Rethinking Aleatoric and Epistemic Uncertainty
Authors: Freddie Bickford Smith
Jannik Kossen
Eleanor Trollope
Mark van der Wilk
Adam Foster
Tom Rainforth
Venue: ICML 2025 poster (2025)
Keywords: aleatoric
epistemic
uncertainty
Primary area: probabilistic_methods

Abstract: The ideas of aleatoric and epistemic uncertainty are widely used to reason about the probabilistic predictions of machine-learning models. We identify incoherence in existing discussions of these ideas and suggest this stems from the aleatoric-epistemic view being insufficiently expressive to capture all the distinct quantities that researchers are interested in. To address this we present a decision-theoretic perspective that relates rigorous notions of uncertainty, predictive performance and statistical dispersion in data. This serves to support clearer thinking as the field moves forward. Additionally we provide insights into popular information-theoretic quantities, showing they can be poor estimators of what they are often purported to measure, while also explaining how they can still be useful in guiding data acquisition.

Rethinking Aleatoric and Epistemic Uncertainty
Freddie Bickford Smith
Jannik Kossen
Eleanor Trollope'
Mark van der Wilk
Adam Foster! ! Tom Rainforth
Abstract
sumptions are made about how uncertainty will decompose on unseen data (Seeböck et al, 2019; Wang & Aitchison, 2021), and misleading connections are drawn between certainty and predictive accuracy (Orlando et al, 2019; Wang et al, 2019). Meanwhile distinct mathematical quantities are used to refer to notionally the same concepts: epistemic uncertainty, for example, has been variously defined us- ing density-based (Mukhoti et al, 2023; Postels et al, 2020), information-be bbased (Gal et al, 2017) and variance-based (Gal, 2016; Kendall & Gal, 2017; McAllister, 2016) quantities.
The ideas of aleatoric and epistemic uncertainty are widely used to reason about the probabilistic predictions of machine-learning models. We iden-
tify incoherence in existing discussions of these ideas and suggest this stems from the aleatoric- epistemic view being insufficiently expressive to capture all the distinct quantities that researchers are interested in. To address this we present a decision-theoreticc perspective that relates rigor- ous notions of uncertainty, predictive performance and statistical dispersion in data. This serves to support clearer thinking as the field moves for- ward. Additionally we provide insights into popu- lar information-theen ttheoretic quantities, showing they can be poor estimators of what they are often pur- ported to measure, while also explaining how they can still be useful in guiding data acquisition.
We suggest this incoherence arises from the aleatoric- epistemic view being too simplistic in the context of ma- chine learning. Researchers are looking for concrete notions of a model's 's predictive uncertainty and how that uncertainty might or might not change with more data (associated with a decomposition into irreducible and reducible components), but also related notions of predictive performance and data dispersion. The alcatoric-epistemic view cannot satisfy all these needs: many concepts stand to be defined, while the view fundamentally only has capacity for two concepts. Yet the current state of play is to nevertheless appeal to the aleatoric-epistemic view, with different researchers using it in different ways. A result of this conceptual overloading is to conflate quantities that ought be recognised as dis- tinct. Far from just matter of semantics, this is having meaningful effect the field's progress: methods are being designed and evaluated based on shaky foundations.
1 Introduction
When making decisions under uncertainty, it can be useful to reason about where that uncertainty comes from (Osband et al, 2023; Wen et al, 2022). Researchers often aim to do this by referring to aleatoric (literal meaning: "relating to chance") and epistemic ("relating to knowledge") uncer- tainty, ideas with a long history in the study of probability (Hacking, 1975). Aleatoric uncertainty is typically associ- ated with statistical dispersion in data (sometimes thought of as noise), while epistemic is associated with a model's in- ternal information state (Hüllermeier& & Waegeman, 2021).
To establish a clearer perspective, we draw on powerful yet underappreciated ideas from decision theory (Dawid, 1998; DeGroot, 1962; Neiswanger al, 2022). Our starting point is final decision of interest with associated loss function. Given this, uncertainty in predictive beliefs can be formalised as the subjective expected loss of acting Bayes- optimally under those beliefs; this generalises quantities like variance and Shannon entropy. From there we show how reasoning about new data gives rise to a notion of expected uncertainty reduction, which we can use to identify a de- composition of uncertainty into irreducible and reducible components. Then we clarify the connection between uncer- tainty, predictive performance and data dispersion, linking to classic decompositions from statistics and information theory. Overall this provides a coherent synthesis of key quantities that researchers are interested in (Figure 1).
Concerningly given their scale of use, these ideas are not be- ing discussed coherently in the literature. The line between model- predictions and data- mata-generating processes is re- peatedly blurred (Amini et al, 2020; Ayhan & Berens, 2018:
Immer et al, 2021; Kapoor et al, 2022; Smith & Gal, 2018: van Amersfoort al, 2020). On top of this, tenuous as-
'University of Oxford. Correspondence to Freddic Bickford Smith <fbickfordsmi th@cs. ox. ac. ack>..kk>
Proceedings of the 42nd International Conference on Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Rethinking Alcatoric and Epistemic Uncertainty
Learning
Reasoning
Decision problem A +
Predictive model pa(#) p(=)).n)
Bayes optimal action Gi = arg min, Eputos ())
Training data Mon Pizxia (81x)/rr
Uncertainty quantification
Predictive uncertainty
Expected uncertainty reduction
Mppl=]] (p. [tto, +]]
(F,)) = A[pa(=]] pprnnss (st //2 [M]P
= Alp[(=)) BA.SE... x.) (x))))))=)))))
Model evaluation
Proper scoring rule. s[pr. - #)
Discrepancy function
Pooat) Paotte [A[p.. x)]
R R/pwoal (=)
Our dlecision-theoreet view coherently relates machine-lcarning concepts that have heen conflated under the alcatoric-epistemic view. We consider taking an action, a E A, in light of imperfect knowledge = E 2. with an action's 's consequences measured by a loss function, E(a, 2). Since is unknown, we use any available training data, yr.. to build a predictive model, p.. (=). with which we can reason over possible values of and thus choose an action. Additionally we can use the model to quantify uncertainty and its reducibility with respect to new data, and we can evaluate the model using ground-trute truth realisation of or reference distribution, (eva
Figure 1
Bridging this generalised perspective back to how aleatoric and epistemic uncertainty have been discussed in past work, we provide new insights BALD, a popular information- theoretic objective for data acquisition (Gal et al, 2017; Houlsby et al, 2011; Lindley, 1956). In particular we high- light that it should be seen not as a direct measure of long- run reducible predictive uncertainty, as has been suggested in the past, but instead as an estimator that can be highly inaccurate. Reconciling this with BALD's practical utility, suggest it is often better understood as approximately measuring short-run reductions in parameter uncertainty. It can therefore be useful, albeit still suboptimal in prediction- oriented settings (Bickford Smith et al, 2023; 2024).
whether that prediction is fundamentally uncertain for the given model class or instead due to lack of data. This breakdown has clear utility in the context of seeking new data that will reduce predictive uncertainty (Bickford Smith et al, 2023; 2024; MacKay, 1992a;b). But it is also relevant elsewhere: in model selection, for example, we might want to quantify a model's scope for improvement by forecasting how its predictions will change given more data (Barbieri & Berger, 2004; Fong & Holmes, 2020; Geisser & Eddy, 1979; Kadane & Lazar, 2004; Laud & Ibrahim, 1995).
Uncertainty that resolves in light of new data can be thought of as "epistemic" in the sense that data conveys knowledge. Intuitively the corresponding irreducible uncertainty seems to be determined by not only the model class but also, among other things, an "inherent" level of uncertainty associated with the data source hand, which is often thought of in terms of randomness chance, hence the word "aleatoric".
Our work thus serves to inform future work in two key ways. On the one hand it sheds light on the contradictions of the aleatoric-episemmic view and presents coherent alternative perspective that allows clearer thinking about uncertainty in machine learning. On the other hand it provides more direct practical insights. It clarifies that what might have seemed like arbitrary choices for a decision-maker can instead be made by following well-defined logic: given some basic components and principles, it becomes clear how we should measure predictive uncertainty, predictive performance and data dispersion, and how we can identify good future train- ing data. It also highlights approximations that often have to be made in practice, revealing scope for suboptimal per- formance and therefore informing future methods research.
While the concepts of aleatoric and epistemic uncertainty had previously been used in machine learning, for example by Lawrence (2012) and Senge et al (2014), their popular- ity grew following work by Gal (2016), Gal et al (2017) and Kendall & Gal (2017). The most widely used math- ematical definitions of these ideas, which we will discuss in Section 4, are the iitoomationnthhoreee quantities used by Gal et al (2017), building on earlier work on Bayesian experimental design (Lindley, 1956) and Bayesian active learning (Houlsby et al, 201 1; MacKay, 1992a;b).
2 Background
A range of perspectives on aleatoric and epistemic uncer- tainty in machine learning have been put forward in recent years. These include a discussion of sources of uncertainty in machine learning (Gruber et al, 2023); case against Shannon entropy as measure of predictive uncertainty
The broad motivation of the aleatoric-epistemic view is to distinguish between different sources of uncertainty. If model's prediction is uncertain, we might want to know
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Rethinking Alcatoric and Epistemic Uncertainty
(Wimmer al, 2023); proposals for alternative information quantities (Schweighofer et al, 2023a;b; 2025); and various other suggestions for how to define uncertainty, such as in terms of class-wise variance (Sale et al, 2023b; 2024b), credal sets (Hofman et al, 2024a; Sale et al, 2023a), dis- tances between probability distributions (Sale et al, 2024a), frequentist risk (Kotelevskii et al, 2022; 2025; Lahlou et al, 2023) and proper scoring rules (Hofman et al, 2024b). As will show, our replacement for the aleatoric-epistemit view unifies and explains many of these ideas.
that minimises this subjective expected loss (Ramsey, 1926; Savage, 1951). Alternative decision-makinn approaches in- clude minimax, which involves minimising the worst-case frequentist risk (von Neumann, 1928; Wald, 1939; 1945).
3.4 Machine learning allows data-driven prediction
Minimising subjective expected loss requires beliefs over 2, and those beliefs can often be informed by some training data, Fi.m ~ Ptrain (y1.m/ /T), where π E II is a policy that controls aspects of data generation. We want notions of uncertainty that reflect how we will actually learn from data, rather than assuming idealised updating that we cannot per- form in practice. We therefore define our predictive model, (2) = p(z; )1:n), to be the output of a generic machine- learning method applied to the training data (and the input of interest there is one) for any given n, which lets us reason about actual changes in uncertainty as n varies.
