Efficient Credal Prediction through Decalibration
Authors: Paul Hofman
Timo Löhr
Maximilian Muschalik
Yusuf Sale
Eyke Hüllermeier
Venue: ICLR 2026 Poster (2026)
Keywords: efficient uncertainty representation
credal sets
relative likelihood
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: Efficient credal prediction based on plausible probability intervals for computationally complex models (e.g. TabPFN, CLIP,…).

Abstract: A reliable representation of uncertainty is essential for the application of modern machine learning methods in safety-critical settings. In this regard, the use of credal sets (i.e., convex sets of probability distributions) has recently been proposed as a suitable approach to representing epistemic uncertainty. However, as with other approaches to epistemic uncertainty, training credal predictors is computationally complex and usually involves (re-)training an ensemble of models. The resulting computational complexity prevents their adoption for complex models such as  foundation models and multi-modal systems. To address this problem, we propose an efficient method for credal prediction that is grounded in the notion of relative likelihood and inspired by techniques for the calibration of probabilistic classifiers. For each class label, our method predicts a range of plausible probabilities in the form of an interval. To produce the lower and upper bounds of these intervals, we propose a technique that we refer to as decalibration. Extensive experiments show that our method yields credal sets with strong performance across diverse tasks, including coverage–efficiency evaluation, out-of-distribution detection, and in-context learning. Notably, we demonstrate credal prediction on models such as TabPFN and CLIP—architectures for which the construction of credal sets was previously infeasible.

Published as a conference paper at ICLR 2026
EFFICIENT CREDAL PREDICTION THROUGH DECALIBRATION
Paul Hofman' 122,, Timo L2hr Maximilian Muschalik 12, Yusuf Sale! Eyke Hüllermeier
"LMU Munich, Muunich Center for Machine Learning (MCML), *DKI (DSA)
"Equal Contribution
{paul. hotman, tmo Loehr, maximi 1 lan www.mmuccaala. yusut, eyxe) Ieu.de
ABSTRACT
A reliable representation of uncertainty is essential for the application of mod- ern machine learning methods in safety-critical settings. In this regard, the use of credal sets (i.e., convex sets of probability distributions) has recently been pro- posed as a suitable approach to representing epistemic uncertainty. However, as with other approaches to epistemic uncertainty, training credal predictors is com- putationally complex and usually involves (re- (re-)training an ensemble of models. The resulting computational complexity prevents their adoption for complex mod- els such foundation models and multi mudti- systems. To address this problem, we propose an efficient method for credal prediction that is grounded in the notion of relative likelihood and inspired by techniques for the calibration of probabilistic classifiers, For each class label, our method predicts range of plausible proba- bilities in the form of an interval. To produce the lower and upper bounds of these intervals, we propose a technique that we refer to as decalibration. Exten- sive experiments show that our method yields credal sets with strong performance across diverse tasks, including eiverage-ffficeen evaluation, out-of- -uitt--distriuttion detection, and in-context learning. Notably, we demonstrate credal prediction on models such as TabPFN and CLIP- arp--aachiecturees for which the construction of credal sets was previously infeasible.
INTRODUCTION
Modern machine learning (ML) is increasingly deployed in domains where decisions carry real con- sequences, from energy systems (Miele et al., 2023) and weather forecasting (Bülte al., 2025) to healthcare (Löhr et al., 2024). In such domains, we need models that not only make accurate predictions, but also express what they do not know. A useful starting point is the distinction be- tween aleatoric and epistemic uncertainty (Hüllermeier & Waegeman, 2021). Aleatoric uncertainty reflects irreducible randomness the data. Epistemic uncertainty reflects limited knowledge and, in principle, can be reduced with more or better information. While standard probabilistic predictors capture the former, representing the latter typically requires higher- formalism.
Credal sets, i.e., (convex) sets of probability distributions, offer such view. Instead of committing single predictive distribution, credal predictor returns a set of plausible distributions, thereby making epistemic uncertainty explicit (Levi, 1978; Walley, 1991). Credal methods have appealing semantics but can be computationally demanding: many pipelines rely on ensembles or approximate posteriors to explore the space of plausible models, which is difficult to justify for large and complex models such as foundation models, CLIP (Radford et al., 2021) or TabPFN (Hollmann et al., 2022).
We take a different route. Building likelihood-based notion of plausibility (Löhr et al., 2025), construct credal predictions from a single trained model by decalibration: we systematically perturb the model's 's logits that the resulting probabilities move away from the maximum -ikeeiiiooood fit while staying within a prescribed relative-likeliee likelihood budget. For each class, this procedure yields plausible probability interval: their product forms credal set that reflects epistemic uncertainty

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Input
Decalibration
Credal Prediction
probabilistic classifier (Cgg,, Neohet,-))
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classes
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photo of [LADEL].
Imagen
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Figure 1: Overview of Efficient Credal Prediction through Decalibration. Given a probabilistic classifier (maximum likelihood estimate), our method decalibrates the predicted distributions by their logits. The resulting credal set contains the ground-truth distribution, as visualized in the credal spider plot (see Appendix C for an explanation). Note that we only show the decalibration of three classes for visualization purposes in practice, all classes are decalibrated.
without retraining (ef. Figure 1). Intuitively, calibration adjusts probabilities be more correct, whereas decalibration explores how far they can be pushed and still remain supported by the data.
In this light, our contributions are as follows. ① A model-agnostic, post-hoc method for credal prediction via decalibration, logit perturbations that produce class-wise plausible probability in- tervals under a likelie- budget, yielding credal sets with the clear semantics "reachable without sacrificing more than chosen fraction of training likelihood." The procedure requires show the relative-likelioodd feasibility set induced by logit shifts is convex (and compact on an identi- fiability hyperplane); that upper class-wise bounds arise from single convex optimization; and that in a one-dimensional, class-specific shift the plausible interval endpoints solve two convex pro- grams with monotone probabilities, implying nested credal sets as the likelihood budget tightens. 3 Empirically, across benchmarks our credal sets achieve favorable coverage eeeerag--fffciennge trade-offs and competitive out-of-dissributie detection while reducing computational cost by orders of magni- tude. The method enables credal prediction for previously out-of-reach models such TabPFN and CLIP, and we introduce credal spider plots to visualize interval- sets beyond three classes.
retraining and only logits, enabling with large pretrained models.
2 Theoretically,
2 CREDAL PREDICTION BASED ON PLAUSIBLE INTERVALS
We assume a supervised classification setting, where X' denotes the instance space, and y {1....,K) K} the finite set of class labels. Further, we assume that the learner has access to (i..d.) training data Dusin (xx ((()) ) H. with hypotheses of the form : X + AK. mapping instances a E X to probability distributions over }: by A denote the set of all probability distributions on the label space y. Our pri- mary concern is predictive uncertainty, i.e., the uncertainty about the predicted label Úg at query point æ,, E X. While probabilistic predictors X' Ak. F. + h(2x,) p(- xg h) do account for aleatoric uncertainty, they do not represent epistemic uncertainty about the predicted probability p(- 2gy h). To make this uncertainty explicit, we move from point predictions in AR to and consider an uncertainty- aware, set-valued predictor H X + K(AK). where K(AK) denotes a suitable family of subsets of the simplex (e.g., nonempty closed convex sets). In this view, the pre- diction at ag is no longer a single vector h((r,)) E A Ax. but a set H (xr..) 2r, C AK. which we refer to as a credal prediction.
X AK. In this paper, we consider a hypothesis space
Credal sets have emerged as a compelling representation of (epistemic) uncertainty in contemporary machine learning research, yet there no consensus on how to construct them in a principled and scalable way. We aim for a construction that is (i) statistically well-founded, (ii) semantically trans- parent, and (iit) computationally feasible for modern large models. Many existing pipelines either rely Bayesian posteriors (Caprio al., 2024a), thus inheriting prior sensitivity and computational burden, or on ad-hoc ensembling and heuristics that offer weak interpretability (Wang et al., 2025a). In contrast, following Löhr et al. (2025), adopt likelihood- based notion of plausibility that is
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prior-free, data-driven, and well established in statistical inference. Concretely, relative (normal- ized) likelihood provides a scalar measure of model plausibility: a model is considered plausible at level E (o, 1] if its likelihood is at least c-fraction of the maximum likelihood in the model class. This yields an o-indexed family of plausibility regions in parameter space, whose images in prediction space induce (class-wise) plausible probability intervals. These intervals serve as the imputs that generate a credal set in the simplex.
With respect to our desiderata (i)-(tti), Löhr et al. (2025) already address (i) and (ti): the likeli- hood ratio supplies a prior-free, data-driven evidential scale that is standard in statistics, and normal- izing by the maximum likelihood yields nested, interpretable a-cuts (Antonucei et al., 2012). This viewpoint is well established, likelihood ratios underpin classical confidence regions and tests and admit clear calibration narrative (e.g., Wilks, 1938; Cox & Hinkley, 1979; Royall, 2017: Edwards, 2018), thereby providing both principledness and transparency. What remains largely open is (Mi)): computational feasibility. For modern large models (foundation models, large language models, and multi-modal systems), retraining ensembles or running costly Bayesian pipelines is often pro- hibitive. This work aims to close this gap by deriving efficient credal predictions while retaining the likelihood- baaddd semanties. We first fix notation and briefly recall the (relative) likelihood- -aaed construction.
Let L: H + [0, oo) denote the (empirical) likelihood of hypothesis Dunin and let
(h) = L(h)
Lhh) [0. 1].
SuPREh
cerdimum-likelihood solution: 7(h) = 1 for any MLE (if it exists) and decreases as the fit worsens; equivalently, log (h) the log- liceliho gap. and -2 log ((h) is the usual likelihood statistic.
Thus 7(h) is the relative likelihood (likelihood ratio) of h with respect
For E (0, 1). define the plausible model set (viz, relative-likelihood a-cut)
C. = ( H 7h)) 2 j.
Given æ E X. the predictive image of Co is Qz.a == {p( x, h) : e C. S Ak. A convenient class-wis wise summary of Qa is given by the marginal extrema
PA (x) = inf (x h).
Pa(x)) = sup (a h), k = 1,...,K. K. heC.
(1)
hec.
We then define the box credal set at x
Da,a == (DEAK:
S (x())) Vk
(2)
By construction, Qr.a S De,a; thus, the box a tractable outer approximation that preserves all classwise extrema. We state a simple, yet illustrative monotonicity property:
Proposition 2.1. If0 < 02 < an < 1, then C., S Con and Qa.ar 22,ay. Thus, for all k;
2 (z: 2 Be (x; a2) and Pa(x;; S (x;
If a muxiimmm-likelihon estimator ML E exists, then Qx,11 (px and (p, (æ; 1), 1),7x(a;; 1)] {px(x, h.L)). As + O, Qa,a + (h(x)) L(h) 0}, and the inter- vals expand accordingly to the coordinatewise infima/suprema over that limit set.
Proposition 2.1 shows that increasing the plausibility threshold a yields nested prediction sets and monotonically tighter classwise intervals. This monotonicity underpins the so-called cover- age-efficiency trade-off used in our evaluation: larger a typically lowers coverage but improves efficiency (smaller sets), allowing a to be tuned to the desired operating point. In this vein, it is natural evaluate set-valued predictions along two axes, coverage and efficiency.
Coverage. Given a set- valued predictor H, coverage is the probability that the ground-truth condi- tional distribution p²( (- x)) is contained in the predicted set:
C(H) El1(p" x) E H[x)
(3)
where the expectation is over the marginal of on X.
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Efficiency. To reward informative (i.e., small) sets, use the complement of the average interval width across classes (positively oriented: higher is better):
F(H) = E (px(2)) -() 2x k=1
(4)
Pragmatically, constructing gelatve--lkkelihoe a-cuts by training ensembles to hit prescribed likeli- hood ratios is principled, but computationally heavy, and hypotheses tend to cluster near the MLE unless ~ 1, which is poor fit for modern large models. To overcome this limitation, an efficient method for credal prediction that is grounded in the notion of relative likelihood and in- spired by techniques for the calibration of probabilistic classifiers, which will call decalibration.
propose
EFFICIENT CREDAL PREDICTION THROUGH DECALIBRATION
We propose decalibration as a single-model route to plausibility: starting from the maximum- likelihood predictor MM, we deliberately distort predicted probabilities, thereby moving from bet- ter predictions (high likelihood) to worse ones (lower likelihood). However, to keep predictions plausible, we make sure to remain within a prescribed relative-lielihoodd budget a E (0. 1]. The probabilities reachable under this budget induce, for any query æ, classwise plausible intervals and hence a box credal set, without retraining or ensembling. Broadly speaking, we answer the follow- ing question: to what extent we decrease increase the predieted probabilities for specific class before reaching a state where predictions become unplausible (have relative likelihood < a)?
