# Second-Order Uncertainty Quantification: Variance-Based Measures

arXiv: https://arxiv.org/abs/2401.00276 · HTML: https://mlsanity.com/paper/2401.00276/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 43 references that are in the map.
- First posted on arXiv 30 Dec 2023.

## Builds on (most cited first, 19 of 19)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [Understanding Measures of Uncertainty for Adversarial Example Detection](https://mlsanity.com/paper/1803.08533/) (2018)
- [Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?](https://mlsanity.com/paper/2209.03302/) (2022)
- [DropConnect Is Effective in Modeling Uncertainty of Bayesian Deep Networks](https://mlsanity.com/paper/1906.04569/) (2019)
- [Prior and Posterior Networks: A Survey on Evidential Deep Learning Methods For Uncertainty Estimation](https://mlsanity.com/paper/2110.03051/) (2021)
- [Evaluating Uncertainty Quantification in End-to-End Autonomous Driving Control](https://mlsanity.com/paper/1811.06817/) (2018)
- [On the Safety of Machine Learning: Cyber-Physical Systems, Decision Sciences, and Data Products](https://mlsanity.com/paper/1610.01256/) (2016)
- [Active Learning for Sequence Tagging with Deep Pre-trained Models and Bayesian Uncertainty Estimates](https://mlsanity.com/paper/2101.08133/) (2021)
- [Second-Order Uncertainty Quantification: A Distance-Based Approach](https://mlsanity.com/paper/2312.00995/) (2023)
- [Bayesian Optimization with Conformal Prediction Sets](https://mlsanity.com/paper/2210.12496/) (2022)
- [Sources of Uncertainty in Supervised Machine Learning -- A Statisticians' View](https://mlsanity.com/paper/2305.16703/) (2023)
- [Engineering Safety in Machine Learning](https://mlsanity.com/paper/1601.04126/) (2016)
- [Evidential Uncertainty Quantification: A Variance-Based Perspective](https://mlsanity.com/paper/2311.11367/) (2023)
- [Robust Bayesian Target Value Optimization](https://mlsanity.com/paper/2301.04344/) (2023)

## Built on by (most cited first, 19 of 19)

- [Rethinking Aleatoric and Epistemic Uncertainty](https://mlsanity.com/paper/2412.20892/) (2024)
- [Is Epistemic Uncertainty Faithfully Represented by Evidential Deep Learning Methods?](https://mlsanity.com/paper/2402.09056/) (2024)
- [A Comprehensive Survey on Evidential Deep Learning and Its Applications](https://mlsanity.com/paper/2409.04720/) (2024)
- [Quantifying Aleatoric and Epistemic Uncertainty with Proper Scoring Rules](https://mlsanity.com/paper/2404.12215/) (2024)
- [On Information-Theoretic Measures of Predictive Uncertainty](https://mlsanity.com/paper/2410.10786/) (2024)
- [Revisiting Essential and Nonessential Settings of Evidential Deep Learning](https://mlsanity.com/paper/2410.00393/) (2024)
- [Integral Imprecise Probability Metrics](https://mlsanity.com/paper/2505.16156/) (2025)
- [Provable Uncertainty Decomposition via Higher-Order Calibration](https://mlsanity.com/paper/2412.18808/) (2024)
- [Reducing Aleatoric and Epistemic Uncertainty through Multi-modal Data Acquisition](https://mlsanity.com/paper/2501.18268/) (2025)
- [Linear Opinion Pooling for Uncertainty Quantification on Graphs](https://mlsanity.com/paper/2406.04041/) (2024)
- [Uncertainty Quantification for Regression: A Unified Framework based on kernel scores](https://mlsanity.com/paper/2510.25599/) (2025)
- [Generalized Regularized Evidential Deep Learning Models: Theory and Comprehensive Evaluation](https://mlsanity.com/paper/2512.23753/) (2025)
- [CUQ-GNN: Committee-based Graph Uncertainty Quantification using Posterior Networks](https://mlsanity.com/paper/2409.04159/) (2024)
- [An Isotropic Approach to Efficient Uncertainty Quantification with Gradient Norms](https://mlsanity.com/paper/2603.29466/) (2026)
- [Tracing Uncertainty in Language Model "Reasoning"](https://mlsanity.com/paper/2605.07776/) (2026)
- [Subjective Risk Decomposition: A New View for Uncertainty Quantification](https://mlsanity.com/paper/2607.15196/) (2026)
- [On Measuring Intrinsic Causal Attributions in Deep Neural Networks](https://mlsanity.com/paper/2505.09660/) (2025)
- [Principled Input-Output-Conditioned Post-Hoc Uncertainty Estimation for Regression Networks](https://mlsanity.com/paper/2506.00918/) (2025)
- [A Risk-Sensitive and Uncertainty-Aware Decision-Making and Control Framework for Safe and Robust Autonomous Driving](https://mlsanity.com/paper/2609.09650/) (2026)

## Abstract (arXiv)

Uncertainty quantification is a critical aspect of machine learning models, providing important insights into the reliability of predictions and aiding the decision-making process in real-world applications. This paper proposes a novel way to use variance-based measures to quantify uncertainty on the basis of second-order distributions in classification problems. A distinctive feature of the measures is the ability to reason about uncertainties on a class-based level, which is useful in situations where nuanced decision-making is required. Recalling some properties from the literature, we highlight that the variance-based measures satisfy important (axiomatic) properties. In addition to this axiomatic approach, we present empirical results showing the measures to be effective and competitive to commonly used entropy-based measures.