3
Key concepts
Our aim in this work is to formalise and link together quanti- ties that have been associated with the ideas of aleatoric and epistemic uncertainty in past work. In particular we look to identify rigorous notion of predictive uncertainty and the extent to which it reduces as more data is observed, and also measures of predictive performance and statistical dis- persion in data. We start by highlighting some foundational concepts that will be used throughout our discussion.
Conventional Bayesian inference-taking a generative model over possible data and conditioning on the observed data, giving Pr. (=) = p(=|v1=>) is one possible updating method. Others include deep learning (LeCun et al, 2015), in-context learning (Brown et al, 2020) and non- Ba-esiaysi ensemble methods (Breiman, 2001). In some cases the pre- dictive distribution is defined as (z) = Po) [po (=))) where 0 ~ Pn (0) = p(0; yim) represents set of stochastic model parameters that we average over at prediction time.
3.1 Reasoning should start with the decision of interest
We consider taking an action, a E A. under imperfect knowl- edge of a ground-truth variable, = E Z. Here could for example be an output relating
given input (if so, the in- put is left implicit in our notation) or parameter in model, and could be a direct prediction of 2, with A = 2 for point prediction, or A = P(Z) for probabilistic prediction. We emphasise our choice to focus on this decision, in deliberate contrast with the more common starting point of learning a model from fixed data. We want a notion of predictive un- certainty that is grounded in actions and their consequences, and we need to reason about different possible datasets to rigorously think about reductions in uncertainty.
Regardless of whether there are stochastic parameters, the updating scheme could be stochastic. To handle this we take the convention that updating stochasticity is implicitly absorbed into yl.m we can consider our machine- machine-learning method to be a deterministic mapping that takes random- number seed as an auxiliary input along with data. Thus, while stochastic updating can be source of variability in how uncertainty reduces, this can be dealt with as part of the variability already present in what data we observe.
3.2 Actions induce losses that reflect preferences
3.5 Bayes optimality is subjective notion
We assume we can measure the consequences of taking action a in light of a realisation of z using loss (or neg- ative utility) function, L A x Z + IR. In principle the specification of é follows directly from having preferences that satisfy basic axioms of rationality (von Neumann & Morgenstem, 1947). In practice it can be hard to know what é should be; options for dealing with this include using an intrinsic loss or random loss (Robert, 1996; 2007).
An action taken by minimising subjective expected loss under (z. is referred to as Bayes optimal (Murphy, 2022); if the action is an estimator of some quantity of interest then it is known as Bayes estimator. The notion of Bayes optimality assumes our beliefs, pr. (z). represent our best knowledge of 2, and l reflects our preferences. It says nothing at all about how well our beliefs match reality, or about the actions we would take if we had different beliefs. Bayes-optimal actions can therefore be suboptimal as judged using realisations of from somewhere other than pn (z), such as a system serving as a source of ground truth.
3.3 Subjective expected loss enables decision-making
Since é is a function of the unknown it cannot be used directly as an objective for selecting an action, a. A prin- cipled solution that we focus on here is to form subjective beliefs over possible values of z (conventionally this be- lief state would be a Bayesian prior or posterior), average over these to form an expected loss, then choose an action
3.6 Predictions often do not match data generation
The correspondence between our predictions, (=). and the data-generating process, Perein (w: | (N<i), Veci), can be
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Rethinking Alcatoric and Epistemic Uncertainty
Aleatoric uncertainty
"captures noise inherent the obsenvations"
"cannot be reduced even more data to be collected" (+)]
Expected parameter conditoo-caniitooall prodictive entropy 2. (e [+9]))
I1 [t) (X))
Epistemic uncertainty
"uncertainty in the model
"can be explained away given
Expected information gain in the model parameters (lp,(s]] (a ([,)))))))))))
parameters"
enough data"
+
(1))
[[]) [=)]
A popular interpretation of alcatoric and epistemic uncertainty in machine learning attaches multiple mathematical quantities to each of the two concepts, rendering it incoherent and thus likely source of conflations in the literature. Here show quotations from Kendall & Gal (2017) expressed mathematically information-thcorete quantities, along with an interpretation of Equation 1 due Gal (2016) and Gal et (2017). Some of the quantities shown can coincide in particular cases but in the general case they are distinct.
Figure 2
weak. One basic reason for this is that they might be defined over different event spaces. We could for example have y/ E 2: perhaps we want to predict a coin's 's bias based on outcomes of coin tosses, or we want predict a variable in one domain (eg, vision) based on data from another domain (eg, text). Even if that is not the case, Pn( is reflection of assumptions and design decisions based on incomplete knowledge (Box, 1976; Kleijn & van der Vaart, 2006), and there is no general guarantee that it will match reality.
et al (2017) made the explicit link to Equation 1, informed by Houlsby et al (201 1). Kendall & Gal (2017) discussed the aleatoric-epistemic view in the context of computer vision.
While that work successfully captured some of the intu- itions from Section 2, we highlight that it also overloaded the ideas of aleatoric and epistemic uncertainty with multi- ple meanings, introducing a number of spurious associations (Figure 2). The competing definitions of aleatoric uncer- tainty conflate H[p (z)]. the entropy of pn (z) as n + oo (this depends on the generrting process, as we will discuss in Section 5.2), with three separate quantities:
3.7 Model evaluation relies on external grounding
Because we expect (2) to be imperfect, we often want to assess it against ground-truth realisation of or a reference distribution, (z). Commonly this distribution- wwhic could for example represent a computer program, human expert or physical sensor-is used as source of evaluation data for computing an estimator of frequentist risk (Berger, 1985), such as the mean squared error. Notably Pz)a((z) could itself be imperfect (Fluri et al, 2023), so designing and interpreting evaluations of this form requires care.
(a) H PPPaainn /T], the entropy in training-data gener- ation. Issue: (z) represents subjective predictive beliefs that need not match Phrain (y1.sl (T) (Section 3.6).
(b) (e)e the entropy in evaluation-dai generation. Issue: Poe (z) represents subjective predictive beliefs that need not match (=) (Sections 3.6 and 3.7).
(c) Ep, (=]0]] the expected conditional predic- tive entropy. Issue: for finite n the expected conditional predictive entropy is only an estimator of H[p and can be highly inaccurate (Section 5.5).
Assessing a popular view
Having outlined intuitive descriptions of aleatoric and epis- temic uncertainty and their motivation in Section 2, we turn to how they have been formalised in machine learning. Aleatoric and epistemic uncertainty are often thought of as additive components of predictive uncertainty. A popular way formalise this for models with stochastic parameters, 0, is to relate three information-theereet quantities:
Meanwhile the multiple definitions of epistemic uncertainty mix up H[px(=)] H(p>o the predictive-entropy reduc- tion from updating on infinite new data, with two quantities:
(a) H pn (0)). the entropy of the model's stochastic parame- ters. Issue: the mapping from parameters to predictions is typically not invertible, so H(p,, (0) will not neces- sarily relate to the reduction in predictive entropy.
EIGe "epistemic"
H(px(())] - (p) (=[p).==00]]
(1)
(b)
(z)x - Ep... (0) [H[,, [2]]] the expected informa- tion gain in the model parameters. Issue: for finite this EIG is only an estimator of (z), H[p (2)), and the estimation error can be large (Section 5.5).
"total"
"aleatoric"
where H denotes Shannon entropy and ETG,, also known as the BALD score (Houlsby et al, 2011), is the expected information gain in 0 from observing z. Gal (2016) stated the "total = alcatoric- - epistemic" relationship and the cor- respondence between Pr. (2) and total uncertainty, while Gal
Other sources of confusion in this aleatoric-epistemii view include an incorrect association between a model's subjec- tive uncertainty and frequentist measures of performance,
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Rethinking Alcatoric and Epistemic Uncertainty
such classification accuracy (Figure 2 in Kendall & Gal (2017)), along with misleading implications about how model's uncertainty will behave with varying (Figure 6. 1- 6. 6.12 in Gal (2016) and Table 3 in Kendall & Gal (2017)) and varying distance from the training data ("Aleatoric uncer- tainty does not increase for out-of-data examples... whereas epistemic uncertainty does" in Kendall Gal (2017)).
depending on their decisions of interest and loss functions. One might use variance (Hastie et al, 2009) while the other uses entropy (Shannon, 1948), as Examples I and show.
Example 1 (Dawid, 1998) Point prediction with A Z and E(a, 2) = (a z)² corresponds to measuring uncer- tainty in our beliefs, Pn(z), using variance.
Proof The optimal action is the mean of pn(:):
An alternative perspective
a = arg min Bp
[(a z)²] (=)(=)]]
We now present a coherent, general synthesis of key ideas used in existing discussions of alcatoric and epistemic uncer- tainty (Figure 1). We begin by reasoning about the subjec- tive expected loss of acting Bayes-optimally under given belief state, which leads to a decision-grounder measure of predictive uncertainty. By thinking about how that uncer- tainty will change in light of new data, we then identify notion of expected uncertainty reduction, which we use to define a decomposition of uncertainty into irreducible and reducible components, and which also has direct practical relevance. Then, shifting our focus to externally grounded evaluation, we highlight the distinction between uncertainty, predictive performance and data dispersion. Finally we re- turn to the BALD score discussed in Section 4, providing insights on its utility as data-acquisition objective.
The subjective expected loss of taking this action is the variance of (z):
h[pm (2)] = Epa(s) [ [[Bp) 2)? (=)+))]
Example 2 (Dawid, 1998) Probabilistic prediction with A P(Z) and E(a, z) log a(z) corresponds to mea- suring uncertainty in our beliefs, (2). using entropy.
Proof The optimal action is (2):
a = arg minaae - (=p = Pn(().
The subjective expected loss of taking this action is the Shannon entropy of pm (z):
5.1 Predictive uncertainty can be derived from the final decision of interest and the associated loss
hp), = pcp pn (=)] Hlp,(=)]
We first deal with the question of how to measure predictive uncertainty, to which many different answers have been put forward (Sections 1 and 2). Revisiting past work, we show that minimising subjective expected loss (Ramsey, 1926; Savage, 1951) directly leads loss-grounded measure of uncertainty that reflects our preferences about model behaviour in the final decision of interest. We thus clarify that a decision- maker does not face an arbitrary choice over uncertainty measures: if they specify a loss function based on their preferences, a rigorous uncertainty measure follows.