Thus, the idea is keep the likelihood semantics but make the search cheap: rather than training many "plausible" models, we start from the MI_E and deliberately push its probabilities toward less likely configurations, while enforcing a global relative-liellihood budget a, as illustrated in Figure 1. This turns the classical likelihood-ratio view ("models within an LR ball are plausible") into post- hoc exploration of the model's output space. As the budget imposed on the training likelihood, any probability vector we produce is still supported by the data up to the chosen evidence level.
Operationally, we implement the exploration with simple, diwdemmnnsionll transforms of the MLE's logits, expressive enough to traverse wide range of alternative class probabilities yet requiring nei- ther retraining nor access to the backbone's gradients. Compared to ensembles that approximate the same plausibility region by re-optimizing parameters, decalibration is orders of magnitude faster and model-agnostic, i.c., it works on top of any pretrained classifier. It is particularly suited to inference- only API-gated systems, foundation models, LLMs, and multimodal encoders, where parameters are frozen or proprietary and retraining, fine-tuning, or ensembling is impractical or disallowed. At the same time, the outputs inherit clear interpretation, "probabilities reachable without losing more than an a-fraction of likelihood", and the resulting intervals are nested as increases.
Among many possible post-hoc maps on probabilities, yet expressive family that operates on logits: we add a global bias vector c E R to every ex- ample's 's logits (both train and test) and then apply softmax. This choice is model-agnostic (no retraining or gradients), preserves the learned representation, and induces a concave change in train- ing log-likelihood, which in turn makes the a-plausible convex. Intuitively, effeetts controlled odds tilts between classes; its softmax invariance under c + + t1 yields natural identifiability hyperplane and keeps optimization well posed as we shall demonstrate now. Formally, consider predictor that produces logits 2(n) e K on the training set and logits x(x) e k for any test point x. Let e = (cI,...,Ck)) E R.K and define, for each training point n and each class
instantiate decalibration with simple
P3 (n) = exp 22)
(n)
+ +.g))
P)(0,c) = exp(2,(2x) cG)
Σ kml exp (m)
2k + +cx)
EK exp Z4 (x) + cc)
Set Al(c) = EN logg (n)
glan (c) log (n) y(m) (0) and F(a) = [CERK: Al(c) > log a Further, note that Al(c + 11) = Al(c) and p, (æ; c + 11) = (=(;;)) for all
Concretely, this decalibration procedure fixes the family and its feasibility region F(a); what fol- lows establishes the key structural properties that make the method tractable: smooth concavity of Al, convexity/compactess of the feasible set, and convex-optimization characterization of the
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upper credal bound (Proposition 3.1). We then specialize to the one dimensional slice t tkk yielding endpoint formulas and simple convex programs for the scalar case (Corollary 3.1).
Proposition 3.1. Let = EKER 1 c =0] 0} be the identifiability hyperplane. Then:
(a) Al is C and concave RK with Vd((c) En= (e - -p(m and
N
v2Dl(c) = E (iag(p( - -p""(c)(" = 0.
Moreover, Ae is strictly concave on S provided at least two classes appear in DD namely, N₁ > for at least two j. where Nj #nn: y = j}. Consequently, F(a) comvex and translation invariant along span[1}. The section Fs(a) == F(a) n S is nonempty and compuct whenever al least two classes appear.
(b) For each fixed x and k; the map e + log c) (2x (a:) ) ck) log Et ea(()) is - and concave on RK with V Vlogp(<z;)) ek p(x;; c) and
Voooph(a;;
- (Diag(pp(a;))) e)) plar; c) p(x;;))
3 0.
In particular, Ca + p=(ax; c) is strictly increasing (holding cj. J # k. fixed), and oj + pi(a; e) is strictly decreasing for k.
(c) The upper credal bound is the value of the convex optimization problem
Px(a)) = sup Ps (=(x;;) cEF(a)
sup exp (logpk(x;c)) = =exp( sup tog cEP(A)
eEF(a)
and the inner problem SuPeEF(a) log (x c) is a concave maximization, i.e., a convex optimization. An optimizer always exists on Fs(a). and unique modulo addition of constants along span
(d) The lower credal bound P. (x) inf inno Pa( c) is, in general, not a convex optimiza- tion problem. Nevertheless, when Fs(a)) is compact, minimizer exists and is atlained at an extreme point of the convex set Fs(a).
We prove Proposition 3.1 in Appendix A. Proposition 3.1 (a) guarantees that the likelihood feasibility region is well-posed convex set (compact on the hyperplane S), optimization over it stable. Moreover, (b) shows the test objective inherits the same curvature structure as the training likelihood. Together these yield (c): the upper credal bound is the value of single convex program with a unique optimizer on S, while (d) clarifies that the lower bound is generally nonconvex and lives on the boundary/extreme points of (a). Practically. the upper credal bounds can be com- puted reliably using convex solvers, while the lower bounds require exploration of the fcasibility set's boundary. This task, however, is not trivial: the lower-bound optimization may feature mul- tiple global extrema. Thoroughly exploring these extrema can become computationally expensive, undermining the main goal of our approach, which is to efficiently compute credal predictions.
To address this challenge we restrict to class-specific biases tcg- This reduces the problem to a tractable one-dimensional slice, where the feasible set becomes simple interval, and the class- k probability varies monotonically with l. Consequently, exact lower and upper bounds can be obtained simply by evaluating the interval endpoints, as we formalize in the following corollary.
Corollary 3.1. Now restrict to shifts of the form Leg, E and define
(t) == Al(to)),
Fa(a) E (1x 2 log a = {t: t. E F(a))
Then the following hold:
All is C and strictly concave R. Consequently F&(a) is nonempty interval; if
0 < No < N, it compact It 1 otherwise it is closed (possibly half-infinite) interval.
(b) For every fixed ær, the map pk (a;; ex) is strictly increasing R.
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(c) With t inf (a) and t sup Fx(a),
Ps (x - Pke (z; t ok).
(x( = pt (x; ox).
(d) The endpoints 1 solve the convex programs
min t s.t. -Px((t) S log a, EER
min (-t) s.t. Atx(t) - log TER
We prove Corollary 3.1 in Appendix A. Algorithmically. the scalar case reduces computing (2x (z), Py. (z)) to finding the two endpoints tx and & of the feasible interval, e.g., by bisection on (tx log a; the bounds are then pı & o&) and pk (a:; E ox). Throughout the empirical evaluation, we focus on the one-dimensional setting, which admits convexity of the lower and up- per probability bounds, resulting the box credal set Dz We defer details about the practical computation of the bounds to Appendix B.
4 EMPIRICAL RESULTS
In this section, we empirically evaluate proposed method with the following research objectives in mind. First, we assess the quality of the uncertainty representation by standard metrics and show its strong performance compared to baselines in Section 4.1. Second, we evaluate the method common downstream tasks and emphasize the competitive performance-whil far more efficient- when compared baselines in Section 4.2. Third, we highlight the distinctive advantage of our method that it can construct uncertainty representations for large architectures such as TabP FN or CLIP, where retraining is infeasible in Sections 4.3 and 4.4.
Thus, we present scenarios where our method (EffCre, see Appendix B for implementation details) newly enables the construction of uncertainty representations and, where possible, we compare to the following suitable baselines, which represent the current ofate-of-the-art in credal prediction:
Credal Wrapper (CreWra) (Wang et al., 2025a). Credal Ensembling (CreEns) (Nguyen al., 2025), Credal Bayesian Neural Networks (CreBNN) (Caprio et al., 2024a), Credal Interval Net (CreNet) (Wang et al., 2025b), and Credal Relative Likelihood (CreRL) (Löhr et al., 2025). The code for all experiments is published in Github repository' and the detailed experimental setup can be found in Appendix D.
4.1 COVERAGE VERSUS EFFICIENCY
We compare our method to the baselines in terms of coverage (3) and efficiency (4). Ideally, a credal predictor generates sets of small size (high efficiency) that cover the ground-truth conditional distribution (high coverage). Moreover, because the relative importance of coverage and efficiency may vary across applications, methods that allow to trade-off one against the other, depending on the setting, are favored.
Setup. We train multilayer perceptron (MLP) on the embeddings of CHIAOSNII and ResNet 8 (He et al., 2016) on CIFAR-10 (Krizhevsky al., 2009) (Schmarje et al., 2022). The models are trained with regular labels and evaluated against ground-truth distributions. Such ground-truth dis- tributions are derived from multiple annotator labels, available through CIFAR- 10H (Peterson et al., 2019) for the CIFAR-16 10 test set, while CHAOSNI.I provides them directly.
Results. We present results for coverage versus the efficiency in Figure 2. The CIFAR-10 dataset shows that our method Pareto dominates CreRL in the high coverage region, while performing sim- ilarly in the medium coverage region. In addition, EffCre Pareto dominates the CreBNN, CreWra, CreNet baselines. For the CHAOSNLI dataset our method performs similarly CreRL in the high coverage region and similar to CreEns in the low coverage region. Whereas the aforementioned baselines can only traverse the low coverage high coverage regions, our method can traverse both regions, allowing a user to specify almost any coverage efficiency value. Furthermore, our method dominates CreBNN with a 0.95, whereas performs similarly to CreNet and CreWra, again with the caveat that these methods are restricted to the low coverage region. For results on an
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CIRAR-1CH
Chaoe
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GreRiam
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Crean
C7
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models
trained
trained
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02
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Figure 3: Out-of-Distrlbution Detection. Performance (AU- ROC, based on epistemic uncer- tainty) as function of required number of models and training time (in hours).
EftCre
CrcRL
CreEns
CreBNN
CreWita
CreNet
Figure 2: Coverage versus Efficiency. Comparison on CIFAR- 10 and CHAOSNLL The plot highlights the Pareto trade- off: higher coverage often requires lower efficiency. EffCre con- sistently advances the Pareto front over baselines.
additional dataset, we refer to Appendix E. Lastly, note that, while in Section 2, we assume >> O, the coverage and efficiency experiments we explicitly use a as a verification that our method produces sufficiently diverse sets. Broadly speaking, if our method is not able to generate dense credal sets for a O, we cannot expect reliably reach the edges of the plausible probability intervals.
4.2 DEN-OO--DSTTIIUTIION DETECTION
Besides coverage and efficiency, the quality of credal predictions. Modern machine learning systems should be able detect whether the data they receive is in-distribution (ID) aut-of-distribuiio (OOD) as strong performance on this task is an indication of good epistemic uncertainty representation. So far, many approaches have been unable to provide epistemic uncertainty representation due to prohibitive computational costs; our method directly addresses this. To demonstrate this, we analyze the trade-off between the training time and performance, in terms of AUROC, for our method and compare it to baselines.
detection is commonly used proxy to evaluate
Setup. We train a ResNet 18 on CIFAR- 10, which serves as the ID data and introduce it to several other datasets that serve as the OOD data. Epistemic uncertainty is quantified based on commonly- used measure from Abellán et al. (2006),
EU((a)) = S(2z) -(Q..)
(5)
where S(Qx) = Im) S(p(- x)) and S( Qxx) defined analogously, are the upper and lower Shannon entropy?, respectively.
Results. We report the OOD detection results alongside training time in Figure and provide additional results and hyperparameter ablations in Appendix F.I. Since any approach requires at least one trained model (e.g., an MLE predictor), our method EffCre comes with no extra training cost, as can be applied post-hoe in a highly efficient manner. Although the baselines achieve slightly higher AUROC scores on the OOD task, they rely on ensembles of models, which demand substantial number of members (e.g.. 10 models) and therefore significantly increase training time. While CreWra, CreEns, CreNet, and CreBNN require full training of each ensemble member, CreRL is slightly more efficient due its carly-stopping criterion. However, our approach EffCre is substantially more efficient compared to all baselines, enabling the application even to large-scale models.
