Subjective Risk Decomposition: A New View for Uncertainty Quantification Raghad Alamri
[email protected] Department of Computer Science, The University of Manchester
Michele Caprio
Department of Computer Science, University of Warwick
Gavin Brown
[email protected] Department of Computer Science, The University of Manchester
arXiv:2607.15196v1 [stat.ML] 16 Jul 2026
Abstract We present a novel viewpoint for uncertainty quantification. Uncertainty measures are not primitives, in need of axioms and argumentation, but instead consequences, of higherlevel modeling 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. From a practical point of view, this suggests a new approach to UQ: given a modeling 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 analyze 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. Keywords: epistemic, aleatoric, subjective risk, bias–variance, mutual information.
1 Introduction A popular framework for uncertainty quantification is the aleatoric/epistemic decomposition introduced by Gal (2016). This adopts a Bayesian viewpoint, where q̄ := q(y | x, D) is the predictive distribution obtained by marginalising parameters θ under a posterior q(θ | D), given observed data D. The entropy of this predictive distribution admits a decomposition, h i H(Ŷ | x, D) = I(Ŷ ; Θ | x, D) + Eθ∼q(θ|D) H(Ŷ | θ, x, D) , whose terms are widely interpreted as epistemic (reducible) and aleatoric (irreducible) uncertainty, respectively. The decomposition itself follows directly from the chain rule of entropy, but it rapidly became one of the most influential frameworks in the field. The success of this formulation inspired a large body of follow-up work. Researchers proposed alternative measures of epistemic and aleatoric uncertainty, motivated by different modelling assumptions, and application domains As a result, the literature now contains many competing definitions of total, epistemic, and aleatoric uncertainty, often motivated from quite different perspectives. ©2026 Alamri, Caprio, and Brown. License: CC-BY 4.0, see https://creativecommons.org/licenses/by/4.0/.
Alamri, Caprio, and Brown
One response to this proliferation has been to define and justify axioms for the UQ terms (Wimmer et al., 2023; Bülte et al., 2025), sparking fascinating community discussion (Various, 2024). Others have questioned whether the A/E dichotomy itself is meaningful, arguing for alternative taxonomies of uncertainty (Valdenegro-Toro and Mori, 2022; Kirchhof et al., 2025), balanced by other voices (Sale et al., 2026) arguing that the dichotomy is alive and well, with issues attributed to limitations of current mathematical frameworks. Most recently, Bickford-Smith et al. (2025) proposed a decision-theoretic perspective of uncertainty based on subjective risk (Savage, 1971). They equate uncertainty with the minimum subjective risk achieved by a Bayes-optimal action, and study its reduction through the acquisition of more data and Bayesian updating. We broaden this perspective by considering subjective risk away from the minimum, and as a random variable influenced by uncertainties such as training data, procedural randomness, and hyperparameters. We show that for the expected subjective risk, a reverse bias–variance decomposition naturally induces measures of uncertainty through the decomposition terms. To illustrate the significance of this proposal, we show how Gal’s classic information theoretic A/E terms arise naturally by decomposing the reverse cross-entropy. Various previously published uncertainty measures can be derived with the same approach, understood as instantiations of the same underlying principle, rather than as competing alternatives. We then extend our viewpoint to learning theory. We introduce and analyse subjective risk analogues of excess risk, approximation error and estimation error, showing how epistemic and aleatoric uncertainty terms relate to standard statistical learning-theoretic notions (Bach, 2024). Thus, we do not suggest (another) definition of aleatoric and epistemic uncertainty; rather, we explain how the current measures can co-exist in a common view, and can be related to standard learning theoretic notions. In short, we propose that uncertainty measures are not primitives in need of axioms and argumentation. Instead, they are consequences of higher level modelling decisions — and can be derived once the modelling and evaluation choices are fixed. Our headline contributions are as follows. Contributions • We show how measures of aleatoric and epistemic uncertainty emerge naturally as components of an expected subjective risk based on a strictly proper loss; • We recover numerous existing measures as special cases. In particular, the informationtheoretic framework (Gal, 2016) arises from the expected subjective log-loss. Under this formulation, epistemic uncertainty is the loss-induced Bregman variance, and aleatoric uncertainty is the expected generalized entropy; • We explain why different uncertainty measures behave differently. Disagreements between entropy-based, variance-based, and divergence-based measures are not contradictions, but instead consequences of loss geometries on the same model; • We define and analyse the subjective risk analogues of common learning-theoretic notions: excess subjective risk, subjective estimation error, and the subjective approximation gap. This serves to highlight how epistemic uncertainty and subjective estimation error are intertwined, and lays a foundation for future study, bridging UQ with classical learning theory. 2
Subjective Risk Decomposition: A New View for Uncertainty Quantification
2 Background: Proper Losses and Bregman Divergences We review the basics of proper losses and Bregman divergences, and establish our notation. Define a convex class of probability distributions P over an outcome space Y, all dominated by a common reference measure ν. We define a loss as a function ℓ : P × Y → R ∪ {+∞}, where the first argument is a model probability distribution, and the second a realised outcome at which we evaluate the loss. The (pointwise) risk of a reported model q ∈ P is its loss in expectation over the true p ∈ P, i.e., Z h i ℓ(q, y) p(y) dν(y). (1) Rp (q) = EY ∼p ℓ(q, Y ) := Y
If Y is at most countable, the expectation is taken as a weighted sum of the loss with respect to the probability mass function. In the remainder of the paper, for ease of notation, we do not distinguish between probability measures, and their density or mass functions. For example, if p, q are categorical, P ν is the counting measure, and ℓ(q, Y ) = − ln q(Y ), this is the cross-entropy, Rp (q) = − y∈Y p(y) ln q(y). The loss ℓ is proper when ∀p, q ∈ P, Rp (p) ≤ Rp (q)
⇐⇒
EY ∼p [ℓ(p, Y )] ≤ EY ∼p [ℓ(q, Y )]
(2)
i.e. the expected loss is minimised when the reported model is q = p. The loss is said to be strictly proper if q = p is the unique minimizer. The minimum achievable expected loss R is called the Bayes risk (or generalized entropy), defined by Hℓ (p) := inf z∈P Rp (z) = Y ℓ(p, y) p(y)dν(y). Assuming ϕ is differentiable on ri(P), then the loss admits the Savage representation (Savage, 1971). This says Rp (q) can be written in terms of the convex function ϕ(p) := −Hℓ (p), as, Rp (q) = −ϕ(q) − ⟨∇ϕ(q), p − q⟩.
(3)
In this form, the regret of a proper loss can be written, Rp (q) − Rp (p) = ϕ(p) − ϕ(q) − ⟨∇ϕ(q), p − q⟩ = Bϕ (p, q).
(4)
This has the form of a Bregman divergence (Bregman, 1967), a broad loss family which includes many popular losses as special cases. A simple rearrangement of equation 4 yields, Rp (q) = Bϕ (p, q) + Hℓ (p).
(5)
That is, a proper risk is a divergence plus a generalized entropy term. The latter is often identified as ‘aleatoric’ (irreducible) risk, and the former as ‘epistemic’ (reducible) risk, an understanding adopted by many authors, e.g. Hofman et al. (2024); Fishkov et al. (2025). When ℓ depends on q only through distributional parameters, the Bregman geometry is induced on the lower-dimensional parameter space rather than the full support of the densities/distributions. For example, ℓ(q, Y ) = (µq − Y )2 yields ϕ(µ) = µ2 and Bϕ (µp , µq ), in which case the proper loss elicits the mean, but not the full distribution. Bregman divergences have two important properties that we rely on. Firstly, Bϕ (p, q) is, in general, asymmetric—the squared/Mahalanobis loss is the only symmetric Bregman divergence (Nielsen and Nock, 2009). Secondly, they admit a bias–variance decomposition (Pfau, 2025). In fact, subject to mild conditions, they are the only class of losses to have this property (Heskes, 2026). In the next section we build upon these properties to introduce our proposal for uncertainty quantification via subjective risk decomposition. 3
Alamri, Caprio, and Brown
3 Our proposal Our proposition is that aleatoric/epistemic uncertainty measures are not primitives that need to be defined through axioms and argumentation. Instead, they are consequences. They are decomposition terms, that can be derived from an expected risk. Our proposal, based on the notion of subjective risk, is a methodology that explains the classic informationtheoretic terms of Gal (2016), as well as numerous other well-known uncertainty measures. 3.1 Subjective vs Objective Risks Our starting point is the decision-theoretic notion of subjective risk, introduced by Ramsey (1926) and formalised by Savage (1971). Rather than evaluating predictions by the true data-generating distribution, subjective risk evaluates them under the agent’s own beliefs. Let P be a convex class of distributions, and H ⊆ P be the agent’s hypothesis class, not necessarily convex.1 Given a model q ∈ H, the subjective risk is defined as follows. Definition 1 (Subjective Risk) Assume a proper loss ℓ : P × Y → R. Given a model q ∈ H ⊆ P, and another arbitrary distribution z ∈ P, the subjective risk for z is defined, Z h i Rq (z) = EŶ ∼q ℓ(z, Ŷ ) := ℓ(z, y) q(y) dν(y). (6) Y
If ℓ is strictly proper, the unique minimizer is z = q, this gives the generalized entropy R Hℓ (q) := Y ℓ(q, y)q(y) dν(y), i.e. the agent is incentivised to truthfully report its belief q. In the original literature of Savage (1971), this concept is known as subjective utility, but we adopt a loss minimization viewpoint for alignment with the modern ML literature. The subjective risk can be contrasted with objective risk, Rp (z) := EY ∼p [ℓ(z, Y )], which evaluates z by averaging loss over the true p. We use the terminology ‘objective risk’ to clearly disambiguate, but it is the conventional ‘risk’ used to evaluate ML models. Analogously to Equation 5, the subjective (proper) risk can be written as a Bregman divergence plus a generalized entropy term, Rq (z) = Bϕ (q, z) + Hℓ (q). (7) For the remainder of the paper we assume ℓ is a strictly proper loss, hence this property holds. For log loss on categorical distributions, the subjective risk for the true p is, X Rq (p) = − q(y) ln p(y) = K(q || p) + H(q). (8) y∈Y
In this case, the subjective risk Rq (p) is the ‘reverse’ cross-entropy, and the right hand side yields the ‘reverse’ KL divergence, plus the Shannon entropy H(q). The forward and reverse KL have notably different behaviours—the forward KL is said to exhibit a meanseeking behaviour, where the model covers the true distribution well on average, whereas reverse KL is mode-seeking, incentivizing the model to put all probability mass on the mode set of p. It is exactly this difference that we will exploit in the coming sections. 1. Convexity (or lack thereof) here has a relevant and natural philosophical interpretation: given two distributions P1 , P2 ∈ P, then any convex combination λP1 + (1 − λ)P2 , with λ ∈ [0, 1], may also be regarded as ‘plausible’. Thus, H may or may not be closed under mixtures of ‘plausible’ distributions.
