Sebastian G. Gruber

Sebastian G. Gruber, Florian Buettner

With model trustworthiness being crucial for sensitive real-world applications, practitioners are putting more and more focus on improving the uncertainty calibration of deep neural networks. Calibration errors are designed to quantify the reliability of probabilistic predictions but their estimators are usually biased and inconsistent. In this work, we introduce the framework of proper calibration errors, which relates every calibration error to a proper score and provides a respective upper bound with optimal estimation properties. This relationship can be used to reliably quantify the model calibration improvement. We theoretically and empirically demonstrate the shortcomings of commonly used estimators compared to our approach. Due to the wide applicability of proper scores, this gives a natural extension of recalibration beyond classification.

Sebastian G. Gruber, Florian Buettner

Reliably estimating the uncertainty of a prediction throughout the model lifecycle is crucial in many safety-critical applications. The most common way to measure this uncertainty is via the predicted confidence. While this tends to work well for in-domain samples, these estimates are unreliable under domain drift and restricted to classification. Alternatively, proper scores can be used for most predictive tasks but a bias-variance decomposition for model uncertainty does not exist in the current literature. In this work we introduce a general bias-variance decomposition for proper scores, giving rise to the Bregman Information as the variance term. We discover how exponential families and the classification log-likelihood are special cases and provide novel formulations. Surprisingly, we can express the classification case purely in the logit space. We showcase the practical relevance of this decomposition on several downstream tasks, including model ensembles and confidence regions. Further, we demonstrate how different approximations of the instance-level Bregman Information allow reliable out-of-distribution detection for all degrees of domain drift.

Sebastian G. Gruber, Pascal Tobias Ziegler, Florian Buettner

The evaluation of image generators remains a challenge due to the limitations of traditional metrics in providing nuanced insights into specific image regions. This is a critical problem as not all regions of an image may be learned with similar ease. In this work, we propose a novel approach to disentangle the cosine similarity of mean embeddings into the product of cosine similarities for individual pixel clusters via central kernel alignment. Consequently, we can quantify the contribution of the cluster-wise performance to the overall image generation performance. We demonstrate how this enhances the explainability and the likelihood of identifying pixel regions of model misbehavior across various real-world use cases.

Sebastian G. Gruber, Florian Buettner

Generative models, like large language models, are becoming increasingly relevant in our daily lives, yet a theoretical framework to assess their generalization behavior and uncertainty does not exist. Particularly, the problem of uncertainty estimation is commonly solved in an ad-hoc and task-dependent manner. For example, natural language approaches cannot be transferred to image generation. In this paper, we introduce the first bias-variance-covariance decomposition for kernel scores. This decomposition represents a theoretical framework from which we derive a kernel-based variance and entropy for uncertainty estimation. We propose unbiased and consistent estimators for each quantity which only require generated samples but not the underlying model itself. Based on the wide applicability of kernels, we demonstrate our framework via generalization and uncertainty experiments for image, audio, and language generation. Specifically, kernel entropy for uncertainty estimation is more predictive of performance on CoQA and TriviaQA question answering datasets than existing baselines and can also be applied to closed-source models.

Teodora Popordanoska, Sebastian G. Gruber, Aleksei Tiulpin et al.

Proper scoring rules evaluate the quality of probabilistic predictions, playing an essential role in the pursuit of accurate and well-calibrated models. Every proper score decomposes into two fundamental components -- proper calibration error and refinement -- utilizing a Bregman divergence. While uncertainty calibration has gained significant attention, current literature lacks a general estimator for these quantities with known statistical properties. To address this gap, we propose a method that allows consistent, and asymptotically unbiased estimation of all proper calibration errors and refinement terms. In particular, we introduce Kullback--Leibler calibration error, induced by the commonly used cross-entropy loss. As part of our results, we prove the relation between refinement and f-divergences, which implies information monotonicity in neural networks, regardless of which proper scoring rule is optimized. Our experiments validate empirically the claimed properties of the proposed estimator and suggest that the selection of a post-hoc calibration method should be determined by the particular calibration error of interest.

Nassim Walha, Sebastian G. Gruber, Thomas Decker et al.