5.2 Decomposing predictive uncertainty requires accounting for the data-generating process
Next, to formalise the popular idea of decomposing predic- tive uncertainty into irreducible and reducible components (Sections and 4), we reason about how predictive uncer- tainty changes in light of new data. We show that, as long as explicitly account for the data-gathering process, we can write down notion of expected uncertainty reduction that is well defined for any method that maps from data to a predictive distribution (Section 3.4). From this we identify a rigorous irreduiibe-reduucbbl decomposition.
If Pr. (2) represents our beliefs over z then we can identify the Bayes-optimal action, ai,, in our final decision of interest by minimising the expected loss under those beliefs:
Characterising the reducibility of uncertainty initially seems as simple as considering new data, Bi.m F(n+ +m)) and measuring the corresponding uncertainty reduction,
a arg minae A (=). z)]
UR& hlpp. hlpx+m (z)],
Now we can reason about the loss we expect (under our be- lief state) ineur by taking this Bayes-optimal action. An important, underappreciated result is that this minimal ex- pected loss provides a way to measure uncertainty in (2) (Dawid, 1998; DeGroot, 1962; Neiswanger et al, 2022):
which we note could be negative. But this uncertainty reduc- tion depends on exactly what the new data is, and that in turn depends on the process by which the data is generated. We therefore need to explicitly account for the -ata-genn process to produce a well-defined notion of reducibility. Any stochasticity in model updating also has to be taken into ac- count, but this can be achieved by absorbing the updating stochasticity into the definition of zo.m (Section 3.4).
hlpp (z)] = Ep..(.))(((a,,, 2)].
A crucial implication of this is that any two decision-makee should not necessarily use the same uncertainty measure,
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Rethinking Alcatoric and Epistemic Uncertainty
Revisiting the data-generating process from Section 3.4, we define the distribution over y; to depend on decisions made by the data-acquisition policy, x, and on the previous data:
this we might also need approximate how our beliefs over update when we obtain new data. In principle the EUR is defined with respect to whichever updating scheme we are using, but in practice the true update can be too expensive to perform within an expectation over new data. This can be addressed by using some gr+mm (2) in place of Pm+ (z). A common approach is to assume a Bayesian belief update given Vi. even if the true update not Bayesian (Bickford Smith al, 2023; 2024; Gal et al, 2017; Kirsch et al, 2019; 2023). It might be, for example, that we want to use a Bayesian model but cannot perform exact inference. If the true update is Bayesian and can be performed exactly then this assumption is of course not an approximation.
Ptrain (Viain(Vi.m ()) = II Perain (o (FV
With this we can define the true expected uncertainty reduc- tion (EUR) in under given policy, म. as
EUR true (I,, m) = E Persin (wi... (T) [UR, (v
(3)
This allows us to shift from thinking about a specific re- alisation of data to the range of possible data that might be generated. Working from this EUR to an uncertainty decomposition, we consider the limit of m + 0c:
Combining model-based data simulation with an approxi- mate updating scheme, we can estimate the true EUR using
true (M, EUR 2
00) (z)
[))] E Periin (vt (z) [x)
EUR est (m',m) = h (p)(=))))))))))))))))))) (=)] E p.......... (=') (1))0)009000000 [M (=)])
reducible
total
Irveducible
The accuracy of this estimator depends on the mismatch between pn (Vimm I=') and Ptrain (T) as well as the mis- match between qn+mm (2) and Pn+m (2). both of which are likely to be greater for larger m. Estimation therefore re- quires careful tradeoffs to mitigate these mismatches.
Thus we see that three components- loss function, a machine-leaming method mapping from data to a predictive distribution, and a data- aata-acqqiiitoon policy-fully specify a rigorous measure of expected uncertainty reduction and an associated rreedubbble-reducbble decomposition. This con- trasts with the decomposition in Equation I, which requires stochastic model parameters and exact Bayesian updating.
The practical relevance of this becomes clearer upon appre- ciating that the EUR estimator generalises a number of exist- ing actuisiquin objectives. Under the exact- Bayesian- updating assumption it is equivalent to what has variously been called the "expected value of [additionna/sampll]]] in- formation" (Bernardo & Smith, 1994; Raiffa & Schlaifer, 1961). the "expected He.A-inforrmtion gain" (Neiswanger et al, 2022) and the "expected decision utility gain" (Huang et al, 2024). From that objective we can then recover the expected information gain in 2, which corresponds to the BALD score (Gal et al, 2017; Houlsby et al, 2011) if 2 = 0 represents a set of stochastic model parameters, or the ex- pected predictive information gain (Bickford Smith et al, 2023) if (I=,, y=) represents a target input and its output. Notably EUR (T'' m) is also closely related to the idea of the martingale posterior (Fong et al, 2023) as Pr. (v. |=') and our updating scheme together imply a joint distribution from which a martingale posterior can be derived.
It is worth noting that in some restricted cases the depen- dency on the data-acquisition policy, T, of the infinite-data terms in this uncertainty decomposition can disappear. For example, if we are in a anpervised-learning setting where the policy's is decisions concern which inputs to acquire labels for, and if we are using a well-specified Bayesian model and exact Bayesian updating, (z) should be independent of π as long as æ produces dense samples across the input space (Kleijn& & van der Vaart, 2012). However, the requirements for this are very strict, with any model misspecification or error in belief updating reintroducing the dependency.
5.3 Practical estimation of expected uncertainty reduction relies on approximations
Now turn estimating expected uncertainty reduction (EUR) in practice. The decomposition in Section 5.2 is well defined but the infinite-data quantities within it are typically not practically obtainable. While this might seem problematic, we suggest the takeaway should in fact be to deemphasise the decomposition in the context of real-world machine learning, where we do not have infinite data. There is more concrete value (eg, for data acquisition model selection) in estimating the EUR in Equation 3 for finite m, which relies on some important practical approximations.
5.4 Model-based uncertainty should be used with care, and externally grounded evaluation is crucial
Next we clarify the relationship between the predictive un- certainty we have discussed so far and quantities commonly associated with it (Sections 2 and 4): measures of predictive performance and data dispersion. We identify how a model could be used to estimate those quantities but emphasise the limitations of that approach, underlining the key role to be played by externally grounded evaluation (Section 3.7).
Since we typically do not know the true data-generating process, Perain (fi.m [T), a core approximation is to use a model over new data, Pa (yim |r'), as a proxy. On top of
We consider assessing a predictive distribution, pa (2), either using single ground-truth value of z or using a reference
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Rethinking Alcatoric and Epistemic Uncertainty
Data, reference distribution and prodictive distributin
Uncortainty and evaluation with loss ((a,2) (a- 2)2
Uncertainty and evaluztion with
loss t(a. max(0. 2) (a 2)2
1.0
training data
1
reference mean predictive mean predictive uncertainty
Volue
Viloe
0.2
0
0.5
dispersion expected score
0.0
0.0
-
0
-5
0
Figure Model-based uncertainty, predictive performance and data dispersion are related but distinct quantities. Here show model's 's predictive distribution over an oulput, z R, corresponding to an input, x E R; we also show reference distribution serving as a source of evaluation data. Using the model alone, we compute measure of predictive uncertainty using a loss function, E(a, 2), where a is an action. This differs from the data dispersion, which describes the reference distribution. Connecting the two distributions, the expected score (lower is better) measures the predictive performance of the model judged using data from the reference distribution.
distribution over z. If we have a single 2, we can evaluate (z) using a proper scoring rule (Savage, 1971) of the form
Proposition 1 (Berger, 1985) Let F be a quantity of interest, and let f(0) represent subjective beliefs over F, derived from pushforward of distribution model parameters O ~ Pa (f). Under quadratic estimation loss, (1 0) (1 - [())², the Bayes estimator of F is = (pp
spnn 2) = Maa,, 2)
where a is defined as in Equation 2 (Dawid, 1998). This scoring rule measures the loss incurred by the Bayes-optimal action under (z) when the ground truth is 2.
Proposition 2 Assume (n(2) Ep., (0) (p)(((())) is model intended to directly approximate (=) Then the model's 's predictive uncertainty, h[p((2)], is a Bayes estimator of Ep..att)(s(Pn: =)], the expected loss from acting Bayes- optimally under (z) when z is in fact drawn from Peval(z).
If we instead have a reference distribution, Peval we can evaluate pa (z) using discrepancy function (Dawid, 1998):
Peva) = Ep (=) [[(aa, =) - El (( =)]
(4)
F pewa [s(pn: (= z)] (PPeaa())) [2)]
(5)
Proof Applying Proposition I with F = Epp (2) [c]]ss(p=> 2)] I
and (0) = F.p. (=10) (=10)) (s(p.; =)] gives vy" h()n(=))
expected score
data dispersion
Assume the same model as in Proposi- tion 2. Then the expected conditional predictive uncertainty, Eo, [p,[([])]], is Bayes estimator of h h(ppewa(z))) the dispersion in data drawn from reference distribution.
Proposition 3
where d'val is defined analogously to 0, but with (2) as the predictive distribution. Equation 4 highlights that the discrepancy is the expected excess loss from acting based on (=( rather than (=vat(=) when (z) repre- sents the ground truth, which is linked to the idea of regret (Szepesvári, 2010). Equation 5 shows that the discrepancy also corresponds the expected score of pr, (z), which mea- sures predictive performance (lower is better), minus the uncertainty measure from Section 5.1 applied to Peval (z), which measures the statistical dispersion in evaluation data drawn from the reference distribution. This relationship gen- eralises classic decompositions from statistics (Rice, 2007) and information theory (Cover & Thomas, 2005), as shown in Examples 3 and 4. Notably the expected score can be seen a form of frequentist risk for an "outer" decision problem; another form also averages over the training data.
Proof Applying Proposition 1 with F = [2]]] and (()) = h/p.. [0) gives 1)* = -Epp.. (=(0))]
Three things are important to note in regard to Propositions 2 and 3. First, while they both suppose that (z) is designed to directly approximate Peval this need not always be the case. Because (eva(()) might itself be imperfect, our evaluation might just be serving to provide a rough signal of the model's predictive performance (Section 3.7). Second, they both assume quadratic estimation loss, which might not reflect our preferences. An alternative loss would result in different Bayes estimators. For example, an absolute loss would lead us to use medians rather than expectations (Berger, 1985). Third, they present estimators that are de- rived from subjective models and that, as a result, have no accuracy guarantee in the general case (Section 3.5).
Because past work has sometimes drawn connections be- tween model-based uncertainty, predictive performance and data dispersion, we now explain how such connections can come about and why in the general case they should not be taken to hold. In particular, if we assume specific model setup and estimation loss then we can derive Bayes estima- tors that generalise the marginal predictive entropy and the expected conditional predictive entropy in Equation 1, but these assumptions will typically not apply and, even if they do apply, the estimators can be highly inaccurate.
