Conformal Prediction

Federated learning (FL) faces challenges in uncertainty quantification (UQ). Without reliable UQ, FL systems risk deploying overconfident models at under-resourced agents, leading to silent local failures despite seemingly satisfactory global performance. Existing federated UQ approaches often address data heterogeneity or model heterogeneity in isolation, overlooking their joint effect on coverage reliability across agents. Conformal prediction is a widely used distribution-free UQ framework, yet its applications in heterogeneous FL settings remains underexplored. We provide FedWQ-CP, a simple yet effective approach that balances empirical coverage performance with efficiency at both global and agent levels under the dual heterogeneity. FedWQ-CP performs agent-server calibration in a single communication round. On each agent, conformity scores are computed on calibration data and a local quantile threshold is derived. Each agent then transmits only its quantile threshold and calibration sample size to the server. The server simply aggregates these thresholds through a weighted average to produce a global threshold. Experimental results on seven public datasets for both classification and regression demonstrate that FedWQ-CP empirically maintains agent-wise and global coverage while producing the smallest prediction sets or intervals.

Gradient-boosted decision trees are among the strongest off-the-shelf predictors for tabular regression, but point predictions alone do not quantify uncertainty. Conformal prediction provides distribution-free marginal coverage, yet standard split conformal uses a single global residual quantile and can adapt poorly to heteroscedasticity. We propose LoBoost, a model-native local conformal method that reuses the fitted ensemble's leaf structure to define a multiscale partition of the feature space. Each input is represented by the sequence of leaves it visits along the boosting path, and matching leaf prefixes define nested groups in which residual quantiles are estimated locally. By reusing the predictive structure already learned by the model, LoBoost requires no auxiliary partition or nuisance model, no retraining, and no additional data split beyond standard conformal calibration. Our theory connects the stability of the fitted ensemble and the geometry of its induced cells to local residual-score homogeneity, providing finite-sample coverage-error control and asymptotic pointwise validity as the local calibration size grows and the cells become sufficiently homogeneous. Experiments show competitive interval quality, low post-hoc calibration costs, and stable behavior across local calibration-size settings.

A central challenge in scientific machine learning (ML) is the correct representation of physical systems governed by multi-regime behaviours. In these scenarios, standard data analysis techniques often fail to capture the nature of the data, as the system's response varies significantly across the state space due to its stochasticity and the different physical regimes. Uncertainty quantification (UQ) should thus not be viewed merely as a safety assessment, but as a support to the learning task itself, guiding the model to internalise the behaviour of the data. We address this by focusing on the Critical Heat Flux (CHF) benchmark and dataset presented by the OECD/NEA Expert Group on Reactor Systems Multi-Physics. This case study represents a test for scientific ML due to the non-linear dependence of CHF on the inputs and the existence of distinct microscopic physical regimes. These regimes exhibit diverse statistical profiles, a complexity that requires UQ techniques to internalise the data behaviour and ensure reliable predictions. In this work, we conduct a comparative analysis of UQ methodologies to determine their impact on physical representation. We contrast post-hoc methods, specifically conformal prediction, against end-to-end coverage-oriented pipelines, including (Bayesian) heteroscedastic regression and quality-driven losses. These approaches treat uncertainty not as a final metric, but as an active component of the optimisation process, modelling the prediction and its behaviour simultaneously. We show that while post-hoc methods ensure statistical calibration, coverage-oriented learning effectively reshapes the model's representation to match the complex physical regimes. The result is a model that delivers not only high predictive accuracy but also a physically consistent uncertainty estimation that adapts dynamically to the intrinsic variability of the CHF.

Hyperdimensional Computing (HDC) offers a computationally efficient paradigm for neuromorphic learning. Yet, it lacks rigorous uncertainty quantification, leading to open decision boundaries and, consequently, vulnerability to outliers, adversarial perturbations, and out-of-distribution inputs. To address these limitations, we introduce ConformalHDC, a unified framework that combines the statistical guarantees of conformal prediction with the computational efficiency of HDC. For this framework, we propose two complementary variations. First, the set-valued formulation provides finite-sample, distribution-free coverage guarantees. Using carefully designed conformity scores, it forms enclosed decision boundaries that improve robustness to non-conforming inputs. Second, the point-valued formulation leverages the same conformity scores to produce a single prediction when desired, potentially improving accuracy over traditional HDC by accounting for class interactions. We demonstrate the broad applicability of the proposed framework through evaluations on multiple real-world datasets. In particular, we apply our method to the challenging problem of decoding non-spatial stimulus information from the spiking activity of hippocampal neurons recorded as subjects performed a sequence memory task. Our results show that ConformalHDC not only accurately decodes the stimulus information represented in the neural activity data, but also provides rigorous uncertainty estimates and correctly abstains when presented with data from other behavioral states. Overall, these capabilities position the framework as a reliable, uncertainty-aware foundation for neuromorphic computing.

