Uncertainty Quantification for Regression: A Unified Framework based on kernel scores

summary

The gist

Regression tasks, notably in safety-critical domains, require reliable uncertainty quantification, yet the literature remains largely classificationfocused The framework provides a principled recipe

In short

The framework proposes a unified method for quantifying uncertainty in regression tasks by basing measures on strictly proper kernel scores. This approach designs new uncertainty metrics whose behavior is controlled by the choice of kernel, offering a principled way to handle structured data and improve out-of-distribution detection.

Key concepts

Uncertainty Decomposition Framework: This framework breaks down total uncertainty into three components: total (overall), aleatoric (inherent noise in data), and epistemic (model uncertainty). It uses strictly proper kernel scores to define these measures, providing a consistent way to analyze prediction reliability.
Kernel Scores as Design Lever: Kernel scores are the core tool. The paper shows that the properties of the chosen kernel directly dictate how the resulting uncertainty measure behaves—for instance, its sensitivity to outliers or its translation invariance. Choosing a specific kernel allows designers to tailor uncertainty measures for specific tasks.
Strictly Proper Kernel Scores: These are mathematical scores derived from kernels that possess crucial properties, such as convexity and translation invariance. These properties ensure that the resulting uncertainty measures satisfy important mathematical conditions, like vanishing epistemic uncertainty when models agree.
Kernel Mean Embeddings (KME): The framework relates kernel scores to the squared distance between kernel mean embeddings of probability distributions in a Hilbert space. This connection helps define tractible uncertainty measures, linking the geometric properties of the data distributions to the resulting uncertainty quantification.

Terminology used across episodes

Reinforcement Learning (RL) — explained in 25 episodes
Sample Complexity — explained in 19 episodes
Federated Learning — explained in 17 episodes
Retrieval-Augmented Generation (RAG) — explained in 17 episodes
Quantum Fisher Information (QFI) — explained in 16 episodes
Interpretability — explained in 15 episodes

This episode discusses

The paper

Uncertainty Quantification for Regression: A Unified Framework based on kernel scores · Read on arXiv

Christopher Bülte, Yusuf Sale, Gitta Kutyniok, Eyke Hülermeier

LMU Munich · Munich Center for Machine Learning (MCML) · DLR-German Aerospace Center · University of Tromsö

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.

Transcript

Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom.

Jane: And I'm Jane, here with Lu, Meng and our in-house large language model, Lalam.

Tom: Today's paper: "Uncertainty Quantification for Regression".

Jane: Regression tasks, notably in safety-critical domains, require reliable uncertainty quantification, yet the literature remains largely classificationfocused The framework provides a principled recipe for designing new uncertainty measures whose behavior,

Tom: First, who's behind it and why it matters.

Paper summary: Tom: Thinking about the title, "Uncertainty Quantification for Regression: A Unified Framework based on kernel scores," it’s a lot of concepts boiled down to one core idea. The authors are essentially providing a general recipe for building uncertainty measures in regression using strictly proper kernel scores <ref:2510.25599#pg1>.

Jane: It means that instead of having to invent a new uncertainty measure every time you tackle a regression task, you just need to select the right kernel and you get the corresponding measure, and its behavior is predictable because of what the kernel does <ref:2510.25599#pg3>.

Lu: The implication is that we have shifted from searching for specific uncertainty representations to designing a systematic construction based on established mathematical concepts like proper scoring rules <ref:2510.25599#pg3>. It’s about connecting the structure of the kernel directly to desired outcomes, like tail sensitivity or out-of-distribution responsiveness <ref:2510.25599#pg1>.

Meng: For deployment teams, this suggests we can move toward more principled uncertainty checks that are tied to concrete mathematical properties derived from the kernel choice rather than just arbitrary heuristics <ref:2510.25599#pg3>. It gives us a foundation for building models that are more robust in ways that we can mathematically analyze <ref:2510.25599#pg3>.

Lalam: The paper shows that this framework works across diverse data modalities and benchmark tasks, with specific scores like Skγ leading in selective prediction and active learning, while SCRPS and SES lead in out-of-distribution detection <ref:2510.25599#pg2>. It’s a comprehensive tool for uncertainty quantification in this domain <ref:2510.25599#pg1>.

Tom: So we have established that kernel scores are a powerful design lever because they inherit the necessary structural properties, and their explicit connections to resulting uncertainty measure behavior—like translation invariance and robustness—are what make this framework so useful <ref:2510.25599#pg3>.

Jane: It’s a principled way to construct these measures that can be used with many different instantiations, which means you can tailor the uncertainty quantification precisely to the specific problem you are facing <ref:2510.25599#pg1>.

Conclusion: Tom: So we've been looking at this paper about uncertainty quantification in regression using these kernel scores, and now we're getting to the end of it <ref:2510.25599#pg1>.

Jane: Right, so basically they put together a whole recipe for measuring total, aleatoric, and epistemic uncertainty all at once based on these kernel scores <ref:2510.25599#pg3>.

Tom: The authors are showing how the choice of the underlying kernel is what actually governs how that uncertainty measure behaves, like its sensitivity to outliers or if it's good at spotting things outside its normal range <ref:2510.25599#pg1>.

Lu: It’s interesting because they take this idea from classification and apply it to multivariate regression, which is where a lot of the real structured data problems are happening <ref:2510.25599#pg3>.

Meng: From my side, what matters is that they give us a systematic way to design these measures for specific tasks without having to reinvent the wheel every time we build a new model <ref:2510.25599#pg3>.

Jane: They're saying it’s not just about picking one uncertainty measure; it’s about choosing the right kernel and that choice dictates all those other properties <ref:2510.25599#pg3>.

Tom: The results are pretty solid, especially with these specific kernel scores like the Energy score showing really good performance in selective prediction <ref:2510.25599#pg4>.

Lu: They even found that the Gaussian kernel score, Skγ, performs really well across all those different experimental setups we looked at <ref:2510.25599#pg4>.

Jane: So to wrap up, this framework turns designing uncertainty measures into selecting a kernel and seeing how that kernel structures the measure itself <ref:2510.25599#pg3>.

Tom: And it shows that these kernel-based rules perform well across all evaluation protocols we tested, which means you get a consistent level of performance regardless of what you're measuring <ref:2510.25599#pg4>.

Jane: But there are still some things to explore, like how we can find the best kernel for a specific problem or maybe look at other types of scores <ref:2510.25599#pg7>.

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