We therefore stress that model-based uncertainty should be considered separate from predictive performance and data dispersion, as shown in Figure (see Appendix C for details). 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.
7

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Rethinking Alcatoric and Epistemic Uncertainty
EIG enrors for discrete data
Prodictive distributions for ciscrete data
0.00?
0.6
n= 10
18
enor
0.001
0.5
10
si C.4-
10
0.000
10a
10
10a
10
10
90 Ran Random-numbber berb seed
50
Number of examples,
EIG errors for continucus data
Precictive distributions for continuous data
Squared (nats'
0.6
5
Pia(z)
(()(2)
0.4
Pip(z)
Pip(2)
Ps(2))
0.0
109
10
10
10
Number of examales,
Random- number seed
Figure 4 BALD's correspondence with infinite-step predictive information gain (a model's long-run reduction in predictive entropy) can be weak, such that it is often better thought of as an estimator of "true" one-step expected information gain in the model parameters. Here show the behaviour of conjugate models trained discrete (top) and continuous data (bottom). For (1, 10, 100, 1000) we computed two estimation errors: e; (EIGa IG, (V ))²² and е사 (EIG, EIG
show these errors (left; mean over 50 random-numher seeds) along with the evolution of the predictive distribution (right). Both estimation errors are due to inaccurately simulating future data, an issue that resolves as n increases and the predictive distribution converges to the mata-generating process.
5.5 Popular information- quantities are best understood as imperfect estimators
parameters, EIG,, from observing is Bayes estimator of the infinite-step predictive information gain, IG= (
Finally we return to the information- -theorettccccc quantities in Equation which have been central to many existing dis- cussions of aleatoric and epistemic uncertainty. Principally we highlight that BALD (that is, EIGg, the expected in- formation gain in set of stochastic model parameters, f) can be understood as an estimator of two separate unknown quantities: the infinite-step information gain in the model predictions and "true" one-step expected information gain in the model parameters. We also suggest the relative mag- nitudes of the two corresponding estimation errors might help explain BALD's utility as Rata-acquisition objective.
In Figure 4 we demonstrate that the approximation FIG, R IG from Proposition 5 can be coarse. These results were produced with extremely simple setups within which we can perform exact inference and we are sure to recover the true data- gentratnng process in the limit of infinite data (see Appendix C for details). We therefore know that the estimation error is due to failure of the model to accurately simulate future data, which in turn is due to being finite.
This behaviour appears to align with existing results (with the caveat that past studies did not match the assumptions of Propositions and 5). Figure 2 in Bickford Smith et al (2024) and Figure 5 in Wimmer et al (2023) show small-n es- timates of FIG, that differ substantially from the changes in predictive entropy that actually occurred in practice. Those results even suggest it would have been more accurate to assume (y):o))) = H[p.. (z)], with H [p,,. (x]vi:c)) = 0, than to estimate it using FIGg- Meanwhile Mucsányi et al (2024) and Valdenegro- & Saromo-Mori (2022) em- phasised the "'ettanleement" of aleatoric- and epistemic- uncertainty estimators, which can be understood as the esti- mators themselves having an "epistemic" component- or, in our terminology. being inaccurate finite-data estimators.
First we show that, under assumptions the data and model that result in convergence to single setting of 0 (Doob, 1949; Freedman, 1963; 1965), BALD can be understood as an estimator of the infinite-step predictive information gain,
IG& (vt.o)) = H(p, (z)] - (1lvtto)))
measuring the reduction in the model's predictive entropy from a Bayesian update on infinite new data, vi..o.
(0) be a com- bination of data sequence and generative model that yield pn (0 + _0_ (0) as m + sc. Then the expected con- ditional predictive entropy, (o) (2]00)) is Bayes estimator of Hpp (= )), the marginal predictive entropy after Bayesian update infinite new data, Viszo
Proposition 4 Let vi.. and Pro (yl0)))) (2(00(((2))))))) (z)0)
This raises the question of why BALD has proven practi- cally useful as a data-acquisition objective in active learning (Gal et al, 2017: Houlsby al, 2011: 1; Osband et al, 2023). our intuition is that BALD's utility stems from its correspon- dence with infinite-step predictive information gain (that is, long-run reduction in a model's predictive entropy) and
Proposition 5 Assume the data and model from Proposi- tion 4. Then the expected information gain in the model
8

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Rethinking Alcatoric and Epistemic Uncertainty
Curated MNIST
Coarse ImageNeet
interest is well-motivated objective for data acquisition, assuming the action of interest is a probabilistic prediction of and we use regative-log-likelltod loss function (Example 2), but it also crucially depends on the model's ability to simulate future data, and it assumes a Bayesian update that might not match the true update (Section 5.3).
Test reg. log liolhhood
random entropy
1.0
1.5
BALD
1.0
0.5
0.5
100
200
300
500
1000
Number of labels
Number of labels
6 Conclusion
Figure 5 BALD outperforms predictive entropy as data- acquisition objective in active learning. even though BALD tends to be worse estimator of long-run predictive information gain in the setups studied. These results were produced using experimen- tal setups descrihed in Bickford Smith (2023; 2024).
We have argued that the deeattric-epistemitn view uncer- tainty does not serve machine-leaming researchers' needs: its lack of expressive capacity has led to conceptual over- loading and confusion. To address this we have presented a decision-theoretie view that unifies many concepts of inter- est to researchers. This provides clarity on five key points:
we consider setups in which the current predictive entropy is a better estimator than BALD, then we would expect that using the predictive entropy as a data-acquisition objec- tive would lead to better predictive performance in active learning. Yet this is not what we see in practice (Figure 5).
(a) Measures of predictive uncertainty need not be an arbi- trary choice but can instead be derived from decision of interest with an associated loss function.
(b) If we explicitly account for how training data is gener- ated, we can identify decomposition of uncertainty into reducible and irreducible components for any method that maps from data to a predictive distribution.
We suggest another perspective on BALD might address this question. In particular it can be understood (given as- sumptions on the data and model) as an estimator of the true one-step expected information gain in the model parameters,
(c) practice we can typically only produce an approx- imate notion of expected uncertainty reduction that relies on a proxy for the true data getta-eenrrting process and possibly also an approximation of model updating.
EIG = H[pm (0)) - Eperan (=) (()=)())))]]]
where the expectation is over observations from the true data-gen tatt-ggenn process, not model-simulated observations.
(d) Predictive uncertainty should be assumed to be sepa- rate from measures of predictive performance and data dispersion, and externally grounded evaluation is there- fore key for building trust in a model's predictions.
Proposition Lei = = +1+ Assume yi.m are independent and identically distributed, with y ~ Perain (y), and assume (2)())))) (2]0)] is a model intended to directly approximate Ptrain Then the expected information gain in the model parameters, FIGo, from observing z is a Bayes estimator of the true one-step expected information gain, EIG where the expectation is with respect to Ptesi((2).
pz(2) Ep,,(0) [p
(e) BALD does not directly measure long-run reducible predictive uncertainty but rather estimates it, and the associated estimation error can be large.
We believe our decision-theoretic view should be used in place of the predominant aleatoric-epistemt view as the field moves forward. Our hope is that it will support more productive discourse and methodological development.
Returning to the same experimental setup as before, we find (Figure 4) that the approximation EIGs R EIG is more accurate than EIG, = (v..)).) In other words, BALD more closely tracks short-run changes in parameter uncer- tainty than it does long-run changes in predictive uncertainty. We do not claim this is general result that will hold in all settings, but it is consistent with BALD being useful as data-acquisition objective. The acquiiiqqin horizons in active learning are typically very short, so it is the short-run notion of information gain that matters, not the asymptotic notion. And while targeting predictions rather than parame- ters can be even more effective (Bickford Smith et al, 2023; 2024), maximising short-run parameter information gain is still often preferable over random acquisition.
We also urge care in using uncertainty and related quantities in practice. A crucial recurring point in this work is that true quantities of interest almost always have be approximated in the real world. Ignoring approximation errors can lead to dangerous assumptions and ineffective methods.
Finally we suggest that future methods research should be guided by practical utility concrete decision problems, such as data acquisition and model selection, rather than more abstract notions of approximation accuracy. Combined with stronger conceptual foundations, renewed emphasis on real-world performance could accelerate progress in the use of probabilistic reasoning in machine learning.
One takeaway here is that information- -iteortion-thhoretiic quantities should not be confused with the quantities they estimate. Another is that expected information gain in a variable of

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Rethinking Alcatoric and Epistemic Uncertainty
Impact statement
Cover & Thomas (2005). Elements of Information Theory. John Wiley and Sons.
This paper presents work whose goal to advance the field of machine learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here.
Dawid (1998). Coherent measures of discrepancy, uncer- tainty and dependence, with applications Bayesian pre- dictive experimental design. Technical report, University College London.
Acknowledgements
DeGroot (1962). Uncertainty, information, and sequential experiments. Annals of Mathematical Statistics.
We thank Philip Dawid for sharing technical report with us. We are also grateful to Andreas Kirsch, Mike Osborne and the anonymous reviewers of this paper for useful discus- sions and feedback. Freddie Bickford Smith is supported by the EPSRC Centre for Doctoral Training in Autonomous Intelligent Machines and Systems (EP/L015597//) Tom Rainforth is supported by EPSRC grant EP/Y037200/1.
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Fong & Holmes (2020). On the marginal likelihood and cross-validation. Biometrika.
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A Examples
Example 3 Evaluating (=) against a reference distribution, (z), with A = 2 and E(a, 2) = (a - =)² corresponds to measuring the predictive performance of Pa(z) using mean squared error, measuring the discrepancy between (z(2) and (2) using squared bias and measuring dispersion in Poval (2) using variance.
Proof Starting from Equation 4 with optimal actions an = [x)((=)[2] = MA and doval [Epevi())(2] = Meval from Example I, the discrepancy between pm (2) and Peval (z) is
d(pm Pevat) (x) (un - =)2 (Meval - -2) = (Jon Mevai)?
Now starting from Equation 5, it can also be written
dp Peval) = (x)) [Mm =) V Peval (s)a
Equating these two expressions for the discrepancy leads to standard bias-variance decomposition:
(x) (Pm 2] (Jin Meval)
V (=)
mean squared error
squared bias
variance
where the mean squared error measures predictive performance and the variance measures data dispersion.
