Subjective Risk Decomposition: A New View for Uncertainty Quantification

Gavin Brown; Michele Caprio; Raghad Alamri

Subjective Risk Decomposition: A New View for Uncertainty Quantification

T0 review · 1 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Epistemic and aleatoric uncertainty measures are not primitives to be axiomatized; they are consequences of choosing a strictly proper loss and decomposing an expected subjective risk.

desk verdict A genuinely unifying framework for UQ via subjective risk decomposition, with clean math behind it—but the epistemic/aleatoric labeling is a semantic choice, not a derivation, so the strongest claims need tempering. read the letter →

arxiv 2607.15196 v2 pith:5SOS3DGD submitted 2026-07-16 stat.ML cs.AIcs.LG

Raghad Alamri , Michele Caprio , Gavin Brown This is my paper

classification stat.MLcs.AIcs.LG

keywords epistemicuncertaintyaleatoricsubjectiveriskBregmandivergencestrictlyproperlossbias–variancedecompositionmutualinformationquantification

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Uncertainty quantification frequently presents epistemic and aleatoric uncertainty as concepts that need separate axiom systems. This paper instead claims they emerge from a single higher-level modeling decision: choose a strictly proper loss and evaluate the agent's subjective risk at the true distribution; decompose that expected risk and the variance term is epistemic uncertainty, the expected generalized entropy is aleatoric. The same recipe reproduces the field's leading measures — mutual information, label-wise, variance-based, ensemble, and data-versus-procedural — so their disagreements are seen as differences in loss geometry, not contradictions. If this is right, UQ acquires a common theoretical foundation and a principled way to design new measures. The paper extends the same subjective-risk view to learning theory, linking the terms to excess risk, approximation error, and estimation error.

What carries the argument

The central object is the Bregman divergence generated by a strictly proper loss through the standard representation of such losses as a convex potential plus an inner-product term — call it the loss-induced divergence. The argument runs on the reverse-argument bias–variance decomposition around the right Bregman centroid q̄ = argmin_z Eθ[Bφ(qθ, z)], the aggregate predictive belief. The divergence is the only loss class with a clean bias–variance split, and its asymmetry separates the 'forward' objective risk from the 'reverse' subjective risk. For the log loss the divergence is the KL divergence, the centroid is the mixture distribution, and the variance term becomes mutual information; for

What would settle it

Measure the Bregman variance term Eθ[Bφ(qθ, q̄)] in a fixed model class while the training set grows, under any strictly proper loss. If this 'epistemic' term ever increases while held-out accuracy and calibration improve, the claim that the variance term is reducible uncertainty — uncertainty that diminishes with data — fails in a directly observable way.

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Extended reading notes

Core claim

The central claim is that epistemic and aleatoric uncertainty are not quantities to be defined by hand; they are terms induced by a higher-level modeling decision. For any strictly proper loss ℓ, the expected subjective risk Eθ[E_{Y∼qθ} ℓ(p, Y)] decomposes as Bφ(q̄, p) + Eθ[Bφ(qθ, q̄)] + Eθ[Hℓ(qθ)], where Bφ is the divergence generated by the loss and q̄ is its right Bregman centroid (the aggregate predictive belief). The second term, the loss-induced Bregman variance, is identified as epistemic uncertainty; the third, the expected generalized entropy, as aleatoric uncertainty. For the log loss on categorical variables this reduces to the reverse cross-entropy and exactly recovers the classi

Load-bearing premise

The load-bearing identification — that the Bregman variance term is epistemic uncertainty and the expected generalized entropy is aleatoric — is a semantic labeling imposed on the decomposition, not a mathematical consequence of it; Theorem 2 alone is an identity about bias, variance, and entropy.

