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.

Credal sets, i.e., closed convex sets of probability measures, provide a natural framework to represent aleatoric and epistemic uncertainty in machine learning. Yet how to quantify these two types of uncertainty for a given credal set, particularly in multiclass classification, remains underexplored. In this paper, we propose a distance-based approach to quantify total, aleatoric, and epistemic uncertainty for credal sets. Concretely, we introduce a family of such measures within the framework of Integral Probability Metrics (IPMs). The resulting quantities admit clear semantic interpretations, satisfy natural theoretical desiderata, and remain computationally tractable for common choices of IPMs. We instantiate the framework with the total variation distance and obtain simple, efficient uncertainty measures for multiclass classification. In the binary case, this choice recovers established uncertainty measures, for which a principled multiclass generalization has so far been missing. Empirical results confirm practical usefulness, with favorable performance at low computational cost.

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 both a fundamental trade-off between performatively changing the world and learning from it, as well as a surprising insight on how to improve generalization guarantees by retraining on performatively distorted samples. We illustrate our bounds using real data on prediction-informed assignments to job trainings.

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) Möbius 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.

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.

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.

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.

Out-of-distribution (OOD) generalization is a defining hallmark of truly robust and reliable machine learning systems. Recently, it has been empirically observed that existing OOD generalization methods often underperform on real-world tabular data, where hidden confounding shifts drive distribution changes that boosting models handle more effectively. Part of boosting’s success is attributed to variance reduction, handling missing variables, feature selection, and connections to multicalibration. This paper uncovers a crucial reason behind its success in OOD generalization: boosting’s ability to infer stable environments robust to hidden confounding shifts and maximize predictive performance within those environments. This paper introduces an information-theoretic notion called $\alpha$-predictive sufficiency and formalizes its link to OOD generalization under hidden confounding. We show that boosting implicitly identifies suitable environments and produces an $\alpha$-predictive sufficient predictor. We validate our theoretical results through synthetic and real-world experiments and show that boosting achieves robust performance by identifying these environments and maximizing the association between predictions and true outcomes.

Imbalanced data is commonly present in real-world applications. While data synthesis can effectively mitigate the data scarcity problem of rare-classes, and LLMs have revolutionized text generation, the application of LLMs to relational/structured tabular data synthesis remains underexplored. Moreover, existing approaches lack an effective feedback mechanism that can guide LLMs towards continuously optimizing the quality of the generated data throughout the synthesis process. In this work, we propose RDDG, Relational Data generator with Dynamic Guidance, which is a unified in-context learning framework that employs progressive chain-of-thought (CoT) steps to generate tabular data for enhancing downstream imbalanced classification performance. RDDG first uses core set selection to identify representative samples from the original data, then utilizes in-context learning to discover the inherent patterns and correlations among attributes within the core set, and subsequently generates tabular data while preserving the aforementioned constraints. More importantly, it incorporates a self-reinforcing feedback mechanism that provides automatic assessments on the quality of the generated data, enabling continuous quality optimization throughout the generation process. Experimental results on multiple real and synthetic datasets demonstrate that RDDG outperforms existing approaches in both data fidelity and downstream imbalanced classification performance. We make our code available at https://github.com/cszhangLMU/RDDG.

We study off-policy evaluation (OPE) under strategic behavior where decision subjects (or agents) respond to a decision maker's policy by strategically modifying their covariates. Such behavior induces a policy-dependent covariate shift, breaking the standard assumption in existing methods that covariates are exogenous to the policy. Related work addresses this challenge by imposing strong assumptions such as repeated interactions or full knowledge of agents' response behavior, substantially limiting its applicability to OPE. In contrast, we consider a one-shot OPE setting where the decision maker has only partial knowledge of the agents' response behavior. Our key insight is that disclosing local information through post-hoc explanations reveals agents' pre-strategic covariates prior to adaptation, mitigating the information loss induced by strategic behavior. Leveraging this structure, we estimate a statistical model for the agents' responses and construct a doubly robust estimator for policy value. By assuming that the agents' cost sensitivity follows a conditional log-normal distribution, we establish consistency of the proposed estimator and validate our approach empirically. More broadly, our results highlight how interaction design can mitigate information asymmetry by revealing otherwise hidden structure in agents' strategic responses.

