Conformal Inference

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

Many learning systems return a single point estimate even when admissible outputs form disconnected or non-convex sets. Under squared loss, an ambiguous conditional distribution can therefore have a Bayes-optimal conditional mean that is invalid. We formalize this failure as conditional mean collapse and introduce Settling, an equilibrium-based inference operator that separates proposal generation, consistency evaluation, and test-time equilibrium selection. The operator treats a mean-seeking proposal as an initialization and refines it toward a locally stable configuration; conditional on initialization, refinement is deterministic. We establish exact-gradient descent, local convergence, and an inexact-gradient robustness condition relevant to learned consistency critics. In a reproducible 100-context geometric diagnostic, the mean-seeking baseline succeeds in 0/100 contexts, stochastic denoising in 100/100, and Settling in 99/100 while producing substantially lower trajectory roughness. A 1,200-run sensitivity study yields 97-100% success across obstacle-jitter ranges up to 0.20 and 94-100% across one-time initialization perturbations from 0.05 to 0.50. Cross-domain panels remain mechanism illustrations; learned high-dimensional validation remains an open empirical test.

Non-intrusive reduced-order models (NIROMs) have become a standard tool for approximating parametric partial differential equations from computer design of experiments while significantly reducing computational costs. However, assessing the reliability of their predictions remains a major challenge, particularly in extrapolation regimes or under limited training data. In this work, we introduce a framework for quantifying model-form uncertainty in NIROMs by combining a perturbative stochastic representation of reduced bases with distribution-free conformal-type methods. Starting from a deterministic reduced basis constructed from snapshot matrices, we model uncertainty through random perturbations defined on the Stiefel manifold, directed along the discarded modes, yielding stochastic reduced-order approximations whose induced variance reflects the basis-truncation error. A transport approximation gives a closed-form posterior variance that separates basis-induced from regression-induced uncertainty, without re-training the underlying Gaussian processes. We include this posterior variance within a conformal risk control calibration framework, that provides prediction sets with coordinate miscoverage guarantees. The calibration factor produced by this framework is itself an interpretable, scalar diagnostic of the quality of the uncertainty estimate. The methodology is evaluated on parametric PDE benchmarks and an industrial tire-manufacturing calendering process. Numerical experiments demonstrate reliable, locally informative uncertainty quantification that goes beyond the Gaussian predictive variance.

Standard model comparison is global, aggregating losses across the covariate space to declare a single winner. This can obscure heterogeneous performance, where different models are preferable in different regions. We introduce conformalized local model comparison, a split-sample framework for constructing calibrated local best-model maps. Given a model comparison score, such as the difference between two squared losses, the method uses three disjoint splits to fit competing models, estimate local centers and scales from out-of-sample scores, and conformally calibrate residual uncertainty. At a target point, the procedure declares a local winner only when a one-sided conformal bound excludes a tie, with the score's sign determining the favored model. We prove finite-sample marginal control for one-sided erroneous declarations on the realized future comparison score, establish pointwise consistency of the localized mean-score estimator away from tie boundaries, show that aggregate comparison can disagree sharply with the prevalence of local superiority, and derive a squared-loss bias--variance decomposition that clarifies how model structure affects local wins. Synthetic and real-data experiments show that the method recovers heterogeneous winner regions, abstains under uncertainty, and yields higher conditional gain than global selection.

Medical vision-language models (VLMs) can retain high observed marginal coverage after clinical shift while substantially under-covering an individual disease class. The affected class varies with acquisition protocol and backbone geometry, so source prevalence does not reliably reveal the failure. Existing localized and tail-aware conformal methods respectively adapt to test neighborhoods and source-frequency tails, leaving held-out class-wise coverage failure unmodeled. We introduce Class-Tail Adaptive Localized Conformal Deferral (CALCoDe), a post-hoc reliability layer for frozen medical VLMs. Cross-fitted validation predictions identify classes at risk of undercoverage, and a disjoint calibration split estimates their class-conditional tail thresholds. CALCoDe combines each protected threshold with a localized conformal threshold using a one-sided maximum. The resulting set contains every label admitted by the localized rule, with additional protection confined to validation-identified classes. An independently calibrated support audit defers cases with insufficient inlier support. Under exchangeability among accepted examples within each protected class, CALCoDe provides finite-sample coverage at the prespecified guard level and contains the corresponding localized conformal sets; coverage on shifted external cohorts is evaluated empirically. Among standard conformal baselines and recent VLM-specific conformal methods evaluated across two dermatology shifts (HAM10000 to ISIC 2019 and HAM10000 to PAD-UFES-20) and four frozen VLM backbones (BiomedCLIP, OpenAI CLIP ViT-B/32, PubMedCLIP ViT-B/32, and MedSigLIP-448), CALCoDe is the only approach whose observed marginal and worst-class accepted coverage both reach 0.95 in all eight settings. On HAM10000 to ISIC 2019, its average worst-class accepted coverage is 0.970, compared with 0.926 for sTACP and 0.864 for LCP-VLM.

