Uncertainty estimation is pivotal in machine learning, especially for classification tasks, as it improves the robustness and reliability of models. We introduce a novel `Epistemic Wrapping' methodology aimed at improving uncertainty estimation in classification. Our approach uses Bayesian Neural Networks (BNNs) as a baseline and transforms their outputs into belief function posteriors, effectively capturing epistemic uncertainty and offering an efficient and general methodology for uncertainty quantification. Comprehensive experiments employing a Bayesian Neural Network (BNN) baseline and an Interval Neural Network for inference on the MNIST, Fashion-MNIST, CIFAR-10 and CIFAR-100 datasets demonstrate that our Epistemic Wrapper significantly enhances generalisation and uncertainty quantification.

Quantifying differences between probability distributions is fundamental to statistics and machine learning, primarily for comparing statistical uncertainty. In contrast, epistemic uncertainty (EU) -- 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 Metric (IPM) 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 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 Imprecision -- 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. Our work advances both theory and practice in IPML, offering a principled framework for comparing and quantifying epistemic uncertainty under imprecision.

The theory of belief functions [162, 67] is a modelling language for representing and combining elementary items of evidence, which do not necessarily come in the form of sharp statements, with the goal of maintaining a mathematical representation of an agent's beliefs about those aspects of the world which the agent is unable to predict with reasonable certainty. While arguably a more appropriate mathematical description of uncertainty than classical probability theory, for the reasons we have thoroughly explored in [50], the theory of evidence is relatively simple to understand and implement, and does not require one to abandon the notion of an event, as is the case, for instance, for Walley's imprecise probability theory [193]. It is grounded in the beautiful mathematics of random sets, and exhibits strong relationships with many other theories of uncertainty. As mathematical objects, belief functions have fascinating properties in terms of their geometry, algebra [207] and combinatorics. Despite initial concerns about the computational complexity of a naive implementation of the theory of evidence, evidential reasoning can actually be implemented on large sample spaces [156] and in situations involving the combination of numerous pieces of evidence [74]. Elementary items of evidence often induce simple belief functions, which can be combined very efficiently with complexity O(n + 1).

Machine learning is increasingly deployed in safety-critical domains where robustness against adversarial attacks is crucial and erroneous predictions could lead to potentially catastrophic consequences. This highlights the need for learning systems to be equipped with the means to determine a model's confidence in its prediction and the epistemic uncertainty associated with it, 'to know when a model does not know'. In this paper, we propose a novel Random-Set Convolutional Neural Network (RS-CNN) for classification which predicts belief functions rather than probability vectors over the set of classes, using the mathematics of random sets, i.e., distributions over the power set of the sample space. Based on the epistemic deep learning approach, random-set models are capable of representing the 'epistemic' uncertainty induced in machine learning by limited training sets. We estimate epistemic uncertainty by approximating the size of credal sets associated with the predicted belief functions, and experimentally demonstrate how our approach outperforms competing uncertainty-aware approaches in a classical evaluation setting. The performance of RS-CNN is best demonstrated on OOD samples where it manages to capture the true prediction while standard CNNs fail.

Probability intervals are an attractive tool for reasoning under uncertainty. Unlike belief functions, though, they lack a natural probability transformation to be used for decision making in a utility theory framework. In this paper we propose the use of the intersection probability, a transform derived originally for belief functions in the framework of the geometric approach to uncertainty, as the most natural such transformation. We recall its rationale and definition, compare it with other candidate representives of systems of probability intervals, discuss its credal rationale as focus of a pair of simplices in the probability simplex, and outline a possible decision making framework for probability intervals, analogous to the Transferable Belief Model for belief functions.

Conditioning is crucial in applied science when inference involving time series is involved. Belief calculus is an effective way of handling such inference in the presence of epistemic uncertainty -- unfortunately, different approaches to conditioning in the belief function framework have been proposed in the past, leaving the matter somewhat unsettled. Inspired by the geometric approach to uncertainty, in this paper we propose an approach to the conditioning of belief functions based on geometrically projecting them onto the simplex associated with the conditioning event in the space of all belief functions. We show here that such a geometric approach to conditioning often produces simple results with straightforward interpretations in terms of degrees of belief. This raises the question of whether classical approaches, such as for instance Dempster's conditioning, can also be reduced to some form of distance minimisation in a suitable space. The study of families of combination rules generated by (geometric) conditioning rules appears to be the natural prosecution of the presented research.

Probability theory is far from being the most general mathematical theory of uncertainty. A number of arguments point at its inability to describe second-order ('Knightian') uncertainty. In response, a wide array of theories of uncertainty have been proposed, many of them generalisations of classical probability. As we show here, such frameworks can be organised into clusters sharing a common rationale, exhibit complex links, and are characterised by different levels of generality. Our goal is a critical appraisal of the current landscape in uncertainty theory.