Given a black-box AI system and a task, at what confidence level can a practitioner trust the system's output? We answer with a reliability level -- a single number per system-task pair, derived from self-consistency sampling and conformal calibration, that serves as a black-box deployment gate with exact, finite-sample, distribution-free guarantees. Self-consistency sampling reduces uncertainty exponentially; conformal calibration guarantees correctness within 1/(n+1) of the target level, regardless of the system's errors -- made transparently visible through larger answer sets for harder questions. Weaker models earn lower reliability levels (not accuracy -- see Definition 2.4): GPT-4.1 earns 94.6% on GSM8K and 96.8% on TruthfulQA, while GPT-4.1-nano earns 89.8% on GSM8K and 66.5% on MMLU. We validate across five benchmarks, five models from three families, and both synthetic and real data. Conditional coverage on solvable items exceeds 0.93 across all configurations; sequential stopping reduces API costs by around 50%.

The recent developments of complex deep learning models have led to unprecedented ability to accurately predict across multiple data representation types. Conformal prediction for uncertainty quantification of these models has risen in popularity, providing adaptive, statistically-valid prediction sets. For classification tasks, conformal methods have typically focused on utilizing logit scores. For pre-trained models, however, this can result in inefficient, overly conservative set sizes when not calibrated towards the target task. We propose DANCE, a doubly locally adaptive nearest-neighbor based conformal algorithm combining two novel nonconformity scores directly using the data's embedded representation. DANCE first fits a task-adaptive kernel regression model from the embedding layer before using the learned kernel space to produce the final prediction sets for uncertainty quantification. We test against state-of-the-art local, task-adapted and zero-shot conformal baselines, demonstrating DANCE's superior blend of set size efficiency and robustness across various datasets.

Concept drift -- the change of the distribution over time -- poses significant challenges for learning systems and is of central interest for monitoring. Understanding drift is thus paramount, and drift localization -- determining which samples are affected by the drift -- is essential. While several approaches exist, most rely on local testing schemes, which tend to fail in high-dimensional, low-signal settings. In this work, we consider a fundamentally different approach based on conformal predictions. We discuss and show the shortcomings of common approaches and demonstrate the performance of our approach on state-of-the-art image datasets.

The merit of Conformal Prediction (CP), as a distribution-free framework for uncertainty quantification, depends on generating prediction sets that are efficient, reflected in small average set sizes, while adaptive, meaning they signal uncertainty by varying in size according to input difficulty. A central limitation for deep conformal classifiers is that the nonconformity scores are derived from softmax outputs, which can be unreliable indicators of how certain the model truly is about a given input, sometimes leading to overconfident misclassifications or undue hesitation. In this work, we argue that this unreliability can be inherited by the prediction sets generated by CP, limiting their capacity for adaptiveness. We propose a new approach that leverages information from the pre-softmax logit space, using the Helmholtz Free Energy as a measure of model uncertainty and sample difficulty. By reweighting nonconformity scores with a monotonic transformation of the energy score of each sample, we improve their sensitivity to input difficulty. Our experiments with four state-of-the-art score functions on multiple datasets and deep architectures show that this energy-based enhancement improves the adaptiveness of the prediction sets, leading to a notable increase in both efficiency and adaptiveness compared to baseline nonconformity scores, without introducing any post-hoc complexity.

Quantifying uncertainty in clinical predictions is critical for high-stakes diagnosis tasks. Conformal prediction offers a principled approach by providing prediction sets with theoretical coverage guarantees. However, in practice, patient distribution shifts violate the i.i.d. assumptions underlying standard conformal methods, leading to poor coverage in healthcare settings. In this work, we evaluate several conformal prediction approaches on EEG seizure classification, a task with known distribution shift challenges and label uncertainty. We demonstrate that personalized calibration strategies can improve coverage by over 20 percentage points while maintaining comparable prediction set sizes. Our implementation is available via PyHealth, an open-source healthcare AI framework: https://github.com/sunlabuiuc/PyHealth.

