Efficient uncertainty quantification (UQ) is essential for trustworthy large-scale learning. Existing UQ methods for regression tasks mainly operate under the assumption that the conditional label marginal satisfies single-peak parametric models, e.g., Gaussians, where the negative log-likelihood function simplifies to the mean square error. However, such single-peak assumptions fail in regression tasks featuring multi-modal distributions. On the other hand, semi-parametric methods which achieve strong regression performance for multi-modal distributions often lack efficient quantification on their prediction variances. In this work, we extend UQ techniques based on Variational Bayesian Inference (VBI) to two widely used semi-parametric regression models that yield histogram-like reconstructions of the conditional label densities: Quantile Regression (QR) and Classification Restoration (CR). Our approach introduces a unified, distribution-agnostic framework that simultaneously achieves accurate estimation of complex conditional distributions and highly efficient UQ. Theoretically, our method is grounded in novel formulations of QR and CR within the VBI framework, yielding analytic Evidence Lower Bounds (ELBO) to streamline training and a closed-form or analytically approximated predictive density for efficient inference. Empirically, we evaluate our methods on three large-scale regression benchmarks with multi-modal label distributions. Our framework outperforms state-of-the-art multi-modal regression baselines, and even matches predictive performance of computationally expensive ensemble models. Furthermore, by leveraging epistemic uncertainty estimation, our approach enables highly data-efficient active learning strategies.
Regression tasks, notably in safety-critical domains, require reliable uncertainty quantification, yet the literature remains largely classification-focused. To address this, we introduce a family of measures for total, aleatoric, and epistemic uncertainty in multivariate regression based on strictly proper kernel scores. The framework provides a principled recipe for designing new uncertainty measures whose behavior, such as tail sensitivity or out-of-distribution responsiveness, is governed by the choice of the underlying kernel, while also encompassing existing measures under a joint analysis. We prove explicit correspondences between properties of the kernel and behavior of resulting uncertainty measures, yielding concrete design guidelines for practitioners. Extensive experiments across structured regression tasks, including spatial and functional domains, demonstrate effectiveness on downstream tasks such as out-of-distribution detection and active learning, and reveal that different kernel choices lead to distinct trade-offs, offering practitioners guidance for task-specific selection.
Uncertainty quantification (UQ) is essential for reliable decision-making in safety-critical applications in probabilistic machine learning. For regression problems, dominant scalar UQ approaches - notably, those based on proper scoring rules - measure uncertainty via pointwise predictive risk. This can lead to counterintuitive results when the target statistic is not the conditional expectation. We propose an alternative framework, in which uncertainty is characterised by the volume of the most probable subset of a distribution's support. QUEST (Quantifying Uncertainty via highest dEnSiTy regions) is a novel approach to UQ based on the concentration of Lebesgue measure at a distribution's peak(s), evaluated at one or more values of a robustness parameter α. We establish connections between our measures and classical statistics from information theory and economics. We show that, unlike popular alternatives based on proper scoring rules, QUEST measures of epistemic and aleatoric uncertainty satisfy a set of axioms adapted from the UQ literature, including monotonicity under distributional spread and invariance to location shifts. Selective prediction benchmarks confirm that QUEST performs favourably against standard measures such as variance and differential entropy.
Active learning for continuous regression has lacked an acquisition function that targets epistemic uncertainty when the predictive distribution is multimodal: variance misses modal disagreement, and information-theoretic targets like BALD are designed for discrete outputs. We introduce a Two-Index framework that makes this separation explicit: one stochastic index selects among competing model hypotheses (epistemic source), while a second governs within-hypothesis randomness (aleatoric source). An entropy decomposition within the framework identifies the mutual information between the output and the epistemic index as a principled acquisition objective, and we prove this quantity vanishes as the model is trained on growing datasets, confirming that it captures exactly the uncertainty data can resolve. Because this mutual information is intractable for continuous outputs, we derive the Mutual Information Lower Bound (MI-LB) acquisition function, a closed-form approximation for Mixture Density Network ensembles. On benchmarks featuring multimodal systems, MI-LB matches or beats every baseline evaluated and is the only method to do so consistently -- geometric and Fisher-based baselines compete only when the input space already encodes the multimodality, and collapse otherwise.
Leonardo Ferreira Guilhoto, Akshat Kaushal, Paris Perdikaris