normalized CP and c) Conformalized Quantile Regression. We further compare these CP-based uncertainty estimates with traditional ensemble-based measures such as standard deviation intervals and ensemble spread. In addition, we investigate the integration of CP-derived uncertainty within the data assimilation cycle through CP perturbations. Our results highlight the strengths and limitations of each approach, providing insight into the effectiveness of CP to complement common ensemble-based uncertainty quantification in simplified atmospheric models.

Ordinal classification (OC) arises in high-stakes domains such as medicine and finance, where uncertainty quantification must account for the severity of ordinal errors. Conformal prediction (CP) provides distribution-free prediction sets with marginal coverage guarantees; however, its practical effectiveness depends critically on the choice of nonconformity function. We introduce a CP method for ordinal classification based on the ranked probability score (RPS), a proper scoring rule defined over cumulative predictive distributions. Although it reflects ordinal risk quite naturally, it has largely been neglected in conformal ordinal prediction (COP). When used as a measure of nonconformity, RPS yields median-centered contiguous prediction sets by construction. The method is model-agnostic, supports both assessed and grouped ordered categorical outcomes, and permits efficient implementation compared to greedy interval selection procedures. Across multiple ordinal image and tabular datasets, RPS-based CP produces contiguous prediction sets and strikes a favorable balance between prediction set width and the magnitude of ordinal miscoverage relative to existing CP methods.

Joint prediction sets for multivariate time series should control a single event while adapting to cross-coordinate dependence. We study filtered conformal ellipsoids: a frozen state-space filter emits a one-step predictive mean and covariance, and split-conformal calibration is applied to the resulting Mahalanobis scores. The filter is used to choose the ellipsoid shape; conformal calibration chooses the scalar radius, so the construction benefits from a learned predictive covariance without relying on Gaussian tail probabilities for coverage. The main difficulty is that filtered scores are dependent and learned recurrent filters need not contract in their raw hidden state; we therefore analyse contraction in an observable predictive-law quotient that identifies hidden states producing the same future sequence of emitted Gaussian laws. Under a stable Bayes Gaussian-projection filter, covariance bounds, and a finite-horizon observability Fisher condition, small excess Gaussian negative log-likelihood implies contraction of the learned emitted laws. Combined with a threshold-autocovariance envelope this yields a Chebyshev-type approximate coverage bound for filtered split-conformal prediction under dependence; a sharper Bernstein-type bound requires an additional geometric-mixing concentration assumption. Under Gaussian oracle realisability we also obtain a near-oracle log-volume comparison within the class of conditionally valid Gaussian ellipsoid rules. We instantiate the framework with a GCN-GRU filter with diagonal-plus-low-rank covariance. On moderate-size graph-native traffic benchmarks (METRLA-$20$ and PEMSBAY-$50$), the learned filter gives sharper at-target ellipsoids than static-covariance and non-filter baselines; at full-graph scale and on non-graph-native datasets, factor and copula baselines can be stronger.

We study uncertainty quantification for aggregated forecasting tasks such as annual totals and year-over-year growth rates. We propose SA-MSCP, a simulation-augmented multi-step split conformal method that generates future paths from cross-validated residuals using a block bootstrap and constructs prediction intervals from empirical quantiles. Experiments show that SA-MSCP improves empirical coverage over a simulated-path baseline for aggregated and growth-rate targets. Our results demonstrate that simulation-enhanced conformal calibration is an effective and general framework for uncertainty quantification in aggregated time-series forecasting.

Conformal prediction gives prediction intervals with finite-sample coverage when the data are exchangeable. Many time-indexed datasets are not exchangeable. They have seasons, recurring regimes, changing frequencies, or other forms of structured dependence. This paper studies a simple way to use that structure. We propose spectral adaptive conformal prediction, a method that forms weighted conformal quantiles using local spectral similarity and then updates the target miscoverage level online. The spectral weights choose calibration residuals that look relevant to the current test point. The adaptive update corrects the long-run miss rate when uncertainty changes over time. We give an approximate coverage result for the fixed spectral weighted quantile and a deterministic long-run calibration result for the adaptive update. Simulations with recurring regimes and slowly changing frequencies, together with three U.S. real-data examples, show that the hybrid method can improve on fixed spectral weighting, while also showing that spectral weighting must be monitored through effective sample size diagnostics.