4
Subjective Risk Decomposition: A New View for Uncertainty Quantification
3.2 Why Subjective Risk is a natural foundation for UQ An important question for UQ is: “when can I trust this model’s prediction?”. BickfordSmith et al. (2025) argue: “Uncertainty alone is not a reliable indicator of whether we can trust a model. Some kind of external grounding is crucial for well-informed practical deployment.” (Bickford-Smith et al., 2025, p.7)
We argue that the subjective risk, augmented with external grounding, is well-aligned with answering this, and provides a natural foundation to understand uncertainty. Subjective risk is not merely an interesting evaluation functional, but has an axiomatic foundation in statistical decision theory. Savage showed that to minimize subjective risk is to behave according to a set of 7 postulates — these encode desirable behaviours, such as avoiding contradictions in the rank-ordering for a set of acts (Savage, 1954). As such, the subjective risk quantifies whether a model’s internal beliefs can be trusted as a basis for rational action, where “rational” is made precise by Savage’s postulates. However rational our models may be, as Bickford-Smith et al. (2025) state, it is crucial to be grounded in reality if we want trustworthy models. This is where we shift from Savage’s classical framework. We do not use subjective risk as a decision-making framework. Instead, we use it as an evaluation functional for the model against reality, EŶ ∼q [ℓ(p, Ŷ )]. This fills the gap between internal rationality and external grounding. We can also make a more pragmatic argument, from the viewpoint of an ML practitioner. The motivation is clearest in the case of reverse cross-entropy and reverse KL-divergence, but the idea is more general. K(q || p) strongly penalizes mass assigned by q where p is small, becoming infinite when q assigns positive mass where p assigns zero mass. Thus, it penalises confident belief in outcomes that are poorly supported by reality—which can be interpreted as measuring a form of predictive hallucination. In the LLM literature, there is growing empirical evidence that hallucinations and reverse KL are strongly linked (Agarwal et al., 2024; Gu et al., 2024; Cao and Kang, 2025). Furthermore, reverse KL regularization is the dominant technique in RLHF (Ziegler et al., 2020; Xiong et al., 2023), forcing the learned policy to stay close to a reference policy. This again can be interpreted as avoiding low-probability tokens that may lead to hallucinated actions. Malinin and Gales (2019) show that using reverse KL as a training criterion for their Prior Networks, results in better estimates of uncertainty and detection of adversarial attacks—that is, inputs that are implausible under reality. We will show how these hallucination-sensitive behaviours extend beyond log-loss, and are interesting for UQ. The objective risk EY ∼p [ℓ(q, Y )] answers the question “how correct is this model?”. The subjective risk EŶ ∼q [ℓ(p, Ŷ )] answers the question “how realistic is this model?”, assigning high2 penalizes when the model predicts outcomes that p regards as implausible.
2. Depending on loss geometry, as we will see later.
5
Alamri, Caprio, and Brown
3.3 Subjective Risk Decompositions as a framework for UQ A primary goal of UQ is to quantify aspects of predictive uncertainty that are relevant to whether a model can be trusted in a given decision problem. We argue that decompositions of the subjective risk provide a principled way to derive such quantities. To support our claim we present the following result. Theorem 2 (Bias-Variance Decomposition of the Subjective Risk) Assume a convex class of distributions P over Y, qθ ∈ H ⊆ P denotes a model defined by parameters θ, and let p ∈ P be the true distribution. Define a strictly proper loss ℓ, whose generalized entropy Hℓ (q) := EŶ ∼q [ℓ(q, Y )] is differentiable on the relative interior of P. The expected subjective risk decomposes as, " # h i h i h i ∗ ∗ Eθ EŶ ∼qθ ℓ(p, Ŷ ) (9) = Bϕ (qR , p) + Eθ Bϕ (qθ , qR ) + Eθ Hℓ (qθ ) , | {z } | {z } | {z } | {z } expected subjective risk
variance
bias
generalized entropy
∗ := arg min where qR z∈P Eθ [Bϕ (qθ , z)] is the right Bregman centroid, and ϕ(q) = −Hℓ (q).
This is a reverse-argument counterpart to the standard bias–variance decomposition. For ℓ as a squared loss, this makes no difference due to symmetry, but for any other loss, the forward/reverse decompositions are around different centroids. The ‘forward’ case is around the left Bregman centroid, qL∗ := arg minz∈P Eθ [Bϕ (z, qθ )] = [∇ϕ]−1 (Eθ ∇ϕ(qθ )). In ∗ := arg min the reverse-argument case as above, it is the right centroid, qR z∈P Eθ [Bϕ (qθ , z)]. For a functional Bregman divergence (Frigyik et al., 2008) the right centroid is the mixture, R ∗ (y) = q̄(y) := qθ (y)q(θ|D) dθ. For a finite-dimensional Bregman divergence, it is i.e. qR the expectation in the relevant (convex) domain. The two directions also differ in their entropy terms. The forward case has Hℓ (p), the ‘noise’ in the true distribution, whereas the reverse case has Eθ [Hℓ (qθ )], the average (generalized) entropy of the models. To illustrate Theorem 2, we give examples of the bias/variance/entropy terms, for common distributional assumptions and losses. We emphasize the significance here, in that we can recover effectively all previous proposals as special cases, from this single viewpoint. The methodology in all Examples is identical: we define H, choose a strictly proper loss ℓ, and apply Theorem 2. Full derivations are presented in Appendix A. Example 1 . Information-theoretic measures (Gal, 2016). Define P = H as the class of categorical distributions over Y = {1, . . . , K}. Let p ∈ P be the true distribution, and qθ a model with unknown parameters θ ∼ q(θ | D). We obtain label predictions Ŷ by sampling, as Ŷ | θ ∼ qθ . We define ℓ(p, Ŷ ) = − ln p(Ŷ ), which induces −Hℓ (p) = ϕ(p) = PK k=1 p(yk ) ln p(yk ). For this ϕ the Bregman is a KL-divergence, and Theorem 2 becomes, " # X −Eθ qθ (y) ln p(y) = K(q̄ || p) + I(Ŷ ; Θ) + H(Ŷ | Θ) (10) y∈Y | {z } | {z } | {z } | {z } expected subjective risk
variance
bias
generalized entropy
where q̄ is the categorical mixture. The variance is exactly the mutual information I(Ŷ ; Θ), and the generalized entropy term is the Shannon conditional entropy H(Ŷ | Θ). 6
Subjective Risk Decomposition: A New View for Uncertainty Quantification
Decomposing the expected reverse cross-entropy yields exactly Gal’s epistemic/aleatoric terms, plus a systematic bias. The epistemic uncertainty is the variance of the expected subjective risk, and the aleatoric uncertainty is the corresponding generalized entropy. In other words: expected subjective risk = bias + epistemic + aleatoric. This single example makes concrete our ‘new view’ of uncertainty quantification. The uncertainty terms here are not primitive objects that need to be defined and debated. Instead, they are components of the expected subjective risk, when appropriately decomposed. The information-theoretic approach generalizes beyond just categorical distributions. Next, we show the case for Gaussians with unknown mean/variance. Example 2. Gaussians with unknown µ, σ 2 (Depeweg et al., 2018). Let H ⊂ P be the class of Gaussians with unknown mean and variance, and denote the true p = N (µp , σp2 ), R and qθ = N (µθ , σθ2 ). Define ℓ(p, Ŷ ) = − ln p(Ŷ ). This induces ϕ(q) = Y q(y) ln q(y) dy, and a reverse KL-divergence between densities. Then, Theorem 2 becomes, # " Z h i qθ (y) ln p(y) dy Eθ − = K(q̄ || p) + I(Ŷ ; Θ) + Eθ 21 ln 2πe σθ2 (11) Y | {z } | {z } | {z } | {z } variance
bias
expected subjective risk
generalized entropy