Example 4 Evaluating Pal (2( against a reference distribution, (=)a(z), with A P(2) and l(a,z) log a(z) corresponds measuring the predictive performance of pn(z) using cross entropy, measuring discrepancy between (n(
and Povat (2) using Kullback-Leibe divergence and measuring dispersion in (2)at(2) using Shannon entropy.
Proof Starting from Equation 4 with optimal actions at PP((z) and deval Peval (Example 2), the discrepancy between (z) and (z is
d(pn: Peval) = E (epca (z) log + Pevma = KL (=) Pm(=))
Now starting from Equation 5, can also be written
d(pn Peval) - (x) [log (ogg)(())))))))))))))))()))))( (2)] H Peval())) H [Peval Pl H[pevat (2))
Equating these expressions for the discrepancy leads standard information-theoreticc decomposition:
H[Peval (2) (x)) (L[Peval [p)())]
Peval (2)
cross entropy
KT. divergence
entropy
where the cross entropy measures predictive performance and the entropy measures data dispersion.
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Rethinking Alcatoric and Epistemic Uncertainty
B Proofs of Propositions 4 to 6
Let Vim and pm (yy (0. (2)0p.)) be a combination of data sequence and generative model that yield Pa (0lliim + 2e (0) as m + Do. Then the expected conditional predictive entropy. Ep,, (2|0)] is Bayes estimator of (zpp (=1:)):)))) the marginal predictive entropy after a Bayesian update on infinite new data, Visc.
Proposition
Proof Since V0000 recovers a single setting of 0, reasoning about equivalent reasoning about yi following the argument presented in Fong al (2023). This allows apply Proposition 1 with F H[pn(21W5oo))) and f()) (=))), which gives 1e Ep. (0) [ (=|0(]
Proposition Assume the data and model from Proposition 4. Then the expected information gain in the model parameters, EIG,, from observing is Bayes estimator of the infinite-step predictive information gain, [V:o))))).
Proof The information gain to be estimated, (Vi:oo)) defined as the reduction in the model's 's predictive entropy from Bayesian update on infinite new data, yi:
IG, (Vioo)) = ((), HIp,, (2lv)oo)))
Combining the known Hzpn(z)) with the Bayes estimator of H(,,, (=lBi.)))) from Proposition 4 gives
EIGg Hpp (=)] Ep.. [Hpp)
as Bayes estimator of [y:ooo))))
Proposition Let 2 V2+1- Assume Yven are independent and identically distributed, with yi ~ Perain (y), and assume (=)= = (0) (z|0)] model intended directly approximate Prrain(z). Then the expected information gain in the model parameters, EIGg, from observing z is a Bayes estimator of the true one-step expected information gain, EIG where the expectation with respect to Ptrai((2).
Proof The expected information gain to be estimated, EIGH is defined as the reduction in the model's parameter entropy from Bayesian update on new data, 2, where is drawn from Perain (2):
EIG = Hl), Eperan (0|=]]
The second term here can be estimated by applying Proposition 1 with Eprain (2) Hp (0) and f(0) = (ele) (0)= The Bayes estimator that results from this, ry" PP H(pp.( (0) 2)]] can be combined with the known current entropy, H[p, (0)), produce
EIGg H pr [0)] (x) H[p,. 2]]
as Bayes estimator of EIGTT
14

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Rethinking Alcatoric and Epistemic Uncertainty
C Implementation details
Code to generate Figures and 4 is available at gi thub. http:///wwiiittooooattaaetaaaaar
C.1
Figure 3
We consider predicting an output, 2 E IR, corresponding to an input, x R. The training data, Yi.. comprises n input-label pairs: y1 (-2, tanh(-2)) and 1/2 (2, tanh(2)). We use this compute a Gaussian process predictive posterior, (=] pl=|a, y)m,), based generative model comprising Gaussian likelihood function, p(=|2, 0) Normal(z (z)),, )²), where 0.1, and Gaussian proccsss prior, ~ GP k), where k(z, x') exp(- - x²) /2). We compare this with reference distribution, Peval (=|2) Normal(y) tanh(x)), o²). Using pm (z|ar) and Peval (z|2), we compute three quantities for E [-8, 8]: the predictive uncertainty, h1pp (2 [2]r)]; the data dispersion, hlPeval (=] and the expected score, E (plm)())e) (p --} We do this for two loss functions: E(a, 2) = (a- =)² and f(a, 2) = max (0, 2) (a)))-(--)))) 2)?
C.2 Figure 4
We consider two cases of predicting = = + a discrete case and a continuous case. the discrete case we have E {0, 1), data generated from Perain Bernoulli (1/7) = 0.5), and the Bayesian generative model is
p(y. n/a, 3) = p(y p(yln)p(zla,, 9)
ply ()) Bernouli(yyn))
p(rla, 9) Beta(n)a, 8).
In the continuous case we have y E R, data generated from Ptrain (y) Normal(y|ı4 1, 1), and the Bayesian generative model is
,la 8, K, m) plylu, A)p(u/m, ())(aa,,
pl A) = Normal(y|u,, x 1
p(u/m, K, A) = Normal (x:))
p(laa, 3) = Gamma (Ala, 9).
In both cases we can compute exact Bayesian posteriors (Murphy, 2022), and there some n for which Pn(2) Perain (z) (Doob, 1949; Freedman, 1963; 1965). For each case we sample four datasets, yim. with y/s ~ Perain (y) and (1, 10, 100, 1000). On each dataset we compute the Bayesian parameter posterior, (0) = p(0lg where 0 (a, $) or (a, B, K, m), and then compute two quadratic estimation errors. The first is the error from approximating the "true" expected information gain in 0 with the standard, model-based expected information gain in A:
Vea-
FIG, EIG
(Hlp,. (0) E pm. (0|= (H(p.(0)) E E (2) (Hp, (0|=)))
Ep (=) [H (0]=)]] E (B]=(
The second is the error from approximating the infinite-step information gain in with the expected information gain in 0:
FIG, - -Gx
(H]p,. - Bp., (0) [H[pm (=]0)]) (H)] H (z))
H[prrain (2)] - Ep. (0) (p, (z]0)]]-
We average over 50 repeats with different random-number seeds.
15


Reviews:
[{"review_id": "bn2vbHvzIW", "reviewer": "Reviewer_GCxo", "summary": "This paper critiques the concepts of aleatoric and epistemic uncertainty in machine learning predictions, identifying inconsistencies and limitations in existing discussions. The authors argue that the traditional aleatoric-epistemic framework is insufficient to capture all relevant aspects of uncertainty in predictive modeling. \n\nTo address these shortcomings, the authors propose a more rigorous framework for model uncertainty based on expected loss values. This approach aims to clarify the concerns discussed and address various aspects of uncertainty more comprehensively, while also capturing existing notions of uncertainty.\n\nThe authors provide experimental insights into the BALD score, a popular information-theoretic quantity. Their findings demonstrate that this metric can sometimes measure something substantially different from what it is commonly perceived to quantify (i.e. the infinite-step predictive information gain). \n\nOverall, the paper challenges existing paradigms in uncertainty quantification for machine learning models and proposes a new perspective based on decision theory to address the identified shortcomings.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review.\n\nWe appreciate your positive feedback on a number of points:\n\n1. Careful writing\n2. Theoretical rigour\n3. Convincing empirical evidence\n4. Clear coverage of prior work\n5. Potential benefit to the community\n\nWe are also grateful that you highlighted some ways to improve the paper.\n\n### **Paper style**\n\n> The paper's focus on conceptual discussion… could potentially position it as more of a position paper… However, the message it conveys is likely to be beneficial for the broader machine learning community.\n> \n\nWe strongly support your emphasis on the benefit the paper could bring to the community, which we believe is ultimately what matters. While we agree that the paper would not have been totally out of place in ICML’s position-paper track, we ultimately felt that the objective and technically precise nature of its core contributions made it a better fit for the standard track, even if it does not fit the common mould within the field, as you note.\n\n### **Future work**\n\n> It might be valuable to consider including in the conclusion a brief exploration of any key open questions or potential research directions that this discussion brings to light.\n> \n\nGreat idea. We will happily add this.\n\nOne exciting direction is deriving new problem-driven data-acquisition objectives using the decision-theoretic approach we demonstrate. In particular, we think our work lays foundations for developing loss-calibrated active-learning methods, which are underappreciated in the literature.\n\nA key open question is how reducible uncertainties should best be estimated. Propositions 1-4 reveal existing estimators to be optimal only if we use a quadratic estimation loss, with further work required to establish optimal estimation strategies in other contexts.\n\n### **Experimental setup**\n\n> I would have preferred if more explanation of the setup was given in the main body, as I had to consult the appendix to be able to understand Figure 3.\n> \n\nThis is useful feedback—thanks. We will happily revisit it.\n\n### **Related paper**\n\n> The paper's discussion of aleatoric and epistemic uncertainty perspectives shares some themes with this ICLR 2025 paper: https://arxiv.org/abs/2412.18808. In particular, Section C.2 of that paper appears relevant to the decision-theoretic perspective in Section 5.1 here.\n> \n\nThanks for drawing our attention to this work. We agree that it is relevant and will cite it.\n\n### **Wording of propositions and proofs**\n\n> I felt the proofs and statements of propositions 2-5 were a bit too brief, sometimes making the discussion hard to follow. It would be useful to concretely restate the quantities referenced in the proposition statements in mathematical terms.\n> \n\n> A number of the proofs (Propositions 2, 3, 5) use the phrase \"...follows from the same working as in...\" perhaps a more standard word choice would be \"reasoning\" or \"argument\" rather than \"working\"?\n> \n\nWe appreciate the feedback. We will update the wording and mathematical statement of all propositions and proofs with a view to making things easier to follow and using standard language.\n\n### **Expected information gain**\n\n> Perhaps I missed it, is $EIG_{\\theta}$ ever defined in the main body?\n> \n\nThanks for flagging this. It is the expected information gain in $\\theta$, where $\\theta$ represents stochastic model parameters, which is the same thing as the BALD score we refer to throughout the paper. We will take care to clarify this.", "claims_and_evidence": "The paper is carefully written, and the theoretical justifications for the paper's argument are rigorous and make sense. \nThe experimental results on the performance of the BALD score offer convincing evidence for the authors' claims about the potential misinterpretation of common information-theoretic quantities. The connection between these experiments and Proposition 5 strengthens the paper's argument.", "methods_and_evaluation_criteria": "The paper's experiments use simple toy models that clearly distill the paper's points and are easy to understand. The BALD score is a popular measure of predictive uncertainty, making it a relevant focus for the experiments.", "theoretical_claims": "While there are no substantial proofs to verify, there are a few small propositions whose proofs seem sound, and overall the analysis of different perspectives on uncertainty in the literature and their connections to each other is clearly stated and theoretically grounded. Please see my comments below about the clarity of the proofs and their associated proposition statements.", "experimental_designs_or_analyses": "I liked the experiments, especially in the context of the result of Proposition 5. However, I would have preferred if more explanation of the setup was given in the main body, as I had to consult the appendix to be able to understand Figure 3.", "relation_to_broader_scientific_literature": "A large number of works have studied aleatoric/epistemic uncertainty and related notions of uncertainty quantification. This paper both adds to the conversation and connects and clarifies the different perspectives prior works take on this topic, making it a valuable addition to the body of research.", "essential_references_not_discussed": "The paper's discussion of aleatoric and epistemic uncertainty perspectives shares some themes with this ICLR 2025 paper: https://arxiv.org/abs/2412.18808. In particular, Section C.2 of that paper appears relevant to the decision-theoretic perspective in Section 5.1 here.", "other_strengths_and_weaknesses": "__Strengths:__\nThe paper is well-written and provides a valuable clarifying perspective on the commonly held views of epistemic and aleatoric uncertainty in machine learning. It also offers a more grounded alternative approach to understanding and quantifying predictive uncertainty.