This paper explores belief inference in credal networks using Dempster-Shafer theory. By building on previous work, we propose a novel framework for propagating uncertainty through a subclass of credal networks, namely chains. The proposed approach efficiently yields conservative intervals through belief and plausibility functions, combining computational speed with robust uncertainty representation. Key contributions include formalizing belief-based inference methods and comparing belief-based inference against classical sensitivity analysis. Numerical results highlight the advantages and limitations of applying belief inference within this framework, providing insights into its practical utility for chains and for credal networks in general.

Property elicitation studies which attributes of a probability distribution can be determined by minimising a risk. We investigate a generalisation of property elicitation to imprecise probabilities (IP). This investigation is motivated by multi-distribution learning, which takes the classical machine learning paradigm of minimising a single risk over a (precise) probability and replaces it with $Γ$-maximin risk minimization over an IP. We provide necessary conditions for elicitability of a IP-property. Furthermore, we explain what an elicitable IP-property actually elicits through Bayes pairs -- the elicited IP-property is the corresponding standard property of the maximum Bayes risk distribution.

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.

Based on existing ideas in the field of imprecise probabilities, we present a new approach for assessing the reliability of the individual predictions of a generative probabilistic classifier. We call this approach robustness quantification, compare it to uncertainty quantification, and demonstrate that it continues to work well even for classifiers that are learned from small training sets that are sampled from a shifted distribution.

The desirable gambles framework provides a foundational approach to imprecise probability theory but relies heavily on linear utility assumptions. This paper introduces {\em function-coherent gambles}, a generalization that accommodates non-linear utility while preserving essential rationality properties. We establish core axioms for function-coherence and prove a representation theorem that characterizes acceptable gambles through continuous linear functionals. The framework is then applied to analyze various forms of discounting in intertemporal choice, including hyperbolic, quasi-hyperbolic, scale-dependent, and state-dependent discounting. We demonstrate how these alternatives to constant-rate exponential discounting can be integrated within the function-coherent framework. This unified treatment provides theoretical foundations for modeling sophisticated patterns of time preference within the desirability paradigm, bridging a gap between normative theory and observed behavior in intertemporal decision-making under genuine uncertainty.

The desirable gambles framework provides a rigorous foundation for imprecise probability theory but relies heavily on linear utility via its coherence axioms. In our related work, we introduced function-coherent gambles to accommodate non-linear utility. However, when repeated gambles are played over time -- especially in intertemporal choice where rewards compound multiplicatively -- the standard additive combination axiom fails to capture the appropriate long-run evaluation. In this paper we extend the framework by relaxing the additive combination axiom and introducing a nonlinear combination operator that effectively aggregates repeated gambles in the log-domain. This operator preserves the time-average (geometric) growth rate and addresses the ergodicity problem. We prove the key algebraic properties of the operator, discuss its impact on coherence, risk assessment, and representation, and provide a series of illustrative examples. Our approach bridges the gap between expectation values and time averages and unifies normative theory with empirically observed non-stationary reward dynamics.

Recently, Cella and Martin proved how, under an assumption called consonance, a credal set (i.e. a closed and convex set of probabilities) can be derived from the conformal transducer associated with transductive conformal prediction. We show that the Imprecise Highest Density Region (IHDR) associated with such a credal set corresponds to the classical Conformal Prediction Region. In proving this result, we relate the set of probability density/mass functions (pdf/pmf's) associated with the elements of the credal set to the imprecise probabilistic concept of a cloud. As a result, we establish new relationships between Conformal Prediction and Imprecise Probability (IP) theories. A byproduct of our presentation is the discovery that consonant plausibility functions are monoid homomorphisms, a new algebraic property of an IP tool.

What does it mean to say that, for example, the probability for rain tomorrow is between 20% and 30%? The theory for the evaluation of precise probabilistic forecasts is well-developed and is grounded in the key concepts of proper scoring rules and calibration. For the case of imprecise probabilistic forecasts (sets of probabilities), such theory is still lacking. In this work, we therefore generalize proper scoring rules and calibration to the imprecise case. We develop these concepts as relative to data models and decision problems. As a consequence, the imprecision is embedded in a clear context. We establish a close link to the paradigm of (group) distributional robustness and in doing so provide new insights for it. We argue that proper scoring rules and calibration serve two distinct goals, which are aligned in the precise case, but intriguingly are not necessarily aligned in the imprecise case. The concept of decision-theoretic entropy plays a key role for both goals. Finally, we demonstrate the theoretical insights in machine learning practice, in particular we illustrate subtle pitfalls relating to the choice of loss function in distributional robustness.

This paper introduces a novel concept of interval probability measures that enables the representation of imprecise probabilities, or uncertainty, in a natural and coherent manner. Within an algebra of sets, we introduce a notion of weak complementation denoted as $\psi$. The interval probability measure of an event $H$ is defined with respect to the set of indecisive eventualities $(\psi(H))^c$, which is included in the standard complement $H^c$. We characterize a broad class of interval probability measures and define their properties. Additionally, we establish an updating rule with respect to $H$, incorporating concepts of statistical independence and dependence. The interval distribution of a random variable is formulated, and a corresponding definition of stochastic dominance between two random variables is introduced. As a byproduct, a formal solution to the century-old Keynes-Ramsey controversy is presented.

Adequate uncertainty representation and quantification have become imperative in various scientific disciplines, especially in machine learning and artificial intelligence. As an alternative to representing uncertainty via one single probability measure, we consider credal sets (convex sets of probability measures). The geometric representation of credal sets as $d$-dimensional polytopes implies a geometric intuition about (epistemic) uncertainty. In this paper, we show that the volume of the geometric representation of a credal set is a meaningful measure of epistemic uncertainty in the case of binary classification, but less so for multi-class classification. Our theoretical findings highlight the crucial role of specifying and employing uncertainty measures in machine learning in an appropriate way, and for being aware of possible pitfalls.