The author contributed to this paper during an internship at the Max Planck Institute for Software Systems.

Surrogate models (including deep neural networks and other machine learning algorithms in supervised learning) are capable of approximating arbitrarily complex, high-dimensional input-output problems in science and engineering, but require a thorough data-agnostic uncertainty quantification analysis before these can be deployed for any safety-critical application. The standard approach for data-agnostic uncertainty quantification is to use conformal prediction (CP), a well-established framework to build uncertainty models with proven statistical guarantees that do not assume any shape for the error distribution of the surrogate model. However, since the classic statistical guarantee offered by CP is given in terms of bounds for the marginal coverage, for small calibration set sizes (which are frequent in realistic surrogate modelling that aims to quantify error at different regions), the potentially strong dispersion of the coverage distribution around its average negatively impacts the relevance of the uncertainty model's statistical guarantee, often obtaining coverages below the expected value, resulting in a less applicable framework. After providing a gentle presentation of uncertainty quantification for surrogate models for machine learning practitioners, in this paper we bridge the gap by proposing a new statistical guarantee that offers probabilistic information for the coverage of a single conformal predictor. We show that the proposed framework converges to the standard solution offered by CP for large calibration set sizes and, unlike the classic guarantee, still offers relevant information about the coverage of a conformal predictor for small data sizes. We validate the methodology in a suite of examples, and implement an open access software solution that can be used alongside common conformal prediction libraries to obtain uncertainty models that fulfil the new guarantee.

Conformal prediction is a model-agnostic approach to generating prediction sets that cover the true class with a high probability. Although its prediction set size is expected to capture aleatoric uncertainty, there is a lack of evidence regarding its effectiveness. The literature presents that prediction set size can upper-bound aleatoric uncertainty or that prediction sets are larger for difficult instances and smaller for easy ones, but a validation of this attribute of conformal predictors is missing. This work investigates how effectively conformal predictors quantify aleatoric uncertainty, specifically the inherent ambiguity in datasets caused by overlapping classes. We perform this by measuring the correlation between prediction set sizes and the number of distinct labels assigned by human annotators per instance. We further assess the similarity between prediction sets and human-provided annotations. We use three conformal prediction approaches to generate prediction sets for eight deep learning models trained on four datasets. The datasets contain annotations from multiple human annotators (ranging from five to fifty participants) per instance, enabling the identification of class overlap. We show that the vast majority of the conformal prediction outputs show a very weak to weak correlation with human annotations, with only a few showing moderate correlation. These findings underscore the necessity of critically reassessing the prediction sets generated using conformal predictors. While they can provide a higher coverage of the true classes, their capability in capturing aleatoric uncertainty and generating sets that align with human annotations remains limited.

Conformal Prediction (CP) is a popular method for uncertainty quantification with machine learning models. While conformal prediction provides probabilistic guarantees regarding the coverage of the true label, these guarantees are agnostic to the presence of sensitive attributes within the dataset. In this work, we formalize \textit{Conformal Fairness}, a notion of fairness using conformal predictors, and provide a theoretically well-founded algorithm and associated framework to control for the gaps in coverage between different sensitive groups. Our framework leverages the exchangeability assumption (implicit to CP) rather than the typical IID assumption, allowing us to apply the notion of Conformal Fairness to data types and tasks that are not IID, such as graph data. Experiments were conducted on graph and tabular datasets to demonstrate that the algorithm can control fairness-related gaps in addition to coverage aligned with theoretical expectations.

Conformal Prediction (CP) is a widely used technique for quantifying uncertainty in machine learning models. In its standard form, CP offers probabilistic guarantees on the coverage of the true label, but it is agnostic to sensitive attributes in the dataset. Several recent works have sought to incorporate fairness into CP by ensuring conditional coverage guarantees across different subgroups. One such method is Conformal Fairness (CF). In this work, we extend the CF framework to the Federated Learning setting and discuss how we can audit a federated model for fairness by analyzing the fairness-related gaps for different demographic groups. Ensuring model fairness is a critical thrust of trustworthy machine learning (ML). ML models, when not calibrated for fairness, are prone to developing biases at each stage of an ML pipeline, as reflected by their predictions Mehrabi et al. (2021). We define bias as disparate performance (i.e., accuracy for classification) between different sub-populations. In the data collection phase, measurement bias may occur due to disproportionate data collection on sub-populations, while representation bias manifests from a lack of training data on specific strata. During training, these biases are inductively learned by the model-leading to incorrect predictions in safety-critical tasks. These models are also susceptible to algorithmic bias, resulting from regularization and optimization techniques during model training, which incorrectly generalize for marginal-ized groups. To mitigate these risks, many ML models must adhere to regulations placed by local governing bodies (Hirsch et al., 2023). Towards model compliance, Komala et al. (2024); Agrawal et al. (2024); Jones et al. (2025) have proposed approaches to enhance model fairness in varying tasks, including federated graph learning and representation learning.