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.

Bayesian online learning promises uncertainty-aware prediction on data streams, but its performance hinges on inferential choices, including learning rates, prior distributions and variational families, which are usually fixed before seeing the stream. We address this by treating Bayesian update rules as experts and aggregating the Bayesian experts according to sequential predictive losses. We prove that the resulting aggregate competes with the best expert in hindsight at an aggregation cost determined by how each expert's per-round performance is evaluated. We instantiate the framework in online conformal inference and Gaussian process regression. The conformal inference application yields a smoothed Bayesian counterpart of adaptive conformal inference with long-run randomized coverage, while the Gaussian process application gives an oracle inequality in cumulative predictive Kullback-Leibler risk and adaptation to unknown Hölder smoothness up to logarithmic factors. Experiments show that the aggregate tracks strong experts without oracle expert selection.

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.

We propose doubly robust adaptive conformal inference (DR-ACI), which constructs prediction intervals for doubly robust pseudo-outcomes under temporal dependence.

Rejecting inputs outside the defined in-distribution (IND) service scope is critical for large language model (LLM) services, where unsupported requests should be filtered before full generation. Existing out-of-distribution (OOD) detectors often rely on final outputs or final-layer representations, leaving unclear where service-boundary signals are most clearly encoded inside the model; they also lack a theoretical guarantee for held-out inputs. In this paper, we introduce SCOPE (Sequential Conformal OOD Probing and Evaluation), a framework that selects a readable hidden layer, constructs a conformal gate with IND calibration, and uses a supermartingale e-process to certify persistent service-boundary evidence. Experiments across multiple LLM backbones and six carefully designed boundary conditions show that SCOPE improves gate-level rejection over standard final-layer detectors, while revealing how different OOD boundaries take different geometric forms in hidden space.

Conformal Bayes combines Bayesian posterior predictives with conformal calibration to produce prediction sets that are both statistically valid and geometrically efficient. We study conformal Bayes under label shift from a unified perspective, identifying two complementary approaches that restore nominal target-domain coverage through importance-weighted conformal calibration but operate through independent mechanisms. \emph{Post-hoc calibration} tilts the posterior predictive toward the target domain and corrects the conformal threshold via an importance-weighted quantile, leaving the parameter posterior unchanged. \emph{In-training adaptation} tilts the parameter posterior itself to the target domain, producing a corrected predictive whose highest predictive density region serves as the highest predictive density (HPD)-based prediction set under the fitted target predictive; efficiency is model-dependent and does not imply finite-sample conditional optimality. Two controlled experiments isolate the regime-dependence of each strategy: in the low-dimensional, well-estimated regime StrategyA produces the narrowest valid intervals, while in the high-dimensional, underdetermined regime StrategyB achieves up to $43\%$ width reduction at unchanged coverage, under the stated source-sampling and label-shift assumptions.

Standard conformal prediction (CP) procedures are typically formulated in terms of p-values, but reliance on p-values alone limits flexibility, for example, when combining dependent evidence across models or data splits. Recent work has explored e-value formulations for conformal inference, yet a direct connection between p- and e-value formulations in CP has been missing, especially regarding their statistical efficiency. We first identify limitations of classical p-to-e calibrators in the CP setting, showing that they are not set-preserving and can lead to overly conservative prediction sets. To address this, we propose a novel P2E calibrator that converts conformal p-values into e-values without altering the prediction set induced by the original conformal p-value. We establish both theoretically and empirically that our calibrator can yield significant efficiency gains over existing p-to-e calibrators. This e-value formulation enables principled use of recent advances in e-value merging and randomization, where we demonstrate its impact in two applications: cross-conformal prediction (CCP), whose variants typically provide only approximate $1-2α$ coverage, and conformal aggregation (CA). In both cases, our e-value-based methods satisfy the desired $1-α$ coverage guarantee while improving efficiency over standard baselines. More broadly, our approach expands the flexibility of CP and opens new directions for efficient, distribution-free uncertainty quantification.