Editorial extensions

If this is right

The classic mutual-information/conditional-entropy A/E split, label-wise uncertainty, variance-based regression terms, deep-ensemble terms, and data-versus-procedural decompositions all arise from one expected-subjective-risk identity, so they stop being rival definitions.
Disagreements among entropy-based, variance-based, and divergence-based measures are explained by the curvature of the loss's potential near the simplex boundary, not by one measure being wrong.
For log loss the mutual information is exactly one component of the subjective estimation error, so the standard epistemic measure systematically under-estimates the reducible part of subjective risk.
The subjective best-in-class model equals the objective one only when the projection of the aggregate predictive belief matches the projection of the true distribution; consistency of the predictive mixture restores this agreement asymptotically.
Practically, new uncertainty measures are obtained by choosing a strictly proper loss and reading off the variance and entropy terms of the decomposition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the paper is right, the debate over which A/E measure to use becomes a debate over which loss geometry to use; the key design question is how sensitive the loss's potential must be near probability-zero events, and that can be tuned.
The reverse-KL connection gives a principled lens for LLM hallucination scores: the expected subjective risk at the true distribution penalizes confident mass in implausible regions, and this suggests loss-specific 'realism' scores beyond log loss that could be tested on hallucination benchmarks.
The paper treats the bias term as external grounding rather than reducible uncertainty; a natural extension is to check whether the Bregman variance term tracks dataset size only when the model class is fixed, while the bias term tracks misspecification.
Because the subjective approximation gap can be negative for restricted hypothesis classes, the common 'total uncertainty = AU + EU' equation is incomplete: the additive view is exactly a de-biased expected subjective risk, and empirical UQ users might gain by reporting all three terms rather than two.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper. First, the core decomposition is real and well-executed: decomposing expected subjective risk under a strictly proper loss into bias, Bregman variance, and generalized entropy, and showing that the variance and entropy terms recover Gal's mutual information/conditional entropy for log-loss and several other published UQ measures, is a genuine unification of a scattered literature. Second, the load-bearing step is not a theorem. The paper equates the variance term with epistemic uncertainty and the generalized entropy with aleatoric uncertainty at §3.3; that equation is a naming decision, not a consequence of the math. The same identity could be read with the bias term as the reducible component, as Jimenez et al. argue. For log-loss the identification matches Gal, so it is strongly motivated there; for arbitrary proper losses it is an interpretive choice.

The paper does many things well. The Bregman three-point identity proof of Theorem 2 is clean, and the examples in Appendix B—label-wise, variance-based, deep ensembles, data/procedural splits—are worked out carefully and genuinely show the framework absorbing known results. The learning-theory bridge is the most original part: the subjective approximation gap, the subjective estimation error, and the corollary that mutual information is a lower bound on subjective estimation error are new and worth engaging with. The paper is also honest in its acknowledgments of prior work on Bregman bias-variance decompositions (Pfau, Heskes), and the derivations are self-contained.

The soft spots are real but not fatal. The strongest claim—that this settles the debate by deriving A/E terms as consequences—outruns what is shown; the semantic mapping must be accepted first. Theorem 8's appendix proof changes notation mid-stream (h(y) becomes rθ(y) without explicit connection) and has assumptions about rθ that were not stated in the theorem. The 'effectively all proposals' claim is not supported by the set of examples, though the examples are substantial. Figure 2 is presented without reproducibility details, which is minor since the theory is the point.

Who is this for? Anyone working on the foundations of uncertainty quantification, and to a lesser extent people building new UQ measures, since the framework gives a principled way to derive them. The reader's conditional verdict is about right. I would send this to peer review—the core derivation deserves referee time, and the paper could be strengthened substantially by addressing the labeling issue and tightening the appendix.

Recommendation: engage with it; the unification is useful even if the philosophical framing needs sharper defense.

Referee Report

1 major / 5 minor

Summary. The paper proposes that epistemic and aleatoric uncertainty measures should be viewed not as primitive quantities requiring axioms, but as consequences of a higher-level modelling choice: the evaluation of a strictly proper loss under a subjective risk. Definition 1 introduces subjective risk R_q(z)=E_{Y~q}[ℓ(z,Y)]; Theorem 2 gives a reverse bias–variance–entropy decomposition of its expectation into Bφ(q̄,p), Eθ[Bφ(qθ,q̄)], and Eθ[Hℓ(qθ)]. For log loss, the decomposition recovers Gal's terms (bias + mutual information + conditional entropy). With other strictly proper losses, the same theorem is shown to recover label-wise uncertainty (Sale et al.), variance-based measures (Kendall and Gal), deep ensembles, and data/procedural uncertainty splits. Sections 4–5 develop subjective-risk analogues of excess risk, approximation error, and estimation error; Theorem 4 establishes non-negativity of the subjective estimation error; Theorem 5 and Corollary 6 link mutual information to subjective estimation error; Theorem 8 argues that EPKL approximates mutual information in a small-disagreement regime. The paper concludes that disagreements among existing UQ measures are explained by loss geometry rather than by incompatibility.