This recovers the Gaussian information-theoretic terms, which extend to regression through the differential entropy (Depeweg et al., 2018; Malinin, 2019). Here, the right centroid R q̄(y) = θ qθ (y)q(θ|D) dθ is a Gaussian mixture, R R which is in P but not H. The variance (epistemic) term is the mutual information, θ Y qθ (y) ln qθ (y)/q̄(y) dydθ; and the aleatoric uncertainty is the expected differential entropy of the Gaussian. Recall from Section 2 that the loss ℓ(q, y) = − ln q(y) and the generalized entropy Hℓ are defined relative to the reference measure ν. A change of ν rescales the loss and the entropy term together, so the decomposition remains consistent. In Example 1 the categorical case, all terms are invariant, because ν is the counting measure and is fixed. However, in the continuous case of Example 2, we have a differential entropy, whose value depends on ν. It is safe to say the information-theoretic approach dominates the literature—it is therefore important that our methodology recovers this in the first instance. Below we present a non-information theoretic example. Example 3. Label-wise Uncertainty (Sale et al., 2024). Let P = H be the class of Bernoullis, and denote p = Ber(µp ), qθ = Ber(µθ ), with uncertainty over θ. Instead of using the log-loss, here we impose a different geometry by choosing a different proper loss. Define ℓ(p, Ŷ ) = (µp − Ŷ )2 , inducing ϕ(µ) = µ2 , and Theorem 2 becomes, h 2 h h i i i Eθ EŶ ∼qθ (µp − Ŷ )2 = Eθ [µθ ] − µp + Eθ (µθ − Eθ [µθ )2 + Eθ µθ · (1 − µθ ) . (12) | {z } | {z } | {z } | {z } expected subjective risk
variance
bias
generalized entropy
These are exactly the “label-wise” uncertainty terms proposed in Sale et al. (2024, Sec 3.2). Many more examples can be found in Appendix B, including cases for ‘variance-based’ vs ‘entropy-based’ methods (Sale et al., 2023), Deep Ensembles (Lakshminarayanan et al., 2017), and cases with multiple uncertainties. These illustrate generality of our approach. 7
Alamri, Caprio, and Brown
3.4 What makes a good uncertainty measure? What make Gal’s mutual information so successful as an uncertainty measure? We showed how it can be derived using the subjective risk perspective, which for log-loss is the reverse cross-entropy. However, its success is not just due to the “reverse” property, but also to the geometry of the proper loss itself. The reverse cross-entropy assigns large loss when the model places mass in areas where the true distribution p has low probability. The same qualitative phenomenon applies for other losses, so long as they exhibit large curvature near the boundary of the probability simplex, as p → 0. For a proper loss, the local geometry of the associated Bregman divergence is governed by the Hessian of the generator ϕ = −Hℓ , via Bϕ (q, p) ≈ 21 (q − p)⊤ ∇2 ϕ(p)(q − p). Note that this is not, in general, the same as the second derivative of the pointwise loss ℓ(p, y). For log loss, for example, ∇2 ϕ(p) scales as 1/pi , whereas ∂ 2 [− log pi ]/∂p2i scales as 1/p2i . Below we plot the log, Brier, and pseudo-spherical losses, alongside the second derivatives ∇2 ϕ(p).
Figure 1: LEFT: Log, Brier, and Pseudo-spherical loss, normalized to a common maximum. RIGHT: Second derivatives of the associated generator function, i.e. ∇2 ϕ(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 derivative) and thus less sensitive to mass placed on events that p regards as implausible. Objective risk answers the question “how correct is this model?”, whilst subjective risk answers “how realistic is this model?”. Large second derivatives near the boundary are what make a subjective risk sensitive to deviations from reality. 3.5 Summary We have presented a novel viewpoint for uncertainty quantification, based on decomposing the subjective risk. This led naturally to many known measures of epistemic and aleatoric uncertainty emerging as decomposition terms, explaining much of the existing literature from a single viewpoint. This provides an interesting point on the additive TU=AU+EU view, debated in the literature. From our work, we see AU+EU is the expected subjective risk with the bias term removed. Those who adopt the additive view are therefore implicitly equating total uncertainty with the ‘de-biased’ expected subjective risk. 8
Subjective Risk Decomposition: A New View for Uncertainty Quantification
4 A Bridge to Statistical Learning Theory Classical statistical learning theory (SLT) has provided a rigorous theoretical foundation for supervised learning, enabling the analysis of generalization, sample complexity, and the trade-offs between model capacity and estimation from limited data. It is desirable for uncertainty quantification to have a similar theoretical foundation. SLT relies on notions such as approximation error, estimation error, and the best-inclass model (Bach, 2024). These notions are defined with respect to the objective risk, Rp (z) := EY ∼p [ℓ(z, Y )]. In this section, we introduce corresponding notions induced by subjective risk, Rq (z) := EŶ ∼q [ℓ(z, Ŷ )] and establish basic properties. We consider this as a first step toward a full learning-theoretic framework for uncertainty quantification. 4.1 Defining a Learning Theory nomenclature for Subjective Risk Define a hypothesis class H ⊆ P, where P is convex, but H is not necessarily convex. As before, we assume a strictly proper loss with associated regularity/differentiability assumptions. In classical SLT, the expected excess risk of qθ ∈ H is defined, with the objective risk as defined throughout the paper, h i h i ∆obj := Eθ Rp (qθ ) − Rp (p) = Eθ Bϕ (p, qθ ) . (13) This quantifies the additional risk (on average w/r uncertainty in θ) that we will have from using a qθ , as opposed to the true p. A corresponding notion can be defined for the subjective risk—the expected excess subjective risk is, h h i i ∆subj := Eθ Rqθ (p) − Rqθ (qθ ) = Eθ Bϕ (qθ , p) . (14) The roles of p, q have been switched, including the ordering of terms in the expression, since by definition, given our proper loss assumption, Rq (q) ≤ Rq (p), for all p, q. Note however that the expression is still a (non-negative) Bregman divergence. SLT defines the best-in-class model as the model in H with the smallest expected risk, n o ∗ (15) qobj := arg min Rp (z) . z∈H
Again, we can define a corresponding subjective version of this as, n o ∗ qsubj := arg min Eθ [Rqθ (z)] z∈H
(16)
3 We note that theh expected i subjective risk is the subjective risk of the Bayesian Model Average, i.e., Eθ Rqθ (z) = Rq̄ (z), which implies q̄ = arg minz∈P { Eθ [Rqθ (z)] }, but this ∗ may not be attainable with non-convex H. Therefore, if the BMA q̄ ∈ H, we have qsubj = q̄, ∗ otherwise qsubj is a (sub-optimal) projection to H.
3. Trivially, Eθ [Rqθ (z)] = Eθ
hR Y
i R ℓ(z, y)qθ (y) dν(y) = Y ℓ(z, y)q̄(y) dν(y) := Rq̄ (z).
9
Alamri, Caprio, and Brown
∗ and q ∗ Given these definitions, it is interesting to consider when qobj subj might coincide—that is, when the argmin of the objective and subjective risks is in fact the same model.
Proposition 3 (Agreement of Objective and Subjective Best-In-Class Models) Assume the strictly proper loss admits the Bregman representation in equation 7, and that the right Bregman projection ΠR H (r) := arg minz∈H Bϕ (r, z) exists and is unique for every distribution r considered below. Then ∗ qobj = ΠR H (p),
∗ qsubj = ΠR H (q̄).
∗ ∗ R Consequently, qobj = qsubj if and only if ΠR H (p) = ΠH (q̄). In particular, if H = P, then ∗ = p and q ∗ ∗ ∗ qobj subj = q̄, and therefore qobj = qsubj if and only if p = q̄.
Now let q̄n denote the predictive mixture obtained from n observations. If q̄n → p and the projection map r 7→ ΠR H (r) is continuous at p, then ∗ R ∗ qsubj,n = ΠR H (q̄n ) −→ ΠH (p) = qobj .
