Verbalizing LLM's Higher-order Uncertainty via Imprecise Probabilities

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 \emph{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.

Incentive Aware AI Regulations: A Credal Characterisation

While high-stakes ML applications demand strict regulations, strategic ML providers often evade them to lower development costs. To address this challenge, we cast AI regulation as a mechanism design problem under uncertainty and introduce regulation mechanisms: a framework that maps empirical evidence from models to a license for some market share. The providers can select from a set of licenses, effectively forcing them to bet on their model's ability to fulfil regulation. We aim at regulation mechanisms that achieve perfect market outcome, i.e. (a) drive non-compliant providers to self-exclude, and (b) ensure participation from compliant providers. We prove that a mechanism has perfect market outcome if and only if the set of non-compliant distributions forms a credal set, i.e., a closed, convex set of probability measures. This result connects mechanism design and imprecise probability by establishing a duality between regulation mechanisms and the set of non-compliant distributions. We also demonstrate these mechanisms in practice via experiments on regulating use of spurious features for prediction and fairness. Our framework provides new insights at the intersection of mechanism design and imprecise probability, offering a foundation for development of enforceable AI regulations.

Instrumental and Proximal Causal Inference with Gaussian Processes

Instrumental variable (IV) and proximal causal learning (Proxy) methods are central frameworks for causal inference in the presence of unobserved confounding. Despite substantial methodological advances, existing approaches rarely provide reliable epistemic uncertainty (EU) quantification. We address this gap through a Deconditional Gaussian Process (DGP) framework for uncertainty-aware causal learning. Our formulation recovers popular kernel estimators as the posterior mean, ensuring predictive precision, while the posterior variance yields principled and well-calibrated EU. Moreover, the probabilistic structure enables systematic model selection via marginal log-likelihood optimization. Empirical results demonstrate strong predictive performance alongside informative EU quantification, evaluated via empirical coverage frequencies and decision-aware accuracy rejection curves. Together, our approach provides a unified, practical solution for causal inference under unobserved confounding with reliable uncertainty.

Performative Learning Theory

Performative predictions influence the very outcomes they aim to forecast. We study performative predictions that affect a sample (e.g., only existing users of an app) and/or the whole population (e.g., all potential app users). This raises the question of how well models generalize under performativity. For example, how well can we draw insights about new app users based on existing users when both of them react to the app's predictions? We address this question by embedding performative predictions into statistical learning theory. We prove generalization bounds under performative effects on the sample, on the population, and on both. A key intuition behind our proofs is that in the worst case, the population negates predictions, while the sample deceptively fulfills them. We cast such self-negating and self-fulfilling predictions as min-max and min-min risk functionals in Wasserstein space, respectively. Our analysis reveals a fundamental trade-off between performatively changing the world and learning from it: the more a model affects data, the less it can learn from it. Moreover, our analysis results in a surprising insight on how to improve generalization guarantees by retraining on performatively distorted samples. We illustrate our bounds in a case study on prediction-informed assignments of unemployed German residents to job trainings, drawing upon administrative labor market records from 1975 to 2017 in Germany.

Lost in Aggregation: The Causal Interpretation of the IV Estimand

Instrumental variable based estimation of a causal effect has emerged as a standard approach to mitigate confounding bias in the social sciences and epidemiology, where conducting randomized experiments can be too costly or impossible. However, justifying the validity of the instrument often poses a significant challenge. In this work, we highlight a problem generally neglected in arguments for instrumental variable validity: the presence of an ''aggregate treatment variable'', where the treatment (e.g., education, GDP, caloric intake) is composed of finer-grained components that each may have a different effect on the outcome. We show that the causal effect of an aggregate treatment is generally ambiguous, as it depends on how interventions on the aggregate are instantiated at the component level, formalized through the aggregate-constrained component intervention distribution. We then characterize conditions on the interventional distribution and the aggregate setting under which standard instrumental variable estimators identify the aggregate effect. The contrived nature of these conditions implies major limitations on the interpretation of instrumental variable estimates based on aggregate treatments and highlights the need for a broader justificatory base for the exclusion restriction in such settings.

When Do Credal Sets Stabilize? Fixed-Point Theorems for Credal Set Updates

Many machine learning algorithms rely on iterative updates of uncertainty representations, ranging from variational inference and expectation-maximization, to reinforcement learning, continual learning, and multi-agent learning. In the presence of imprecision and ambiguity, credal sets -- closed, convex sets of probability distributions -- have emerged as a popular framework for representing imprecise probabilistic beliefs. Under such imprecision, many learning problems in imprecise probabilistic machine learning (IPML) may be viewed as processes involving successive applications of update rules on credal sets. This naturally raises the question of whether this iterative process converges to stable fixed points -- or, more generally, under what conditions on the updating mechanism such fixed points exist, and whether they can be attained. We provide the first analysis of this problem and illustrate our findings using Credal Bayesian Deep Learning as a concrete example. Our work demonstrates that incorporating imprecision into the learning process not only enriches the representation of uncertainty, but also reveals structural conditions under which stability emerges, thereby offering new insights into the dynamics of iterative learning under imprecision.