As Large Language Models (LLMs) are increasingly integrated in diverse applications, obtaining reliable measures of their predictive uncertainty has become critically important. A precise distinction between aleatoric uncertainty, arising from inherent ambiguities within input data, and epistemic uncertainty, originating exclusively from model limitations, is essential to effectively address each uncertainty source. In this paper, we introduce Spectral Uncertainty, a novel approach to quantifying and decomposing uncertainties in LLMs. Leveraging the Von Neumann entropy from quantum information theory, Spectral Uncertainty provides a rigorous theoretical foundation for separating total uncertainty into distinct aleatoric and epistemic components. Unlike existing baseline methods, our approach incorporates a fine-grained representation of semantic similarity, enabling nuanced differentiation among various semantic interpretations in model responses. Empirical evaluations demonstrate that Spectral Uncertainty outperforms state-of-the-art methods in estimating both aleatoric and total uncertainty across diverse models and benchmark datasets.

Sebastian G. Gruber

In this PhD thesis, we propose a novel framework for uncertainty quantification in machine learning, which is based on proper scores. Uncertainty quantification is an important cornerstone for trustworthy and reliable machine learning applications in practice. Usually, approaches to uncertainty quantification are problem-specific, and solutions and insights cannot be readily transferred from one task to another. Proper scores are loss functions minimized by predicting the target distribution. Due to their very general definition, proper scores apply to regression, classification, or even generative modeling tasks. We contribute several theoretical results, that connect epistemic uncertainty, aleatoric uncertainty, and model calibration with proper scores, resulting in a general and widely applicable framework. We achieve this by introducing a general bias-variance decomposition for strictly proper scores via functional Bregman divergences. Specifically, we use the kernel score, a kernel-based proper score, for evaluating sample-based generative models in various domains, like image, audio, and natural language generation. This includes a novel approach for uncertainty estimation of large language models, which outperforms state-of-the-art baselines. Further, we generalize the calibration-sharpness decomposition beyond classification, which motivates the definition of proper calibration errors. We then introduce a novel estimator for proper calibration errors in classification, and a novel risk-based approach to compare different estimators for squared calibration errors. Last, we offer a decomposition of the kernel spherical score, another kernel-based proper score, allowing a more fine-grained and interpretable evaluation of generative image models.

Han Zhou, Sebastian G. Gruber, Teodora Popordanoska et al.

Several variants of reweighted risk functionals, such as focal loss, inverse focal loss, and the Area Under the Risk--Coverage Curve (AURC), have been proposed for improving model calibration, yet their theoretical connections to calibration errors remain unclear. In this paper, we revisit a broad class of weighted risk functions commonly used in deep learning and establish a principled connection between calibration error and selective classification. We show that minimizing calibration error is closely linked to the selective classification paradigm and demonstrate that optimizing selective risk in low-confidence region naturally leads to improved calibration. This loss shares a similar reweighting strategy with dual focal loss but offers greater flexibility through the choice of confidence score functions (CSFs). Our approach uses a bin-based cumulative distribution function (CDF) approximation, enabling efficient gradient-based optimization without requiring expensive sorting and achieving $O(nK)$ complexity. Empirical evaluations demonstrate that our method achieves competitive calibration performance across a range of datasets and model architectures.

Jiameng Li, Teodora Popordanoska, Aleksei Tiulpin et al.

Ratio-based biomarkers -- such as the proportion of necrotic tissue within a tumor -- are widely used in clinical practice to support diagnosis, prognosis, and treatment planning. These biomarkers are typically estimated from soft segmentation outputs by computing region-wise ratios. Despite the high-stakes nature of clinical decision making, existing methods provide only point estimates, offering no measure of uncertainty. In this work, we propose a unified confidence-aware framework for estimating ratio-based biomarkers. Our uncertainty analysis stems from two observations: i) the probability ratio estimator inherently admits a statistical confidence interval regarding local randomness (bias and variance), ii) the segmentation network is not perfectly calibrated. We conduct a systematic analysis of error propagation in the segmentation-to-biomarker pipeline and identify model miscalibration as the dominant source of uncertainty. We leverage tunable parameters to control the confidence level of the derived bounds, allowing adaptation towards clinical practice. Extensive experiments show that our method produces statistically sound confidence intervals, with tunable confidence levels, enabling more trustworthy application of predictive biomarkers in clinical workflows.