Predictions in the form of sets of probability distributions, so-called credal sets, provide a suitable means to represent a learner's epistemic uncertainty. In this paper, we propose a theoretically grounded approach to credal prediction based on the statistical notion of relative likelihood: The target of prediction is the set of all (conditional) probability distributions produced by the collection of plausible models, namely those models whose relative likelihood exceeds a specified threshold. This threshold has an intuitive interpretation and allows for controlling the trade-off between correctness and precision of credal predictions. We tackle the problem of approximating credal sets defined in this way by means of suitably modified ensemble learning techniques. To validate our approach, we illustrate its effectiveness by experiments on benchmark datasets demonstrating superior uncertainty representation without compromising predictive performance. We also compare our method against several state-of-the-art baselines in credal prediction.

Quantifying differences between probability distributions is fundamental to statistics and machine learning, primarily for comparing statistical uncertainty. In contrast, epistemic uncertainty--due to incomplete knowledge--requires richer representations than those offered by classical probability. Imprecise probability (IP) theory offers such models, capturing ambiguity and partial belief. This has driven growing interest in imprecise probabilistic machine learning (IPML), where inference and decision-making rely on broader uncertainty models--highlighting the need for metrics beyond classical probability. This work introduces the Integral Imprecise Probability Metric (IIPM) framework, a Choquet integral-based generalisation of classical Integral Probability Metrics (IPMs) to the setting of capacities--a broad class of IP models encompassing many existing ones, including lower probabilities, probability intervals, belief functions, and more. Theoretically, we establish conditions under which IIPM serves as a valid metric and metrises a form of weak convergence of capacities. Practically, IIPM not only enables comparison across different IP models but also supports the quantification of epistemic uncertainty (EU) within a single IP model. In particular, by comparing an IP model with its conjugate, IIPM gives rise to a new class of EU measures--Maximum Mean Imprecisions (MMIs)--which satisfy key axiomatic properties proposed in the uncertainty quantification literature. We validate MMI through selective classification experiments, demonstrating strong empirical performance against established EU measures, and outperforming them when classical methods struggle to scale to a large number of classes.

Reliable deployment of machine learning models requires reasoning under epistemic uncertainty--the ability to recognize when the operating distribution has shifted beyond the scope of what was encountered during training. This challenge is central to test-time adaptation (TTA), a paradigm in which a model pretrained on source distribution Ps receives unlabeled data from a target distribution Pt = Ps at deployment time. Existing TTA methods (Wang et al., 2021; Niu et al., 2023; Zhang et al., 2022a; Yuan et al., 2023; Su et al., 2022) improve accuracy under distribution shift by adapting model parameters using statistics computed from test batches, but they provide no formal guarantees about when predictions should be trusted or how much risk degrades as a function of shift magnitude. This gap is particularly concerning in safety-critical applications such as autonomous driving, medical imaging, and financial risk assessment, where a model that silently degrades under distribution shift can cause significant harm. The inability to quantify how wrong a model's predictions might be in an unseen environment fundamentally limits its trustworthy deployment.

Credal sets, i.e., closed convex sets of probability measures, provide a natural framework to represent aleatoric and epistemic uncertainty in machine learning. Yet how to quantify these two types of uncertainty for a given credal set, particularly in multiclass classification, remains underexplored. In this paper, we propose a distance-based approach to quantify total, aleatoric, and epistemic uncertainty for credal sets. Concretely, we introduce a family of such measures within the framework of Integral Probability Metrics (IPMs). The resulting quantities admit clear semantic interpretations, satisfy natural theoretical desiderata, and remain computationally tractable for common choices of IPMs. We instantiate the framework with the total variation distance and obtain simple, efficient uncertainty measures for multiclass classification. In the binary case, this choice recovers established uncertainty measures, for which a principled multiclass generalization has so far been missing. Empirical results confirm practical usefulness, with favorable performance at low computational cost.

Credal sets are sets of probability distributions that are considered as candidates for an imprecisely known ground-truth distribution. In machine learning, they have recently attracted attention as an appealing formalism for uncertainty representation, in particular, due to their ability to represent both the aleatoric and epistemic uncertainty in a prediction. However, the design of methods for learning credal set predictors remains a challenging problem. In this paper, we make use of conformal prediction for this purpose. More specifically, we propose a method for predicting credal sets in the classification task, given training data labeled by probability distributions. Since our method inherits the coverage guarantees of conformal prediction, our conformal credal sets are guaranteed to be valid with high probability (without any assumptions on model or distribution). We demonstrate the applicability of our method on ambiguous classification tasks for uncertainty quantification.

Statistical learning theory is the foundation of machine learning, providing theoretical bounds for the risk of models learned from a (single) training set, assumed to issue from an unknown probability distribution. In actual deployment, however, the data distribution may (and often does) vary, causing domain adaptation/generalization issues. In this paper we lay the foundations for a `credal' theory of learning, using convex sets of probabilities (credal sets) to model the variability in the data-generating distribution. Such credal sets, we argue, may be inferred from a finite sample of training sets. Bounds are derived for the case of finite hypotheses spaces (both assuming realizability or not), as well as infinite model spaces, which directly generalize classical results.

Credal predictors are models that are aware of epistemic uncertainty and produce a convex set of probabilistic predictions. They offer a principled way to quantify predictive epistemic uncertainty (EU) and have been shown to improve model robustness in various settings. However, most state-of-the-art methods mainly define EU as disagreement caused by random training initializations, which mostly reflects sensitivity to optimization randomness rather than uncertainty from deeper sources. To address this, we define EU as disagreement among models trained with varying relaxations of the i.i.d. assumption between training and test data. Based on this idea, we propose CreDRO, which learns an ensemble of plausible models through distributionally robust optimization. As a result, CreDRO captures EU not only from training randomness but also from meaningful disagreement due to potential distribution shifts between training and test data. Empirical results show that CreDRO consistently outperforms existing credal methods on tasks such as out-of-distribution detection across multiple benchmarks and selective classification in medical applications.

However, the design of methods for learning credal set predictors remains a challenging problem. In this paper, we make use of conformal prediction for this purpose.

This paper presents an innovative approach to classification tasks called Credal Deep Ensembles (CreDEs), ensembles of novel Credal-Set Neural Networks (CreNets), aiming to improve EU quantification in the framework of credal inference.

Statistical learning theory is the foundation of machine learning, providing theoretical bounds for the risk of models learned from a (single) training set, assumed to issue from an unknown probability distribution.