Proof By the proper risk decomposition, Rp (z) = Bϕ (p, z) + Hℓ (p), where the entropy ∗ = arg min R term does not depend on z. Hence qobj z∈H Bϕ (p, z) = ΠH (p). Similarly, we have ∗ Eθ [Rqθ (z)] = Rq̄ (z) = Bϕ (q̄, z) + Hℓ (q̄), and therefore qsubj = arg minz∈H Bϕ (q̄, z) = ΠR H (q̄). The exact characterization follows immediately. If H = P, strict propriety implies that Bϕ (r, z) is uniquely minimized at z = r. The asymptotic statement follows from the continuity of ΠR H. The proposition distinguishes two notions of optimality. The objective best-in-class model is the element of H that best approximates reality p, whereas the subjective best-inclass model best approximates the aggregate predictive belief q̄. Their coincidence means that internal model beliefs and external reality select the same optimal model within the available hypothesis class. This is desirable when subjective risk is intended to support reliable prediction or action under the true data-generating distribution: minimizing subjective risk then targets the same model as minimizing objective risk. When H = P, coincidence requires the strongest possible form of agreement, namely q̄ = p. For a restricted hypothesis class, however, p and q̄ may differ while still having the same projection onto H. Thus, the result separates disagreement between the distributions themselves from disagreement that actually changes the selected model. The asymptotic statement is particularly useful: if the predictive mixture is consistent and the projection is stable, then the subjective learning target converges to the objective target. In that case, subjective and objective learning are asymptotically aligned. Conversely, if the two projections differ, the subjective criterion targets a different model from the one preferred under reality. This mismatch is driven by the discrepancy between q̄ and p, which is measured globally by the bias term Bϕ (q̄, p). However, a non-zero bias does not necessarily imply different best-in-class models when H is restricted, since two distinct distributions may still have the same projection onto H.
10
Subjective Risk Decomposition: A New View for Uncertainty Quantification
4.2 Approximation and Estimation from the Subjective Risk Classical learning theory decomposes the expected excess (objective) risk as, h i ∗ ∗ ∆obj = Eθ Rp (qθ ) − Rp (qobj ) + Rp (qobj ) − Rp (p). | {z } | {z } estimation error
(17)
approximation error
Here, both terms4 are non-negative.
The estimation error is the part of the excess risk which can be attributed to insufficient data. We compare models with different parameters θ, but cannot distinguish their performance on a small dataset—in effect, an instance of the multiple hypothesis testing problem. The estimation error is there because we could ∗ not identify the best-in-class model qobj ∈ H. The approximation error is, by contrast, independent of data. It quantifies the notion of model capacity for us—and is non-zero when H is not sufficiently expressive to capture the true p. A well-appreciated tenet of learning theory is the trade off between these terms. As we increase the size of H, approximation error decreases monotonically, but estimation error may grow, as it becomes harder to find the best-in-class model with limited data. We can define a corresponding subjective risk version of this decomposition as, i h h i ∗ ∗ ) + Eθ Rqθ (qsubj ∆subj = Eθ Rqθ (p) − Rqθ (qsubj (18) ) − Rqθ (qθ ) | {z } | {z } subjective approximation gap
subjective estimation error
We can observe two important differences from the objective risk case. Firstly, the approximation part can be negative—as such we name it a ‘gap’ rather than an ‘error’. Secondly, both terms depend on the model parameters θ—this is because qθ is key to the evaluation protocol in the subjective case. We note that the subjective approximation gap can be written as a difference of two Bregman divergences, h i ∗ ∗ Eθ Rqθ (p) − Rqθ (qsubj ) = Bϕ (q̄, p) − Bϕ (q̄, qsubj ). (19) ∗ ∗ , is far from the optimal q̄, then the gap will be negative. On the hand, if q̄ = qsubj If qsubj as in the case of a convex hypothesis class, the subjective approximation gap is non-negative, and in fact equal to the bias term Bϕ (q̄, p). In contrast to the subjective approximation gap, the subjective estimation error is nonnegative, as shown below. h i ∗ Theorem 4 The subjective estimation error, Eθ Rqθ (qsubj ) − Rqθ (qθ ) , is non-negative.
Proof Using the proper risk decomposition Rqθ (z) = Bϕ (qθ , z) + Hℓ (qθ ), it can be written ∗ ∗ Eθ Rqθ (qsubj ) − Rqθ (qθ ) = Eθ Bϕ (qθ , qsubj ) ≥ 0. (20)
4. We follow the convention of Bottou and Bousquet (2007) in that the first term in the decomposition is called ‘estimation error’ as opposed to ‘expected estimation error’.
11
Alamri, Caprio, and Brown
4.3 Relation to the Bias–Variance Decomposition Brown and Ali (2024) identified the relationships between bias–variance decompositions and approximation/estimation error, for the case of excess objective risk. We can do the same here, for excess subjective risk. In the objective risk, Brown and Ali (2024) showed the relationships as, h i ∗ ∗ ∆obj = Eθ Rp (qθ ) − Rp (qL∗ ) + Rp (qL∗ ) − Rp (qobj ) − Rp (p) (21) ) + Rp (qobj {z } | {z } | estimation error
|
{z
}
variance
|
approximation error
{z
}
bias
The corresponding subjective version of this is, i i i h h h ∗ ∗ ) + Eθ Rqθ (qsubj ∆subj = Eθ Rqθ (p) − Rqθ (qsubj ) − Rqθ (q̄) + Eθ Rqθ (q̄) − Rqθ (qθ ) | {z } | {z } subjective approximation gap
|
subjective estimation error
{z
}
bias
|
{z
variance
} (22)
From here, we can see an important conclusion. Theorem 5 For the log proper loss, ℓ(z, y) = − ln z(y), the variance of the subjective risk (one part of the subjective estimation error) is the mutual information, that is, Gal’s epistemic uncertainty I(Ŷ ; Θ). Proof Using the proper risk decomposition Rqθ (z) = Bϕ (qθ , z) + Hℓ (qθ ), it can be written Eθ [Rqθ (q̄) − Rqθ (qθ )] = Eθ [Bϕ (qθ , q̄)] ≥ 0. (23) R Choosing the log loss induces the generalized entropy Hℓ (qθ ) = − Y qθ (y) ln qθ (y) dy, whose corresponding Bregman divergence is the (reverse) KL divergence, Eθ [Bϕ (qθ , q̄)] = Eθ [KL(qθ || q̄)] = I(Ŷ ; Θ).
(24)
The mutual information is only one part of the subjective estimation error—but remaining i ∗ ∗ part is also non-negative, i.e., Eθ Rqθ (qsubj ) − Rqθ (q̄) = Eθ Bϕ (q̄ , qsubj ) ≥ 0. The proof of this claim is similar to other results in this section, though we can also see non-negativity via Eθ Rqθ (q̄) = Rq̄ (q̄) := inf z∈P Rq̄ (z). Corollary 6 Mutual information as an epistemic uncertainty measure is a systematic under-estimate of the true subjective estimation error. Consequently, mutual information captures only one component of subjective estimation error, that represented by posterior disagreement. It ignores the additional error induced by restricting predictions to the hypothesis class. If we take subjective estimation error to be the ‘true’ reducible/epistemic component of the subjective risk, then mutual information should therefore be interpreted as a lower bound on epistemic uncertainty, rather than as its complete characterization. 12
Subjective Risk Decomposition: A New View for Uncertainty Quantification
5 Discussion We present a brief discussion of relations to recent work. 5.1 Bickford-Smith et al. (2025) The closest work to our own is Bickford-Smith et al. (2025), who identify incoherence in existing discussions across the UQ literature. They state that the community is looking for “Researchers are looking for concrete notions of a model’s predictive uncertainty and how that uncertainty might or might not change with more data [...] but also related notions of predictive performance and data dispersion. The aleatoric-epistemic view cannot satisfy all these needs: many concepts stand to be defined, while the view fundamentally only has capacity for two concepts.” (Bickford-Smith et al., 2025, pg. 1)
To remedy this, they suggest a decision-theoretic perspective. In particular, they use the subjective expected loss of acting Bayes-optimally under a model’s beliefs. They proceed to define epistemic uncertainty as how much this is expected to reduce once future data is observed under Bayesian updating. To account for predictive performance and data dispersion, they separately examine the (objective) risk and entropy of the true p. In summary, they treat uncertainty, external evaluation, and data dispersion as separate objects. We instead evaluate the subjective risk at the true distribution, Rq (p) = EŶ ∼q [ℓ(p, Ŷ )]. We consider it as a quantity that varies with the modelling process, and decompose its expectation. Hence with our approach, the external grounding is built into the risk itself: the bias term accounts for external grounding, while the variance and generalized entropy terms recover the uncertainty quantities. 5.2 Jimenez et al. (2026) Jimenez et al. (2026) present a position paper, arguing that epistemic uncertainty estimation methods are fundamentally incomplete. As Shaker and Hüllermeier (2020) before them, they discuss uncertainty in the context of statistical learning theory. Critically, they only consider existing theory, i.e., properties of the objective risk. They utilise results from Brown and Ali (2024) to argue that certain aspects are missing from the community’s understanding of uncertainty. In particular, they consider model bias as a reducible part of epistemic uncertainty that current estimators fail to capture, and conclude on this basis that epistemic estimates are incomplete. We instead view bias as evidence of external grounding rather than an uncertainty component. This distinction is crucial, since a model can have completely certainty, but also be completely wrong. Finally, in their conclusion, they state, “epistemic uncertainty is generally not well defined and is still a heuristic concept for machine learning models”