\n\n__Weaknesses:__\nThe paper's focus on conceptual discussion, coupled with limited technical results and experiments confined to synthetic data, could potentially position it as more of a position paper, making it better suited to a different venue. However, the message it conveys is likely to be beneficial for the broader machine learning community.", "other_comments_or_suggestions": "- I felt the proofs and statements of propositions 2-5 were a bit too brief, sometimes making the discussion hard to follow. It would be useful to concretely restate the quantities referenced in the proposition statements in mathematical terms. \n- Perhaps I missed it, is $EIG_{\\theta}$ ever defined in the main body?\n- It might be valuable to consider including in the conclusion a brief exploration of any key open questions or potential research directions that this discussion brings to light.\n- A number of the proofs (Propositions 2, 3, 5) use the phrase \"...follows from the same working as in...\" perhaps a more standard word choice would be \"reasoning\" or \"argument\" rather than \"working\"?", "questions_for_authors": "I have no specific questions, but please address any comments in my review that may indicate a misunderstanding of the paper's key points.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "eHyfeRVNt1", "reviewer": "Reviewer_MLbH", "summary": "The paper examines the concepts of aleatoric and epistemic uncertainty. It highlights inconsistencies in existing discussions of these concepts, attributing them to the limited expressiveness of the aleatoric-epistemic framework in capturing the diverse uncertainty quantities. To address this, the authors propose a decision-theoretic perspective on prediction, deriving formal definitions of model-based uncertainty and statistical dispersion in data. This new framework aims to provide a clearer foundation for future discourse in the field. Additionally, the paper investigates popular information-theoretic quantities, revealing their limitations as estimators of intended metrics while demonstrating their potential utility in guiding data acquisition processes.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review.\n\nYour critical feedback is much appreciated. We hope our responses and new demonstrative plots help alleviate your concerns.\n\n### **Loss functions and uncertainty**\n\n> \"Given this, we argue that a principled notion of predictive uncertainty cannot be detached from this loss\"… I remain unable to fully grasp the rationale and necessity for adopting this viewpoint as the foundation.\n> \n\nThis is useful feedback—thanks. We will clarify our case.\n\nOur core argument is that notions of uncertainty should be derived from the task we ultimately wish to perform, rather than being chosen in abstract.  The loss is the measure of success on our final task, and our analysis shows that we can then directly derive rigorous notions of uncertainty from this loss using a framework of rational actions, namely minimising the expected loss.\n\nThe necessity of adopting this viewpoint as a foundation then manifests in a variety of ways, such as ensuring uncertainties are consistent with rational actions, having an end-goal-driven approach so that uncertainties reflect what we actually care about, and allowing data acquisition to be performed in a meaningful way that targets the problem of interest.\n\n### **Practical implications**\n\n> The practical applicability of this work remains ambiguous. What tangible benefits might arise…?\n> \n\nThanks for highlighting this. We will happily add more discussion on the practical benefits. We also plan to add a case study that helps show the implications of our work in a practical setting (see \"Case study\" in our response to Reviewer Atnu).\n\nOne takeaway is to stop using off-the-shelf uncertainty measures and instead derive the one that will be most useful in a given decision problem (eg, data acquisition). We show how to do this.\n\nAnother is that we cannot think simply in terms of \"Analyzing the sources of uncertainty\" in the way the current literature does, with such decompositions themselves being highly subjective in practice. It is hard to characterise the application value of the work in the context of the aleatoric-epistemic viewpoint, as we are ultimately arguing that this is fundamentally flawed.\n\nOther takeaways relate to estimation of uncertainty in practice. Part of this is promoting caution in using common estimators; part is encouraging alternatives. For example, standard estimators can be so inaccurate that we are better off directly using a numerical estimate based on our prior knowledge (see \"Imperfect estimators\" in our response to Reviewer 43ee).\n\n### **Concrete algorithms and empirical evidence**\n\n> If the authors aim to substantiate the superiority of their decision-theoretic perspective over conventional approaches, the development of concrete algorithms accompanied by comprehensive experimentation is indispensable\n\n> the experimental section of the paper is severely lacking, with insufficient empirical validation to support the claims\n\nWe feel that this misses the primary contribution of our work, which is not algorithmic but instead in showing the inconsistencies and flaws in the current foundations on which uncertainty-quantification algorithms are usually based, as well as providing a more rigorous and principled foundation.  As noted by other reviewers, this makes it an unusual paper, but we do not believe every paper should be about proposing a new algorithm.\n\nBy extension, it is not clear what empirical experimentation it would actually make sense to add to the paper, beyond confirming the one empirical claim we make about how the BALD estimator should be understood. We do not agree that there claims being made that are lacking empirical validation, but if there are any experiments you think are missing we will do our best to add them.\n\nTo try and provide more clarity on how the work can be used for concrete algorithms, we have also added a new case study in our code base (see \"Case study\" in our response to Reviewer Atnu). This shows how using a non-standard loss can significantly impact how the uncertainty should be measured.\n\n### **Title**\n\n> This title lacks distinctiveness\n> \n\nWe believe the broad span and foundational nature of our work actually requires a title like the one we use. Our paper brings together ideas from much of the ongoing discussions of aleatoric and epistemic uncertainty in the literature and aims to change the way people think about these concepts themselves.\n\n### **Key concepts**\n\n> Is the author intending for the concepts in this section to be treated as assumptions that must be satisfied, or as prerequisite knowledge that readers are expected to possess?\n> \n\nThe concepts in the section are intended to be didactic rather than assumptions we are making.  The section is also covering some of our problem formulation and synthesising relevant facts from past work, but its core is to introduce the key ideas that underpin our formulations.\n\n### **Citations**\n\nWe will revisit all of the citations you raise.", "claims_and_evidence": "Yes. The authors provide proofs for the examples and propositions. Nevertheless, as I noted in the Weaknesses, I find certain arguments and motivations to be insufficiently persuasive.", "methods_and_evaluation_criteria": "No.", "theoretical_claims": "Yes. The article incorporates a substantial number of mathematical expressions; however, it does not involve highly complex derivation or proof processes.", "experimental_designs_or_analyses": "Yes. I think the experiments are not sufficient. See Weaknesses for details.", "relation_to_broader_scientific_literature": "This study examines the limitations of the traditional aleatoric-epistemic framework and introduces a distinctive decision-theoretic perspective. It further highlights that widely used information-theoretic quantities may serve as inadequate estimators of the constructs they are commonly assumed to measure.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:\n1. The existing aleatoric-epistemic framework indeed suffers from conceptual ambiguity, coupled with evident terminological misuse in the design of current methodologies. This constitutes a compelling and worthwhile topic for discussion within the field of uncertainty estimation.\n\n2. I agree with the opinions in Section 5.4. I think the analysis really provides some interesting insights.\n\nWeakness:\n1. Some of the article’s arguments are somewhat vague and difficult to comprehend, or not convincing enough. For instance:\n\nLine 126: \"This assumption is equivalent to stipulating basic axioms of rationality (Ramsey, 1926; von Neumann & Morgenstern, 1947).\"\nThe connection between the cited references and the preceding context remains unclear. Furthermore, the specific mechanism by which this equivalence is ensured is not elucidated.\n\nLine 212: \"Given this, we argue that a principled notion of predictive uncertainty cannot be detached from this loss\".\nI find this perspective both difficult to comprehend and lacking in credibility. Even after thoroughly reviewing the entire article, I remain unable to fully grasp the rationale and necessity for adopting this viewpoint as the foundation. The authors may need to provide a more comprehensive explanation to justify this position and clarify its significance in the context of the study.\n\n2. I find the presentation in Section 3: Key Concepts confusing. Is the author intending for the concepts in this section to be treated as assumptions that must be satisfied, or as prerequisite knowledge that readers are expected to possess?\n\n3. The practical applicability of this work remains ambiguous. What tangible benefits might arise from replacing the traditional aleatoric-epistemic view with the proposed decision-theoretic perspective at the application level? Could it facilitate more accurate analysis of uncertainty sources? Analyzing the sources of uncertainty is a primary developmental goal of the aleatoric-epistemic framework, and the application value of this direction should be explored in that context. However, the experimental section of the paper is severely lacking, with insufficient empirical validation to support the claims.\n\n4. As I read through Section 5.3 and realized that the authors’ primary contributions concluded at that point, I experienced a slight sense of surprise and disappointment. I had anticipated the emergence of a concrete uncertainty decoupling algorithm guided by the authors’ novel perspective. It is possible that this reflects a limitation in my own understanding, but I currently hold the view that, while this work offers ample and innovative theoretical analysis, it remains incomplete in its present form. I recognize one of the authors’ assertions: that a unique decomposition of uncertainty into reducible and irreducible components is often unattainable. However, I contend that this should not serve as a justification for forgoing the exploration of specific algorithms. Under the traditional aleatoric-epistemic framework, researchers have pursued a variety of uncertainty decoupling algorithms, even if these are not always precise. If the authors aim to substantiate the superiority of their decision-theoretic perspective over conventional approaches, the development of concrete algorithms accompanied by comprehensive experimentation is indispensable.\n\n5. Inappropriate or Inaccurate Literature Citations: (1) Page 1, Line 14: The citation (Kendall & Gal, 2017) merely introduces a variance-based measure for epistemic uncertainty within the related work section. Utilizing it as an example in this context appears somewhat inappropriate. (2) Page 1, Line 18: The reference (van Amersfoort et al., 2020) does not employ distance-based measures to quantify epistemic uncertainty. The study explicitly states that it does not engage in the decoupled analysis of aleatoric and epistemic uncertainty, rendering this citation potentially misleading.\n\n6. Inappropriate Submission Title: The author should focus more precisely on the unique contributions of the paper when selecting the title, rather than adopting a generic phrase such as \"Rethinking Aleatoric and Epistemic Uncertainty.