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.

X Y the available training set, diverse, ranging from single point estimates N being the number of training instances. In Bayesian (often averaged over prediction samples) to Neural Networks (BNNs) (Buntine and Weigend, 1991; predictive distributions, to set-valued or Neal, 2012; Jospin et al., 2022; Kingma and Welling, credal-set representations. We propose a novel 2013), this uncertainty is explicitly represented through unified evaluation framework for uncertaintyaware posterior predictive distributions over the parameter classifiers, applicable to a wide range space. In Deep Ensembles (DEs) (Lakshminarayanan of model classes, which allows users to tailor et al., 2017), a predictive distribution is formed by the trade-off between accuracy and precision aggregating the individual predictions generated by of predictions via a suitably designed performance multiple independently trained models.

We provide a version for lower probabilities of Monge's and Kantorovich's optimal transport problems. We show that, when the lower probabilities are the lower envelopes of $\epsilon$-contaminated sets, then our version of Monge's, and a restricted version of our Kantorovich's problems, coincide with their respective classical versions. We also give sufficient conditions for the existence of our version of Kantorovich's optimal plan, and for the two problems to be equivalent. As a byproduct, we show that for $\epsilon$-contaminations the lower probability versions of Monge's and Kantorovich's optimal transport problems need not coincide. The applications of our results to Machine Learning and Artificial Intelligence are also discussed.

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.

Recently, Halpern and Leung suggested representing uncertainty by a set of weighted probability measures, and suggested a way of making decisions based on this representation of uncertainty: maximizing weighted regret. Their paper does not answer an apparently simpler question: what it means, according to this representation of uncertainty, for an event E to be more likely than an event E'. In this paper, a notion of comparative likelihood when uncertainty is represented by a set of weighted probability measures is defined. It generalizes the ordering defined by probability (and by lower probability) in a natural way; a generalization of upper probability can also be defined. A complete axiomatic characterization of this notion of regret-based likelihood is given.

Recently, Halpern and Leung suggested representing uncertainty by a weighted set of probability measures, and suggested a way of making decisions based on this representation of uncertainty: maximizing weighted regret. Their paper does not answer an apparently simpler question: what it means, according to this representation of uncertainty, for an event E to be more likely than an event E'. In this paper, a notion of comparative likelihood when uncertainty is represented by a weighted set of probability measures is defined. It generalizes the ordering defined by probability (and by lower probability) in a natural way; a generalization of upper probability can also be defined. A complete axiomatic characterization of this notion of regret-based likelihood is given.

Jeffrey's rule has been generalized by Wagner to the case in which new evidence bounds the possible revisions of a prior probability below by a Dempsterian lower probability. Classical probability kinematics arises within this generalization as the special case in which the evidentiary focal elements of the bounding lower probability are pairwise disjoint. We discuss a twofold extension of this generalization, first allowing the lower bound to be any two-monotone capacity and then allowing the prior to be a lower envelope.

In Moral, Campos (1991) and Cano, Moral, Verdegay-Lopez (1991) a new method of conditioning convex sets of probabilities has been proposed. The result of it is a convex set of non-necessarily normalized probability distributions. The normalizing factor of each probability distribution is interpreted as the possibility assigned to it by the conditioning information. From this, it is deduced that the natural value for the conditional probability of an event is a possibility distribution. The aim of this paper is to study methods of transforming this possibility distribution into a probability (or uncertainty) interval. These methods will be based on the use of Sugeno and Choquet integrals. Their behaviour will be compared in basis to some selected examples.

Shafer's theory of belief and the Bayesian theory of probability are two alternative and mutually inconsistent approaches toward modelling uncertainty in artificial intelligence. To help reduce the conflict between these two approaches, this paper reexamines expected utility theory-from which Bayesian probability theory is derived. Expected utility theory requires the decision maker to assign a utility to each decision conditioned on every possible event that might occur. But frequently the decision maker cannot foresee all the events that might occur, i.e., one of the possible events is the occurrence of an unforeseen event. So once we acknowledge the existence of unforeseen events, we need to develop some way of assigning utilities to decisions conditioned on unforeseen events. The commonsensical solution to this problem is to assign similar utilities to events which are similar. Implementing this commonsensical solution is equivalent to replacing Bayesian subjective probabilities over the space of foreseen and unforeseen events by random set theory probabilities over the space of foreseen events. This leads to an expected utility principle in which normalized variants of Shafer's commonalities play the role of subjective probabilities. Hence allowing for unforeseen events in decision analysis causes Bayesian probability theory to become much more similar to Shaferian theory.

Lower and upper probabilities, also known as Choquet capacities, are widely used as a convenient representation for sets of probability distributions. This paper presents a graphical decomposition and exact propagation algorithm for computing marginal posteriors of 2-monotone lower probabilities (equivalently, 2-alternating upper probabilities).