We propose PRISM-FCP (Partial shaRing and robust calIbration with Statistical Margins for Federated Conformal Prediction), a communication-efficient Byzantine-robust federated conformal prediction framework that uses partial model sharing to mitigate stochastic model-poisoning attacks during training and histogram-based filtering to mitigate adversarial calibration submissions. Existing robust FCP approaches primarily address adversarial behavior during calibration, leaving training-stage poisoning to separate robust-training mechanisms. PRISM-FCP instead considers the coupling between the two stages. During training, clients partially share updates by transmitting only $M$ of $D$ parameters per round. Under the stated stochastic attack model, this attenuates the expected energy of each Byzantine client's perturbation contribution to the aggregate by a factor of $M/D$ relative to full sharing. When this benefit outweighs the optimization slowdown caused by partial updates, it can reduce training error and improve interval efficiency. During calibration, the server uses client-provided characterization vectors to filter suspected Byzantine clients before estimating the conformal quantile from the retained clients. Experiments on synthetic benchmarks and the UCI Superconductivity and YearPredictionMSD datasets, including Gaussian, ALIE, and sign-flipping training-stage attacks, demonstrate near-nominal empirical coverage and favorable communication--performance tradeoffs in the studied settings.

Conformal prediction gives exact finite-sample coverage guarantees under exchangeability, but deployed systems are judged by more than coverage alone. For a fixed calibrated rule reused over a finite operational window, stakeholders also care about deployment-facing quantities such as commitment frequency, deferral, and decisive error exposure. These are not determined by coverage: calibration choices with similar coverage can still induce materially different operational profiles. We study this characterization gap in a scoped setting: binary split conformal prediction under exchangeability with a fixed deployed rule. We introduce the Small-Sample Beta Correction (SSBC) which gives finite-sample coverage semantics for the deployed rule: it inverts the Beta/Beta--Binomial law governing calibration-conditional coverage to map a user request $(α^\star,δ)$ to the least conservative calibration grid point with calibration-conditional PAC semantics for the realized deployed rule. Calibrate-and-Audit then fixes the rule by calibration and uses an independent audit split to estimate the induced region--class label table, a reusable summary from which deployment-facing Key Performance Indicators (KPIs) follow by projection. Under this design, fixed operational rates admit exact finite-sample Binomial inference, while Beta--Binomial envelopes serve as practical predictive summaries for future windows. The induced partition also exposes regime boundaries, Pareto-relevant tradeoffs, and inverse-pricing questions for fixed downstream conventions. Simulations validate the SSBC semantics and compare audit-based summaries with leave-one-out planning proxies; molecular toxicity data provide an audit-based empirical example, and a solubility case study illustrates scenario planning once coverage semantics are fixed.

Medical vision-language models (VLMs) are strong zero-shot recognizers for medical imaging, but their reliability under domain shift hinges on calibrated uncertainty with guarantees. Split conformal prediction (SCP) offers finite-sample coverage, yet prediction sets often become large (low efficiency) and class-wise coverage unbalanced-high class-conditioned coverage gap (CCV), especially in few-shot, imbalanced regimes; moreover, naively adapting to calibration labels breaks exchangeability and voids guarantees. We propose **LATA** (Laplacian-Assisted Transductive Adaptation), a training- and label-free refinement that operates on the joint calibration and test pool by smoothing zero-shot probabilities over an image-image k-NN graph using a small number of CCCP mean-field updates, preserving SCP validity via a deterministic transform. We further introduce a failure-aware conformal score that plugs into the vision-language uncertainty (ViLU) framework, providing instance-level difficulty and label plausibility to improve prediction set efficiency and class-wise balance at fixed coverage. **LATA** is black-box (no VLM updates), compute-light (windowed transduction, no backprop), and includes an optional prior knob that can run strictly label-free or, if desired, in a label-informed variant using calibration marginals once. Across three medical VLMs and nine downstream tasks, **LATA** consistently reduces set size and CCV while matching or tightening target coverage, outperforming prior transductive baselines and narrowing the gap to label-using methods, while using far less compute. Comprehensive ablations and qualitative analyses show that **LATA** sharpens zero-shot predictions without compromising exchangeability.