4.3 IN-CONTEXT LEARNING WITH TABPFN
To highlight the ability of method to be applied in post-hoc man- ner, requiring only logits, we apply it to a foundation model for tabular data.
"Shannon entropy: S(p(- x)) - (2( log (m)) with O by definition.
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TabPFN (Hollmann et al., 2025) is prior-fitted transformer, trained on a large number of synthetic datasets. It uses in-context learning, based all training data and additional exemplary instances, to make predictions, while not requiring any gradient- based changing of its weights. Therefore, the base- lines used in the experiments in Sections 4.1 and 4.2 cannot be applied as they require training (an ensem- ble). which, besides being challenging due com- putational cost, also requires the original training data, which we do not have access to.
TabPFN EffCre
0.8
Emiciency
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0.8
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Coverage
Active In- In-Context Learning (ID 46941)
ENCre te (ertmay)
Entre qee one)
HIF [nandom)
Setup Coverage Versus Efficiency. To illustrate the proper uncertainty representation generated by using our method top of TabPFN, compute the coverage and efficiency of the predicted credal sets by applying it to all multi-class datasets from the TABARENA benchmark (Erickson et al., 2025). Since these datasets do not provide ground-truth conditional distributions, we propose a simple way create semi-synuteen synihhytthetic ground-truth distributions allow evaluation by coverage and efficiency. Details about this experiment and the process of creating such distributions can be found in Appendix E.3 and Appendix D.5, respectively.
Accuracy
0.35
0.65
300
500
7.0
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Figure 4: EffCre used with TabPFN. Top: Coverage versus efficiency perfor- mance all multi-class TABARENA datasets. Bottom: Active In-Context Learning perfor- mance versus the random baseline.
Results Coverage Versus Efficiency. We show the coverage and efficiency results in Figure 4, confirming that uncertainty representations obtained by applying our method to TabPFN provide small sets that often include the ground-truth distribution.
In addition, we perform active in-context learning, which has become an important task in the con- text of foundation models since labeling represents the limiting factor to leveraging pre-trained mod- els effectively. Ideally, a mooell-equippe with a reliable (epistemic) uncertainty representation- able to sample informative instances, which, when used for in-context learning, improve the perfor- mance more than a random sample of instances would.
Setup Active Learning. Specifically, quantify epistemic uncertainty using (5) and addition- ally use a measure based on zero-one-loss, which has been show to perform well on similar tasks (Hofman et al., 2024). Concretely,
max EU(Qz) p(1=), (l,
max px(x)) Parg max F/, (m) (x).
(6)
We perform this task for number of TABARENA datasets using 0.8. For more details regarding the experimental setup and additional results, we refer to Appendix E.3.
Results Active Learning. We present the results in Figure 4. This highlights the ability of our method to represent its epistemic uncertainty well, in order to sample the most informative instances accordingly. An ablation in the active in-context learning setting can be found in Appendix F.2.
4.4 ZERO-SHOP SHOT CLASSIFICATION WITH CLIP-BASED MODELS
We demonstrate the flexibility of our method by creating credal sets for lision-language models (VLMs), including CLIP (Radford et al., 2021), ) LigLIP (Zhai et al., 2023), SigLIP- (Tschan- et al., 2025), and Bi BiomeCCLIP (Zhang et al., 2024).
Setup Coverage Versus Efficiency. To demonstrate the proper uncertainty representation generated by using method top of CLIP-based models-Something that is computationally prohibitive for the baselines-we compute the coverage and efficiency of the predicted credal sets by applying CIFAR-10. 10, using CIFAR- 10H as ground-trutt truth distributions. Therefore, we turn the models into zero-shot classifiers by reformulating the label set into natural-language templates and comparing
"The datasets included in TabArena vO.1. This collection may be subject change.
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Input to CLIP
CLIP Credal Sets
Input to CLIP
CLIP Credal Sets
ship
MiE Credal
A photo o (LABRL].
photo of [LABEL].
Figure 5: Credal Prediction with CLIP. Examples from CIFAR- 10H with high epistemic uncer-
tainty (Ieft) and high aleatoric uncertainty (right) as predicted by applying our method on CLIP.
the resulting text embeddings with image embeddings (see Appendix D.6 for details and perfor- mance results).
Results Coverage Versus Efficiency. Figure 6 shows performance on the coverage- coverage-efficiency trade- off, with our method performing well, being able to reach regions with high coverage and high efficiency for CLIP, Si gI.I, and S S......-.
To further highlight the usefulness of our method, we compare the predicted credal sets to human un- certainty patterns. To visualize credal sets beyond three classes, we propose credal spider plots where each axis corresponds to a class and intervals mark upper and lower probabilities (see Appendix C for detailed guide). Profiles such as MILE predictions or ground-truth distributions can be overlaid for direct comparison.
CIFAR-10
Effickency
Modob
22
Setup Qualitative Evaluation. For the visual eval- uation, we apply our method on top of CLIP- based models and predict credal sets for CIFAR-10 10 while using CIFAR- 10H to reference ground-truth distributions. We sort and present instances from 10-ARs10-H's test split based on the aleatoric and epistemic uncertainties represented by the predicted credal sets of the CLIP-based models.
Figure 6: EffCre used with CLIP-based models. We demonstrate the performance of EffCre by applying it to CLIP,, Si gI.L.P, and Si gLIP-2 without seraining-ssometiing that can't be done by the baselines.
Results Qualitative Evaluation. Figure 5 presents two instances from CIFAR-1O-HI's test split: the left image is misclassified by the MLE due to the unusual context of the ship being out of water in a dock. Our method reflects the resulting epistemic uncertainty with plausible intervals across all classes and higher probability intervals for class ship, truck, and automobile. The right image shows an animal in an ambiguous pose: the ground distribution splits mass between dog and cat, representing aleatoric uncertainty. Our method represents this uncertainty well, covering the true distribution even though the MLE misclassifies the image as cat. Additional multilingual examples and further results are shown in Appendix E.4.
5 RELATED WORK
Credal sets originate in learning, credal sets offer an appealing way to represent epistemic uncertainty, motivating work on (credal) uncertainty quantification (Hüllermeier et al., 2022; Sale et al., 2023; Hofman et al., 2024; Chau et al., 2025a), calibration (Jürgens et al., 2025; Chau et al., 2025b), self-supervision (Lienen & Hüllermeier, 2021), and learning theory (Caprio et al., 2024b). A consensus on construction has not emerged, current practice spans a variety of designs that trade theoretical guarantees against practicality in different ways. Some methods siiiee validity guarantees (Javanmardi et al., 2024). Others form sets by aggregating multiple predictors, whether standard deep ensembles (Wang al., 2025a; Nguyen et al., 2025), interval-
literature (Levi, 1978; Walley, 1991). In machine
conformal prediction obtain credal sets with

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head networks trained with tailored losses (Wang et al., 2025b), or Bayesian ensembles built from posterior samples (Caprio et al., 2024a). Recently, Löhr et al. (2025) adopted a selative-likeliban criterion (Bimbaum, 1962; Antonucci et al., 2012; Senge et al., 2014) to form credal sets in machine learning settings.
Ensemble training underpins much of the prior work, but for large model architectures, retraining even once-is rarely feasible. This, in turn, has led to lightweight alternatives for representing uncertainty. One line of work focuses on single forward pass methods estimate uncertainty (van Amersfoort et al., 2020; Mukhoti et al., 2023), e.g., via distance-based features (Liu et al., 2020) or evidential Dirichlet heads (Sensoy al., 2018; Amini al., 2020), though recently evidential variants were criticized (Bengs et al., 2023; Jürgens et al., 2024). In contrast, approximate Bayesian inference techniques such as Laplace approximations (Daxberger et al., 2021; Weber et al., 2025) and variational last-layer or sub-ensemble approaches (Valdenegro- Toroo 201 9; Kristiadi et al., 2020; Harrison et al., 2024) reduce cost by training only limited parts of the network. Similarly, others compress ensembles by sharing parameters (Durasov et al., 2021; Laurent al., 2023) or by distill- ing ensemble knowledge into a single model (Malinin et al., 2020; Penso et al., 2022). Another line of work focuses large models such diffusion or language models and explores low-rank adap- tation to efficiently build ensembles for uncertainty quantification (Berry et al., 2024; Yang et al., 2024; Wang et al., 2024). Yet, computationally efficient methods for credal prediction remain absent from the literature.
6 DISCUSSION
We presented a post-hoc, model-agnostic method for credal prediction that captures epistemic un- certainty as class-wise plausible probability intervals derived from relative likelihood. The key idea, decalibration, perturbs a trained model's logits under global likelihood-ratio budget, thereby ex- ploring less-likely yet still plausible predictions without retraining. We formally analyze decalibra- tion and show that the logit-shift feasibility set is convex (compact on an identifiability hyperplane). In the one-logit (class-specific) case, each interval endpoint is obtained by solving a small convex program, readily handled by off-the-shelf optimizers. Empirically, our method matches or surpasses baselines on awverage-ffficenng and competitive for OOD detection, while cutting computation by orders of magnitude. Because it is post-hoc and needs only logits, we apply it to large pretrained models-including TabPEN and CLIP-for which ensemble retraining is impractical.
Limitations and Future Work. We primarily deploy the one-logit (class-specific) variant of our logit-shift family. The fully coupled case remains open; upper bounds still reduce to a convex program, whereas lower bounds are non-convex. Developing reliable relaxations, certificates, or approximation schemes, and clarifying their statistical trade-offs, is promising direction. Open- vocabulary, multimodal models such CLIP raise additional challenges. Because the label set is chosen at inference time, uncertainty should reflect not only prediction but also label selection and prompt choice. Designing credal formalisms and evaluation protocols for this setting is an important avenue for future work.
Reproducibility Statement. We are committed to ensuring the reproducibility of our results. To this end, we provide our code in the following Github repository https: //gi thub. com/ The theoretical results in Section 3 are ac- companied by proofs in Appendix A and, where necessary, the assumptions have been discussed. The full experimental setup, used to produce the results presented in Section 4 and Appendices and F, is provided in Appendix D. In particular, we diseuss details about datasets, including the transformation performed on the input to models and the creation of (semi-synthetic) ground truth distributions in Appendices D.2 and D.5. The models we use, and our implementation of them, are discussed in detail in Appendix D.I. We elaborate on the implementation of all baselines in Ap- pendix D.3 and details regarding the practical implementation of our method, that are not discussed in the main paper, are provided in Appendix B.
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ACKNOWLEDGMENTS
Timo Löhr and Maximilian Muschalik gratefully acknowledge funding by the Deutsche Forschungs- gemeinschaft (DFG, German Research Foundation): TRR 318/3 2026 438444522. Yusuf Sale is supported by the DAAD program Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the Federal Ministry of Education and Research.
REFERENCES
Joaquín Abellán, George J. Klir, and Serafín Moral. Disaggregated total uncertainty measure for credal sets. Int. J. Gen. Syst.. 35(1):29-44) 44099-442222 2006.
Alexander Amini, Wilko Schwarting, Ava Soleimany, and Daniela Rus. Deep evidential regression. In Hugo Larochelle, Marc' Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin (eds.), Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020. NeurlPS 2020, December 6-12, 2020, virtual, 2020.
Alessandro Antonucci, Marco E. G. Cattaneo, and Giorgio Corani. Likelihood- based robust clas- sification with bayesian networks. In Salvatore Greco, Bernadette Bouchon Meennier, Giulianella Coletti, Mario Fedrizzi, Benedetto Matarazzo, and Ronald R. Yager (eds.), Advances in Computa- fional Intelligence 14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Batt Based Systems, IPMU 2012, Catania, Italy, July 9-13, 2012, Pro- ceedings, Part III, volume 299 of Communications in Computer and Information Science, pp- 491-500. Springer, 2012.