We posit that our work has resolved this—providing well-defined terms, with solid links to standard notions from learning theory, and showing how they generate numerous existing proposals as special cases. 13
Alamri, Caprio, and Brown
5.3 Fishkov et al. (2025) Just as in our work, Fishkov et al. (2025), building on Kotelevskii et al. (2025) also derive uncertainty measures from proper scoring rules and risk decomposition. Their starting point however is what we would refer to as ‘objective’ risk, evaluating expected loss against the true distribution. They consider Bayesian approximations to the risk expressions, substituting various estimates for the unavailable true distribution. Since the approximations can be inserted into either argument of the induced divergence, this generates several candidate measures. By enumerating all combinations of argument orderings, they recover many previously proposed epistemic and aleatoric terms as special cases. The “all combinations” strategy produces the measures, but does not say which is appropriate for a given modelling scenario, or explain why they behave differently. We instead fix the modelling and evaluation choices first and obtain a single bias–variance–entropy decomposition in which each term has a clear role. The particular measure that appears is then a consequence of the loss geometry rather than a free choice, and disagreements between measures are explained directly by differences in loss curvature (See subsection 3.4). In the following section, we examine the different epistemic uncertainty measures that arise from varying the arguments and their order. 5.4 The Various Proposals for Epistemic Uncertainty The seminal work of Gal (2016) introduced the world to mutual information as a measure of epistemic uncertainty. In particular, written in the form I(Ŷ ; Θ) = Eθ [K(qθ || q̄)], it has an appealing structure that has led to multiple very similar proposals, from various authors. Here we briefly review these, showing relations to our work. For simplicity, we restrict discussion to the case of KL divergences on categorical distributions, but the conclusions hold more generally for arbitrary Bregman divergences. As a preliminary step, we remind the reader of the definitions of h right andi left centroids, defined ∗ := arg min briefly earlier. The right centroid is, qR E c∈P θ K(qθ || c) = Eθ [qθ ], and the left centroid is, qL∗ := arg minc∈P Eθ K(c || qθ ) ∝ exp Eθ ln qθ . The left centroid here is a normalized geometric mean, denoted below as q̊. Note that this is the left centroid only for KL, and for general Bregman divergences qL∗ is a quasi-arithmetic mean (Nielsen and Nock, 2009). Given this, we can state the various epistemic uncertainty definitions—see Table 1. Name Mutual Information Expected Pairwise KL (EPKL) Reverse Mutual Information (RMI) Modified Bregman information Reverse modified Bregman information
KL form Eθ K(qθ || q̄) Eθ Eθ′ [K(qθ || qθ′ )] Eθ K(q̄ || qθ ) Eθ K(qθ || q̊) Eθ K(q̊ || qθ )
First suggested by Gal (2016) Malinin (2019) Malinin and Gales (2021) Kotelevskii et al. (2025) Kotelevskii et al. (2025)
Table 1: KL-based proposals for epistemic uncertainty. 14
Subjective Risk Decomposition: A New View for Uncertainty Quantification
These measures follow a common template, with each a variant of the original mutual information form. The template form is Eθ K(a, b) , and explores all possible substitutions of arguments a, b —e.g. with a = q̊, and b = qθ , we get the ‘reverse modified Bregman information’. As mentioned, exactly this, the strategy of ‘all possible combinations’ is adopted by Fishkov et al. (2025). We note that only two of the terms are ‘standard’ variances in the sense that they are a Jensen gap. These are the first and last rows above: Eθ [K(qθ || q̄)], and Eθ [K(q̊ || qθ )]. As we showed earlier, the mutual information comes from a bias–variance decomposition of the subjective risk. The term Eθ [K(q̊ || qθ )] is the corresponding variance from the objective risk decomposition. These ‘standard’ Bregman variances are also special in that they are guaranteed to be finite. Reversing the order of arguments, as in RMI and Modified Bregman information, yields non-standard quantities that can be infinite in some cases. The Expected Pairwise KL, Eθ Eθ′ [K(qθ || qθ′ )] , was proposed as an epistemic uncertainty by Malinin (2019) and later analysed in depth by Schweighofer et al. (2023), who found it to have several favourable properties. It is particularly interesting in that it is guaranteed to be finite, and makes intuitive sense as a measure of uncertainty—taking all possible pairwise disagreements. Theorem 7 shows that this is in fact one term in an expansion of the mutual information, and in many cases, the dominant term. Theorem 7 (Asymmetry decomposition of mutual information) Let qθ := q(· | θ) with θ ∼ q(· | D). Then h i 1 h i 1 I(Ŷ ; Θ) = Eθ,θ′ K(qθ || qθ′ ) + Eθ K(qθ || q̄) − K(q̄ || qθ ) , (25) 2 2 | {z } | {z } symmetric component
asymmetric component
i.i.d.
where q̄ = Eθ [qθ ] and θ, θ′ ∼ q(· | D) Thus, the mutual information contains a symmetric component, and an asymmetric one. The following theorem characterises when the asymmetric term is negligible. Theorem 8 (EPKL is a 2nd-order approximation of Mutual Information) When the distributions involved are close, i.e. Z qθ (y) = qθ′ (y) 1 + ϵh(y) , h(y)qθ′ (y) dν(y) = 0, with ϵ small, the symmetric component dominates: h i 1 I(Ŷ ; Θ) = Eθ,θ′ K(qθ || qθ′ ) + O(ϵ3 ). 2
(26)
This follows because each K(qθ , qθ′ ) in the symmetric term is of order ϵ2 , while each asymmetric difference K(qθ , q̄) − K(q̄, qθ ) is of order ϵ3 . When ϵ is small, this implies, h i Eθ,θ′ K(qθ || qθ′ ) ≈ 2I(Ŷ ; Θ). (27) Proof: See Appendix C. The claim is supported by the empirical results in Schweighofer et al. (2023, Figure 2); we replicate selected cases in Figure 2. 15
Alamri, Caprio, and Brown
Beta(5,5)
U[0,1]
MI = 0.07 EPKL = 0.14 RMI = 0.08
0
0.5 Bernoulli parameter
Beta(0.4,0.4)
MI = 0.28 EPKL = 0.72 RMI = 0.44
1
0
MI = 0.50 EPKL = 1.80 RMI = 1.30
0.5 Bernoulli parameter
1
0
0.5 Bernoulli parameter
1
expected value of under the posterior
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, axiomatized, and defended, we view them as consequences of a more fundamental modelling choice: the risk under which predictive beliefs are evaluated. Our starting point was Savage’s notion of subjective risk under a strictly proper loss. We adopt this not as a decision-making framework, but as an evaluation functional against reality. Such a loss induces a Bregman divergence geometry, and the corresponding reverse bias–variance decomposition separates the expected subjective risk into systematic bias, variance between predictive distributions, and their generalized entropy. These terms give intuitive meanings to epistemic and aleatoric uncertainty. The epistemic term is the lossinduced Bregman variance of the random predictive distribution, while the aleatoric term is its expected generalized entropy. The validity of this claim is evidenced by exactly recovering numerous previously published proposals as special cases, e.g. (Gal, 2016; Sale et al., 2024; Bülte et al., 2025; Kendall and Gal, 2017; Fishkov et al., 2025; Jimenez et al., 2026). Therefore, these measures should not be understood as competing alternatives, or mutually incompatible definitions: they arise from different losses, geometries, and assumptions. We proceeded to explore connections with classical statistical learning theory, defining subjective risk analogues of notions such as approximation and estimation error, and their relation to bias–variance decompositions. This yielded various insights, including how existing measures of epistemic uncertainty can fit into a classical learning theoretic framework. In conclusion, uncertainty measures need not be regarded as primitives requiring independent justification. Instead, they emerge naturally from modelling and evaluation choices through decompositions of subjective risk. Beyond unifying existing uncertainty measures, our perspective shows intriguing connections to statistical learning theory, with analogous definitions of approximation and estimation but for subjective risk. Our hope is that this ‘new view’ will help establish a deeper theoretical foundation for uncertainty quantification.