\" This title lacks distinctiveness, as the reconsideration of aleatoric and epistemic uncertainty has been an ongoing topic of discussion for an extended period.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "See Weaknesses", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OCeD5zlHTa", "reviewer": "Reviewer_43ee", "summary": "This paper argues that the current view on the decomposition of uncertainty into (reducible) epistemic and (non-reducible) aleatoric uncertainty is not only insufficient but also inappropriate from the theoretical viewpoint. They argue that the whole notion of predictive uncertainty should be grounded in a loss function over actions, entering into an argument of rational behavior.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review.\n\nWe are happy to see you highlight a number of positive aspects of the paper:\n\n1. Clear writing\n2. Reasonable theory\n3. Useful experimental results\n4. Clear contextualisation within the literature\n5. Refreshing style of contribution\n\nWe also recognise there are some things you think should be improved.\n\n### **Clarity and space**\n\n> Even though I find the whole work compelling and quite well formalized, I am inclined to say that the paper somewhat falls short in its goal to bring clarity. The paper is excellently written, but it is extremely dense and technical with a lot of notation, which is understandable and necessary, but it makes the paper quite difficult to read\n> \n\nWe agree that the paper is necessarily idea-dense in order to communicate the full picture with appropriate nuance. Happily we think the paper’s clarity could quite easily be improved using the extra page allowed in the camera-ready paper, some careful rearrangement of the content, and extra efforts to highlight the key takeaways.\n\nA key improvement we can make is to provide visual examples of the points we are making (see “Case study” in our response to Reviewer Atnu). Another thing we will try is presenting Examples A1-A3 and B1-B3 together: then the logic of the main technical content will be uninterrupted and the progression across the examples will be clearer. Finally we will make more use of the appendix to provide more verbose explanations of the trickier concepts covered in the paper.\n\n### **Paper style**\n\n> This paper is really different from standard papers… Personally I find this very refreshing, but it is problematic even only fairly review this paper into the new ICML review scheme…\n> \n\nIt is great to hear that you find our contribution refreshing. We believe it could provide real value to the community, especially after we update for improved clarity based on your feedback. As noted above, we do feel there should be sufficient space to convey our message more clearly.\n\n### **Imperfect estimators**\n\n> I am not sure what to think of this type of issue that \"... is only an estimator of [the true uncertainty quantity]\".\n> \n\nYou are right: estimation is what we have to do in practice. Two points are worth highlighting.\n\nFirst, past work has often discussed common estimators as if they are the quantities they are in fact only approximating. We think emphasising the potential inaccuracy of the estimators has value in resolving misconceptions and supporting more clear-eyed use of common quantities.\n\nSecond, by establishing that these are only estimators, our work reveals the scope for alternative estimators that might be superior. For example, it might be clear that the infinite-step irreducible predictive uncertainty should be effectively zero, which a practitioner can use directly; Figure 2 in https://arxiv.org/abs/2404.17249 shows this can be better than using standard estimators. From a more theoretical perspective, an important implication of Propositions 1-4 is that the standard uncertainty estimators we consider are only optimal if we use a quadratic estimation loss, with other losses yielding different optimal estimators. Our exposition provides a template for deriving better estimators.\n\n### **Unique uncertainty decomposition**\n\n> I don't see what is wrong about a decomposition that still involves the expected value of those additional data points.\n> \n\nThe key point here is that stochasticity in the data makes the decomposition depend on the data-generating process: any notion of irreducible and reducible uncertainty becomes conditional on it. This means we cannot talk about *the* decomposition into irreducible and reducible uncertainty: there are effectively endless possible decompositions because the data-generating process itself depends on design decisions we make (eg, how inputs are sampled or selected). Even if we fix a design policy (https://arxiv.org/abs/1604.08320), the corresponding true data-generating process is still unknown and so we are approximating the true expected decomposition with our model of the data-generating process. We will make updates to clarify this in the paper.\n\nAnother thing we would be happy to expand on in the paper (if it is of interest) is the existence of families of distinct data-generating processes that produce the same uncertainty decomposition given a long enough rollout of data generation. For example, it is possible for different experimental-design policies to be equivalent in the extent to which they will reduce uncertainty over a given rollout length.\n\n### **Training-data entropy**\n\n> line 149 left: Why would the Shannon entropy of the training data be n log 2?\n> \n\nWe have $2^n$ possible outcome sequences, $y_{1:n}$, each with probability $1/2^n$. The entropy is\n\n$$\n\\mathrm{H}[p_\\mathrm{train}(y_{1:n})] = -\\sum_{i=1}^{2^n} p_\\mathrm{train}(y_{1:n}^{(i)}) \\log p_\\mathrm{train}(y_{1:n}^{(i)}) = -\\sum_{i=1}^{2^n} 2^{-n} \\log 2^{-n} = n \\log 2.\n$$", "claims_and_evidence": "I would argue that this is one of the few papers I have seen that does not make a technical but rather a didactical claim: That they provide clarity on three fronts: (i) that a loss function drives the principled treatment of uncertainty, (ii) that decomposition of uncertainty is usually not possible, and (iii) that model-based uncertainty can be interpreted as estimating the model predictive performance on unseen data, which is however not a theoretical substitute for external grounding. Even though I find the whole work compelling and quite well formalized, I am inclined to say that the paper somewhat falls short in its goal to bring clarity. The paper is excellently written, but it is extremely dense and technical with a lot of notation, which is understandable and necessary, but it makes the paper quite difficult to read, and in many parts, it is necessary to read several times and even then it does not become entirely clear (see also my questions below). I think that this paper would need at least one additional page but probably more to give justice to its own ambition of really bringing more clarity. This way, the paper above all raises awareness but, at least in my case, not really much clarity.", "methods_and_evaluation_criteria": "n/a (interestingly)", "theoretical_claims": "There are some few propositions, most of which I checked the proofs and found them reasonable. These propositions are cool and should be there. They are not a game changer for the paper in either direction though, and I wouldn't even call them \"claims\".", "experimental_designs_or_analyses": "There is a minimalistic experimental design around BALD for illustrative purposes, which is nice and would also be necessary in other parts of the paper.", "relation_to_broader_scientific_literature": "Excellent", "essential_references_not_discussed": "None I am aware of", "other_strengths_and_weaknesses": "None", "other_comments_or_suggestions": "This paper is really different from standard papers one reviews at ICML since there is no technical method, no result tables, bold figures etc. Personally I find this very refreshing, but it is problematic even only fairly review this paper into the new ICML review scheme, which is entirely oriented towards papers with tables with bold numbers.  I appreciate the attempt to try something different. Frankly I also have no suggestion of what to take out from the paper since everything appears relevant; there seems just to be too little space available to properly convey the message.", "questions_for_authors": "- line 149 left: Why would the Shannon entropy of the training data be n log 2?\n- I am not sure what to think of this type of issue that \"... is only an estimator of [the true uncertainty quantity]\". I mean the authors surely will agree that we will always only be able to estimate the true uncertainty, whichever it is, at least in what *you* here define as aleatoric and epistemic uncertainty. So in the light of an impossibly perfect estimate, what would be even the desireable objective in your viewpoint?\n- I am also not sure that I follow the argument of unique decomposability of the expected uncertainty reduction only if the additional training samples are determined without any noise (bottom right of page 5). I don't see what is wrong about a decomposition that still involves the expected value of those additional data points. At least this part doesn't become clear to me.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YPjCkp822c", "reviewer": "Reviewer_Atnu", "summary": "This paper revisits the concepts of aleatoric and epistemic uncertainty in machine learning. It identifies inconsistencies in how these uncertainties are commonly discussed. The authors argue that traditional definitions are overly simplistic, causing confusion. They propose a decision-theoretic approach to clarify uncertainty, defining it through the expected loss of optimal predictions. This framework allows a clearer distinction between uncertainty that can and cannot be reduced by additional data. Additionally, the authors critique commonly used information-theoretic measures like entropy and BALD scores. They demonstrate these metrics are useful practically, but often inaccurately estimate uncertainty. The paper emphasizes that subjective uncertainty estimates from models should not replace objective external evaluations. Overall, the authors aim to provide a unified, clear foundation for future research on uncertainty estimation.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review.\n\nWe are pleased to see positive feedback on multiple aspects of the paper:\n\n1. Strong motivation\n2. Interesting use of decision theory\n3. Comprehensive view on reasoning and learning\n4. Clear writing, proofs and overall presentation\n\nWe also appreciate your thoughtful, in-depth conceptual comments along with useful pointers for lower-level improvements.\n\n### **Loss functions**\n\n> I am afraid the authors introduce another such (and equally questionable) dichotomy by fundamentally differentiating between objective model evaluation and subjective uncertainty quantification.\n> \n\nYou raise an astute point that is more aligned with our perspective than you might think.\n\nWhile we agree that a loss function is typically an imprecise abstraction of our internal desires for how a system should behave (and thus there is a subjective element in its construction), we argue that this falls under our problem definition because it relates to our definition of optimality in an ideal world, and not how that optimality is achieved. Core to our arguments is the idea that all uncertainty characterisation should start from the problem definition; our technical contributions are then based on how this can be done in a rigorous way. This results in a goal-driven notion of uncertainty that is rigorous given a problem definition (note that this setup is very standard across machine learning and not just the uncertainty-quantification literature).\n\nAnother key point is that even if we are unsure about how best to set up our loss function, this is not an uncertainty that arises from a *lack of information*. There is therefore no notion of changes in uncertainty from having access to more data or knowing the underlying data-generating process, which is ultimately what we wish to characterise in our decompositions. The choice of loss function is thus objective from a statistical-terminology perspective, even if it is a subjective decision (in the lay sense) in practice. We believe this use of terminology reflects a long history within decision theory, perhaps most prominently in the work of Savage (1971), who linked subjective beliefs to externally grounded evaluations through scoring rules.