Viktor Bengs, Eyke Hüllermeier, and Willem Waegeman. On second-order scoring rules for epis- temic uncertainty quantification. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett (eds.), International Conference Machine Learning, ICML 2023, 23-29 July 2023, Honolulu, Hawaii, USA, volume 202 of Proceedings of Machine Learning Research, pp. 2078 2091. PMLR, 2023.
Lucas Berry, Axel Brando, and David Meger. Shedding light on large generative networks: Es- timating epistemic uncertainty in diffusion models. In Conference on Uncertainty in Artificial Intelligence, 2024.
Allan Birnbaum. On the foundations of statistical inference. Journal of the American Statistical Association, 57(298):269-300 1962.
Bernd Bischl, Giuseppe Casalicchio, Taniya Das, Mattiis Feurer, Sebastian Fischer, Pieter Gijs- bers, Subhaditya Mukherjee, Andreas C Müller, László Németh, Luis Oala, Lennart Purucker, Sahithya Ravi, Jan N van Rijn, Prabhant Singh, Joaquin Vanschoren, Jos van der Velde, and Mar- cel Wever. Openml: Insights from 10 years and more than thousand papers. Patterns, 6(7): 101317, 2025.
Christopher Bülte, Nina Horat, Julian Quinting, and Sebastian Lerch. Uncertainty quantification for data-driven weather models. Artificial Intelligence for the Earth Systems, 2025.
Michele Caprio, Souradeep Dutta, Kuk Jin Jang, Vivian Lin, Radoslav Ivanov, Oleg Sokolsky, and Insup Lee. Credal bayesian deep learning. Trans. Mach. Learn. Res., 2024, 2024a.
Michele Caprio, Maryam Sultana, Eleni Elia, and Fabio Cuzzolin. Credal learning theory. In Amir Globersons, Lester Mackey, Danielle Belgrave, Angela Fan, Ulrich Paquet. Jakub M. Tomezak, and Cheng Zhang (eds.), Advances in Neural Information Processing Systems 38: Annual Con- ference on Neural Information Processing Systems 2024, NewrIPS 2024, Vancouver, BC, Canada, December 10 15, 2024, 2024b.
Siu Lun Chau, Michele Caprio. and Krikamol Muandet. Integral imprecise probability metrics. arXiv preprint arXiv:2505.161 16/56, 2025a
Siu Lun Chau, Antonin Schrab, Arthur Gretton, Dino Sejdinovic, and Krikamol Muandet. Credal two-sample tests of epistemic uncertainty. Yingzhen Li, Stephan Mandt, Shipra Agrawal, and Mohammad Emtiyaz Khan (eds.), International Conference Artificial Intelligence and

---

Published as a conference paper at ICLR 2026
Statistics, AISTATS 2025, Mai Khao, Thailand, 3-5 May 2025, volume 258 of Proceedings of Machine Learning Research, pp. 127-135. PMLR, 2025b.
David Roxbee Cox and David Victor Hinkley. Theoretical statistics. CRC Press, 1979.
Erik A. Daxberger, Agustinus Kristiadi, Alexander Immer, Runa Eschenhagen. Matthias Bauer, and Philipp Hennig. Laplace redux effortless bayesian deep learning. In Marc' Aurelio Ranzato, Alina Beygelzimer, Yann N. Dauphin, Percy Liang, and Jennifer Wortman Vaughan (eds.), Ad- vances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems 2021, NeurlPS 2021, December 6-14, 2021, virtual, pp. 20089-20103, 2021.
Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE Computer Society Conference on Computer Vision and Puttern Recognition (CVPR 2009), 20-25 June 2009, Miami, Florida, USA, pp. 248-255. IEEE Computer Society, 2009.
Nikita Durasov, Timur M. Bagautdinov, Pierre Baqué, and Pascal Fua. Masksembles for uncer- tainty estimation. In IEFE Conference on Computer Vision and Pattern Recognition, CVPR 2021, virtual, June 19-25, 2021, pp. 13539-13548. Computer Vision Foundation / IEEE, 2021.
Anthony William Fairbank Edwards. Likelihood. In The New Palgrave Dictionary of Economics, pp. 7857-7860. Springer, 2018
Nick Erickson, Lennart Purucker, Andrej Tschalzev, David Holzmüller, Prateek Mutalik Desai, David Salinas, and Frank Hutter. Tabarena: A living benchmark for machine learning on tab- ular data. CoRR, abs/2506.16791, 2025.
James Harrison, John Willes, and Jasper Snoek. Variational bayesian last layers. In The Twelfth International Conference on Learning Representations, ICLR 2024, Vienna, Austria, May 7-11, 2024. OvenReview.net, 2024.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. 2016 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2016, Las Vegas, NV. USA, June 27-30, 2016, pp. 770-778. IEEE Computer Society, 2016.
Paul Hofman, Yusuf Sale, and Eyke Hüllermeier. Quantifying aleatoric and epistemic uncertainty: A credal approach. In ICML. 2024 Workshop on Structured Probabilistic Inference & Generative Modeling, 2024.
Noah Hollmann, Samuel Müller, Katharina Eggensperger, and Frank Hutter. Tabpfn: A transformer that solves small tabular classification problems in a second. arXiv preprint S2Vv:2207.01844 2022.
Noah Hollmann, Samuel Müller, Lennart Purucker, Arjun Krishnakumar, Max Körfer, Shi Bin Hoo, Robin Tibor Schirmeister, and Frank Hutter. Accurate predictions small data with tabular foundation model. Nat., 637(804433199-326 2025.
Eyke Hüllermeier and Willem Wacgeman. Alcatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods. Machine learning, 11033)457-506, 2021.
Eyke Hüllermeier, Sébastien Destercke, and Mohammad Hossein Shaker. Quantification of credal uncertainty in machine learning: A critical analysis and empirical comparison. In James Cussens and Kun Zhang (eds.), Uncertainty in Artificial Intelligence, Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence, UAI 2022, 1-5 August 2022, Eindhoven, The Netherlands, volume 180 of Proceedings of Machine Learning Research, pp. 548-557. PMLR, 2022.
Alireza Javanmardi, David Stutz, and Eyke Hüllermeier. Conformalized credal set predictors. In Amir Globersons, Lester Mackey, Danielle Belgrave, Angela Fan, Ulrich Paquet, Jakub M. Tom- czak, and Cheng Zhang (eds.), Advances in Neural Information Processing Systems 38: An- nual Conference on Neural Information Processing Systems 2024, NeurIPS 2024, Vancouver, BC, Canada, December 10 15, 2024, 2024.
12

---

Published as a conference paper at ICLR 2026
Mira Jürgens, Nis Meinert, Viktor Bengs, Eyke Hüllermeier, and Willem Waegeman. Is epistemic uncertainty faithfully represented by evidential deep learning methods? In Forty-first Interna- tional Conference Machine Learning. ICML 2024, Vienna, Austria, July 21-27, 2024. Open- Review.net, 2024.
Mira Jürgens. Thomas Mortier, Eyke Hüllermeier, Viktor Bengs, and Willem Waegeman, A calibra- tion test for evaluating set-based epistemic uncertainty representations. CoRR, abs/2502.16299, 2025.
Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Yoshua Bengio and Yann LeCun (eds.), 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA. May 7-9, 2015. Conference Track Proceedings, 2015.
Agustinus Kristiadi, Matthias Hein, and Philipp Hennig. Being bayesian, even just bit, fixes overconfidence in relu networks. In Proceedings of the 37th International Conference on Machine Learning. ICML 2020. 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning Research, pp. 5436-5446 PMLR, 2020.
Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of 2009.
features from tiny images.
Olivier Laurent, Adrien Lafage, Enzo Tartaglione, Geoffrey Daniel, Jean-Mare Martinez, Andrei Bursuc, and Gianni Franchi. Packed ensembles for efficient uncertainty estimation. In The Eleventh International Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenReview.net, 2023.
Isaac Levi. On indeterminate probabilities. In Foundations and Applications of Decision Theory: Volume Theoretical Foundations, pp. 233-261. Springer, 1978.
Yucen Lily Li, Daohan Lu, Polina Kirichenko, Shikai Qiu, Tim G. J. Rudner, C. Bayan Bruss, and Andrew Gordon Wilson. Out-of-distribution detection methods answer the wrong questions. CoRR, 0bs/2507.0183/ 2025.
Julian Lienen and Eyke Hüllermeier. Credal self-supervised learning. In Marc' Aurelio Ranzato, Alina Beygelzimer, Yann N. Dauphin, Percy Liang, and Jennifer Wortman Vaughan (eds.), Ad- vances Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems 2021, NeurlPS 2021, December 6-14, 2021, virtual, pp. 14370-14382, 2021.
Jeremiah 7. Liu, 7i Lin, Shreyas Padhy, Dustin Tran, Tania Bedrax- and Balaji Lakshmi- narayanan. Simple and principled uncertainty estimation with deterministic deep learning via distance awareness. In Hugo Larochelle, Marc' Aurelio Ranzato, Raia Hadsell, Maria-Floni Balcan, and Hsuan-Tien Lin (eds.), Advances in Neural Information Processing Systems 33: An- nual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020.
Timo Löhr, Michael Ingrisch, and Eyke Hüllermeier. Towards aleatoric and epistemic uncertainty in medical image classification. In International Conference on Artificial Intelligence Medicine, pp. 155. Springer, 2024.
Timo Löhr, Paul Hofman, Felix Mohr, and Eyke Hüllermeier. Credal prediction based on relative likelihood. CoRR, abs/2505.22332. 2025.
Ilya Loshchilov and Frank Hutter. SGDR: stochastic gradient descent with warm restarts. In 5th International Conference Learning Reprexentations, ICIR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings. OpenReview.net, 2017.
Andrey Malinin, Bruno Mlodozeniec, and Mark J. F. Gales. Ensemble distribution distillation. In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020. OpenReview.net, 2020.
Valentin Margraf, Marcel Wever, Sandra Gilhuber, Gabriel Marques Tavares, Thomas Seidl, and Eyke Hüllermeier. Alpbench: A benchmark for active learning pipelines on tabular data. CoRR, abs/2406. 17322, 2024.
13

---

Published as a conference paper at ICLR 2026
Eric Stefan Miele, Nicole Ludwig, and Alessandro Corsini. Multi-horizon wind power forecasting using multi-modal apatio-tempore neural networks. Energies, 16(8):3522, 2023.
Jishnu Mukhoti,, Andreas Kirsch, Joost van Amersfoort, Philip H. S. Torr and Yarin Gal. Deep deterministic uncertainty: A new simple baseline. In IEEE/CVF Conference Computer Vision and Puttern Recognition, CVPR 2023, Vancouver, BC, Canada, June 17-24, 2023, pp. 24384- 24394, IEEE, 2023.
Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y. Ng. Reading digits in natural images with unsupervised feature learning. In NIPS Workshop on Deep Learning and Unsupervised Feature Learning 2011, 2011.
Vu-Linh Nguyen, Sébastien Destercke, and Eyke Hüllermeier. Epistemic uncertainty sampling. In Petra Kralj Novak, Tomislav Smuc, and Saso Dzeroski (eds.). Discovery Science 222nd Interna- tional Conference, DS 2019, Split, Croatia, October 28-30, 2019, Proceedings, volume 1 11828 of Lecture Notes in Computer Science, pp. 72 86. Springer, 2019.
Vu-Linh Nguyen, Haifei Zhang, and Sébastien Destercke. Credal ensembling in multi-class classi- fication. Machine Learning, 114(1): 2025.
Yixin Nie, Xiang Zhou, and Mohit Bansal. What can we learn from collective human opinions on natural language inference data? In Bonnie Webber, Trevor Cohn, Yulan Hc, and Yang Liu (eds.), Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing, EMNLP 2020, Online, November 16-20, 2020, pp. 9131-9143, Association for Computational Linguisties, 2020.
Rafal Obuchowicz, Mariusz. Oszust, and Adam Piórkowski. Interobserver variability in quality assessment of magnetic resonance images. BMC Medical Imaging, 20(1):109, 2020.
Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Pretten- hofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research, 12:2825- 2830, 2011.
Coby Penso, Idan Achituve, and Ethan Fetaya. Functional ensemble distillation. In Sanmi Koyejo, S. Mohamed, A. Agarwal, Danielle Belgrave, K. Cho, and A. Oh (eds.), Advances in Neural In- formation Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, NeurIPS 2022, New Orleans, LA, USA, November 28- December 9, 2022, 2022.
