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Second-Order Uncertainty Quantification: A Distance-Based Approach
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Proposes axiomatic criteria (A0–A8) for total, aleatoric and epistemic uncertainty of second-order distributions and a distance-based framework that instantiates them with the Wasserstein distance, with closed forms and proofs for Dirichlet second-order distributions.
Problem
Entropy-based uncertainty measures (TU, AU, EU) for second-order distributions in classification, as used in evidential deep learning and Bayesian inference, have been criticized for violating desirable theoretical properties, and no alternative with guarantees existed.
Approach
Define TU, AU, EU as minimal distances (Wasserstein/Kantorovich-Rubinstein with trivial label metric, i.e. total variation) from the second-order distribution Q to reference sets of second-order distributions lacking the respective uncertainty type, and prove that this instantiation satisfies a set of axioms A0–A8 extending earlier criteria.
Key results
- Proves the Wasserstein-based TU, AU, EU measures satisfy all axioms A0–A8 (Theorem 1), with TU and AU having closed forms 1 - max_y E[p(y)] and 1 - E[max_y p(y)].
- Shows all measures are bounded by (K-1)/K, and EU = K/(K-1) cannot coexist with AU > 0 (Corollary).
- For Dirichlet second-order distributions, derives closed-form TU and a unique constrained minimizer for EU via Lagrangian; illustrates on 3-class examples without empirical benchmarks.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Wasserstein distance in the distance-based uncertainty measures | Second-order predictive uncertainty measures · Classification uncertainty criteria A0–A8 | criteria satisfied | Helped |
| Use the default entropy, conditional-entropy, and mutual-information measures | Second-order predictive uncertainty measures · Classification uncertainty criteria A5–A6 | criteria satisfied | Hurt |
Surprising findings
- Maximal epistemic uncertainty is achieved only when aleatoric uncertainty is zero, and vice versa.
Limitations
- Purely theoretical; no systematic experimental comparison in evidential deep learning settings.
- Results are restricted to classification with the trivial label metric; other metrics and distances not analysed.
- EU for Dirichlet requires solving a constrained optimization problem with no closed form; TU's expectation for EU/AU is approximated by Monte Carlo.
Experimental setup
Baselines: entropy-based TU/AU/EU (mutual information decomposition) Sale et al. alternative cross-entropy-based measures
Builds on: Evidential Deep Learning Shannon entropy mutual information decomposition of uncertainty Wasserstein distance (optimal transport)
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