Sample predictions
$S$ stochastic forward passes produce probability vectors on the simplex.
UAI 2026
Not Just How Much,
But Where
Decomposing Epistemic Uncertainty into Per-Class Contributions
Mutual information says how uncertain a model is. The per-class vector $\mathbf{C}(x)$ reveals which classes drive that uncertainty.
34.7%
lower selective risk than MI on diabetic retinopathy
56.2%
lower selective risk than the per-class variance baseline
2 / 2
OoD benchmarks where $\sum_k C_k$ achieves the highest AUROC
The central problem
A single uncertainty score can hide the failure mode that matters most.
In asymmetric classification, ignorance involving a benign class is not equivalent to ignorance involving a safety-critical class. Scalar MI collapses both cases into the same number.
Same scalar summary
MI = 0.024
Catastrophic miss: Grade 3 predicted as Grade 0
versus
Nearly identical scalar summary
MI = 0.027
Severity underestimate: Grade 3 predicted as Grade 2
Their scalar MI is nearly identical, but their per-class epistemic signatures are not.
01 / Method
Decompose MI into class-level contributions.
From $S$ stochastic predictions, estimate each class mean $\mu_k$ and variance $\operatorname{Var}[p_k]$. A second-order Taylor expansion of entropy gives a simple, additive decomposition.
Per-class epistemic contribution
$$C_k(x) = \frac{1}{2}\frac{\operatorname{Var}[p_k](x)}{\mu_k(x)}$$
Additive connection
$$\sum_{k=1}^{K} C_k(x) \approx \operatorname{MI}(y;\boldsymbol{\omega}\mid x)$$
Estimate moments
Compute the class-wise mean, variance, covariance, and third central moment.
Attribute uncertainty
Each $C_k$ identifies the share of epistemic uncertainty associated with class $k$.
Why the normalization matters
Raw variance is suppressed exactly where rare classes live.
On the probability simplex, $\operatorname{Var}[p_k]\leq\mu_k(1-\mu_k)$. As $\mu_k\rightarrow0$, raw variance must vanish even when posterior disagreement remains important.
$\mu_k \rightarrow 0$
rare class
Raw variance
upper bound $\rightarrow 0$
$C_k$
upper bound $\rightarrow \tfrac{1}{2}$
The $1/\mu_k$ term is not an ad hoc correction.
It arises from the entropy Hessian. A given amount of probability variance carries more information-theoretic weight for a low-probability class.
This makes $C_k$ comparable across rare and common classes, while preserving the additive connection to MI.
02 / Reliability
The method says when its approximation should be trusted.
Boundary correction improves sensitivity to rare classes, but also makes the Taylor approximation more sensitive to higher-order skewness. The paper pairs $C_k$ with a diagnostic.
Skewness diagnostic
$$\rho_k(x)=\frac{|m_{3,k}|}{3\mu_k\operatorname{Var}[p_k]}$$
$\rho_k \ll 1$
quadratic approximation is reliable
$\rho_k > 0.3$
interpret $C_k$ with caution
Correlation-aware fallback
CBEC detects active confusion across a safety boundary.
$$\operatorname{CBEC}(x)=\sum_{i\in\mathcal{S}}\sum_{j\in\mathcal{C}}
\sqrt{C_iC_j}\max(0,-\rho_{ij})$$
Use $C_{\text{crit\_max}}$ when critical-class $C_k$ values are reliable. CBEC is the robust alternative when skewness degrades the Taylor approximation.
Axiomatic profile of $\sum_k C_k$
A0 non-negativity
A1 zero iff no disagreement
A3 increases under mean-preserving spread
A2 and A5 inherited trade-offs
The A5 violation is the mechanism that counteracts boundary suppression: sensitivity to $\mu_k$ prevents rare-class contributions from being forced to zero.
03 / Evidence
Class-aware uncertainty changes the decisions a model can support.
The paper evaluates selective prediction, OoD detection, and sensitivity to controlled label noise.
A
Selective prediction / diabetic retinopathy
Defer when uncertainty involves a clinically critical class.