Joshua C. Peterson, Ruairidh M. Battleday, Thomas L. Griffiths, and Olga Russakovsky. Human uncertainty makes classification more robust. In 2019 IEEF/CVE International Conference on Computer Vision, ICCV 2019, Seoul, Korea (South), October 27 November 2, 2019, pp. 9616 9625. IEFE, 2019.
Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agar- wal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever. Learning transferable visual models from natural language supervision. Proceed- ings of the 38th International Conference on Machine Learning, ICMI. 2021, 18-24 July 202 I, Virtual Event, volume 139 of Proceedings of Machine Learning Research, pp. 8748 8763. PMLR, 2021.
Richard Royall. Statistical evidence: a likelihood paradigm. Routledge, 2017.
Yusuf Sale, Michele Caprio, and Eyke Hüllermeier. the volume of credal set good measure for epistemic uncertainty? In Uncertainty in Artificial Intelligence, pp. 7795-1804 PMLR, 2023.
Lars Schmarje, Vasco Grossmann, Claudius Zelenka, Sabine Dippel, Rainer Kiko, Mariusz Oszust, Matti Pastell, Jenny Stracke, Anna Valros, Nina Volkmann, and Reinhard Koch. one annotation enough? A data-centric image classification benchmark for noisy and ambiguous label estima- tion. In Sanmi Koyejo. S. Mohamed, A. Agarwal, Danielle Belgrave, K. Cho, and A. Oh (eds.), Advances in Neural Information Processing Systems 35: Annual Conference Neural Informa- tion Processing Systems 2022, NeurIPS 2022, New Orleans, LA, USA, November 28- December 9, 2022, 2022.
14

---

Published as a conference paper at ICLR 2026
Robin Senge, Stefan Bösner, Krzysztof Dembezynski, Jörg Haasenritter, Oliver Hirsch, Norbert Nonner-Banzhol and Eyke Hüllermeier. Reliable classification: Learning classifiers that distin- guish aleatoric and epistemic uncertainty. Inf. Sci., 255: 2014.
Murat Sensoy, Lance M. Kaplan, and Melih Kandemir. Evidential deep learning to quantify clas- sification uncertainty. In Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Garnett (eds.), Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurlPS 2018, December 3-8, 2018, Montréal, Canada, pp. 3183-3193, 2018.
Michael Tschannen, Alexey Gritsenko, Xiao Wang, Muhammad Ferjad Naeem, Ibrahim Alabdul- mohsin, Nikhil Parthasarathy, Talfan Evans, Lucas Beyer, Ye Xia, Basil Mustafa, Olivier Hénaff, Jeremiah Harmsen, Andreas Steiner, and Xiaohua Zhai. SigLIP 2: Multilingual vision-language encoders with improved semantic understanding. localization, and dense features. preprint 14T7:22022.17786, 2025.
Matias Valdenegro-Tone Toro. Deep sub-ensembles for fast uncertainty estimation in image classification. CoRR, abs/1910.08168, 2019.
Joost Amersfoort, Lewis Smith, Yee Whye Teh, and Yarin Gal. Uncertainty estimation using a single deep deterministic neural network. In Proceedings of the 37th International Conference on Machine Learning, ICMI. 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning Research, pp. 9690-9700. PMLR, 2020.
Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Courna- peau, Evgeni Burovski. Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson K. Jarrod Millman, Nikolay Mayorov, Andrew R. Nel- son, Eric Jones, Robert Kern, Eric Larson, C J Carey, ilhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, Paul van Mul- bregt, and SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17:261-272, 2020. doi: 10.1038/s41 1992-019-066---2
Peter Walley. Statistical reasoning with imprecise probabilities. 1991.
Kaizheng Wang, Fabio Cuzzolin, Keivan Shariatmadar, David Moens, and Hans Hallez. Credal wrapper of model averaging for uncertainty estimation in classification. In The Thirteenth In- ternational Conference on Learning Representations, ICLR 2025, Singupore, April 24-28, 2025. OpenReview.net, 2025a.
Kaizheng Wang, Keivan Shariatmadar, Shireen Kudukkil Manchingal, Fabio Cuzzolin, David Moens, and Hans Hallez. Creinns: Credal-set interval neural networks for uncertainty estima- tion in classification tasks. Neural Networks, 185:107198, 2025b.
Yibin Wang, Haizhou Shi, Ligong Han, Dimitris N. Metaxas. and Hao Wang. Blob: Bayesian low-rank adaptation by backpropagation for large language models. In Amir Globersons, Lester Mackey, Danielle Belgrave, Angela Fan, Ulrich Paquet, Jakub M. Tomezak, and Cheng Zhang (eds.), Advances in Neural Information Processing Systems 38: Annual Conference on Neural Information Processing Systems 2024, NeurIPS 2024, Vancouver, BC, Canada, December 10- 15, 2024, 2024.
Tobias Weber, Bálint Mucsányi, Lenard Rommel, Thomas Christie, Lars Kasüschke, Marvin Pförtner, and Philipp Hennig. laplax laplace approximations with JAX. CORR, ab//250..101333 2025.
Samuel Wilks. The large-sample distribution of the likelihood ratio for testing composite hypothe- ses. The annals of mathematical statistics, 9())660-62, 1938.
Han Xiao, Kashif Rasul, and Roland Vollgraf. bushion-mmiste a novel image dataset for benchmark- ing machine learning algorithms. CoRR, abs/ 0774777 2017.
Adam X. Yang. Maxime Robeyns, Xi Wang, and Laurence Aitchison. Bayesian low-rank adaptation for large language models. In The Twelfth International Conference on Learning Representattons, ICLR 2024, Vienna, Austria, May 7-11, 2024. OpenReview.net, 2024.
15

---

Published as a conference paper at ICLR 2026
Xiaohua Zhai, Basil Mustafa, Alexander Kolesnikov, and Lucas Beyer. Sigmoid loss for language image pre-training. In IEEE/CVE International Conference on Computer Vision, ICCV 2023, Paris, France, October 1-6, 2023, pp. 11941-11952. IEEE, 2023.
Sheng Zhang, Yanbo Xu, Naoto Usuyama, Hanwen Xu, Jaspreet Bagga, Robert Tinn, Sam Pre- ston, Rajesh Rao, Mu Wei, Naveen Valluri, Cliff Wong, Andrea Tupini, Yu Wang, Matt Mazzola, Swadheen Shukla, Lars Liden, Jianfeng Gao, Angela Crabtree, Brian Piening, Carlo Bifulco, Matthew P. Lungren, Tristan Naumann, Sheng Wang, and Hoifung Poon. A multimodal biomed- ical foundation model trained from fifteen million image-text pairs. NEJM AI, 2(1), 2024.
Bolei Zhou, Agata Lapedriza, Aditya Khosla, Aude Oliva, and Antonio Torralba. Places: A 10 million image datahase for scene recognition. IEEE Trans. Pattern Anal, Mach. Intell. 40(6): 1452-1464, 2018.
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ORGANIZATION OF THE APPENDIX
Appendix A provides proof for Proposition 2.1 and Propo- sition 3.1, followed by detailed description of implementation in Appendix B. Appendix C describes the newly introduced credal spider plots in detail, before we give details about the differ- ent setups of our experiments in Appendix D. We finish with additional results in Appendix E and two ablation studies in Appendix F.
We structure the appendix as follows:
Proofs
18
Implementation Details
20
C Guide on Interpreting Credal Spider Plots
21
Experimental Setup
22
D.1 Models
22
D.2 Datasets
23
D.3 Baselines
25
D.4
Compute Resources
26
D.5
Generating Semi-Synthetic Ground- Distributions
26
D.6
Turning CL.IP-based Models into Zero-Shot Classifiers
26
Additional Experimental Results
28
E.1
Coverage versus Efficiency
28
E.2
Out-of-Distribution Detection
................................
28
E.3
In-Context Learning with TabPFN
................................
30
E,4
Zero-Shot Classification with CLIP-Based Models
31
Ablations
35
F1
Number of Ensemble Members in Out-of-Distribuun Detection
35
F2 a- Values for Active In-Context Learning...
35
F33 Accuracy and Expected Calibration Score Evaluation for single models
35
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A PROOFS
Proof of Proposition 2.1. Write ((h) = L(h)/ sup DEN L(g) E 10, Co = { E H >(h) 2 a), yeP and 2n.a = {p(-1 x, h) h E ). If0 < a2 S a I 1 and h E Ca. then 7 (h) 2 ai 2 02 hence h Cag. Thus Co, S Caz and, by applying the prediction map, 2 C 2z.ng. Consequently, for each class k. Pa. a1) infnec., p=(a,h) 2 infneC., P& h) = PA (2; a2), and similarly (x; a1) supheCay (x( h) S suphec., (a h) = (2;
an MLE f ML exist, then (/h")) = and C₁ is the set of MLEs. In particular, if the (predictive) MLE is unique at (e.g., the MLE is unique, or all MLES agree at œ), then
2x.1 {p(- x h } and Ip, 12: 1)] {px ()) }.
As a +. 0, the sets Ca increase to {h E H L(h) > 0), hence Qa,a t {pl. x h) L(h) > 0}- For an increasing family of sets, coordinate- infima over 2x decrease the infimum over the union, and suprema increase to the supremum. Therefore.
2 (2) 4
inf pk (2, h), (h:L(h)>00)
(=(2;)) f sup (a (1,,)), (h:((k)20)
as claimed. This completes the proof.
Proof of Proposition 3.1. (a) For each n, define (n(c) = llogg (n)
y(m (c). Then
K
Φn (c) = (Cgge)) c) Σ exp(2) (m) +c) + const(>!!))
log
hence th 0000 Direct differentiation yields
Von(c) = Byt - (c),
V (p (c) = -Diaaia =- -p
(c)) 3 0,
so each on is concave, and thus Af(c) = Σn (x(c)-E Sn (0) is C and concave. The Hessian matrices in the sum are positive semi-definite with nullspace containing span [1), since (Diag(p)- pp" ) 0; therefore V²)((c) 0 on S provided at least two classes appear. Concavity implies that every level-set {e Al(e) 2 } is convex. Non-emptiness follows from A! = 0 2 log a.
To compactness of (s( when least two classes appear, fix d E ES) (0) and consider ld with + 0o. Then
Al(td) Σ
(eem), td) log
+eds + const
Nydj
log
(n) ttd: + const.
As t 00, log E, et 22(l)+ tmax di + 0(1), hence
Al(td) = Σ Nydj N Nmmxdd) + O(1).
Because d S and least two de differ, have Σ, Nydj < N max; d. Thus Al(td) along every ray in S. proving coercivity on S, and hence compactness of Fs(a).
(b) Follows by differentiating
Ph (x; = (Zx(0) cx) log
(a))+t
The gradient and Hessian stated, and negative demi-eefiniteness of the Hessian shows cavity. The coordinate-we monotonicity is immediate from ò log Pr/Oc; = S.x - (x; c) together with p. (2;; e) (0, 1).
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(c) Since log (=: is concave and F(a) is convex, suPe logpp(z;c) c) is concave maxi- mization, i.e., convex optimization problem. Existence of optimizer Fs(a)) follows from compactness; invariance along span [1] yields uniqueness only modulo translations by 1. The equality between supps and exp(sup log pk) follows from strict monotonicity of the exponential.
(d) Because (o; is concave, minimizing Pk is equivalent to minimizing a concave func- tion, which is not convex optimization problem in general. Nevertheless, on the compact convex set (a) minimizer exists and is attained at an extreme point by standard results on concave functions over convex compact sets.
This completes the proof.
Proof of Corollary 3.1. Fix E (I,...,K] and restrict the multivariate objects Proposition 3.1 to the affine line tek, R.
(a) Since A!(-) is concave on R2 by Proposition 3.1(a), its restriction + A& (t) is concave on R; non-emptiness follows from Al, (0) 0 2 log Moreover, if < NN < N, then (bx(t) + -00 (terms with y(") & decay like t). so Fx( a compact interval itz t. If N& E N0, the same tail check shows F2 (a) is
as - +oo (terms with g(7) * decay like -t) and as
closed (possibly half-in infinitei interval.