Exact Shapley Attributions in Quadratic-time for FANOVA Gaussian Processes

Shapley values are widely recognized as a principled method for attributing importance to input features in machine learning. However, the exact computation of Shapley values scales exponentially with the number of features, severely limiting the practical application of this powerful approach. The challenge is further compounded when the predictive model is probabilistic - as in Gaussian processes (GPs) - where the outputs are random variables rather than point estimates, necessitating additional computational effort in modeling higher-order moments. In this work, we demonstrate that for an important class of GPs known as FANOVA GP, which explicitly models all main effects and interactions, *exact* Shapley attributions for both local and global explanations can be computed in *quadratic time*. For local, instance-wise explanations, we define a stochastic cooperative game over function components and compute the exact stochastic Shapley value in quadratic time only, capturing both the expected contribution and uncertainty. For global explanations, we introduce a deterministic, variance-based value function and compute exact Shapley values that quantify each feature's contribution to the model's overall sensitivity. Our methods leverage a closed-form (stochastic) M\"{o}bius representation of the FANOVA decomposition and introduce recursive algorithms, inspired by Newton's identities, to efficiently compute the mean and variance of Shapley values. Our work enhances the utility of explainable AI, as demonstrated by empirical studies, by providing more scalable, axiomatically sound, and uncertainty-aware explanations for predictions generated by structured probabilistic models.

When Shift Happens - Confounding Is to Blame

Distribution shifts introduce uncertainty that undermines the robustness and generalization capabilities of machine learning models. While conventional wisdom suggests that learning causal-invariant representations enhances robustness to such shifts, recent empirical studies present a counterintuitive finding: (i) empirical risk minimization (ERM) can rival or even outperform state-of-the-art out-of-distribution (OOD) generalization methods, and (ii) its OOD generalization performance improves when all available covariates, not just causal ones, are utilized. Drawing on both empirical and theoretical evidence, we attribute this phenomenon to hidden confounding. Shifts in hidden confounding induce changes in data distributions that violate assumptions commonly made by existing OOD generalization approaches. Under such conditions, we prove that effective generalization requires learning environment-specific relationships, rather than relying solely on invariant ones. Furthermore, we show that models augmented with proxies for hidden confounders can mitigate the challenges posed by hidden confounding shifts. These findings offer new theoretical insights and practical guidance for designing robust OOD generalization algorithms and principled covariate selection strategies.

Kernel Quantile Embeddings and Associated Probability Metrics

Embedding probability distributions into reproducing kernel Hilbert spaces (RKHS) has enabled powerful nonparametric methods such as the maximum mean discrepancy (MMD), a statistical distance with strong theoretical and computational properties. At its core, the MMD relies on kernel mean embeddings to represent distributions as mean functions in RKHS. However, it remains unclear if the mean function is the only meaningful RKHS representation. Inspired by generalised quantiles, we introduce the notion of kernel quantile embeddings (KQEs). We then use KQEs to construct a family of distances that: (i) are probability metrics under weaker kernel conditions than MMD; (ii) recover a kernelised form of the sliced Wasserstein distance; and (iii) can be efficiently estimated with near-linear cost. Through hypothesis testing, we show that these distances offer a competitive alternative to MMD and its fast approximations.

Computing Exact Shapley Values in Polynomial Time for Product-Kernel Methods

Kernel methods are widely used in machine learning due to their flexibility and expressive power. However, their black-box nature poses significant challenges to interpretability, limiting their adoption in high-stakes applications. Shapley value-based feature attribution techniques, such as SHAP and kernel-specific variants like RKHS-SHAP, offer a promising path toward explainability. Yet, computing exact Shapley values remains computationally intractable in general, motivating the development of various approximation schemes. In this work, we introduce PKeX-Shapley, a novel algorithm that utilizes the multiplicative structure of product kernels to enable the exact computation of Shapley values in polynomial time. We show that product-kernel models admit a functional decomposition that allows for a recursive formulation of Shapley values. This decomposition not only yields computational efficiency but also enhances interpretability in kernel-based learning. We also demonstrate how our framework can be generalized to explain kernel-based statistical discrepancies such as the Maximum Mean Discrepancy (MMD) and the Hilbert-Schmidt Independence Criterion (HSIC), thus offering new tools for interpretable statistical inference.