Significance. If the semantic identification is accepted, the paper offers a genuinely unifying framework: a single risk decomposition generates many existing aleatoric/epistemic measures as special cases, with no fitted parameters, and it connects these measures to standard learning-theoretic quantities. Theorems 2, 4, 5, and Corollary 6 are mathematically sound, and the examples are worked out in enough detail to be checked. The main caveat is that the epistemic/aleatoric interpretation is imposed rather than derived; the paper's contribution is therefore best read as a unifying identification, not a derivation. With that framing, this could be an influential conceptual contribution to the UQ literature.

major comments (1)

[Abstract; §3.3, Eq. (10)] The paper's central claim — that aleatoric and epistemic uncertainty measures are 'consequences' rather than primitives — is stronger than what Theorem 2 establishes. Theorem 2 is a bias–variance–entropy identity; the identification of the Bregman variance term as epistemic and the expected generalized entropy as aleatoric is asserted after Eq. (10) ('The epistemic uncertainty is the variance... and the aleatoric uncertainty is the corresponding generalized entropy'). The same identity is compatible with other semantic assignments, e.g. viewing the bias term as the reducible component, as argued by Jimenez et al. (2026), which the paper rejects in §5.2 on conceptual grounds. This is not an internal mathematical error, but it is load-bearing: without this mapping, the headline result reduces to 'a strictly proper expected subjective risk decomposes into three nonnegative terms.' I recomme

minor comments (5)

[§5.4, Theorem 8; Appendix C] The statement and proof of Theorem 8 use inconsistent notation. The theorem parameterizes perturbations as qθ(y)=qθ′(y)(1+εh(y)) with ∫h(y)qθ′(y)dν=0, but the proof introduces a reference q0 and functions rθ satisfying ∫q0 rθ=0 and Eθ[rθ]=0, then derives expansions (55)–(56) in terms of rθ. No bridge between h and rθ is given. The theorem as stated is therefore not proved; the proof should be rewritten in a single parameterization (e.g. qθ=q0(1+εrθ), with h identified as rθ−rθ′ or as rθ, depending on the intended statement).
[§5.4, Theorem 7 statement] Typo: the sentence after Eq. (25) says the mutual information 'contains an asymmetric component, and an asymmetric one'; the second occurrence should be 'symmetric'.
[Appendix C, proof of Theorem 7] The proof is essentially a rearrangement of the known identity EPKL = MI + RMI (Malinin and Gales, 2021) rather than a derivation from first principles. Please cite that identity explicitly at the point of use and clarify which parts are being proved.
[Appendix B, Example 5, Eq. (52)] There is a bracket typo in the variance term: 'E_Z[µZ − E_Z[µZ])²' should read 'E_Z[(µZ − E_Z[µZ])²]'.
[§4.1, after Eq. (16)] The sentence 'the expected subjective risk is 3 the subjective risk of the Bayesian Model Average' appears to have a superscript/marker artifact; please fix the formatting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivations are self-contained and recover existing UQ measures without fitting or self-citation load-bearing steps.

full rationale

The paper's central derivation is Theorem 2, which decomposes the expected subjective risk E_theta[E_{Y~q_theta}[ell(p,Y)]] into bias B_phi(q*_R,p), variance E_theta[B_phi(q_theta,q*_R)], and generalized entropy E_theta[H_ell(q_theta)]. The proof is given in Appendix A and uses only the proper-loss/Bregman representation, the three-point identity, and Jensen-style centroid arguments. No parameter is fitted to data and then renamed as a prediction; no quantity defined in terms of the target result is used to derive it. The examples recover previously published uncertainty measures (Gal mutual information/conditional entropy, Sale label-wise terms, Kendall-Gal variance terms, Huang data/procedural splits, deep ensembles) as special cases, and these are external benchmarks rather than assumptions of the derivation. The only potentially circular-looking move is the interpretive labeling in §3.3, where the Bregman variance term is called epistemic and the expected generalized entropy is called aleatoric. This labeling is asserted rather than derived, and one could alternatively read the bias term as a reducible component, as Jimenez et al. do. But this is a semantic/interpretive choice, not an equation reducing to itself or a fitted input being called a prediction. No load-bearing step depends on a self-citation: Brown and Ali (2024) appears only as a comparison for approximation/estimation relationships, and the relevant identities in Section 4 are proven in the text. External citations such as Pfau (2025) and Heskes (2026) provide background properties of Bregman divergences, not the paper's own conclusions. Under the stated rules, this is a normal non-circular case: score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No constants are fitted; the choice of loss ℓ and hypothesis class H are modelling decisions rather than fitted parameters. The central derivation uses standard Bregman/proper-loss machinery plus the paper's interpretive identification of variance with epistemic uncertainty. No new physical entities are introduced.