We present an online, distribution-free framework for controlling the Conditional Value-at-Risk (CVaR), extending conformal tail risk control to non-stationary and adversarial environments. Unlike classical risk control methods, which rely on stationarity or linearity of expectation, our approach provides provable safety guarantees for a nonlinear tail risk functional under arbitrary data-generating processes that may drift or shift strategically over time. By leveraging deep connections between conformal tail risk control, online learning, and the variational representation of CVaR introduced by Rockafellar and Uryasev, we develop a novel procedure for online CVaR control with adversarial regret guarantees. The proposed method operates without assumptions on the underlying data-generating process, making it broadly applicable in modern high-stakes deployment settings. We prove that the realized empirical CVaR is asymptotically controlled at the target level, and that the resulting control is asymptotically tight up to a finite-sample conservatism gap. We demonstrate the effectiveness of our approach on portfolio risk management and toxicity mitigation for Large Language Models (LLMs), where rare but catastrophic failures dominate system risk.

This paper addresses structured out-of-distribution (OOD) testing in high-stakes machine learning applications. Traditional conformal methods rely on joint exchangeability, making it difficult to incorporate auxiliary information such as spatiotemporal or grouping structures. To overcome this limitation, we propose the structure-adaptive conformal q-value (SCQ), a significance index that integrates individual test evidence with structural patterns. We also develop pseudo-score-guided transductive automated model selection (P-TAMS), which adapts conformalized model selection to structured OOD testing across a toolbox of candidate models. Together, SCQ and P-TAMS form a unified framework under pairwise exchangeability, providing finite-sample error-rate control, improved power, and enhanced interpretability. Experiments on simulated and real data demonstrate that the proposed approach controls the false discovery rate and performs well across diverse settings.

Recent advances in uncertainty quantification increasingly emphasise the distinction between aleatory and epistemic uncertainty in machine learning, motivating the need for more unified frameworks. However, despite much progress in producing reliable predictions, existing methods often lack rigorous guarantees when generalising beyond the training domain. We propose a conformalised imprecise inference framework for robust extrapolation, which is model-agnostic and augments predictive models with imprecision and distance awareness. The proposed approach yields imprecise predictions (probability boxes) that remain valid under distributional shift, maintaining coverage while adaptively expanding uncertainty in extrapolation regimes. Experiments on synthetic and benchmark datasets demonstrate improved robustness and reliable coverage compared to standard probabilistic approaches, particularly under limited data.

We study counterfactual distribution learning for high-dimensional outcomes whose counterfactual law may concentrate near lower-dimensional structure. Standard isotropic smoothing treats all ambient directions equally, leading to unfavorable scaling and unstable local inference. We propose two diffusion-guided estimators based on semiparametric debiasing: diffusion-informed smoothing for counterfactual densities and diffusion-informed score smoothing for counterfactual scores. The estimators combine causal nuisance adjustment with geometry-adaptive localization driven by diffusion score information, removing first-order nuisance bias while aligning smoothing with local outcome geometry. We establish asymptotic expansions, risk bounds, and inference procedures for smoothed density and score-based targets, with ambient density inference obtained under additional approximation conditions. Under structural geometry conditions, the leading stochastic error is governed by an effective dimension induced by the diffusion-guided kernel, rather than by the ambient dimension. Semi-synthetic experiments based on CelebA show steeper error decay for geometry-adaptive methods, supporting the proposed effective-dimension theory.

