An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression

summary

The gist

The gist: This work provides a formal way of representing uncertainty in continuous space using a general parametric formulation and proposes axioms to rigorously assess total, aleatoric, and

In short

This work introduces a formal framework to represent uncertainty in continuous regression using a general parametric family of distributions and proposes six axioms to rigorously assess uncertainty measures. It compares entropy-based and variance-based quantification methods, revealing that while variance-based measures satisfy most axioms, entropy-based ones have limitations regarding non-negativity.

Key concepts

Parametric Framework: This is a way to mathematically define the uncertainty in a regression model. Instead of just predicting a single value, it assumes the uncertainty comes from an unknown parameter vector $\theta$. The actual predictive distribution $P(x)$ is then defined as a mixture over all possible parameter settings $\theta$, linking the input data $x$ to the underlying model parameters.
Total Uncertainty (TU): Total uncertainty represents the overall randomness in a prediction. It is quantified using entropy-based measures, which measure how much information is needed to describe the predictive distribution. It combines both aleatoric and epistemic sources of uncertainty into one comprehensive measure.
Axioms: These are six formal rules proposed to guide the design and evaluation of uncertainty measures in continuous regression. They include properties like translation invariance and scale-order equivariance, which help determine if a specific measure is mathematically sound or well-behaved across different scenarios.

Terminology used across episodes

Reinforcement Learning (RL) — explained in 24 episodes
Retrieval-Augmented Generation (RAG) — explained in 16 episodes
Interpretability — explained in 15 episodes
Sample Complexity — explained in 15 episodes
Federated Learning — explained in 14 episodes
Generalization — explained in 13 episodes

This episode discusses

The paper

An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression · Read on arXiv

LMU Munich · Munich Center for Machine Learning (MCML) · DLR-German Aerospace Center · University of Tromsø · German Research Center for Artificial Intelligence (DFKI, DSA)

Transcript

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

Tom: Today's paper: "An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression".

Jane: The gist: This work provides a formal way of representing uncertainty in continuous space using a general parametric formulation and proposes axioms to rigorously assess total, aleatoric,

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

Paper summary: Tom: So, wrapping up this discussion on "An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression". The core contribution here is providing that formal uncertainty representation framework and the set of axioms to rigorously assess total, aleatoric, and epistemic uncertainty measures.

Jane: Basically, it gives us a structured way to look at how we quantify uncertainty when dealing with continuous regression problems by defining what those measures should actually do.

Lu: The authors showed that when you compare variance-based measures against the axioms, they fulfill most of them, but also that they have limitations regarding epistemic uncertainty.

Meng: And the entropy-based measures have their own set of trade-offs, showing where they can be negative or scale dependent in their ordering.

Lalam: The paper establishes a formal foundation for uncertainty representation and quantification in the supervised regression setting by introducing this parametric framework and those axioms to guide the design of new measures.

Tom: It’s a principled methodology that helps us evaluate existing measures, highlighting both their strengths and where they fall short based on these formal constraints.

Jane: It gives us a clear roadmap for designing better uncertainty quantification methods by showing what properties an uncertainty measure should actually possess in this specific mathematical context.

Conclusion: Tom: So we're wrapping up this look at "An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression." Basically, the authors gave us a formal way to check if these different ways of measuring uncertainty—entropy versus variance—actually work together in a regression setting.

Jane: Right. They set up this whole framework with six rules, or axioms, to make sure whatever measure we use is actually behaving correctly across different kinds of data and scenarios.

Lu: It's really interesting how they force the measures to obey these structural rules, like translation invariance and scale-order equivariance in a continuous space. They’re essentially trying to build a mathematical house for uncertainty that doesn't collapse when you move things around or change the size of your data.

Meng: From an engineering standpoint, it tells us which measure is more reliable when we’re actually building something. It points out that while variance-based measures hit most of these rules, they still have a specific gap regarding epistemic uncertainty—that part about what we *don't* know about the model itself.

Lalam: I see a lot of potential here for how AI systems can get more reliable when making predictions. If we can rigorously assess total versus aleatoric versus epistemic uncertainty, those models become much better at telling us when they’re guessing based on data versus when they’re genuinely unsure about the underlying rules.

Tom: Exactly. It moves us past just picking a measure that looks good on paper and gives us a way to check its fundamental structure against these formal constraints.

Jane: So, the authors showed that one approach has some clear strengths and some known weaknesses, which is super important for anyone trying to choose a method for their own work.

Lu: It's not about finding the perfect measure yet; it's about having a language to properly critique them and see what they’re actually measuring in the continuous regression world.

Meng: And this framework could guide us toward creating new uncertainty measures that satisfy all these rules, which is a big step for practical application.

Tom: It sets up a really solid foundation for how we think about the uncertainty in predictive models moving forward. We need to see how this plays out in real-world deep learning applications next.

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