Conformal prediction (CP) offers distribution-free uncertainty quantification for machine learning models, yet its interplay with fairness in downstream decision-making remains underexplored. Moving beyond CP as a standalone operation (procedural fairness), we analyze the holistic decision-making pipeline to evaluate substantive fairness-the equity of downstream outcomes. Theoretically, we derive an upper bound that decomposes prediction-set size disparity into interpretable components, clarifying how label-clustered CP helps control method-driven contributions to unfairness. To facilitate scalable empirical analysis, we introduce an LLM-in-the-loop evaluator that approximates human assessment of substantive fairness across diverse modalities. Our experiments show that label-clustered CP often provides a favorable balance between utility and substantive fairness, while reducing set-size disparities in line with our theory. Finally, we empirically show that equalized set sizes, rather than coverage, strongly correlate with improved substantive fairness, enabling practitioners to design more fair CP systems. Our code is available at https://github.com/layer6ai-labs/llm-in-the-loop-conformal-fairness.

We study online conformal prediction for non-stationary data streams subject to unknown distribution drift. While most prior work studied this problem under adversarial settings and/or assessed performance in terms of gaps of time-averaged marginal coverage, we instead evaluate performance through training-conditional cumulative regret. We specifically focus on independently generated data with two types of distribution shift: abrupt change points and smooth drift. When non-conformity score functions are pretrained on an independent dataset, we propose a split-conformal style algorithm that leverages drift detection to adaptively update calibration sets, which provably achieves minimax-optimal regret. When non-conformity scores are instead trained online, we develop a full-conformal style algorithm that again incorporates drift detection to handle non-stationarity; this approach relies on stability - rather than permutation symmetry - of the model-fitting algorithm, which is often better suited to online learning under evolving environments. We establish non-asymptotic regret guarantees for our online full conformal algorithm, which match the minimax lower bound under appropriate restrictions on the prediction sets. Numerical experiments corroborate our theoretical findings.

Conformal prediction gives finite-sample coverage guarantees for regression, but most standard constructions are designed for Euclidean output spaces. When the response lies on a Riemannian manifold, Euclidean residuals and coordinate-based regions can ignore the geometry that defines meaningful error. We propose adaptive geodesic conformal prediction, a simple framework that builds nonconformity scores from geodesic distances and normalizes them with a cross-validated estimate of local prediction difficulty. On the sphere, this produces geodesic caps whose area is independent of position, while their radii still adapt to heteroscedastic noise. In both a synthetic sphere experiment and an IGRF-14 geomagnetic field forecasting task, the adaptive method preserves valid marginal coverage, reduces variation in conditional coverage, and improves worst-case coverage relative to non-adaptive and coordinate-based baselines.

Conformal prediction (CP) offers distribution-free marginal coverage guarantees under an exchangeability assumption, but these guarantees can fail if the data distribution shifts. We analyze the use of pseudo-calibration as a tool to counter this performance loss under a bounded label-conditional covariate shift model. Using tools from domain adaptation, we derive a lower bound on target coverage in terms of the source-domain loss of the classifier and a Wasserstein measure of the shift. Using this result, we provide a method to design pseudo-calibrated sets that inflate the conformal threshold by a slack parameter to keep target coverage above a prescribed level. Finally, we propose a source-tuned pseudo-calibration algorithm that interpolates between hard pseudo-labels and randomized labels as a function of classifier uncertainty. Numerical experiments show that our bounds qualitatively track pseudo-calibration behavior and that the source-tuned scheme mitigates coverage degradation under distribution shift while maintaining nontrivial prediction set sizes.

Accurate production forecasts are essential for the integration of renewable energy sources into the power grid. This paper illustrates how to obtain probabilistic forecasts of wind power generation using gradient boosting trees and an ensemble of weather forecasts. To this end, we perform a comparative analysis across three state-of-the-art probabilistic prediction methods-conformalized quantile regression, natural gradient boosting and conditional diffusion models-all of which can be combined with tree-based machine learning. The methods are validated using four years of data for all Belgian offshore wind farms. We benchmark the models against the power curve and a calibrated wake model as well as a probabilistic method using stochastic variational Gaussian process regression. The tree-based models significantly reduce the mean absolute error in comparison to the deterministic baselines. Additionally, all three methods outperform the Gaussian process baseline in probabilistic skill, while two out of the three also improve point forecast accuracy. The conditional diffusion model attains the best performance, with improvements of 5% in mean absolute error and 12% in continuous rank probability score compared to the probabilistic baseline. Last, the results indicate an average improvement in point forecast accuracy of 17% by using an ensemble of weather forecasts instead of a single provider.