Estimating treatment effects from observational data requires choosing an adjustment set, but valid adjustment depends on an unknown causal graph. Graph misspecification can cause under-coverage, while graph-agnostic conformal wrappers may regain nominal coverage only through large padding. We introduce CausalGuard, a structure-weighted conformal framework that calibrates after aggregating graph-conditional doubly robust pseudo-outcomes. Candidate DAGs are proposed from an LLM-derived edge prior, pruned by conditional-independence tests, and reweighted by Bayesian Information Criterion. A composite nonconformity score then calibrates the posterior-weighted pseudo-outcome. CausalGuard provides distribution-free finite-sample marginal coverage for this aggregated pseudo-outcome; under causal identification, overlap, conditional-mean nuisance stability, and concentration on target-aligned valid adjustment strategies, its conditional mean converges to the true Conditional Average Treatment Effect. Across five benchmarks, CausalGuard attains mean coverage above the nominal 90% level for the directly evaluable target and reduces width when graph-agnostic conformal baselines require large padding. Stress tests show that CausalGuard suppresses invalid collider adjustment and remains stable under misspecified priors when the retained candidate set is data-supported.

Modern applications of conformal inference to multiple testing problems, such as outlier detection and candidate selection, often involve selecting test samples whose conformal p-values fall below a threshold. The quality of such methods is often measured by the false discovery proportion (FDP), defined as the fraction of incorrect selections. Existing approaches typically control the expected value of the FDP, using methods such as the Benjamini-Hochberg procedure. This approach fails to provide high-probability bounds on the realized false discovery proportion and invalidates statistical guarantees if the rejection threshold is selected after inspecting the data. This paper establishes finite-sample, distribution-free upper bounds on the FDP that hold simultaneously over all possible rejection thresholds, enabling arbitrary post hoc selection of the threshold. Simultaneous validity is achieved by constructing a high-probability envelope for the empirical distribution function of null conformal p-values by sampling from their joint distribution. Furthermore, our framework allows practitioners to modulate the envelope's shape, thereby producing tight bounds in rejection regions of primary interest. We use this flexible approach to derive simultaneous FDP upper bounds for both outlier detection and conformal selection. We demonstrate through synthetic and real-data experiments that the resulting bounds are both valid and substantially less conservative than those derived from existing approaches.

We adapt split conformal prediction and adaptive conformal inference (ACI) to continuous AI agent evaluation, providing distribution-free coverage guarantees for forecasted quality scores. Conformal intervals achieve calibration error below 0.02 across all nominal levels at the 24h horizon, while ACI correctly widens intervals by 35% following agent releases then reconverges. We further develop compositional uncertainty bounds for multi-agent pipelines (validated via simulation across inter-stage correlations rho in [-0.5, 0.9]), a conformal abstention rule for pairwise rankings with controlled false-ranking rate, and FDR-corrected abstention for leaderboard-scale multiple testing. Evaluating 50 agents via 18 real-time signals collected hourly, we show that per-agent conditional coverage is well-concentrated around the nominal level (mean 80.4%, 90% of agents within [72%, 90%]), and that cross-source sentiment divergence predicts ranking instability (r=0.64, p<0.01). A circularity-controlled validation confirms the framework captures signal beyond benchmarks (rho_s=0.52, p<0.01, n=35). Code and data are released under CC BY 4.0.

We address the problem of conformal selection, where an agent must select a minimal subset of options to ensure that at least one success'' is identified with a pre-specified target probability $φ$. While traditional online conformal prediction focuses on maintaining validity for the observed sequence, minimizing the resource cost (efficiency) of such selections, especially under limited feedback, remains a significant challenge. In this work, we consider settings with the most limited bandit'' feedback, and demonstrate that the simple Adaptive Conformal Inference (ACI) update rule, when applied to the appropriate control parameter or dual variable, is both adversarially valid, ensuring the success target is met on average for any input sequence (and hence under distribution shifts), and stochastically efficient, achieving sublinear efficiency regret for $i.i.d.$ inputs against an appropriate stochastic benchmark. We show such guarantees under canonical models capturing bandit and semi-bandit feedback to the agent via a unifying algorithmic technique, and analytic framework involving Lyapunov functions. Our approach handles more complex settings than prior work, while requiring significantly less feedback, and our results provide a new theoretical bridge between efficient online learning with limited feedback and distribution-free uncertainty quantification.

Conformal prediction methods provide statistically rigorous marginal coverage guarantees for machine learning models, but such guarantees fail to account for algorithmic biases, thereby undermining fairness and trust. This paper introduces a fair conformal inference framework for classification tasks. The proposed method constructs prediction sets that guarantee conditional coverage on adaptively identified subgroups, which can be implicitly defined through nonlinear feature combinations. By balancing effectiveness and efficiency in producing compact, informative prediction sets and ensuring adaptive equalized coverage across unfairly treated subgroups, our approach paves a practical pathway toward trustworthy machine learning. Extensive experiments on both synthetic and real-world datasets demonstrate the effectiveness of the framework.