16
Subjective Risk Decomposition: A New View for Uncertainty Quantification
References Rishabh Agarwal, Nino Vieillard, Yongchao Zhou, Piotr Stanczyk, Sabela Ramos Garea, Matthieu Geist, and Olivier Bachem. On-policy distillation of language models: Learning from self-generated mistakes. In The Twelfth International Conference on Learning Representations, 2024. URL https://openreview.net/forum?id=3zKtaqxLhW. Francis Bach. Learning theory from first principles. MIT press, 2024. Freddie Bickford-Smith, Jannik Kossen, Eleanor Trollope, Mark Van Der Wilk, Adam Foster, and Tom Rainforth. Rethinking aleatoric and epistemic uncertainty. In International Conference on Machine Learning, 2025. URL https://proceedings.mlr.press/v267/ bickford-smith25a.html. Léon Bottou and Olivier Bousquet. The tradeoffs of large scale learning. Advances in neural information processing systems, 20, 2007. Lev M Bregman. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR computational mathematics and mathematical physics, 7(3):200–217, 1967. Gavin Brown and Riccardo Ali. Bias-variance is not the same as approximation-estimation. In Transactions on Machine Learning Research, 2024. Christopher Bülte, Yusuf Sale, Timo Löhr, Paul Hofman, Gitta Kutyniok, and Eyke Hüllermeier. An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression. arXiv preprint arXiv:2504.18433, 2025. Yihan Cao and Yanbin Kang. On LLM knowledge distillation - a comparison between forward KL and reverse KL. In The Fourth Blogpost Track at ICLR 2025, 2025. URL https://openreview.net/forum?id=jGVCs8gomF. Stefan Depeweg, Jose-Miguel Hernandez-Lobato, Finale Doshi-Velez, and Steffen Udluft. Decomposition of uncertainty in bayesian deep learning for efficient and risk-sensitive learning. In International conference on machine learning, pages 1184–1193. PMLR, 2018. Alexander Fishkov, Kajetan Schweighofer, Mykyta Ielanskyi, Nikita Kotelevskii, Mohsen Guizani, and Maxim Panov. Uncertainty quantification for regression using proper scoring rules, 2025. URL https://arxiv.org/abs/2509.26610. Béla A Frigyik, Santosh Srivastava, and Maya R Gupta. Functional bregman divergence and bayesian estimation of distributions. IEEE Transactions on Information Theory, 54 (11):5130–5139, 2008. Yarin Gal. Uncertainty in Deep Learning. PhD thesis, University of Cambridge, 2016. Stuart Geman, Elie Bienenstock, and René Doursat. Neural networks and the bias/variance dilemma. Neural computation, 4(1):1–58, 1992. 17
Alamri, Caprio, and Brown
Sebastian G. Gruber and Florian Buettner. Uncertainty estimates of predictions via a general bias-variance decomposition, 2023. URL https://arxiv.org/abs/2210.12256. Yuxian Gu, Li Dong, Furu Wei, and Minlie Huang. MiniLLM: Knowledge distillation of large language models. In International Conference on Learning Representations, 2024. URL https://openreview.net/forum?id=5h0qf7IBZZ. Neha Gupta, Jamie Smith, Ben Adlam, and Zelda E Mariet. Ensembles of classifiers: a bias-variance perspective. Transactions on Machine Learning Research, 2022. ISSN 2835-8856. URL https://openreview.net/forum?id=lIOQFVncY9. Tom Heskes. Bias-variance decompositions: the exclusive privilege of bregman divergences, 2026. URL https://arxiv.org/abs/2501.18581. Paul Hofman, Yusuf Sale, and Eyke Hüllermeier. Quantifying aleatoric and epistemic uncertainty with proper scoring rules, 2024. URL https://arxiv.org/abs/2404.12215. Ziyi Huang, Henry Lam, and Haofeng Zhang. Quantifying epistemic uncertainty in deep learning. arXiv preprint arXiv:2110.12122, 2021. Ziyi Huang, Henry Lam, and Haofeng Zhang. Efficient uncertainty quantification and reduction for over-parameterized neural networks, 2023. URL https://arxiv.org/abs/ 2306.05674. Sebastian Jimenez, Mira Juergens, and Willem Waegeman. Position: Epistemic uncertainty estimation methods are fundamentally incomplete. In Forty-third International Conference on Machine Learning Position Paper Track, 2026. URL https: //openreview.net/forum?id=g598HZM6ib. Alex Kendall and Yarin Gal. What uncertainties do we need in bayesian deep learning for computer vision? In Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, page 5580–5590, Red Hook, NY, USA, 2017. Curran Associates Inc. ISBN 9781510860964. Michael Kirchhof, Gjergji Kasneci, and Enkelejda Kasneci. Reexamining the aleatoric and epistemic uncertainty dichotomy. In The Fourth Blogpost Track at ICLR 2025, 2025. URL https://openreview.net/forum?id=lE7WZ2DpMq. Nikita Kotelevskii, Vladimir Kondratyev, Martin Takáč, Eric Moulines, and Maxim Panov. From risk to uncertainty: Generating predictive uncertainty measures via bayesian estimation. In The Thirteenth International Conference on Learning Representations, 2025. URL https://openreview.net/forum?id=cWfpt2t37q. Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. In I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017. URL https://proceedings.neurips.cc/paper_files/paper/2017/file/ 9ef2ed4b7fd2c810847ffa5fa85bce38-Paper.pdf. 18
Subjective Risk Decomposition: A New View for Uncertainty Quantification
Andrey Malinin. Uncertainty Estimation in Deep Learning with application to Spoken Language Assessment. PhD thesis, University of Cambridge, 2019. URL https: //www.repository.cam.ac.uk/handle/1810/298857. Andrey Malinin and Mark Gales. Uncertainty estimation in autoregressive structured prediction. In International Conference on Learning Representations, 2021. Andrey Malinin and Mark JF Gales. Reverse kl-divergence training of prior networks: Improved uncertainty and adversarial robustness. In Neural Information Processing Systems, 2019. Frank Nielsen and Richard Nock. Sided and symmetrized bregman centroids. IEEE Transactions on Information Theory, 55(6):2882–2904, 2009. doi: 10.1109/TIT.2009.2018176. David Pfau. A generalized bias-variance decomposition for bregman divergences, 2025. URL https://arxiv.org/abs/2511.08789. Frank P Ramsey. Truth and probability. In Readings in formal epistemology: Sourcebook, pages 21–45. Springer, 1926. Yusuf Sale, Paul Hofman, Lisa Wimmer, Eyke Hüllermeier, and Thomas Nagler. Secondorder uncertainty quantification: Variance-based measures, 2023. URL https://arxiv. org/abs/2401.00276. Yusuf Sale, Paul Hofman, Timo Löhr, Lisa Wimmer, Thomas Nagler, and Eyke Hüllermeier. Label-wise Aleatoric and Epistemic Uncertainty Quantification. In The 40th Conference on Uncertainty in Artificial Intelligence, 2024. Yusuf Sale, Nikita Kotelevskii, Maxim Panov, and Eyke Hüllermeier. The AleatoricEpistemic Dichotomy of Uncertainty is Meaningful and Indispensable for Machine Learning. SSRN, 2026. URL https://ssrn.com/abstract=6783338. Leonard J Savage. Elicitation of personal probabilities and expectations. Journal of the American Statistical Association, 66(336):783–801, 1971. L.J. Savage. The Foundations of Statistics. John Wiley and Sons, 1954. doi: https: //doi.org/10.1002/nav.3800010316. Kajetan Schweighofer, Lukas Aichberger, Mykyta Ielanskyi, and Sepp Hochreiter. Introducing an improved information-theoretic measure of predictive uncertainty. arXiv preprint arXiv:2311.08309, 2023. Mohammad Hossein Shaker and Eyke Hüllermeier. Aleatoric and epistemic uncertainty with random forests. In Advances in Intelligent Data Analysis XVIII: 18th International Symposium on Intelligent Data Analysis, IDA 2020, Konstanz, Germany, April 27–29, 2020, Proceedings 18, pages 444–456. Springer, 2020. Matias Valdenegro-Toro and Daniel Saromo Mori. A deeper look into aleatoric and epistemic uncertainty disentanglement. In 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pages 1508–1516. IEEE, 2022. 19
Alamri, Caprio, and Brown
Various. X discussion thread on uncertainty and machine learning. https://x.com/ BlackHC/status/1817556167687569605, July 2024. X (formerly Twitter) thread, accessed 2026-05-29. Lisa Wimmer, Yusuf Sale, Paul Hofman, Bernd Bischl, and Eyke Hüllermeier. Quantifying aleatoric and epistemic uncertainty in machine learning: Are conditional entropy and mutual information appropriate measures? In Proceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence, volume 216 of Proceedings of Machine Learning Research, pages 2282–2292. PMLR, 31 Jul–04 Aug 2023. URL https://proceedings. mlr.press/v216/wimmer23a.html. Wei Xiong, Hanze Dong, Chenlu Ye, Ziqi Wang, Han Zhong, Heng Ji, Nan Jiang, and Tong Zhang. Iterative preference learning from human feedback: Bridging theory and practice for rlhf under kl-constraint. arXiv preprint arXiv:2312.11456, 2023. Daniel M. Ziegler, Nisan Stiennon, Jeffrey Wu, Tom B. Brown, Alec Radford, Dario Amodei, Paul Christiano, and Geoffrey Irving. Fine-tuning language models from human preferences, 2020. URL https://arxiv.org/abs/1909.08593.
20
Subjective Risk Decomposition: A New View for Uncertainty Quantification
Appendix A. Proofs for section 3 Theorem 2, the ‘reverse-argument’ BV decomposition, uses the divergence/entropy decomposition of a strictly proper loss to obtain a Bregman divergence, then applies the rightcentroid decomposition from Pfau (2025) to obtain bias and variance components. We present the full proof here in our notation for completeness. Proof of Theorem 2. Since ℓ is strictly proper, its generalized entropy Hℓ is strictly concave, and ϕ := −Hℓ is strictly convex. Also, as ϕ is differentiable on the relative interior of P, this gives the Bregman representation in equation 7. Hence, Rqθ (p) = EŶ ∼qθ [ℓ(p, Ŷ )] = Bϕ (qθ , p) + Hℓ (qθ ), and assuming expectations are finite, Eθ Rqθ (p) = Eθ Bϕ (qθ , p) + Eθ Hℓ (qθ ) .