Transductive conformal prediction addresses the simultaneous prediction for multiple data points. Given a desired confidence level, the objective is to construct a prediction set that includes the true outcomes with the prescribed confidence. We demonstrate a fundamental trade-off between confidence and efficiency in transductive methods, where efficiency is measured by the size of the prediction sets. Specifically, we derive a strict finite-sample bound showing that any non-trivial confidence level leads to exponential growth in prediction set size for data with inherent uncertainty. The exponent scales linearly with the number of samples and is proportional to the conditional entropy of the data. Additionally, the bound includes a second-order term, dispersion, defined as the variance of the log conditional probability distribution. We show that this bound is achievable in an idealized setting. Finally, we examine a special case of transductive prediction where all test data points share the same label. We show that this scenario reduces to the hypothesis testing problem with empirically observed statistics and provide an asymptotically optimal confidence predictor, along with an analysis of the error exponent.

Conformal prediction (CP) is a powerful framework for quantifying uncertainty in machine learning models, offering reliable predictions with finite-sample coverage guarantees. When applied to classification, CP produces a prediction set of possible labels that is guaranteed to contain the true label with high probability, regardless of the underlying classifier. However, standard CP treats classes as flat and unstructured, ignoring domain knowledge such as semantic relationships or hierarchical structure among class labels. This paper presents hierarchical conformal classification (HCC), an extension of CP that incorporates class hierarchies into both the structure and semantics of prediction sets. We formulate HCC as a constrained optimization problem whose solutions yield prediction sets composed of nodes at different levels of the hierarchy, while maintaining coverage guarantees. To address the combinatorial nature of the problem, we formally show that a much smaller, well-structured subset of candidate solutions suffices to ensure coverage while upholding optimality. An empirical evaluation on three new benchmarks consisting of audio, image, and text data highlights the advantages of our approach, and a user study shows that annotators significantly prefer hierarchical over flat prediction sets.

Conformal predictors are machine learning algorithms developed in the 1990's by Gammerman, Vovk, and their research team, to provide set predictions with guaranteed confidence level. Over recent years, they have grown in popularity and have become a mainstream methodology for uncertainty quantification in the machine learning community. From its beginning, there was an understanding that they enable reliable machine learning with well-calibrated uncertainty quantification. This makes them extremely beneficial for developing trustworthy AI, a topic that has also risen in interest over the past few years, in both the AI community and society more widely. In this article, we review the potential for conformal prediction to contribute to trustworthy AI beyond its marginal validity property, addressing problems such as generalization risk and AI governance. Experiments and examples are also provided to demonstrate its use as a well-calibrated predictor and for bias identification and mitigation.

Machine learning (ML) models always make a prediction, even when they are likely to be wrong. This causes problems in practical applications, as we do not know if we should trust a prediction. ML with reject option addresses this issue by abstaining from making a prediction if it is likely to be incorrect. In this work, we formalise the approach to ML with reject option in binary classification, deriving theoretical guarantees on the resulting error rate. This is achieved through conformal prediction (CP), which produce prediction sets with distribution-free validity guarantees. In binary classification, CP can output prediction sets containing exactly one, two or no labels. By accepting only the singleton predictions, we turn CP into a binary classifier with reject option. Here, CP is formally put in the framework of predicting with reject option. We state and prove the resulting error rate, and give finite sample estimates. Numerical examples provide illustrations of derived error rate through several different conformal prediction settings, ranging from full conformal prediction to offline batch inductive conformal prediction. The former has a direct link to sharp validity guarantees, whereas the latter is more fuzzy in terms of validity guarantees but can be used in practice. Error-reject curves illustrate the trade-off between error rate and reject rate, and can serve to aid a user to set an acceptable error rate or reject rate in practice.