We study the efficient computation of Shapley values for \emph{product games} -- cooperative games in which the coalition value factorizes as a product of per-player terms. Such games arise in machine learning explainability whenever the value function inherits a multiplicative structure from the underlying model, as in kernel methods with product kernels and tree-based models. Our key result is that the Shapley value of each player in a product game admits an exact one-dimensional integral representation: the weighted sum over exponentially many feature coalitions collapses to the integral of a degree-$(d-1)$ polynomial over $[0,1]$, where $d$ is the total number of features. This yields a Gauss--Legendre quadrature scheme that is \emph{provably exact} whenever the number of nodes satisfies $m_q \geq \lceil d/2 \rceil$, and otherwise provides a \emph{near-exact} approximation with error provably decaying geometrically in $m_q$. In practice, a few hundred nodes can achieve highly precise estimates even with thousands of features. Building on this formulation, we derive a numerically stable implementation via log-space evaluation, together with an efficient parallel implementation based on associative scan primitives that achieves $O(d\,m_q)$ total work and $O(\log d)$ parallel time. Experiments show that \textsc{QuadraSHAP} is the fastest numerically stable method across all tested configurations.

Learning robust models under distribution shifts between training and test datasets is a fundamental challenge in machine learning. While learning invariant features across environments is a popular approach, it often assumes that these features are fully observed in both training and test sets-a condition frequently violated in practice. When models rely on invariant features absent in the test set, their robustness in new environments can deteriorate. To tackle this problem, we introduce a novel learning principle called the Sufficient Invariant Learning (SIL) framework, which focuses on learning a sufficient subset of invariant features rather than relying on a single feature. After demonstrating the limitation of existing invariant learning methods, we propose a new algorithm, Adaptive Sharpness-aware Group Distributionally Robust Optimization (ASGDRO), to learn diverse invariant features by seeking common flat minima across the environments. We theoretically demonstrate that finding a common flat minima enables robust predictions based on diverse invariant features. Empirical evaluations on multiple datasets, including our new benchmark, confirm ASGDRO's robustness against distribution shifts, highlighting the limitations of existing methods. Code: https://github.com/MLAI-Yonsei/SIL-ASGDRO.

The quality of probabilistic forecasts is crucial for decision-making under uncertainty. While proper scoring rules incentivize truthful reporting of precise forecasts, they fall short when forecasters face epistemic uncertainty about their beliefs, limiting their use in safety-critical domains where decision-makers (DMs) prioritize proper uncertainty management. To address this, we propose a framework for scoring imprecise forecasts -- forecasts given as a set of beliefs. Despite existing impossibility results for deterministic scoring rules, we enable truthful elicitation by drawing connection to social choice theory and introducing a two-way communication framework where DMs first share their aggregation rules (e.g., averaging or min-max) used in downstream decisions for resolving forecast ambiguity. This, in turn, helps forecasters resolve indecision during elicitation. We further show that truthful elicitation of imprecise forecasts is achievable using proper scoring rules randomized over the aggregation procedure. Our approach allows DM to elicit and integrate the forecaster's epistemic uncertainty into their decision-making process, thus improving credibility.

The technique of data augmentation (DA) is often used in machine learning for regularization purposes to better generalize under i.i.d. settings. In this work, we present a unifying framework with topics in causal inference to make a case for the use of DA beyond just the i.i.d. setting, but for generalization across interventions as well. Specifically, we argue that when the outcome generating mechanism is invariant to our choice of DA, then such augmentations can effectively be thought of as interventions on the treatment generating mechanism itself. This can potentially help to reduce bias in causal effect estimation arising from hidden confounders. In the presence of such unobserved confounding we typically make use of instrumental variables (IVs)—sources of treatment randomization that are conditionally independent of the outcome. However, IVs may not be as readily available as DA for many applications, which is the main motivation behind this work. By appropriately regularizing IV based estimators, we introduce the concept of *IV-like (IVL)* regression for mitigating confounding bias and improving predictive performance across interventions even when certain IV properties are relaxed. Finally, we cast parameterized DA as an IVL regression problem and show that when used in composition can simulate a worst-case application of such DA, further improving performance on causal estimation and generalization tasks beyond what simple DA may offer. This is shown both theoretically for the population case and via simulation experiments for the finite sample case using a simple linear example. We also present real data experiments to support our case.