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

Conformal prediction is a framework that provides valid uncertainty quantification for general models with exchangeable data. However, in the online learning and time-series settings, exchangeability is not satisfied. Existing online conformal methods, such as adaptive conformal inference (ACI), can achieve long-run validity, yet they remain inefficient under covariate heterogeneity because they rely on global calibration. We propose \emph{Online Localized Conformal Prediction (OLCP)}, which combines online adaptation with covariate-dependent localization to better reflect heterogeneity. To reduce sensitivity to the localization bandwidth, we further develop \emph{OLCP-Hedge}, which performs bandwidth selection as an online expert aggregation problem using a constrained online convex optimization framework. Importantly, we provide coverage guarantees for both algorithms and demonstrate through simulations and real-data experiments that the proposed methods attain valid long-run coverage with narrower prediction sets than existing baselines.

Conformal prediction (CP) provides powerful, distribution-free prediction sets, but its guarantees rely on the exchangeability of training and test data, which is often violated in practice due to covariate shifts. While weighted conformal prediction (WCP) is designed to handle such shifts, it can suffer from significant undercoverage when the density ratio between the distributions is unbounded and/or must be learned. This is because of both overfitting in learning the density ratio, and high variance in estimating the nonconformity score threshold. To address this, we introduce clipped least-squares importance fitting (CLISF) as a reduced-variance method for density ratio estimation. Specifically, we show that density ratios learned using CLISF, when plugged into WCP, have bounded expected undercoverage. Furthermore, we show that the undercoverage can be corrected by running WCP with a slightly inflated coverage target; crucially, we are able to estimate the required level of inflation from the data. We provide the first theoretical guarantees for weight clipping in conformal inference, achieving dataset-conditional coverage with a sample complexity that does not blow up with the higher moments of the true density ratio -- a key limitation of prior work. We verify our results on real-world benchmarks and synthetic data.

The Super Learner (SL) is a widely used ensemble method that combines point predictions from a library of learners based on their predictive performance. Interval predictions are of considerable practical interest because they allow uncertainty in predictions produced by an individual learner or an ensemble to be quantified. Several methods have been proposed for constructing interval predictions based on the SL, however, these approaches are typically justified using asymptotic arguments or rely on computationally intensive procedures such as the bootstrap. Conformal prediction (CP) is a machine learning framework for constructing prediction intervals with finite-sample and asymptotic coverage guarantees under mild conditions. We propose coupling CP with the SL through a natural construction that mirrors the original SL framework, using individual learner weights and combining learner-specific conformity scores via a weighted majority vote. We characterize the properties of the resulting SL-based prediction intervals for continuous outcomes. We cover settings under exchangeability, potential violations of exchangeability, and data-generating mechanisms exhibiting heteroscedasticity, sparsity, and other forms of distributional heterogeneity. A comprehensive simulation study shows that the conformalized SL achieves valid finite-sample coverage with competitive performance relative to the true data-generating mechanism. A central contribution of this work is an application to predicting creatinine levels using socio-demographic, biometric, and laboratory measurements. This example demonstrates the benefits of an ensemble with carefully selected learners designed to capture key aspects of complex regression functions, including non-linear effects, interactions, sparsity, heteroscedasticity, and robustness to outliers.

Safety filters have been shown to be effective tools to ensure the safety of control systems with unsafe nominal policies. To address scalability challenges in traditional synthesis methods, learning-based approaches have been proposed for designing safety filters for systems with high-dimensional state and control spaces. However, the inevitable errors in the decisions of these models raise concerns about their reliability and the safety guarantees they offer. This paper presents Adaptive Conformal Filtering (ACoFi), a method that combines learned Hamilton-Jacobi reachability-based safety filters with adaptive conformal inference. Under ACoFi, the filter dynamically adjusts its switching criteria based on the observed errors in its predictions of the safety of actions. The range of possible safety values of the nominal policy's output is used to quantify uncertainty in safety assessment. The filter switches from the nominal policy to the learned safe one when that range suggests it might be unsafe. We show that ACoFi guarantees that the rate of incorrectly quantifying uncertainty in the predicted safety of the nominal policy is asymptotically upper bounded by a user-defined parameter. This gives a soft safety guarantee rather than a hard safety guarantee. We evaluate ACoFi in a Dubins car simulation and a Safety Gymnasium environment, empirically demonstrating that it significantly outperforms the baseline method that uses a fixed switching threshold by achieving higher learned safety values and fewer safety violations, especially in out-of-distribution scenarios.

