Prediction sets provide a means of quantifying the uncertainty in predictive tasks. Using held out calibration data, conformal prediction and risk control can produce prediction sets that exhibit statistically valid error control in a computationally efficient manner. However, in the standard formulations, the error is only controlled on average over many possible calibration datasets of fixed size. In this paper, we extend the control to remain valid with high probability over a cumulatively growing calibration dataset at any time point. We derive such guarantees using quantile-based arguments and illustrate the applicability of the proposed framework to settings involving distribution shift. We further establish a matching lower bound and show that our guarantees are asymptotically tight. Finally, we demonstrate the practical performance of our methods through both simulations and real-world numerical examples.

As large language models (LLMs) are increasingly deployed in risk-sensitive applications such as real-world open-ended question answering (QA), ensuring the trustworthiness of their outputs has become critical. Existing selective conformal prediction (SCP) methods provide statistical guarantees by constructing prediction sets with a constrained miscoverage rate for correct answers. However, prior works unrealistically assume that admissible answers for all instances can be obtained via finite sampling, even for open-ended QA scenarios that lack a fixed and finite solution space. To address this, we introduce a two-stage risk control framework comprising abstention-aware sampling and conformalized filtering (SAFER). Firstly, on a held-out calibration set, SAFER calibrates a sampling budget within the maximum sampling cap, using the Clopper-Pearson exact method at a user-desired risk level (i.e., the maximum allowable miscoverage rate of the sampling sets). If the risk level cannot be satisfied within the cap, we abstain; otherwise, the calibrated sampling budget becomes the minimum requirements at test time. Then, we employ calibration instances where correct answers are attainable under the calibrated budget and apply the conformal risk control method to determine a statistically valid uncertainty threshold, which filters unreliable distractors from the candidate set for each test data point. In this stage, SAFER introduces an additional risk level to guide the calculation of the threshold, thereby controlling the risk of correct answers being excluded. Furthermore, we show that SAFER is compatible with various task-specific admission criteria and calibration-test split ratios, highlighting its robustness and high data efficiency.

This paper introduces a boosted conformal procedure designed to tailor confor-malized prediction intervals toward specific desired properties, such as enhanced conditional coverage or reduced interval length. We employ machine learning techniques, notably gradient boosting, to systematically improve upon a predefined conformity score function. This process is guided by carefully constructed loss functions that measure the deviation of prediction intervals from the targeted properties. The procedure operates post-training, relying solely on model predictions and without modifying the trained model (e.g., the deep network). Systematic experiments demonstrate that starting from conventional conformal methods, our boosted procedure achieves substantial improvements in reducing interval length and decreasing deviation from target conditional coverage.

This study introduces a significance testing-enhanced conformal prediction (CP) framework to improve trustworthiness of large language models (LLMs) in multiple-choice question answering (MCQA). While LLMs have been increasingly deployed in disciplinary QA scenarios, hallucination and nonfactual generation substantially compromise response reliability. Although CP provides statistically rigorous marginal coverage guarantees for prediction sets, and significance testing offers established statistical rigor, their synergistic integration remains unexplored. To mitigate hallucination and factual inaccuracies, our framework integrates $p$-value computation with conformity scoring through self-consistency resampling of MCQA responses. This approach calculates option frequencies to address LLMs' black-box nature, subsequently constructing prediction sets via null hypothesis testing ($\mathcal{H}_0$) with empirically derived $p$-values. Evaluations on MMLU and MMLU-Pro benchmarks using off-the-shelf LLMs demonstrate: (1) The enhanced CP achieves user-specified empirical miscoverage rates; (2) Test-set average prediction set size (APSS) decreases monotonically with increasing risk levels ($α$), validating APSS as an effective uncertainty metric. This work establishes a principled statistical framework for trustworthy LLM deployment in high-stakes QA applications.

Large Language Models (LLMs) have shown remarkable progress in multiple-choice question answering (MCQA), but their inherent unreliability, such as hallucination and overconfidence, limits their application in high-risk domains. To address this, we propose a frequency-based uncertainty quantification method under black-box settings, leveraging conformal prediction (CP) to ensure provable coverage guarantees. Our approach involves multiple independent samplings of the model's output distribution for each input, with the most frequent sample serving as a reference to calculate predictive entropy (PE). Experimental evaluations across six LLMs and four datasets (MedMCQA, MedQA, MMLU, MMLU-Pro) demonstrate that frequency-based PE outperforms logit-based PE in distinguishing between correct and incorrect predictions, as measured by AUROC. Furthermore, the method effectively controls the empirical miscoverage rate under user-specified risk levels, validating that sampling frequency can serve as a viable substitute for logit-based probabilities in black-box scenarios. This work provides a distribution-free model-agnostic framework for reliable uncertainty quantification in MCQA with guaranteed coverage, enhancing the trustworthiness of LLMs in practical applications.

