Imprecise Probability

In social learning, agents learn not only from direct evidence but also through interactions with their peers. We investigate the role of imprecision in such interactions and ask whether it can improve the effectiveness of the collective learning process. To that end we propose a model of social learning where beliefs are equivalent to formulas in a propositional language, and where agents learn from each other by combining their beliefs according to a fusion operator. The latter is parametrised so as to allow for different levels of imprecision, where a more imprecise fusion operator tends to generates a more imprecise fused belief when the two combined beliefs differ. In this context we describe both difference equation models and agent-based simulations of social learning under a variety of conditions and with different initial biases. The results presented suggest that for populations with a strong initial bias towards incorrect beliefs some level of imprecision in fusion can improve learning accuracy across a range of learning conditions. Furthermore, such benefits of imprecision are consistent with a stability analysis of the fixed points of the proposed difference equation models.

Quantitative verification of neural networks requires reasoning about probabilities under substantial uncertainty in both input distributions and their dependence structure. In realistic settings, this information is often only partially specified, and assuming precise probabilistic models can lead to unreliable results. We propose a sound framework for quantitative verification under imprecise probabilistic information, combining interval belief structures to represent marginal uncertainty with imprecise copulas to model uncertain dependence. We develop a propagation method for imprecisely coupled interval belief structures through feed-forward neural networks. Using mixed imprecise copula volumes, we derive sound push-forward constructions through affine transformations and activation functions. The resulting output can provide guaranteed lower and upper bounds on probabilistic safety properties, valid for all probability models compatible with the specified imprecise inputs.

Value-of-information (VOI) analysis is usually conducted under a single probability measure. However, in practice, the available evidence often pins the measure down only to a set. Consequently, under a set of probability measures, VOI requires different formulations. First, we explicate a rule-specific VOI that fixes a decision rule for acting under imprecision (such as Gamma-maximin) and measures what the information is worth to a decision maker who uses that rule. Second, we derive a fixed-measure envelope that evaluates the classical VOI functional over all admissible precise measures. We formalize this distinction and explicate its consequences for the expected perfect, partial, and sample information. The expected value of perfect information is concave over the credal set. Hence, when the set is generated by finitely many measures, its lower envelope endpoint is obtained exactly from the generators, while its upper endpoint may be interior and is computed by a finite linear program. The Gamma-maximin value, in contrast, can exceed the entire envelope, so a rule-specific value is not recovered from the envelope's endpoints. A continuity bound limits how much the VOI can change as the measure varies, and we identify when the partial- and sample-information endpoints can still be obtained from the generators. Because the single-measure VOI must itself be estimated, the procedure we give combines standard estimators for it with a search over the credal set. By using a worked decision problem, we show how the two quantities separate conclusions that hold across every admissible measure from conclusions that depend on one unidentified choice of measure.

A non-agentic "oracle" that reports probabilities of future events is performative: once its answer is learned and acted upon, it can change the very probability it was asked to report. Performativity is not in itself the difficulty -- one consults an oracle precisely in order to be informed, and hence influenced, by it. The difficulty is agency. The requirement that a report be self-consistent, still holding once announced, may be met by many different values -- the classical non-uniqueness of self-fulfilling prophecies -- and any rule the system uses to choose among them is a lever for goal-directed steering. We remove the choice rather than the performativity. Reporting a credal set instead of a single probability distribution, we lift the reaction to an isotone operator on the complete lattice of closed credal sets, whose fixed points are self-consistent, and report its Knaster--Tarski least fixed point as a canonical, rule-determined answer; a variant reports instead the least fixed point that contains every self-consistent point estimate. We prove existence, self-consistency, and nonemptiness; show that the construction reduces to the classical point answer when the question is non-performative; and show that for a binary event the answer is, under a natural hull-factoring assumption, an interval.

Recent advances in uncertainty quantification increasingly emphasise the distinction between aleatory and epistemic uncertainty in machine learning, motivating the need for more unified frameworks. However, despite much progress in producing reliable predictions, existing methods often lack rigorous guarantees when generalising beyond the training domain. We propose a conformalised imprecise inference framework for robust extrapolation, which is model-agnostic and augments predictive models with imprecision and distance awareness. The proposed approach yields imprecise predictions (probability boxes) that remain valid under distributional shift, maintaining coverage while adaptively expanding uncertainty in extrapolation regimes. Experiments on synthetic and benchmark datasets demonstrate improved robustness and reliable coverage compared to standard probabilistic approaches, particularly under limited data.