Current instance segmentation models achieve high performance on average predictions, but lack principled uncertainty quantification: their outputs are not calibrated, and there is no guarantee that a predicted mask is close to the ground truth. To address this limitation, we introduce a conformal prediction algorithm to generate adaptive confidence sets for instance segmentation. Given an image and a pixel coordinate query, our algorithm generates a confidence set of instance predictions for that pixel, with a provable guarantee for the probability that at least one of the predictions has high Intersection-Over-Union (IoU) with the true object instance mask. We apply our algorithm to instance segmentation examples in agricultural field delineation, cell segmentation, and vehicle detection. Empirically, we find that our prediction sets vary in size based on query difficulty and attain the target coverage, outperforming baselines (naive best parameter and morphological dilation-based methods). We provide versions of the algorithm with asymptotic and finite sample guarantees. Our work is the first to capture structural uncertainty in instance segmentation by constructing confidence sets of diverse segmentation predictions.

Prediction-set size offers a coarse summary of conformal uncertainty. We develop a richer perspective through the geometry of credal sets: families of probability distributions compatible with conformal prediction (CP) regions. We establish this representation for split CP, extending a correspondence recently identified for full CP. Building on this representation, we derive analytical characterisations and exact algorithms for several widely used credal uncertainty measures, reducing potentially exponential computations to linear time after sorting the conformal p-values. We further unify several of these measures with classical conformal summaries by integrating set-size penalties based on Tsallis entropies across error levels. We also prove an impossibility theorem: for common split-conformal regression constructions, uncertainty summaries of the induced credal sets cannot distinguish between inputs. Classification experiments on selective prediction and distribution-shift detection show that these summaries often recover useful ranking information lost by prediction-set size at a fixed error level, enabling finer-grained uncertainty assessment. These results establish credal geometry as a principled framework for quantifying conformal uncertainty and understanding its limitations.

Full conformal prediction is a framework that implicitly formulates distribution-free confidence prediction regions for a wide range of estimators. However, a classical limitation of the full conformal framework is the computation of the confidence prediction regions, which is usually impossible since it requires training infinitely many estimators (for real-valued prediction for instance). The main purpose of the present work is to describe a generic strategy for designing a tight approximation to the full conformal prediction region that can be efficiently computed. Along with this approximate confidence region, a theoretical quantification of the tightness of this approximation is developed, depending on the smoothness assumptions on the loss and score functions. The new notion of thickness is introduced for quantifying the discrepancy between the approximate confidence region and the full conformal one.

In many fairness and distribution robustness problems, one has access to labeled data from multiple source distributions yet the test data may come from an arbitrary member or a mixture of them. We study the problem of constructing a conformal prediction set that is uniformly valid across multiple, heterogeneous distributions, in the sense that no matter which distribution the test point is from, the coverage of the prediction set is guaranteed to exceed a pre-specified level. We first propose a max-p aggregation scheme that delivers finite-sample, multi-distribution coverage given any conformity scores associated with each distribution. Upon studying several efficiency optimization programs subject to uniform coverage, we prove the optimality and tightness of our aggregation scheme, and propose a general algorithm to learn conformity scores that lead to efficient prediction sets after the aggregation under standard conditions. We discuss how our framework relates to group-wise distributionally robust optimization, sub-population shift, fairness, and multi-source learning. In synthetic and real-data experiments, our method delivers valid worst-case coverage across multiple distributions while greatly reducing the set size compared with naively applying max-p aggregation to single-source conformity scores, and can be comparable in size to single-source prediction sets with popular, standard conformity scores.