Lowest AUSC: 0.285
$C_{\text{crit\_max}}$ dominates across the full coverage range.
| Family | Policy | AUSC ↓ | Critical FNR @80% ↓ |
|---|---|---|---|
| Scalar | Entropy | 0.604 ± 0.022 | 0.401 ± 0.016 |
| Scalar | MI | 0.436 ± 0.019 | 0.339 ± 0.014 |
| Per-class variance | Sale_EU_crit | 0.650 ± 0.013 | 0.409 ± 0.016 |
| Per-class $C_k$ | $C_{\text{crit\_sum}}$ | 0.327 ± 0.017 | 0.321 ± 0.014 |
| Per-class $C_k$ | $C_{\text{crit\_max}}$ | 0.285 ± 0.016 | 0.302 ± 0.013 |
| Correlation-aware | CBEC | 0.416 ± 0.020 | 0.335 ± 0.014 |
Grade 3 has mean probability near $0.06$. Raw variance is therefore heavily suppressed; the entropy-derived $1/\mu_k$ weighting recovers the critical-class signal.
Across inference methods, the preferred policy follows the diagnostic: direct $C_k$ targeting under reliable posteriors, CBEC when skewness is high.
B
Interpretability / error signatures
Nearly identical MI can conceal different confusion pathways.
Cross-boundary epistemic confusion is 2.7× stronger than within-group confusion.
C
Out-of-distribution detection
The per-class view shows where distributional shift concentrates.
Best AUROC on both tasks
All-class aggregation captures shifts whose locus is not known in advance.
| Method | FashionMNIST → KMNIST | MIMIC-III ICU → Newborn | ||
|---|---|---|---|---|
| AUROC ↑ | OoD / ID ratio | AUROC ↑ | OoD / ID ratio | |
| Neg. MSP | 0.665 ± 0.013 | 2.07 | 0.688 ± 0.030 | 1.24 |
| MI | 0.724 ± 0.009 | 5.92 | 0.802 ± 0.004 | 1.61 |
| $EU_{\text{var}}$ | 0.710 ± 0.010 | 5.92 | 0.778 ± 0.015 | 1.71 |
| $\sum_k C_k$ | 0.735 ± 0.009 | 6.43 | 0.815 ± 0.017 | 1.62 |
On MIMIC-III, the OoD/ID contribution ratio is $2.15\times$ for survival but $1.30\times$ for mortality. The larger shift signal resides in the non-critical class.
This is why critical-class targeting helps selective prediction, while all-class aggregation is necessary for general OoD detection.
D
Sensitivity to data quality
Posterior and training regime shape uncertainty at least as strongly as the metric.
19 / 20
End-to-end settings where $\sum_k C_k$ is less entangled than MI.
End-to-end Bayesian training
$|R_{\mathrm{rel}}| \ll 0.3$
Both metrics remain near-perfectly disentangled.
Frozen-backbone transfer learning
$0.74$ to $1.97$
Entanglement increases by over an order of magnitude.
04 / What the paper changes
The value of $\mathbf{C}(x)$ is not that it replaces MI as a scalar.
It enables class-specific questions that scalar uncertainty cannot express: which classes drive uncertainty, whether confusion crosses a safety boundary, and where a distribution shift enters the label space.
Selective prediction
Target the uncertainty connected to the costly failure.
OoD diagnosis
Inspect whether shift is uniform or concentrated in particular classes.
Posterior scrutiny
Use $\rho_k$ and attribution behavior to examine the quality of uncertainty propagation.
Across all tasks, how uncertainty is propagated through the network matters as much as how it is measured.
Limitations and next steps
A diagnostic decomposition, with a clear frontier.
The paper is explicit about where the approximation loosens and what must be studied next.
High skewness
The additive approximation loosens for low-probability classes with high skewness. $\rho_k$ diagnoses this but does not correct it.
High-cardinality tasks
The $1/\mu_k$ normalization introduces $O(K^2)$ aggregate scaling, motivating truncation or reweighting.
Broader posteriors
Future evaluation should include Laplace approximations, SWAG, structured prediction, and low-rank ensembles.
Class-targeted decisions
Per-class attribution can support active learning, structured deferral, and richer safety-aware uncertainty profiles.
Resources
Paper materials
Paper, poster, citation, implementation, and project links.
Citation
@inproceedings{
Toure2026not,
title={Not Just How Much, But Where: Decomposing Epistemic Uncertainty into Per-Class Contributions},
author={Mame Diarra Toure, David A. Stephens},
booktitle={Forty-Second Annual Conference on Uncertainty in Artificial Intelligence},
year={2026},
url={https://openreview.net/forum?id=cxuWscJmAr}
}