(b) From Proposition 3. (b),, V Vlogpa(<;;)) c) = c - p(æ; c). Along the line c = t.k.
log logpe(2;:) (ek - p( exs)) T ek = pk (ax; 1):
dt
hence (x; (m; t) p1 t)) > O. Thus pa t) is strictly increasing.
(c) Since (a(a)) is an interval and t + (x; t) is strictly increasing, the infimum/supremum over (a) are attained the endpoints:
Ps (x)) - pa (a; (x;
Px - pa (x;(;; 1t)-
(d) The feasible set can be written as {t R - (t)(t) < - log a). where A is convex. Minimizing & (resp. -) over this convex set convex program whose optimizer exactly the left (resp. right) endpoint tx (resp. ( In the half-infinite cases N₂ E {0, N) the same conclusions hold with the appropriate limits f -00, f = too (so pm (2; t = 0 or Pa (a t 1).
This completes the proof.
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B IMPLEMENTATION DETAILS
In this section, we provide detailed description of how our method EffCre is practically imple- mented, including the optimization procedure, the evaluation of probability intervals, and the steps taken to ensure computational efficiency.
As discussed in Section given the logits of the MLE, our method essentially solves two con- vex optimization problems per class to determine the boundaries boundaries-nameely namely, the lower and upper ofobabilities-of a plausible interval according to the relative likelihood constraint. Specifically, for each class logit of the MLE, we add value to the logit to perturb the resulting probability, thereby deriving bound for the plausible probability interval as a result of the optimization. In practice, we use the mi minimiie ze function from SciPy (Virtanen et al., 2020) to optimize this value, with the rela- tive likelihood threshold as a constraint, an initial solution of O, and bounds set to (-10000, 10000). Roughly speaking, each optimization produces a constant that is added to single class logit, giving the lower (or upper) bound of the plausible probability interval for that class, which used to con- struct the box credal set D As result, applying method to dataset requires solving 2K convex optimization problems for each value of a.
The constants obtained by our method, EffCre, then be used to evaluate method test data instances, thus, constructing probability intervals, and thereby credal sets. Each interval bound is directly associated with specific relative likelihood, which served as the constraint during the opti- mization. For models we trained ourselves, we use the training dataset to evaluate the log-lik log-likelihood and compare it with that of the maximum likelihood estimator (MLE) predictor to compute the rela- tive likelihood, as described in Section 2. When the original training data is unavailable, we instead use subset of the target dataset to compute the relative likelihood budget. For example, in the case of CLIP, we do not have access to the original training data, which spans many benchmark dataset in addition to a large sample of images from the internet. Since we want to make credal predictions for CIFAR- 10, we use the respective train split of CIFAR- 10 to compute the (relative) log- lielhooss in order to solve the optimization problem described above. Credal predictions are then made using the respective test split of the dataset.
In general, our setup allows for straightforward computation alpha-cuts once results for multiple alpha values have been obtained, task made feasible by the efficiency of our method. The implementation is simple and intuitive; for further clarity, we refer to the func- tion classwi se addi ng optim logit in the code https: / tht.....tt....gg..g..gi..... com/ hwwp//whoorr f efflccient-credal-preeec
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C GUIDE ON INTERPRETING CREDAL SPIDER PLOTS
So far, the quantitative evaluation of credal sets has mainly been restricted to the three class setting, due to the inability visualize credal sets in (K - -1)-simplex for K 3. As many machine learn- ing problems involve more then three classes and as a visual representation of the output of models can give useful insight, it is important be able to have such a visual representation. To enable this, we propose credal spider plots- these plots offer an intuitive way to evaluate the interval- based credal sets. Given an instance that we want evaluate, we plot the different classes as variables in the spider diagram and generate bars, starting at the lower probability and ending at the upper probability, for each class, which represent the (plausible) probability intervals. The ground-truth distributions is then plotted as multiple dots (depending probability mass) on the radii corresponding to the given probability mass of a class. As our method relies on the maximum likelihood estimate (MLE), additionally plot the MLE in similar way.
the number of classes with non-zero
In Section 4.4, we sort instances in descending order by the aleatoric and epistemic uncertainty associated with the predicted credal set using measures by Abellán al. (2006) that have been proposed on the basis of a number of suitable axioms. Specifically,
S(Om)
S(2x) + +S((2)) -(Qxx)
TU().))
AU((.))
EU((=))
Therefore, maximum aleatoric uncertainty (lower entropy) will manifest itself in the credal spider plot for classes as having intervals that include 1/K for all K classes. The maximum epistemic uncertainty (difference upper and lower entropy) is obtained by having similar plausible intervals as for alcatoric uncertainty. but additionally, the plausible interval for (at least) one class should admit a probability of 1. Besides, this instance-win coverage and efficiency can also easily be observed from the credal spider plot by evaluating whether the ground-trutt point fall into the plausible intervals and by considering the average length of the aforementioned intervals, respectively.
Stund
BYGEM
Oraal Publ
Figure 7: Example credal spider plots for CLIP, SigLIP, and SigLIP-2. used for illustration purposes. The credal spider plots includes the maximum likelihood estimate, the plausible intervals, and the ground-truth distribution.
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D EXPERIMENTAL SETUP
D.1 MODELS
Multilayer Perceptron We train multilayer perceptron on the ChaosNLI dataset. The model con-
sists of four linear layers with dimensions [768- 256 64 - 16 3) employing ReLU activations for all hidden layers, while the output layer uses Softmax function produce probability distribu- tions from the logits, For training, we adopt hyperparameters similar to those identified as optimal in Javanmardi al. (2024) (see Table 1).
ResNet18 For experiments on CIFAR-10, we employ the ResNet-18 implementation and train- ing configuration from https: //github.bb/////// com/ kuangliu/pytorerre cifar. This variant is tailored to CIFAR-10 10 and is trained entirely from scratch, without ImageNet pretraining.
TabPFN TabPFN (Tabular Prior-data Fitted Network) (Hollmann et al., 2022) is a transformer- based foundation model developed for supervised classification and regression tasks on small to medium-sized tabular datasets, typically up to 10, 000 samples and 500 features. Pre-trained on approximately 130 million synthetic datasets generated via structural causal models, TabPFN learns to approximate Bayesian inference through single forward pass, eliminating the need for task- specific tuning. It adeptly handles numerical and categorical features, missing values, and outliers. We use TabPFN for all experiments with tabular data as presented in Section 4.3. The model is publicly accessible under a custom license based Apache 2.0, which includes enhanced attribution requirement.
CLIP CLIP (Contrastive Language Image Pretraining) (Radford al., 2021) is multimodal neural network that learns visual concepts from natural language supervision. Trained on 400 mil- lion image-text pairs sourced from the internet, CLIP can understand images in the context of nat- ural language prompts, enabling zero-shot classification across various tasks without task-specific tuning. employs a vision transformer architecture to process images and causal language model to process text, aligning both modalities in a shared embedding space. This design allows CLIP to generalize wide range of visual tasks by interpreting textual descriptions directly. We em- ploy CLIP to assess our method's performance in zero-shot classification tasks, demonstrating its applicability to large-scale models without the need for task-specific training. The model is publicly available under the MIT License, permitting both academic and commercial use.
SigLIP SigLIP (Sigmoid Loss for Language-Image Pretraining) (Zhai et al., 2023) is a multi- modal vasion-language model that enhances the CLIP framework by employing a pairwise sigmoid loss function instead of the traditional softmax loss. This modification allows for more efficient scaling to larger batch sizes while maintaining SigLIP utilizes separate image and text encoders to generate representations for both modalities, aligning them in shared embedding space. The model has demonstrated superior performance in zero-shot image classification tasks compared to CLIP, achieving an ImageNet zero-shot accuracy of 84.5% with batch size of 32, 000. Same for CLIP, we demonstrate with SigLIP the ability to construct credal sets based on large-scale models. SigLIP is publicly available under the Apache 2.0 license, facilitating research and application in various domains.
improving performance at smaller batch sizes.
SigLIP-2 SigLIP-2 (Sigmoid Loss for Language-Image Pretraining 2) (Tschannen et al., 2025) multilingual vision- lasoun-anguaae encoder. Building upon the original SigLIP, SigLIP-2 in- tegrates advanced pretraining losses -self-distillation and masked prediction), and online data curation-to enhance semantic un- derstanding, localization, and dense feature extraction. The model demonstrates improved perfor- mance in zero-shot classification, image-text retrieval, and transfer tasks, particularly when extract- ing visual representations for Vision-Language Models. Notably, SigLIP introduces dynamic resolution variant, NaFlex, which supports multiple resolutions and preserves the native aspect ratio, making it suitable for applications sensitive to image dimensions. We use SigLIP- 2 in similar fashion as SigLIP and CLIP for large-scale experiments. The model is publicly available under the Apache 2.0 license, facilitating research and application across various domains.
iisquuss-iccluding captioning- based pretraining, self-supervised
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BioMedCLIP BiomedCLIF (Zhang al., 2024) is multimodal biomedical foundation model, pretrained on the PMC-15M dataset- a collection of 15 million figure-caption pairs extracted from over 4.4 million scientific articles in PubMed Central. Utilizing PubMedBERT the text encoder and Vision Transformer as the image encoder, B omedCIIP tailored for biomedical vision- language processing through domain-specific adaptations. It has demonstrated state-of-the-an afrtth-rrt per- formance across various biomedical tasks, including cross-modal retrieval, image classification, and visual question answering, outperforming previous models such as Bio radiology-specifi tasks like RSNA pneumonia detection. The model is publicly available under the Apache 2.0 li- cense, facilitating research and application in the biomedical domain.
Hyperparameters For certain experiments in our empirical evaluation, dation) models, which do not require training and thus do not need hyperparameter specifications. In contrast, for the goverage-efficieme train models from scratch using a dataset-specific set of hyperparameters, summarized in Table 1. Multiple configurations were evaluated, and the best-perforning setup was selected individually for each dataset. To ensure comparability, all methods our approach as well as the baselines share the same hyperparameter settings within given dataset. The only exception is CreBNN, which, when trained with the Adam optimizer (Kingma & Ba. 2015), requires KL-divergence penalty of le and weight decay set to zero. When instead using SGD combined with a cosine annealing learning rate schedule (Loshchilov & Hutter, 2017), CreBNN additionally needs a momentum of 0.9 achieve stable training.
pre-trained (foun-
experiments CIFAR-10, ChaosNLI, and QualityMRI,
Table 1: Hyperparameters used for each dataset.
Hyperparameter
ChaosNLI
CIFAR-10
QualityMRI
Model Epochs
FCNet
ResNet18
ResNet18
300
200
200
Learning rate
0.01
0.1
0.01
Weight decay
0.0
0.0005
0.0005
Optimizer
Adam
SGD
SGD
Ensemble members
20
20
20
LR scheduler
CoseanAnne
CosineAnnelling
D.2 DATASETS
ChaosNLI ChaosNLI, introduced by Nie et al. (2020), is a large-scale dataset created to inves- tigate human disagreement in natural language inference (NLI). It includes 100 annotations per example for 3, 113 instances from SNLI and MNLI, as well as 1, 532 examples from the aNLI dataset, totaling around 464, 500 annotations. In line with Javanmardi al. (2024), we focus only on the SNLI and MNLI portions, which we refer to simply as ChaosNLI for convenience. Each entry provides rich metadata, including a unique identifier, the count of labels assigned by annotators, the majority label, the full label distribution, the distribution's 's entropy, the original text, and the original label from the source dataset. ChaosNLI facilitates detailed study of variability in human judgments, highlighting examples where disagreement is high and illustrating the limitations of treating the ma- jority label as definitive ground truth. The dataset is publicly accessible under the CC BY-NC 4.0 License. For our experiments, https: / //githuu com/ all iizezaa-aa javanmardi coonooraaloooooooooooooccooa mwnfformml-reeaalll. creeall-eaall--e- sets, with fur- ther details on their generation provided by Javanmardi al. (2024).
use the precomputed 4is-dimensional embeddings, available at
QualityMRI Introduced by Obuchowicz et al. (2020), the Quality dataset is part of the Data- Centric Image Classification (DCIC) Benchmark, which studies the role of dataset quality in shaping model performance. It consists of 310 magnetic resonance images that cover different quality levels, providing resource for assessing MRI image quality. The dataset is distributed under the Creative Commons BY-SA 4.0 License.