(28)
(29)
∗ = arg min ∗ By definition qR z∈P Eθ [Bϕ (qθ , z)], which is the mixture qR = q̄ = Eθ [qθ ]. The Bregman three-point identity (Nielsen and Nock, 2009),
Bϕ (qθ , p) = Bϕ (qθ , q̄) + Bϕ (q̄, p) + ∇ϕ(q̄) − ∇ϕ(p), qθ − q̄ .
(30)
Taking the expectation Eθ , the inner-product term vanishes since Eθ [qθ − q̄] = 0, giving Eθ Bϕ (qθ , p) = Bϕ (q̄, p) + Eθ Bϕ (qθ , q̄) , (31) | {z } | {z } bias
variance
which is the “reverse-argument” bias–variance decomposition in Pfau (2025). Substituting equation 31 into equation 29 yields the stated decomposition, Eθ EŶ ∼qθ [ℓ(p, Ŷ )] = Bϕ (q̄, p) + Eθ Bϕ (qθ , q̄) + Eθ Hℓ (qθ ) . {z } | {z } {z } | {z } | | expected subjective risk
bias
variance
(32)
generalized entropy
Derivations to support Example 1. Proof Let ℓ(p, Ŷ ) = − ln p(Ŷ ) with Ŷ ∼ qθ , so the pointwise subjective risk of reporting p under belief qθ is X Rqθ (p) = EŶ ∼qθ ℓ(p, Ŷ ) = −EŶ ∼qθ ln p(Ŷ ) = − qθ (y) ln p(y). (33) y∈Y
Its generalized entropy is, Hℓ (qθ ) := Rqθ (qθ ) = −
X
qθ (y) ln qθ (y) = H(qθ ),
(34)
y∈Y
the P Shannon entropy of the categorical qθ . Hence the generator is ϕ(q) = −Hℓ (q) = y∈Y q(y) ln q(y), the negative Shannon entropy. The risk decomposes into a generalized entropy and a Bregman divergence (see Equation 7): Rqθ (p) = Bϕ (qθ , p) + Hℓ (qθ ). 21
(35)
Alamri, Caprio, and Brown
As Hℓ (q) is the negative Shannon entropy, the induced Bregman divergence is the reverse KL, Bϕ (qθ , p) = K(qθ || p). Take expectation Eθ of equation 35 and apply the bias–variance decomposition where q̄ := Eθ [qθ ] is the categorical mixture, Eθ EŶ ∼qθ [− ln p(Ŷ )] = K(q̄ || p) + Eθ [K(qθ || q̄)] + Eθ H(qθ ) (36) | {z } | {z } | {z } | {z } variance
bias
expected risk
generalized entropy
The variance term is the mutual information. With Ŷ | θ ∼ qθ and marginal Ŷ ∼ q̄, Eθ K(qθ || q̄) = H(q̄) − Eθ Hℓ (qθ ) = H(Ŷ ) − H(Ŷ | Θ) = I(Ŷ ; Θ), P where H(q̄) := − y q̄(y) ln q̄(y). Thus the variance is the mutual information I(Ŷ ; Θ), understood as epistemic, and the generalized entropy Eθ [Hℓ (qθ )] = H(Ŷ | Θ), the Shannon conditional entropy, as aleatoric.
Derivations to support Example 2. Proof Let ℓ(p, Ŷ ) = − ln p(Ŷ ) with Ŷ ∼ qθ , so the pointwise subjective risk of reporting p under belief qθ is Z Rqθ (p) = EŶ ∼qθ ℓ(p, Ŷ ) = −EŶ ∼qθ ln p(Ŷ ) = − qθ (y) ln p(y) dy. (37) Y
Its generalized entropy is, Z Hℓ (qθ ) := Rqθ (qθ ) = −
Y
qθ (y) ln qθ (y) dy = 21 ln(2πe σθ2 ),
(38)
R the differential entropy of a Gaussian. Thus, ϕ(q) = −Hℓ (q) = Y q(y) ln q(y) dy, the negative differential entropy. The risk decomposes into a generalized entropy and a functional Bregman divergence (see Equation 7): Rqθ (p) = Bϕ (qθ , p) + Hℓ (qθ )
(39)
as Hℓ (q) is the negative differential entropy, The induced Bregman divergence is the reverse KL. Take expectation Eθ of equation 39 and apply the bias–variance decomposition where q̄ := Eθ [qθ ] is a mixture of distributions, Eθ EŶ ∼qθ [− ln p(Ŷ )] = K(q̄ || p) + Eθ [K(qθ || q̄)] + Eθ 21 ln(2πe σθ2 ) (40) | {z } | {z } | {z } | {z } expected risk
variance
bias
generalized entropy
The variance term is the mutual information. With Ŷ | θ ∼ qθ and marginal Ŷ ∼ q̄, Z Z Eθ K(qθ || q̄) = Eθ qθ ln qθ − q̄ ln q̄ = H(q̄)−Eθ Hℓ (qθ ) = H(Ŷ )−H(Ŷ | Θ) = I(Ŷ ; Θ), where H(q̄) := −
R
q̄(y) ln q̄(y)dy.
22
Subjective Risk Decomposition: A New View for Uncertainty Quantification
Derivations to support Example 3. Label-wise UQ (Sale et al., 2024) treats each label separately as a present/absent event. The indicator of a single label is a Bernoulli variable, so its uncertainty is fully described by Ber(µ). We show the squared loss example but the idea is more general. Proof As in Example 4, the loss is ℓ(p, Ŷ ) = (µp − Ŷ )2 , which induces the same Bregman generator ϕ(µ) = µ2 and hence the same divergence Bϕ (µθ , µp ) between the means. The bias and variance components are therefore identical to Example 2. However, the generalised entropy is the predictive variance, which for the Bernoulli distribution is 2 (41) Hℓ (µθ ) := Rqθ (qθ ) = EŶ ∼qθ (µθ − Ŷ )2 = EŶ ∼qθ Ŷ 2 − EŶ ∼qθ [Ŷ ] Since Ŷ ∼ Ber(µθ ) takes values in {0, 1}, we have Ŷ 2 = Ŷ , so this gives, Hℓ (µθ ) = µθ − µ2θ = µθ · (1 − µθ ).
23
(42)
Alamri, Caprio, and Brown
Appendix B. Further Examples of Bias/Variance for Subjective Risk Example 4 . Variance-based measures (Kendall and Gal, 2017). Let P be the class of Gaussians with fixed variance σ 2 = 12 , and denote the true p = N (µp , 12 ), and qθ = N (µθ , 12 ). Define ℓ(p, Ŷ ) = (µp − Ŷ )2 . Ignoring affine terms, this induces ϕ(µ) = µ2 , and Theorem 2 becomes, 2 h h i i + Eθ (µθ − Eθ [µθ )2 + Eθ EŶ ∼qθ (µp − Ŷ )2 = Eθ [µθ ] − µp σq2 (43) | {z } | {z } {z } | |{z} expected subjective risk
variance
bias
generalized entropy
These are the “variance-based” terms studied by many authors for UQ in regression, e.g. Kendall and Gal (2017); Depeweg et al. (2018); Bülte et al. (2025); Jimenez et al. (2026). The variance of µθ , also written as V arθ (EŶ ∼qθ [Ŷ ]), is again understood as epistemic, and the generalized entropy Eθ [V ar(Ŷ | Θ = θ)] as aleatoric. Note, this is constant w/r θ, and so equal to our assumed σq2 = 12 . The “total uncertainty’ (TU) is their sum, V ar(Ŷ ). Furthermore we note that, since we assume σp2 = σq2 , this decomposition is exactly equivalent to the classic squared loss bias–variance decomposition (Geman et al., 1992), with µθ as the predictor and Y ∼ p as the true label. Proof Let ℓ(p, Ŷ ) = (µp − Ŷ )2 with Ŷ ∼ qθ , so the pointwise subjective risk of reporting p under belief qθ is Rqθ (p) = EŶ ∼qθ ℓ(p, Ŷ ) = EŶ ∼qθ (µp − Ŷ )2 . (44) Its generalized entropy is, 2 Hℓ (qθ ) := Rqθ (qθ ) = EŶ ∼qθ (µθ − Ŷ )2 = EŶ ∼qθ [Ŷ 2 ] − EŶ ∼qθ [Ŷ ] = σq2 = 21 ,
(45)
the predictive variance of Ŷ ∼ qθ , which under the fixed-variance assumption is constant in θ. Since −Hℓ (µθ ) equals µ2θ up to an affine term, the generator is ϕ(µ) = −Hℓ (µ) = µ2 , and the induced Bregman divergence is the squared difference of means, Bϕ (µθ , µp ) = (µθ −µp )2 . The risk decomposes into a generalized entropy and a Bregman divergence: Rqθ (p) = Bϕ (µθ , µp ) + Hℓ (qθ ).