This paper introduces the MPS (Model Prediction Set), a novel framework for online model selection for nonstationary time series. Classical model selection methods, such as information criteria and cross-validation, rely heavily on the stationarity assumption and often fail in dynamic environments which undergo gradual or abrupt changes over time. Yet real-world data are rarely stationary, and model selection under nonstationarity remains a largely open problem. To tackle this challenge, we combine conformal inference with model confidence sets to develop a procedure that adaptively selects models best suited to the evolving dynamics at any given time. Concretely, the MPS updates in real time a confidence set of candidate models that covers the best model for the next time period with a specified long-run probability, while adapting to nonstationarity of unknown forms. Through simulations and real-world data analysis, we demonstrate that MPS reliably and efficiently identifies optimal models under nonstationarity, an essential capability lacking in offline methods. Moreover, MPS frequently produces high-quality sets with small cardinality, whose evolution offers deeper insights into changing dynamics. As a generic framework, MPS accommodates any data-generating process, data structure, model class, training method, and evaluation metric, making it broadly applicable across diverse problem settings.

Online conformal prediction enables the runtime calibration of a pre-trained artificial intelligence model using feedback on its performance. Calibration is achieved through set predictions that are updated via online rules so as to ensure long-term coverage guarantees. While recent research has demonstrated the benefits of incorporating prior knowledge into the calibration process, this has come at the cost of replacing coverage guarantees with less tangible regret guarantees based on the quantile loss. This work introduces intermittent mirror online conformal prediction (IM-OCP), a novel runtime calibration framework that integrates prior knowledge, while maintaining long-term coverage and achieving sub-linear regret. IM-OCP features closed-form updates with minimal memory complexity, and is designed to operate under potentially intermittent feedback.

Conformal prediction has emerged as a general and versatile framework for constructing prediction sets in regression and classification tasks (Shafer and Vovk, 2008). Unlike traditional methods, which often depend on rigid distributional assumptions, conformal prediction transforms point predictions from any prediction (or black-box) algorithm into prediction sets that guarantee valid finite-sample marginal coverage. Originally introduced by Vovk et al. (2005), it has become increasingly influential, with numerous methods and extensions being proposed since its introduction. In particular, full conformal prediction by Vovk et al. (2005), demonstrates favorable properties regarding the coverage and the size of the prediction set. However, these advantages are counterbalanced by a substantial computational cost, which limits its practical application.

Multimodal Large Language Models (MLLMs) exhibit promising advancements across various tasks, yet they still encounter significant trustworthiness issues. Prior studies apply Split Conformal Prediction (SCP) in language modeling to construct prediction sets with statistical guarantees. However, these methods typically rely on internal model logits or are restricted to multiple-choice settings, which hampers their generalizability and adaptability in dynamic, open-ended environments. In this paper, we introduce TRON, a two-step framework for risk control and assessment, applicable to any MLLM that supports sampling in both open-ended and closed-ended scenarios. TRON comprises two main components: (1) a novel conformal score to sample response sets of minimum size, and (2) a nonconformity score to identify high-quality responses based on self-consistency theory, controlling the error rates by two specific risk levels. Furthermore, we investigate semantic redundancy in prediction sets within open-ended contexts for the first time, leading to a promising evaluation metric for MLLMs based on average set size. Our comprehensive experiments across four Video Question-Answering (VideoQA) datasets utilizing eight MLLMs show that TRON achieves desired error rates bounded by two user-specified risk levels. Additionally, deduplicated prediction sets maintain adaptiveness while being more efficient and stable for risk assessment under different risk levels.

Uncertainty quantification for time series nowcasting and forecasting is crucial in many areas such as climate science, epidemiology, industrial engineering, and macroeconomics. Ideally, the forecaster would generate a prediction interval at each time period that is calibrated in the sense that the fraction of intervals covering the true outcomes is approximately equal to the target coverage level in the long run. Classical approaches for generating prediction intervals are mostly model-based Box and Jenkins [1976], Engle [1982a], Stock and Watson [2010], Brown [1964], Jorda [2005]. However, time series models are often mis-specified due to nonstationarity or changing environments. As a result, the model-based prediction intervals tend to be poorly calibrated (see for instance the gray curves in Figure 1). Moreover, many forecasters have upgraded their workflows by incorporating black-box machine learning algorithms [e.g. Taylor and Letham, 2018, Makridakis et al., 2018, Herzen et al., 2022], for which valid uncertainty quantification proves to be challenging.