Uncertainty quantification is a key aspect in many tasks such as model selection/regularization, or quantifying prediction uncertainties to perform active learning or OOD detection. Within credal approaches that consider modeling uncertainty as probability sets, upper entropy plays a central role as an uncertainty measure. This paper is devoted to the computational aspect of upper entropies, providing an exhaustive algorithmic and complexity analysis of the problem. In particular, we show that the problem has a strongly polynomial solution, and propose many significant improvements over past algorithms proposed for 2-monotone lower probabilities and their specific cases.

This paper develops a measure-theoretic framework establishing when and how a possibilistic representation of incomplete knowledge contracts into a probabilistic representation of intrinsic stochastic variability. Epistemic uncertainty is encoded by a possibility distribution and its dual necessity measure, defining a credal set bounding all probability measures consistent with current evidence. As evidence accumulates, the credal set contracts. The epistemic collapse condition marks the transition: the Choquet integral converges to the Lebesgue integral over the unique limiting density. We prove this rigorously (Theorem 4.5), with all assumptions explicit and a full treatment of the non-consonant case. We introduce the aggregate epistemic width W, establish its axiomatic properties, provide a canonical normalization, and give a feasible online proxy resolving a circularity in prior formulations. Section 7 develops the dynamics of epistemic contraction: evidence induces compatibility, compatibility performs falsification, posterior possibility is the min-intersection of prior possibility and compatibility, and a credibility-directed flow governs support geometry contraction. This is not belief updating. It is knowledge contraction. Probability theory is the limiting geometry of that process. The UKF and ESPF solve different problems by different mechanisms. The UKF minimizes MSE, asserts truth, and requires a valid generative model. The ESPF minimizes maximum entropy and surfaces what evidence has not ruled out. When the world is Gaussian and the model valid, both reach the same estimate by entirely different routes -- convergent optimality, not hierarchical containment. We prove this (Theorem 9.1) and compare both on a 2-day, 877-step orbital tracking scenario. Both achieve 1-meter accuracy. The UKF is accurate but epistemically silent. The ESPF is accurate and epistemically honest.

Despite the growing demand for eliciting uncertainty from large language models (LLMs), empirical evidence suggests that LLM behavior is not always adequately captured by the elicitation techniques developed under the classical probabilistic uncertainty framework. This mismatch leads to systematic failure modes, particularly in settings that involve ambiguous question-answering, in-context learning, and self-reflection. To address this, we propose novel prompt-based uncertainty elicitation techniques grounded in imprecise probabilities, a principled framework for repesenting and eliciting higher-order uncertainty. Here, first-order uncertainty captures uncertainty over possible responses to a prompt, while second-order uncertainty (uncertainty about uncertainty) quantifies indeterminacy in the underlying probability model itself. We introduce general-purpose prompting and post-processing procedures to directly elicit and quantify both orders of uncertainty, and demonstrate their effectiveness across diverse settings. Our approach enables more faithful uncertainty reporting from LLMs, improving credibility and supporting downstream decision-making.

The rapid proliferation of AI applications has intensified debate on effective regulation of these black-box services. Effective regulation must balance two competing goals: (1) deterring non-compliant providers from entering the market, while (2) retaining compliant ones. We call this ideal the perfect market outcome (PMO). Regulators face two compounding obstacles that make PMO difficult to achieve: providers hold private information and can act strategically to evade compliance, while any evidence drawn or derived from a finite sample carries statistical uncertainty in proving non-compliance. As this information asymmetry and statistical uncertainty is inherent to any effective regulation, we formalise them through a mechanism design framework that explicitly accounts for such statistical uncertainty. This yields a sharp characterisation: a mechanism achieves PMO if and only if the set of non-compliant evidence distributions forms a closed, convex set of probability measures, known in imprecise probability as a credal set. This result serves as a diagnostic tool to determine whether PMO is achievable under a given regulation. We further show that PMO-achieving mechanisms can be constructed from a collection of hypothesis tests, and validate our theoretical contributions through experiments on spurious-feature and fairness-based regulations.