(46)
Take expectation Eθ of equation 46 and apply the bias–variance decomposition where µ̄ := Eθ [µθ ]; the cross term vanishes since Eθ [µ̄ − µθ ] = 0, Eθ EŶ ∼qθ (µp − Ŷ )2 = (µ̄ − µp )2 + Eθ (µθ − µ̄)2 + σq2 . (47) | {z } | {z } |{z} | {z } expected risk
bias
variance
generalized entropy
Many ML models have multiple sources of uncertainty—due to uncertainty in training data, initial parameters, procedural training issues, etc. Huang et al. (2021) showed how to decompose epistemic uncertainty into procedural and data uncertainty, separating these sources of uncertainty. We can again explain this within our framework. In addition to the bias–variance decomposition, we need the Bregman Law of Total Variance (Gupta et al., 2022), to obtain a finer-grained variance decomposition where appropriate. 24
Subjective Risk Decomposition: A New View for Uncertainty Quantification
Theorem 9 (Bregman Law of Total Variance (Gupta et al., 2022) ) Let Bϕ be a Bregman divergence generated by a strictly convex, differentiable function ϕ. Assume a convex class of distributions P over an outcome space Y. Let qZ ∈ P be a model induced by a joint random variable Z = (A, B). For the reverse-argument Bregman divergence, the ∗ variance EZ Bϕ (qZ , qR ) can be decomposed by the law of total variance as follows: h i h i h i ∗ ∗ ∗ ∗ EZ Bϕ (qZ , qR ) = EA Bϕ (qB|A , qR ) + EA EB|A Bϕ (qZ , qB|A ) . (48) | {z } | {z } | {z } total variance
between-group variance
within-group variance
∗ := arg min E [B (q , c)] is the right centroid wrt the joint random variable Z. The Here qR c Z ϕ Z ∗ conditional right centroid is qB|A := arg minc EB|A [Bϕ (qZ , c)]. This result is due to Gupta et al. (2022), with the functional form we use given by Gruber and Buettner (2023).
Proof The result follows by applying the three-point identity at the conditional centroid and taking iterated expectations, ∗ ∗ ∗ ∗ ∗ ∗ ∗ Bϕ (qZ , qR ) = Bϕ (qZ , qB|A ) + Bϕ (qB|A , qR ) + ∇ϕ(qB|A ) − ∇ϕ(qR ), qZ − qB|A . ∗ ] = 0, Take EB|A the inner-product term vanishes since EB|A [qZ − qB|A ∗ ∗ ∗ ∗ EB|A Bϕ (qZ , qR ) = Bϕ (qB|A , qR ) + EB|A Bϕ (qZ , qB|A ) .
Taking EA and using the tower rule EZ = EA EB|A gives ∗ ∗ ∗ ∗ , qR ) + EA EB|A [Bϕ (qZ , qB|A )] . EZ Bϕ (qZ , qR ) = EA Bϕ (qB|A | {z } | {z } | {z } total variance
between-group variance
(49)
(50)
(51)
within-group variance
If our subjective risk has a joint variable Z, the decomposed variance term from Theorem 2 ∗ )]. We can then apply the LTV above, to recover the procedural/data will be EZ [Bϕ (qZ , qR uncertainty split of Huang et al. (2021, 2023). Example 5 . Data vs Procedural Uncertainty (Jimenez et al., 2026, Eq. 7). Let P be the class of Gaussians with fixed variance σ 2 = 12 . Define the joint random variable Z = (θ, D), where θ captures procedural randomness (e.g. weight initialization) and D captures dataset variations. Denote the true p = N (µp , 12 ), and qZ = N (µZ , 21 ). Define ℓ(p, Ŷ ) = (µp − Ŷ )2 . Ignoring affine terms, this induces ϕ(µ) = µ2 , and Theorem 2 becomes, h 2 h i 2i 2 EZ EŶ ∼qZ (µp − Ŷ ) = EZ [µZ ] − µp + EZ µZ − EZ [µZ ) + σq2z , (52) | {z } | {z } | {z } | {z } expected subjective risk
variance
bias
generalized entropy
where, h h h h i i i EZ µZ − EZ [µZ )2 = ED (Eθ|D [µZ ] − EZ [µZ )2 + ED Eθ|D (µZ − Eθ|D [µZ ] )2 . (53) | | {z } {z } | {z } variance
data uncertainty
25
procedural uncertainty
Alamri, Caprio, and Brown
Proof This is Example 4 with the expectation taken wrt a joint variable Z = (θ, D), so the squared loss gives ϕ(µ) = µ2 , bias (EZ [µZ ] − µp )2 , variance EZ [(µZ − EZ [µZ ])2 ], and the constant generalized entropy σq2Z . The only new step is to decompose the variance by its source. Applying the Bregman law of total variance (Theorem 9) to Z = (A, B) = (D, θ) splits it into a between-group term over datasets and a within-group term over procedural randomness, EZ (µZ − EZ [µZ ])2 = ED (Eθ|D [µZ ] − EZ [µZ ])2 + ED Eθ|D [(µZ − Eθ|D [µZ ])2 ] , (54) | {z } | {z } data
procedural
recovering the data/procedural decomposition discussed in Huang et al. (2021), Huang et al. (2023), and Jimenez et al. (2026).
Example 6 . Deep Ensembles. Bülte et al. (2025) propose an axiomatic framework, with Deep Ensembles (Lakshminarayanan et al., 2017) as an exemplar. Assume the same setting as Example 4, but now with unknown variance and a finite set of m models: each θi 1 Pm . induces qθi = N (µi , σi2 ), so the expectation over θ becomes a finite average m i=1 # " m i h 1 X 2 EŶ ∼qθ (µp − Ŷ ) = i m
m
1 X µi − µ p m
i=1
|
{z
expected subjective risk
}
!2
i=1
|
{z
m
1 X µi m
i=1
}
bias
m
1 X 2 + µi − m |
!2
i=1
{z
variance
m
+
1 X 2 σi m i=1
}
|
{z
}
generalized entropy
The epistemic/aleatoric terms are exactly those derived in Bülte et al. (2025, Section 4). Proof This is Example 4 with the posterior over θ replaced by the distribution over a finite 1 Pm and each member set of m members, so every expectation Eθ becomes the average m i=1 2 2 contributes qθi = N (µi , σi ). Allowing σi to vary across members is the only difference from Example 4.
26
Subjective Risk Decomposition: A New View for Uncertainty Quantification
Appendix C. Proofs for section 5 We prove the relation between mutual information and EPKL (Schweighofer et al., 2023). Proof of Theorem 7. Multiplying the expression by 2 and rearranging gives h i 2 · I(Ŷ ; Θ) = Eθ,θ′ K(qθ || qθ′ ) + Eθ K(qθ || q̄) − K(q̄ || qθ ) I(Ŷ ; Θ) = Eθ,θ′ K(qθ || qθ′ ) − Eθ K(q̄ || qθ ) Rearranging this final equation yields exactly Malinin and Gales (2021, Eq. 6), RM I = EP KL − M I, which completes the proof. Proof of Theorem 8. Let (Y, A, λ) be a measure space, and let q0 be a strictly positive probability density with respect to λ. For each θ, suppose Z qθ (y) = qθ′ (y) 1 + ϵh(y) , h(y)qθ′ (y) dν(y) = 0, where the perturbations satisfy ∥rθ ∥∞ ≤ M almost surely, for |ϵ|M < 1, and Z q0 (y)rθ (y) dλ(y) = 0 for every θ. Assume also that Eθ [rθ (y)] = 0, for q0 -almost every y. Then, q̄(y) := Eθ [qθ (y)] = q0 (y). Let θ, θ′ be independent draws from the posterior over parameters, and define EPKLϵ := Eθ,θ′ K(qθ || qθ′ ) and Aϵ := Eθ [K(qθ || q̄) − K(q̄ || qθ )]. Then, as ϵ → 0, Z 2 2 EPKLϵ = ϵ Eθ q0 (y)rθ (y) dλ(y) + O(ϵ3 ), (55) whereas
ϵ3 Aϵ = Eθ 6
Z
q0 (y)rθ (y) dλ(y) + O(ϵ4 ). 3
(56)
Consequently, if Z V2 := Eθ
q0 (y)rθ (y) dλ(y) > 0, 2
then
Aϵ = O(ϵ). EPKLϵ Therefore the asymmetry correction is negligible relative to the expected pairwise KL in the small-disagreement regime. In particular, since 1 1 I(Ŷ ; Θ) = EPKLϵ + Aϵ , 2 2 we have 1 I(Ŷ ; Θ) = Eθ,θ′ K(qθ || qθ′ ) + O(ϵ3 ). (57) 2 Equivalently, Eθ,θ′ K(qθ || qθ′ ) = 2I(Ŷ ; Θ) + O(ϵ3 ).
27