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.

Uncertainty Quantification (UQ) presents a pivotal challenge in the field of Artificial Intelligence (AI), profoundly impacting decision-making, risk assessment and model reliability. In this paper, we introduce Credal and Interval Deep Evidential Classifications (CDEC and IDEC, respectively) as novel approaches to address UQ in classification tasks. CDEC and IDEC leverage a credal set (closed and convex set of probabilities) and an interval of evidential predictive distributions, respectively, allowing us to avoid overfitting to the training data and to systematically assess both epistemic (reducible) and aleatoric (irreducible) uncertainties. When those surpass acceptable thresholds, CDEC and IDEC have the capability to abstain from classification and flag an excess of epistemic or aleatoric uncertainty, as relevant. Conversely, within acceptable uncertainty bounds, CDEC and IDEC provide a collection of labels with robust probabilistic guarantees. CDEC and IDEC are trained using standard backpropagation and a loss function that draws from the theory of evidence. They overcome the shortcomings of previous efforts, and extend the current evidential deep learning literature. Through extensive experiments on MNIST, CIFAR-10 and CIFAR-100, together with their natural OoD shifts (F-MNIST/K-MNIST, SVHN/Intel, TinyImageNet), we show that CDEC and IDEC achieve competitive predictive accuracy, state-of-the-art OoD detection under epistemic and total uncertainty, and tight, well-calibrated prediction regions that expand reliably under distribution shift. An ablation over ensemble size further demonstrates that CDEC attains stable uncertainty estimates with only a small ensemble.

Uncertainty quantification is essential for deploying reliable Graph Neural Networks (GNNs), where existing approaches primarily rely on Bayesian inference or ensembles. In this paper, we introduce the first credal graph neural networks (CGNNs), which extend credal learning to the graph domain by training GNNs to output set-valued predictions in the form of credal sets. To account for the distinctive nature of message passing in GNNs, we develop a complementary approach to credal learning that leverages different aspects of layer-wise information propagation. We assess our approach on uncertainty quantification in node classification under out-of-distribution conditions. Our analysis highlights the critical role of the graph homophily assumption in shaping the effectiveness of uncertainty estimates. Extensive experiments demonstrate that CGNNs deliver more reliable representations of epistemic uncertainty and achieve state-of-the-art performance under distributional shift on heterophilic graphs.

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.

Recently, Bjøru et al. proposed a novel divide-and-conquer algorithm for bounding counterfactual probabilities in structural causal models (SCMs). They assumed that the SCMs were learned from purely observational data, leading to an imprecise characterization of the marginal distributions of exogenous variables. Their method leveraged the canonical representation of structural equations to decompose a general SCM with high-cardinality exogenous variables into a set of sub-models with low-cardinality exogenous variables. These sub-models had precise marginals over the exogenous variables and therefore admitted efficient exact inference. The aggregated results were used to bound counterfactual probabilities in the original model. The approach was developed for Markovian models, where each exogenous variable affects only a single endogenous variable. In this paper, we investigate extending the methodology to \textit{semi-Markovian} SCMs, where exogenous variables may influence multiple endogenous variables. Such models are capable of representing confounding relationships that Markovian models cannot. We illustrate the challenges of this extension using a minimal example, which motivates a set of alternative solution strategies. These strategies are evaluated both theoretically and through a computational study.

Deep ensembles (DE) have emerged as a powerful approach for quantifying predictive uncertainty and distinguishing its aleatoric and epistemic components, thereby enhancing model robustness and reliability. However, their high computational and memory costs during inference pose significant challenges for wide practical deployment. To overcome this issue, we propose credal ensemble distillation (CED), a novel framework that compresses a DE into a single model, CREDIT, for classification tasks. Instead of a single softmax probability distribution, CREDIT predicts class-wise probability intervals that define a credal set, a convex set of probability distributions, for uncertainty quantification. Empirical results on out-of-distribution detection benchmarks demonstrate that CED achieves superior or comparable uncertainty estimation compared to several existing baselines, while substantially reducing inference overhead compared to DE.

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.