Conformal prediction (CP) constructs uncertainty sets for model outputs with finite-sample coverage guarantees. Yet ranking scores is straightforward only when they are scalar-valued, limiting CP to real-valued scores or ad-hoc one-dimensional reductions. Vector-valued scores arise naturally in multi-output regression and model aggregation, where each predictor in an ensemble provides its own score. Optimal transport (OT) defines vector ranks and center-outward multivariate quantile regions, though generally with asymptotic coverage guarantees. Applying a fixed transport map learned from calibration data to a new point introduces an uncontrolled approximation error. We restore finite-sample, distribution-free coverage by conformalizing vector-valued OT quantile regions. Each candidate's rank is defined by transporting the calibration scores augmented with that candidate's score, preserving the symmetry needed for validity. This appears to require a continuum of OT problems. However, we prove that the optimal assignment is piecewise constant across a fixed polyhedral partition of score space. This lets us characterize the entire prediction set in $O(n^3)$ time, matching the cost of a single assignment solve. It also addresses a limitation of prediction sets: they indicate which outcomes are plausible, but not their relative likelihood. In one dimension, conformal predictive distributions (CPDs) fill this gap by producing a predictive distribution with finite-sample calibration. Extending CPDs beyond one dimension remained an open problem. We construct, to our knowledge, the first multivariate CPDs with finite-sample calibration: a center-outward predictive distribution whose derived uncertainty regions have conformal coverage. We present both conservative and exact randomized versions; the latter generalizes the classical Dempster-Hill procedure.

Conformal prediction has emerged as a powerful framework for constructing distribution-free prediction sets with guaranteed coverage assuming only the exchangeability assumption. However, this assumption is often violated in online environments where data distributions evolve over time. Several recent approaches have been proposed to address this limitation, but, typically, they slowly adapt to distribution shifts because they update predictions only in a forward manner, that is, they generate a prediction for a newly observed data point while previously computed predictions are not updated. In this paper, we propose a novel online conformal inference method with retrospective adjustment, which is designed to achieve faster adaptation to distributional shifts. Our method leverages regression approaches with efficient leave-one-out update formulas to retroactively adjust past predictions when new data arrive, thereby aligning the entire set of predictions with the most recent data distribution. Through extensive numerical studies performed on both synthetic and real-world data sets, we show that the proposed approach achieves coverage close to the nominal level while reducing predictive interval width by up to approximately 30% compared to existing online conformal prediction methods, demonstrating improved statistical efficiency alongside faster adaptation.

This article considers an online version of conformal inference, called adaptive conformal inference [ACI] and introduced by Gibbs and Candès (2021): prediction sets are issued sequentially, after observing features and before the outcomes are revealed. These sets are evaluated both in terms of validity (the fraction of rounds where the outcome was lying in the prediction set) and efficiency (the average lengths of the prediction sets). The two criteria point to different directions (validity favors larger sets). We also target a wide range of scenarios, with exchangeable data and arbitrary data (lack of any stochastic guarantees) as two extremes. A series of existing strategies for ACI typically guarantee that empirical coverage converges to the desired level for arbitrary sequences, but they generally lack simultaneous efficiency guarantees. To provide a unified study, we first formulate ACI as a repeated two-player game with finite action sets and vector-valued payoffs encoding validity and efficiency. Building on this reformulation, we introduce a strategy based on Blackwell approachability and on its opportunistic extension by Bernstein et al. (2014) that ensures validity while adapting the efficiency of the prediction intervals to the underlying degree of stochasticity of the opponent player. The resulting guarantee is "best of many worlds": it recovers the relevant efficiency guarantees in exchangeable and adversarial settings, and provides guarantees in intermediate settings that arise in typical applications such as the forecasting of time series.

Active Test-Time Adaptation (ATTA) improves model robustness under domain shift by selectively querying human annotations at deployment, but existing methods use heuristic uncertainty measures and suffer from low data selection efficiency, wasting human annotation budget. We propose Conformal Prediction Active TTA (CPATTA), which first brings principled, conformal uncertainty with coverage-aware online calibration into ATTA. CPATTA employs smoothed conformal scores with a top-$K$ certainty measure, an online weight-update algorithm driven by pseudo coverage, a domain-shift detector that adapts human supervision, and a staged update scheme that balances human-labeled and model-labeled data. Extensive experiments demonstrate that CPATTA consistently outperforms the state-of-the-art ATTA methods by around 5% in accuracy.

In high-stakes scenarios, such as medical imaging applications, it is critical to equip the predictions of a regression model with reliable confidence intervals. Recently, Conformal Prediction (CP) has emerged as a powerful statistical framework that, based on a labeled calibration set, generates intervals that include the true labels with a pre-specified probability. In this paper, we address the problem of applying CP for regression models when the calibration set contains noisy labels. We begin by establishing a mathematically grounded procedure for estimating the noise-free CP threshold. Then, we turn it into a practical algorithm that overcomes the challenges arising from the continuous nature of the regression problem. We evaluate the proposed method on two medical imaging regression datasets with Gaussian label noise. Our method significantly outperforms the existing alternative, achieving performance close to the clean-label setting.