We study explanation design in algorithmic decision making with strategic agents, individuals who may modify their inputs in response to explanations of a decision maker's (DM's) predictive model. As the demand for transparent algorithmic systems continues to grow, most prior work assumes full model disclosure as the default solution. In practice, however, DMs such as financial institutions typically disclose only partial model information via explanations. Such partial disclosure can lead agents to misinterpret the model and take actions that unknowingly harm their utility. A key open question is how DMs can communicate explanations in a way that avoids harming strategic agents, while still supporting their own decision-making goals, e.g., minimising predictive error. In this work, we analyse well-known explanation methods, and establish a necessary condition to prevent explanations from misleading agents into self-harming actions. Moreover, with a conditional homogeneity assumption, we prove that action recommendation-based explanations (ARexes) are sufficient for non-harmful responses, mirroring the revelation principle in information design. To demonstrate how ARexes can be operationalised in practice, we propose a simple learning procedure that jointly optimises the predictive model and explanation policy. Experiments on synthetic and real-world tasks show that ARexes allow the DM to optimise their model's predictive performance while preserving agents' utility, offering a more refined strategy for safe and effective partial model disclosure.

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.

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.

Quantifying differences between probability distributions is fundamental to statistics and machine learning, primarily for comparing statistical uncertainty. In contrast, epistemic uncertainty---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 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~(EU) within a single IP model. In particular, by comparing an IP model with its conjugate, IIPM gives rise to a new class of epistemic uncertainty measures---Maximum Mean Imprecision (MMI) ---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 Imprecise Probabilistic Machine Learning, offering a principled framework for comparing and quantifying epistemic uncertainty under imprecision.

Kernel methods are widely used in machine learning and statistics for their flexibility and expressive power, yet their black-box nature limits adoption in high-stakes applications. Shapley value-based attribution methods such as SHAP, and kernel-specific adaptations including RKHS-SHAP, provide a principled framework for explainability – but exact computation of Shapley values is generally intractable, forcing existing approaches to rely on approximations that incur unavoidable estimation error. We introduce PKeX-Shapley, an algorithm that exploits the multiplicative structure of product kernels to compute exact Shapley values for all d features in quadratic time in d. The method rests on a distribution-free removal operator intrinsic to the product-kernel structure: removing a feature replaces its kernel factor with the multiplicative identity. This yields a parameter-free value function – requiring no sampling and no density estimation – and uniquely determines a functional decomposition of the model. Building on this value function, we develop shared recursive formulations that evaluate all feature attributions jointly, achieving amortized linear time per feature with numerical stability. Beyond predictive modeling, the framework extends to widely used kernel-based discrepancies such as the Maximum Mean Discrepancy (MMD) and the Hilbert-Schmidt Independence Criterion (HSIC), providing new tools for interpretable statistical analysis.

We introduce credal two-sample testing, a new hypothesis testing framework for comparing credal sets -- convex sets of probability measures where each element captures aleatoric uncertainty and the set itself represents epistemic uncertainty that arises from the modeller's partial ignorance. Compared to classical two-sample tests, which focus on comparing precise distributions, the proposed framework provides a broader and more versatile set of hypotheses. This approach enables the direct integration of epistemic uncertainty, effectively addressing the challenges arising from partial ignorance in hypothesis testing. By generalising two-sample test to compare credal sets, our framework enables reasoning for equality, inclusion, intersection, and mutual exclusivity, each offering unique insights into the modeller's epistemic beliefs. As the first work on nonparametric hypothesis testing for comparing credal sets, we focus on finitely generated credal sets derived from i.i.d. samples from multiple distributions -- referred to as credal samples. We formalise these tests as two-sample tests with nuisance parameters and introduce the first permutation-based solution for this class of problems, significantly improving existing methods. Our approach properly incorporates the modeller's epistemic uncertainty into hypothesis testing, leading to more robust and credible conclusions, with kernel-based implementations for real-world applications.