Conformal prediction (CP) has attracted broad attention as a simple and flexible framework for uncertainty quantification through prediction sets. In this work, we study how to deploy CP under differential privacy (DP) in a statistically efficient manner. We first introduce differential CP, a non-splitting conformal procedure that avoids the efficiency loss caused by data splitting and serves as a bridge between oracle CP and private conformal inference. By exploiting the stability properties of DP mechanisms, differential CP establishes a direct connection to oracle CP and inherits corresponding validity behavior. Building on this idea, we develop Differentially Private Conformal Prediction (DPCP), a fully private procedure that combines DP model training with a private quantile mechanism for calibration. We establish the end-to-end privacy guarantee of DPCP and investigate its coverage properties under additional regularity conditions. We further study the efficiency of both differential CP and DPCP under empirical risk minimization and general regression models, showing that DPCP can produce tighter prediction sets than existing private split conformal approaches under the same privacy budget. Numerical experiments on synthetic and real datasets demonstrate the practical effectiveness of the proposed methods.

Large Language Models (LLMs) are increasingly deployed as autonomous agents capable of reasoning, planning, and acting within interactive environments. Despite their growing capability to perform multi-step reasoning and decision-making tasks, internal mechanisms guiding their sequential behavior remain opaque. This paper presents a framework for interpreting the temporal evolution of concepts in LLM agents through a step-wise conformal lens. We introduce the conformal interpretability framework for temporal tasks, which combines step-wise reward modeling with conformal prediction to statistically label model's internal representation at each step as successful or failing. Linear probes are then trained on these representations to identify directions of temporal concepts - latent directions in the model's activation space that correspond to consistent notions of success, failure or reasoning drift. Experimental results on two simulated interactive environments, namely ScienceWorld and AlfWorld, demonstrate that these temporal concepts are linearly separable, revealing interpretable structures aligned with task success. We further show preliminary results on improving an LLM agent's performance by leveraging the proposed framework for steering the identified successful directions inside the model. The proposed approach, thus, offers a principled method for early failure detection as well as intervention in LLM-based agents, paving the path towards trustworthy autonomous language models in complex interactive settings.

Selective conformal prediction can yield substantially tighter uncertainty sets when we can identify calibration examples that are exchangeable with the test example. In interventional settings, such as perturbation experiments in genomics, exchangeability often holds only within subsets of interventions that leave a target variable "unaffected" (e.g., non-descendants of an intervened node in a causal graph). We study the practical regime where this invariance structure is unknown and must be estimated from data. Our main result quantifies how coverage degrades when the estimated safe calibration set accidentally includes interventions that affect the target, and gives a conservative correction when an upper bound on this error is available. Rather than learning a full causal graph, we learn only the intervention-target relationships needed to choose calibration interventions. We give algorithms for this partial learning task and evaluate them on synthetic structural equation models and Replogle K562 CRISPR-interference data, where the experiments illustrate synthetic gains from selective calibration and finite-sample tradeoffs on real perturbation screens.

An agent must try new behaviors to explore and improve. In high-stakes environments, an agent that violates safety constraints may cause harm and must be taken offline, curtailing any future interaction. Imitating old behavior is safe, but excessive conservatism discourages exploration. How much behavior change is too much? We show how to use any safe reference policy as a probabilistic regulator for any optimized but untested policy. Conformal calibration on data from the safe policy determines how aggressively the new policy can act, while provably enforcing the user's declared risk tolerance. Unlike conservative optimization methods, we do not assume the user has identified the correct model class nor tuned any hyperparameters. Unlike previous conformal methods, our theory provides finite-sample guarantees even for non-monotonic bounded loss functions, and it introduces a new policy control setting. Our experiments on applications ranging from natural language question answering to biomolecular engineering show that safe exploration is not only possible from the first moment of deployment, but can also improve performance.

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.