CIFAR-10 CIFAR-10, introduced by Krizhevsky et al. (2009) and Geoffrey Hinton in 2009, is a widely adopted benchmark in machine learning and computer vision. It consists of 60, 000 color
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images with resolution of 32 32 pixels, evenly divided among 10 classes: airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck. The dataset is split into 50, 000 training images and 10, 000 test images, organized into five training batches and single test batch, each containing 10, 000 images. CIFAR-10 is publicly available and has been extensively used for training and evaluating machine learning models. While the original dataset does not explicitly define a license, versions distributed through platforms such as TensorFlow datasets are provided under the Creative Commons Attribution 4.0 License.
CIFAR-10H CIFAR-10H provides human-generated soft labels for the 10, 000 images in the CIFAR-10 test set, reflecting the variability in human judgments for image classification. Intro- duced by Peterson al. (2019), the dataset contains 511, 400 annotations from 2, 571 workers on Amazon Mechanical Turk, with each image receiving around 51 labels. Each annotation assigns an image one of the ten CIFAR-10 classes, allowing the creation of probability distribution over labels for every image. CIFAR-1OH is publicly available under the Creative Commons BY-NC-SA 4.0 License.
CIFAR-100 CIFAR-100, introduced by Krizhevsky et al. (2009), consists of 60, 000 color images at resolution of 32 x 32 pixels, organized into 100 classes with 600 images per class. Each im- age carries both a "fine" label, indicating its specific class, and "coarse" label corresponding to one of 20 broader superclasses. The dataset is divided into 50, 000 training images and 10, 000 test images. CIFAR- 100 is derived from the Tiny Images dataset and is widely used for benchmarking image classification models. While the original dataset does not define a license, versions distributed through platforms such TensorFlow Datasets are available under the Creative Commons Attribu- tion 4.0 License.
SVHN The SVHN dataset, introduced by Netzer et al. (2011), contains over 600, 000 32 x 32 RGB images of digits (0-9) extracted from real-world house numbers in Google Street View. It is organized into three subsets: 73, 257 images for training, 26, 032 for testing, and an additional set of 531, 131 images for extended training. SVHN is intended for digit recognition tasks and requires minimal preprocessing. Although the original dataset does not specify license, versions distributed through platforms like TensorFlow Datasets available under the Creative Commons Attribution 4.0 License.
Places365 Places365, introduced by Zhou et al. (2018), is a large-scale dataset for scene recog- nition, comprising 1.8 million training images spanning 365 scene categories. The validation set contains 50 images per category, while the test set includes 900 images per category. An expanded variant, Mices366-Challlene-2000
incorporates an additional 6.2 million images and 69 new scene categories, bringing the total to million images across 434 categories. Although the original dataset does not specify license, versions distributed through platforms such as TensorFlow Datasets are available under the Creative Commons Attribution 4.0 License.
FMNIST Fashion-MNISS (FMNIST), introduced by Xiao al. (2017), contains 70, 000 grayscale images of Zalando products, each sized 28 x 28 pixels and categorized into 10 classes, including T-shirt/top. Trouser, and Sneaker. The dataset divided into 60, 000 training images and 10, 000 test images, and it is commonly used as a modern replacement for the original MNIST dataset machine learning benchmarks. FMNIST is publicly released under the MIT License.
ImageNet ImageNet, introduced by Deng et al. (2009), is a large-scale image dataset structured according to the WordNet hierarchy, comprising over 14 million images spanning more than 20, 000 categories. Its ILSVRC subset, commonly referred as ImageNet- contains 281, 167 training images, 50, 000 validation images, and 100, 000 test images across 1, 000 classes. The dataset is freely accessible to researchers for non-commercial purposes.
TabArena Benchmark Data TabArena (Erickson et al., 2025) is continuously maintained benchmarking system designed for evaluating tabular machine learning models. It comprises 51 manually curated datasets representing real-world tabular tasks, including both classification and regression problems. Each dataset has been evaluated across 9 to 30 different splits, ensuring ro- bust performance assessments. The datasets encompass a diverse range of domains, such as finance,
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healthcare, and e-commerce, providing comprchensive foundation for benchmarking various ma- chine learning models. This diversity ensures that evaluations reflect the complexities and nuances found in real-world tabular data scenarios. We use 7 different datasets with IDs 46906, 46930, 46941, 46958, 46960, 46963, 46980 from the benchmark to validate our method in different ex- periments. Each of the datasets contains classification task with 2 or more classes and a number of instances between 898 and 12, 684. The datasets are publicly accessible and released under the Apache 2.0 license, ensuring permissive use and redistribution for research purposes.
D.3 BASELINES
Below, we provide detailed descriptions of the baseline implementations used in this paper, relying on the implementations provided by Löhr et al. (2025).
Credal Prediction based on Relative Likelihood (CreRL) The implementation of Credal Pre- diction based on Relative Likelihood (Löhr et al., 2025) is provided in https://g/ thub. com/ www.ooo--reaaaieeeeeeeeeeeeeeeee
We use the provided code to perform all experiments involving CreRL. Similar to our method, the CreRL defines plausibility in terms of the relative likelihood of a model. One significant difference that, while CreRI. tries to find sufficiently diverse hypotheses that satisfy the relative likelihood criterion, therefore having to train an ensemble of model, we directly obtain the plausible probability intervals by varying the logits of the maximum likelihood estimate.
iimoverse/
Credal Wrapper (CreWra) The Credal Wrapper (Wang et al., 2025a) was initially implemented in TensorFlow, but then reimplemented in PyTorch to ensure compatibility with other baselines. It follows a standard ensemble learning approach, training multiple models independently. Like our method, the Credal Wrapper constructs credal sets using class-wise upper and lower probability bounds, making it well-aligned with our implementation.
Credal Ensembling (CreEns,,) Our implementation adheres closely to the specifications outlined in Nguyen al. (2025). The method extends standard ensemble training. adapting the inference stage by ranking predictions according to a distance metric and including only the top a% of closest predictions when forming the credal sets. In our experiments, we employ the Euclidean distance and test multiple a values.
Credal Deep Ensembles (CreNet) Since the official Credal Deep Ensembles implementation was provided only in TensorFlow, it was reimplemented by Löhr et al. (2025) in PyTTrcc to integrate seamlessly with other baselines. The version maintains the key design choices of the original, espe- cially regarding the architecture and loss function. In particular, the models' final linear layers are replaced with head that outputs classes values corresponding to upper and lower probability bounds, followed by batch normalization and the custom IntSoftmax activation. The loss function applies standard cross-entropy the upper bounds, while for the lower bounds, gradients are prop- agated only for the 8% of samples exhibiting the largest errors, in line with Wang et al. (2025b). For our experiments, we adopt 8 0.5 as recommended in the original work.
Credal Bayesian Deep Learning (CreBNN) The method proposed by Caprio et al. (2024a) was reimplemented by Löhr al. (2025) using only the high-level description from the original work. The ensemble consists of Bayesian neural networks (BNNs), each trained via variational inference with distinct priors: the prior means μ are drawn from [-1, 1] and the standard deviations a from [0.1, 2). ensuring diverse prior set. At inference time, we sample once from each BNN to generate finite collection of probability distributions, and the credal set is defined as the convex hull of these samples.
Evidential Deep Learning Our implementation of Evidential Deep Learning follows (Sensoy et al., 2018) as closely as possible. We use a single model and include a SoftPlus activation function after the last layer ensure the output is non-negative. We use the Type II Maximum Likelihood as a loss function and the KL-divergence as a regularization term as described in (Sensoy et al., 2018). The regularization term is scaled by X; min(1, 1, i/10) epoch as also done in the original work.
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At inference time, the model predicts the evidence for each class, which can then be used to compute the parameters the corresponding Dirichlet distribution.
Deep Deterministic Uncertainty For the Deep Deterministic Uncertainty method (Mukhoti et al., 2023), we used the original implementation provided onegafragger/DDU, This approach uses single model to reason about uncertainty. In the original paper, the authors apply additional techniques techniquss-includnng includdig spectral normalization and resid- ual connections- to encourage son, we omit these techniques to ensure a fair comparison, as integrating such modifications into pre-trained model would require re-training the model. Thus, we rely the identical, trained ResNet 18, 8, which is also used for the other experiments. At inference time, epistemic uncertainty can be quantified through density estimation in space: normal distribution is fit the em- beddings of training data for each class, and epistemic uncertainty is computed on the basis of the likelihood of new embeddings under this distribution.
com/
regularized embeddings in feature-space space. For compari-
D.4 COMPUTE RESOURCES
All experiments in this work were conducted using the computing resources listed in Table 2, with an estimated total GPU usage of approximately 820 hours.
Table 2: Specifications of Computing Resources
Component
Specification
CPU
AMD EPYC MILAN 7413 Processor, 24C/48T 2.65GHz 128MB L.3 Cache 2 NVIDIA A40 (48 GB GDDR each)
GPU
RAM
128 GB (4x 32GB) DDR4-3200MHz ECC DIMM
Storage
2 480GB Samsung Datacenter SSD PM893
D.5 GENERATING SEMI-SYNTHETIC GROUND-TRUN DISTRIBUTIONS
Due to a lack of ground-truth distributions, the evaluation of credal predictors remains non-trivial. While number of datasets have (test) set that includes multiple human annotations such the ones used in this work Therefore, we work-most most of the commonly-use ground-truth distributions. The general idea is as follows: given a training set Dirain we either train a model retrieve strong model from model hub. The trained or retrieved model is then considered to be the ground-truth model h² and ground-truth distributions may be generated by collecting the predicted distributions p(- he) based on instances from the Drain or Diest The model that is to be evaluated (in terms of coverage and efficiency) is then trained on the same instances E Durinn but the labels are sampled from p(- x h"). Thereafter, the model can be evaluated using the test set.
benchmarking datasets do not provide these. simple method to generate semi-synthetic datasets that include (conditional)
For example, in Section 4.3, we train a Ra RandomForest with the default parameters from scikit- learn (Pedregosa et al., 2011) with the exception of maximum depth; this is set to 5 to prevent the predicted distributions too "peaked". The RandomF orest is assumed to be the ground-truth model ht and it's prediction for an instance is taken to be the ground-truth conditional distribution p(- x, ht). The TabPFN model is then trained on the same instances æ, but the labels y are realizations sampled from the distribution p(- h²). The model is then evaluated on the test set, for which the ground -ruth distributions are also generated by the RandomE Porest.rrst.t.oooo h*
We refer to this semi-synthetic because, while the generated distribution is not (necessarily) the ground-truth, under the assumption that the used model is sufficiently well-trained, they should be "close" to the ground-truth.
D.6 TURNING CLIP-BASED MODELS INTO ZERO-SHOT CLASSIFIERS
To demonstrate the usefulness and flexibility of our method for producing credal sets for any black- box model structure without the need for retraining, we apply it to multi-modal cL.IP-based models.
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Contrastive Language Image Pretraining (CLIP) (Radford al., 2021) introduced a mechanism to pre-train models that share embeddings across two modalities. The training data consists large corpus of images and their corresponding descriptions (e.g,, captions alternative text from web- sites). The central idea is to align each image with its textual description: images and their captions should be close in the emhedding space, while mismatched pairs should be far apart. To achieve this, two moodaltt-sppeciii encoders are trained to produce embeddings of equal dimension, from which a similarity score (e.g., cosine similarity) computed. Captions that accurately describe an image re- ceive high similarity scores, whereas unrelated captions receive low scores. This training paradigm and model architecture have since been refined by subsequent works, yielding better-pent peatter-prfoomingg or more specialized models. For example, Bi omedCLIP (Zhang et al., 2024), trained on biomedical data from PubMed, achieves superior performance on medical tasks. Similarly, the SigLIP (Zhai et al., 2023) and S Sig.IP-- (Tschannen et al., 2025) families adapt the training procedure and ex- tend the datasets to include multilingual text sources, resulting in improved performance on general tasks (Zhai et al., 2023; Tschannen et al., 2025).