Off-policy evaluation (OPE) methods estimate the value of a new reinforcement learning (RL) policy prior to deployment. Recent advances have shown that leveraging auxiliary datasets, such as those synthesized by generative models, can improve the accuracy of OPE methods. Unfortunately, such auxiliary datasets may also be biased, and existing methods for using data augmentation within OPE lack principled uncertainty quantification. In high stakes domains like healthcare, reliable uncertainty estimates are important for ensuring safe and informed deployment of RL policies. In this work, we propose two methods to construct valid confidence intervals for OPE with data augmentation. The first provides a confidence interval over $V^π(s)$, the policy value conditioned on an initial state $s$. To do so we introduce a new conformal prediction method suitable for Markov Decision Processes (MDPs) with continuous state spaces, extending prior work to higher-dimensional settings. Second, we consider the more common task of estimating the average policy performance over all initial states, $V^π$; we introduce a method that draws on ideas from doubly robust estimation and prediction powered inference. Across simulators spanning inventory management, robotics, healthcare, and a real healthcare dataset from MIMIC-IV, we find that our methods can effectively leverage auxiliary data and consistently produce confidence intervals that cover the ground truth policy values, unlike previously proposed methods. Our work enables a future in which OPE can provide rigorous uncertainty estimates for high-stakes domains.

This paper introduces a framework for uncertainty quantification in regression models defined on metric spaces. Using a proposed notion of homoscedasticity, we define a conformal prediction algorithm that provides finite-sample marginal coverage guarantees and fast convergence rates to the oracle prediction region. For heteroscedastic settings, we introduce a kNN procedure that yields locally adaptive prediction radii in general metric spaces. Although this procedure does not provide the same finite-sample guarantees as the conformal algorithm, it is designed to improve local coverage calibration without imposing smoothing assumptions. Both procedures are compatible with a broad range of regression algorithms and scale to large datasets, allowing practitioners to use their preferred models and incorporate domain-specific knowledge. Building on the heteroscedastic $k$NN approach, we also develop a flexible sequential extension for metric-space-valued time series based on nearest-neighbor expert aggregation. We establish the consistency of the proposed estimators under minimal conditions. Finally, we illustrate the practical utility of our framework in personalized medicine applications involving random objects such as probability distributions and graph Laplacians.

Regression problems with continuous bounded outcomes frequently arise in statistical and machine learning applications, such as the analysis of rates and proportions. A central challenge in this setting is predicting the response at a new covariate value. Most of the existing literature has focused either on point prediction or on interval prediction based on asymptotic approximations. We develop conformal prediction intervals for bounded outcomes within the framework of transformation regression models, encompassing widely used models such as beta regression and logit-normal regression. We construct non-conformity scores based on model-aligned residuals and identify a quantile-residual score that is particularly well suited to bounded outcomes, bridging normalized conformal prediction and distributional conformal prediction. This score accounts for both the heteroscedasticity inherent in such data and the asymmetry that emerges near the boundaries of the response space. We establish marginal validity and asymptotic conditional validity for both full and split conformal prediction, holding under model misspecification. A comprehensive simulation study confirms that both methods empirically attain valid finite-sample coverage, including cases under model misspecification. A real-data application demonstrates their practical performance against bootstrap-based alternatives.

Conformal prediction (CP) is a widely used frequentist framework to quantify uncertainty by constructing prediction sets with user-specified marginal coverage guarantees. In practice, CP is typically applied on top of probabilistic classifiers, which are able to express aleatoric but not epistemic uncertainty. In this paper, we consider the question of how to optimally employ CP on top of a more expressive formalism, namely credal sets, which can express both aleatoric and epistemic uncertainty. More specifically, we propose probabilistic Bernoulli prediction sets (BPS) and derive a variant that achieves conditional coverage for valid credal sets while remaining minimal in expected size. We then address the more realistic scenario in which the validity of the credal sets is not guaranteed. Assuming access to calibration data with ground-truth distributions over labels, we apply conformal risk control to BPS and derive a PAC-style guarantee: with high probability over the data, the achieved conditional coverage is at least the desired level. We validate our theoretical findings empirically over various datasets.