\n\nGiven the above, we do not feel it is problematic to condition our analysis on a real-valued loss, but we do agree that the importance and “subjectivity” of choosing the loss function warrants more discussion in the paper, and we will happily add this in our update.\n\n### **Practical implications**\n\n> The paper could benefit from some more constructive and practical recommendations for ML practitioners.\n> \n\nGreat point. We have laid out some points in “Practical implications” in our response to Reviewer MLbH.\n\n### **Case study**\n\n> the paper's accessibility to the (applied) ML community could be improved by e.g. a more tangible case study\n> \n\nThis is an excellent idea.\n\nIn our code repo (link below) we have added a new practical demonstration of some of the key ideas from the paper. We look at Gaussian-process regression and show how the loss function affects three things:\n\n1. The model’s subjective uncertainty, $h_\\ell[p_n(z)]$\n2. The model’s discrepancy, $d(p_n, p_\\mathrm{eval})$, with respect to a reference system\n3. Which data gets prioritised during data acquisition\n\nWe also highlight the crucial distinction between model-based uncertainty quantification and externally grounded evaluation using a scoring rule or discrepancy function.\n\nWe do this by comparing a standard quadratic loss against a weighted quadratic loss that encodes a preference for accuracy on larger values of $z$ ($z$ might be a variable we need to be large, such as the solubility of a candidate drug molecule).\n\nWe plan to add some of these plots to the paper, along discussion relating it to practical applications.\n\n### **Code**\n\n> It appears the author did not share code\n> \n\nThank you for flagging this oversight. We have made our code available at https://anonymous.4open.science/r/rethinking-aleatoric-epistemic-00DF.\n\n### **Experimental evidence**\n\n> Experimental evidence is only anecdotal and limited to BALD.\n> \n\nWe recognise that our experiment is simple, but we contest the characterisation of our evidence as anecdotal. As explained in the paper, our experiment is carefully chosen to support our empirical claim, complementing results from past work.\n\n### **Existing results**\n\n> I also encourage the authors to clearly note some examples… as standard textbook derivations\n> \n\nGood point. We will flag existing results clearly.\n\n### **Terminology**\n\n> The paper occasionally uses different terms interchangeably without explicitly stating equivalences clearly.\n> \n\nThis is useful feedback. We will standardise the terminology.\n\n### **Related work**\n\nThanks for these pointers. We will happily add citations.\n\n### **Typos**\n\nWe appreciate you pointing these out. We will fix them.", "claims_and_evidence": "Examples are supported by simple proofs. Experimental evidence is only anecdotal and limited to BALD.", "methods_and_evaluation_criteria": "The paper's conceptual nature implies no comprehensive benchmark study is required. Proofs are clear and easy to follow. Experimental illustration could be enhanced.", "theoretical_claims": "Yes, the proofs are correct as far as I could check. This comes at little suprise, as the claims are mostly based on previous work and well-known results.", "experimental_designs_or_analyses": "It appears the author did not share code to reproduce experimental evidence, violating open science standards.", "relation_to_broader_scientific_literature": "I welcome a) the decision-theoretic perspective on predictive UQ/model uncertainty and b) the shift away from (mostly purely probabilistic) UQ towards a more comprehensive view on modeling and reasoning in machine learning. Of course, a) allows for b), but I consider both parts of independent interest and valuable contributions on its own. Generally, I like the writing. The presentation is clear and the setup is well-motivated.  \n\nHaving said this, I identify two main concerns with the current state of the paper. While the first one is a conceptional caveat that might not be resolved so easily, the second refers to the presentation and should be fairly easy to address by minor revisions. \n\n1. As strong as I support the authors in arguing against the simplistic aleatoric-epistemic dichotomy, I am afraid the authors introduce another such (and equally questionable) dichotomy by fundamentally differentiating between objective model evaluation and subjective uncertainty quantification. Let me explain in detail. The authors start by emphasizing the basic rationality axioms by VNM and Ramsey but to my great regret, they fail to expand on them nor explicitly use them. The decision-theoretic embedding would allow for a rigoros study of how the choice of axioms affects the utility (i.e., loss function) in machine learning. Instead, the authors naively equate the predictive task with a loss function. I appreciate how the authors discuss the subjectivity of UQ and expose it to be based on the internal belief state of a model which does not necessarily need to correspond to what the authors call p_{eval}, the law of interest for evaluation. However, they miss the opportunity of doing the same for the choice of the loss function. The authors write, \"Bayes optimality simply means we are taking an action that reflects our belief state; it says nothing at all about how well that belief state matches reality.\" I could not agree more, but the same applies to the loss. For instance, consider the desiderata of a loss function to reflect multiple users' preferences (total orders). It has been long recognised in the social choice literature (e.g. Arrow's famous impossibility result) that no total order exists to aggregate such preferences under reasonable assumptions on the aggregation. See https://apps.dtic.mil/sti/tr/pdf/AD0708563.pdf for more background. In other words, by naively conditioning their decision-theoretic analysis on a real-valued loss (i.e., stipulating a total order), the authors exclude all subjectivity in how such a loss function arises. I consider this to be somewhat inconsistent with the differentiation between p_eval and p_train. Why should a machine learning model be evaluated with respect to different probability distributions but always with respect to the same loss? It was already recognised by Abraham Wald in his seminal 1949 paper \"Statistical Decision Functions\" (Ann. Math Stat.), that in the case where P_eval is not known and the decision maker has only ordinal preferences, rationality axioms imply a classical maximin instead of the Bayes criterion. Notably, these order theoretic deliberations on the loss aren't purely theoretical. They have been applied in the ML community recently. For instance, https://proceedings.mlr.press/v216/jansen23a/jansen23a.pdf and https://papers.nips.cc/paper_files/paper/2024/hash/b1f140eeee243db24e9e006481b91cf1-Abstract-Conference.html derived optimal procedures for multivariate random variables with locally varying scale of measurement (e.g. cardinal (e.g. autmoatic evaluation of ML model) and ordinal (e.g. human ranking of ML model's output like in RLHF) criteria). In summary, just like \"probability does not exist\" (de Finetti), i.e., there is no objective probability, there is no objective loss tied to a predictive task. I thus encourage the authors to reconsider the \"objective\" terminology and reframe their concept as e.g. addressing probabilistic vs. non-probabilistic uncertainty and not subjective vs. objective reasoning. This is further supported by the fact that there workarounds avoiding the sensitivity of Bayes-actions towards the choice of prior. Besides classical objective Bayesian approach (see the cited Berger book), these workarounds comprise \"prior near ignorance\" credal sets of priors (i.e. convex sets of probability measures that represent partial ignorance), see e.g. https://alessiobenavoli.com/research/prior-near-ignorance/\n\n2. The paper could benefit from some more constructive and practical recommendations for ML practitioners. This is not my area of expertise, but I feel the paper's accessibility to the (applied) ML community could be improved by e.g. a more tangible case study. Why not expand the experiments on BALD to a more comprehensive setup involving both data acquisition and prediction?", "essential_references_not_discussed": "I do not think the following references are strictly essential, but certainly tangential and thus worth a read.\n\n@ \"Training data need not be direct examples of the predictive task\": This is reminiscent of the notion of \"Institutional Separation\" introduced in this 2024 ICML paper: https://arxiv.org/abs/2404.04669\n\n@ \"Reasoning about new data can (but will not always) yield a unique uncertainty decomposition\": The Bayesian selection of new data was discussed here (https://www.jmlr.org/papers/v24/21-1067.html) and here (https://arxiv.org/abs/2406.12560). On a slightly different note, the paper (as far as I could check) focuses on acquiring new data exclusively in the context of active learning. It has recently been noted that the decision-theoretic embedding of data acquisition extends beyond active learning and comprises e.g. self-training in SSL, boosting, and bandits: https://proceedings.neurips.cc/paper_files/paper/2024/hash/0337b41b4e8b2eb5d7ab161ffd42cf3b-Abstract-Conference.html (For an example of Bayes-optimal acquisition in SSL, see https://proceedings.mlr.press/v216/rodemann23a/rodemann23a.pdf)", "other_strengths_and_weaknesses": "The paper occasionally uses different terms interchangeably without explicitly stating equivalences clearly. For example, \"statistical dispersion in data\" and \"noise\" are used interchangeably. It would be clearer to pick one term and consistently use it after an initial definition. I also encourage the authors to clearly note some examples (e.g., variance and entropy derivations) as standard textbook derivations to avoid confusion about novelty.", "other_comments_or_suggestions": "Minor remarks/typos:\n\n- \"foundations ideas\" →  \"foundational ideas\"\n\n- \"this spurious associations\" →  \"these spurious associations\"\n\n- \"implicitly using favouring\" → either \"implicitly using\" or \"favouring.\"\n\n-  \"defined to conditional\" →  \"defined conditional\" or \"defined to be conditional.\" ?\n\n- \"lead to different reduction\" →  \"lead to different reductions.\"\n\n-  \"Variously referred in\" →  \"variously referred to in.\" ?\n\n- \"thus that objective reasoning\" → better phrased as \"thus objective reasoning.\"\n\n- \"show that recovers\" → missing \"this,\" should be \"show that this recovers.\"\n\n-  \"down to extremely pure, unambiguous case\" → should be plural, \"cases.\"\n\n- \"over horizon of m observations\" → \"over a horizon...\"\n\n- \"Justifying the need for external grounding.\" → This sentence fragment should be combined with the preceding sentence.\n\n- \"We average over 50 repeats with different random seeds.\" →  \"We average over 50 random seeds.\"", "questions_for_authors": "See above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper argues that the epistemic and aleatoric uncertainty dichotomy, as codified in the existing uncertainty quantification literature, is "insufficiently expressive" and incoherent. The authors instead argue for an alternative task-oriented perspective rooted in Bayesian decision theory. They discuss what tasks yield uncertainty decompositions and the effects of finite data on such uncertainty estimates. The authors finally analyze the popular BALD uncertainty decomposition as a motivating example.

This paper is an unconventional ICML submission. Though the argument is rigorous and well rooted in decision-theoretic principles, it is mainly philosophical, as the theory is limited to propositions, and there is little experimental evidence. Moreover, there are no "practical" takeaways for researchers or practitioners, as the paper is essentially a negative result. Nevertheless, the argument presented in this paper is important and will be of high interest to the uncertainty quantification community. Because the paper's argument is well-grounded in statistical principles (and not merely a subjective argument like a position paper) I recommend acceptance into the main ICML track.

For the camera ready, I suggest that the authors address the minor clarity issues brought up during the review process.
Decision: Accept (poster)