assumptions (5)

domain assumption The generalized entropy Hℓ(q)=E_{Y~q}[ℓ(q,Y)] is differentiable on the relative interior of P, so the Savage/Bregman representation of the proper loss holds.
Used in Theorem 2 and throughout §3 to write Rq(p)=Bφ(q,p)+Hℓ(q). Cited to Savage (1971); standard for proper losses.
standard math The right Bregman centroid minimizing Eθ[Bφ(qθ,z)] is the mixture qbar=Eθ[qθ].
Invoked in Theorem 2 and Examples 1/2 to identify the bias and variance terms. Relies on Frigyik et al. (2008) for functional Bregman divergences.
standard math The Bregman three-point identity holds and cross terms vanish after taking expectations since Eθ[qθ−qbar]=0.
Core algebraic step in the proofs of Theorem 2 and Theorem 9.
ad hoc to paper The Bregman variance term is identified as epistemic uncertainty and the expected generalized entropy as aleatoric uncertainty.
This semantic mapping is the paper's proposal, not a mathematical consequence; the same decomposition could be labeled differently (e.g. bias-as-epistemic in Jimenez et al. 2026). It is load-bearing for the claim that UQ measures are consequences.
domain assumption Existence and uniqueness of right Bregman projections ΠR_H(r), and continuity of the projection map at p for the asymptotic statement.
Required for Proposition 3 and the learning-theoretic decompositions (Eqs 19–22); not proven in the paper.

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Cite this review

Pith. "Pith review of Subjective Risk Decomposition: A New View for Uncertainty Quantification." pith.science (2026). https://pith.science/paper/5SOS3DGD

@misc{pith2026260715196,
  author       = {Pith},
  title        = {Pith review of: Subjective Risk Decomposition: A New View for Uncertainty Quantification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5SOS3DGD}},
  note         = {Machine review of arXiv:2607.15196}
}

read the original abstract

We present a novel viewpoint for uncertainty quantification. Uncertainty measures are not primitives, in need of axioms and argumentation, but instead consequences, of higher-level modelling decisions. We show how epistemic and aleatoric uncertainty measures can be derived via decomposition of a subjective risk, based on a strictly proper loss. Reverse cross entropy provides a prominent example, where decomposition recovers the classic information-theoretic uncertainty terms. The same approach recovers numerous measures previously proposed across the UQ literature, providing them a common theoretical foundation. This suggests a new approach to UQ: given a modelling scenario and strictly proper loss, the corresponding epistemic and aleatoric terms are induced by the subjective-risk decomposition. We then extend our view to learning theory: we introduce and analyse subjective risk analogues of excess risk, approximation error and estimation error, and identify the connections to UQ. We consider this a first step towards a full learning-theoretic framework for uncertainty quantification.

Figures

Figures reproduced from arXiv: 2607.15196 by the authors.

**Figure 1.** LEFT: Log, Brier, and Pseudo-spherical loss, normalized to a common maximum. RIGHT: Second derivatives of the associated generator function, i.e. ∇2ϕ(p). The case of log and pseudo-spherical (with β = 1.5) rise rapidly as p approaches zero, affording them sensitivity to probability mass being placed on implausible events, the far left side of the x-axis. The Brier (squared loss) has a slower growth (constant second … view at source ↗

**Figure 2.** Different posteriors q(θ | D) for the parameter θ of a Bernoulli. As the variance of q(θ | D) increases (i.e. ϵ grows from left to right), the 2nd-order approximation of Theorem 8, EPKL ≈ 2 × MI, becomes less accurate. 6 Conclusion We have presented a new view of uncertainty quantification. Rather than treating aleatoric and epistemic uncertainty as primitive quantities that must be independently defined, axiomatiz… view at source ↗

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Reviewed August 1, 2026 · model on record in the stance chip above.