Zero-Shot Prediction. Zero-shot image classification with CLIP-based models proceeds by refor- mulating the label into natural-language lemplates. For each candidate class, short descriptive text is created (c.g., the template "a photo of a [label]" yields "a photo of dog" or "a photo of cat"). These textual descriptions are embedded by the text encoder, while the imput image is embed- ded by the image encoder. The similarity between the image embedding and each text embedding then computed, typically using cosine similarity. The resulting similarity values can be treated as logits, where the highest-scoring label determines the predicted class. Importantly, this formu- lation also makes straightforward to restrict classification to any subset of labels without training classifier, since can simply retain and compare the logits corresponding to the labels of interest. This procedure enables CLIP-based models to serve as flexible, task-agnostic classi- fiers without requiring any additional training, and has proven effective across diverse downstream domains (Radford et al., 2021; ; Zhang et al., 2024; Zhai et al., 2023; T'ccannen et al., 2025).
Templates for Multi-Lingual Datasets. Zero-shot classification can be extended multi- lulti-lingual datasets by translating labels into the target language and constructing corresponding templates. For example, in our experiments we used the English template "This is photo of [label]" alongside Swahili template "Hii ni picha [label]", allowing classification in either language. Models such as S gL.IP-2 (Tschannen al., 2025), trained on multilingual data, further improve robustness in this setting.
To illustrate the effectiveness of different CLIP-based models in our setting, report their zero-shot classification accuracy CIFAR-10, ImageNet, and DermMNIST (see Table 3). The results show that while standard CLIP performs strongly on general-purpose datasets, specialized variants such as Bi BiomedCLIP yield improved performance on domain-specifi tasks, and recent multilingual models like SigLIP and SigLIP-- further enhance accuracy on broad benchmarks.
Model Performance.
Table 3: Zero-shot classification accuracy (%) of CLIP-based models on CIFAR-10 (EN = English, SW = Swahili, FR French, ZH Chinese), ImageNet, and DermaMNIST.
Model
CIFAR-10
ImageNet
DermaMNIST
EN
SW
FR
ZH
CLIP 88.97%
9.11%
85.97%
33.73%
57.14%
24.74%
SigLIP
92.17%
15.33%
84.05%
91.28%
72.83 %
8.28%
SigLIP2
93.91%
10.21%
92.21% %
93.85%
69.87%
11.67%
BiomedCLIP
45.89
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E ADDITIONAL EXPERIMENTAL RESULTS
E.1 COVERAGE VERSUS EFFICIENCY
In addition to the datasets provided Section 4.1, we present an additional comparison to the base- lines in the form of the QUALITYMRI dataset. Figure 8 shows that approach Pareto dominates the
QualityMRI
1.0
1.0
0.8
0.8
Efficiency
0.6
0.6
a
Ours
0.4
EffCre
0.4
Baselines
CreRL
0.2
CreEns
0.2
CreWra
CreBNN
0.0
CreNet
0.0
0.2
0.4
1.0
Coverage
Figure 8: Coverage Efficiency. Our method, EffCre, is compared to baselines QUALITYMRI.
CreRL method, while having a similar coverage and efficiency to CreBNN for a = 0.4. However, our method allows a trade-off between coverage and efficiency beyond that, allowing the explo- ration of regions with a better efficiency or better coverage. Our method is Pareto incomparable to the CreEns, because CreEns does not reach the high coverage region (while having higher efficiency, whereas our method does (while having lower efficiency). It should be noted that reaching the high coverage area, as our method does, especially important in medical settings, is the case for the QualityMRI dataset.
E.2
DETECTION
For this experiment, trained ResNet18 on CIFAR-10, which serves the in-distribution dataset. At evaluation time, we consider both the in-distribution data and five out-of- distribution datasets to compute epistemic uncertainty values with our method. These values are then used to separate ID from OOD samples, with performance measured via AUROC. In addition to the results in Section 4.2, Table 5 reports AUROC scores across different values for our method, and Table states the corresponding training and inference times for each method. Furthermore, Appendix F.I presents an ablation study on the effect of the ensemble size for OOD detection performance.
Table 4: Training and inference time in seconds for models trained on CIFARIO. Mean with standard de- viation over three runs. Computed based ensembles with 10 members.
Method
Training time
Inference time
EffCre CreRL
2136.33 + 1.70
1.50 + 0.02
12675.84 + 412.68
11.46 + 0.12
CreWra
21363.34 33.99
11.44 +: 0.19
CreEns
21363.34 33.99
11.44 + 0.23
CreNet
24996.65 = 180.12
11.41 + 0.18
CreBNN
29796.67 = 12.94
12.74 1.1
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Table Out-of-Distribution Detection.
Method
SVHN
Places365
CIFAR-100
FMNIST
ImageNet
RffCreo.o
0.408:10.006
0.408
0.480=10.005
0.486:100002
0.481 40.002
EffCreo.2
0.47410.003
0.474110.001
0.473:10.002
0.47411.002
0.473
EffCreg.4
0.303 1..10000100
0.389 100072
0.335.10.040
0.32510.057
0.338 1.388.0.0000 035
EffCreo.6
0.415=+0.010
0.428=0.007
0.440=#00017
0.4044+0002
0.435=0.000 008
BffCrep.s
0.744:+0.008
0.721=0.000
0. 0.720==0.00F
0.733#+00012
0.700=0.00 008
EffCrep.y
0.85411.0005
0.827 000006
0.822:10.004
0.860
0.796
EffCrep.95
0.885:10.008
0.862
0.854+1..00s
0.907
0.826=10.00
EffCree.o
0.894+000015
0.886
0.868410.005
0.933+0.010
0.844
CreRLo.98 0.95
0.917=0.015
0.910#0.001
0.901
0.945
0.878
Wra
0.957=+0.00
0.916=0.000
0.9 0.916=000000
0.952#0.000
0.890=0.000 000
CreEnsg.o
0.955.10.000
0.913 0.000
0.91410.0001
0.949
0.888110 0..000
CreBNN
0.907 1.0006
0.885 0.0002
0.88011..002
0.935
0.859
CreNet
0.943++0.00
0.918=0.000
0.912++0.000
0.951+0.002
0.884+0.000
In the main paper, we focused exclusively on comparing credal predictors in order to ensure a con- sistent evaluation of methods within a single framework (that of credal predictors). This allows us to isolate the effect of the credal predictor from the influence of other factors such as the uncertainty measure the base model. Here, additionally compare method to other methods that allow for uncertainty quantification with a single model. In particular, we compare EffCre to evidential deep learning (EDL) (Sensoy al., 2018) and deep deterministic uncertainty (DDU) quantifica- tion (Mukhoti et al., 2023). For evidential deep learning, the epistemic uncertainty quantification is computed
K EU
S
k= (= + denotes the sum of the predicted pa- rameters of the Dirichlet distribution for an input for æ. Deep deterministic uncertainty quantifies epistemic uncertainty on the basis of the likelihood of the embedding of an input
where K the number of classes and S = >
EU Σ a(e k))(k).
where q represents the density function of normal distribution. The implementation details described in Appendix D.3. The results are presented Table 6. When compared to EDL, our
Table 6: Out-of-DDisrrbbutio Detection.
Method
SVHN
Places365
CIFAR-100
FMNIST
ImageNet
EDL
0.938=0.010
0.889=0.00
0.887+0.00 0011
0.940 000
0.866#0.001
DDU
0.973+0.001
0.969=0.001
0.873
0.892+0.010
0.969+0.000
method (EffCrei.o) performs on par with EDL on Places365, while being slightly outperformed on the remaining datasets. However, it is important to emphasize that EDL requires re-training the model with a specific activation and loss function, hence it cannot be applied directly in standard settings, specifically if the training data is not available. In contrast, our method can be applied without re-training, making compatible with broader range of settings such as the ones using TabPFN and CLIP presented in the manuscript. Our method outperforms DDU on FMNIST, while being weaker on other datasets. Moreover, while DDU also a post-training method, requires access to the embeddings generated by the model, whereas our method does not. EffCre, by operating on logits, can be applied on top black-box models, which enables it to be directly applied in more general settings, such as large language models, which form interesting direction for future work. Overall, there no clear winner across the evaluated methods, and drawing definitive conclusions remains challenging. Besides the stark differences in the working of the
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Active In-Context Learning (ID 46963)
Active In-Context Learning (ID 46930)
0.92
0.96
0.90
0.88
0.94
Accuracy
0.86
Accuracy
0.84
0.82
0.90
0.80
EffCre (entropy)
0.88
EffCre (entropyl
0.78
EffCre (zero-one)
EffCre (zero-one)
MLE (random)
MLE Irandom)
0.76
0.86
200
400
600
800
250
1000 1250 1500 1750 Number of Instances
500
750
Number of Instances
Figure 9: Active In-Context Learning with TabPFN, Performance on TabArena datasets 46963, and 46930 versus the random baseline.
methods, the OOD detection task itself comes with numerous caveats (Li et al., 2025), meaning that performance on this task can only be taken as proxy of the quality of the epistemic uncertainty representation.
Additionally, although these methods also rely on a single (re-trained) model, they differ from the other (credal) approaches in that they do not produce credal sets. We consider this distinction to be particularly important, as the credal set predictors quantify fundamentally different form of epistemic uncertainty than DDU. Indeed, the credal set represents epistemic uncertainty with respect to the predicted probability distribution, which will directly affect the subsequent decision- making. DDU, however, quantifies a form of epistemic uncertainty about the "familiarity" of an input, derived from its density relative to the training data. is not immediately clear how this should influence the decision-maln making process that follows.
E.3 IN-CONTEXT LEARNING WITH TABPFN
We compute coverage and efficiency for our method used with TabPFN with all multi-class TABARENA datasets. As discussed in Section 4.3, the datasets do not come with ground-truth dis- tributions. Therefore, we constuct semi-synthethic distributions that serve as the ground-truth. In Appendix D.5, give a detailed explanation of our approach. Note that consider the resulting distributions to only be proxy of the "true" ground-truth distributions. In addition to computing the coverage and efficiency, that will follow, this is done by first splitting the data, in stratified manner, to have a 0.3 test split. The remaining 0.7 split is then split into initial training and the sampling pool, again stratified, such that the initial training set contains 2K instances, where K is the number of classes. At every iteration, the predictor "allowed" to sample 2K instances from the pool, based on its epistemic uncertainty, which is common setup in active learning (Nguyen et al., 2019; Margraf et al., 2024). This is done until the pool exhausted- and, hence, until the performance converges what would be obtained with a traditional train-test split. The goal is thus, to select at every iteration samples that are most informative, i.c. the samples that will give the greatest performance increase at that it- eration. In addition to the results in Section 4.3, we present active in-context learning results for two additional TABARENA datasets: 46963 and 46930. Figure shows the results for our method using epistemic uncertainty sampling based on (5) and (6) compared to the random baseline. Conform the results presented in Section 4.3, our method applied to TABPEN provides a valuable advantage over the random baseline in terms of accuracy.
perform active in-context learning. In Section 4.3 and the results
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E.4 ZERO-SHOP PERO-SHOT CLASSIHICATION WITH CLIP-BASED MODELS
Extending on the examples shown in Section 4.4, we create additional credal spider plots for CLIP- based models in Figures 10 12. Figure 11 highlights challenging natural images with high uncer- tainty in CLIP, Figure 12 examines medical images from DERMAMNIST, and Figure 10 analyzes cross-lingual predictions on CIFAR-10. Together. these visualizations complement the quantitative results reported in Table 3 by showcasing how credal spider plots reveal distinct uncertainty patterns that align with the models' performance across domains.
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DINR1O label: bird
CLP
CLP
SiLLIF2
(English)
(French)
CLP
Uack aech
SigLPP
SIyLI2
701
(Chinese)
(Swahili)
Figure 10: Credal spider plots for an image of a bird with CLIP, SigLIP, and SigLIP-2 across different languages. In English, all models confidently predict the image as bird. In French, SigLIP-2 maintains the correct maximum likelihood prediction but shows increased uncertainty toward cat. In Chinese, CLIP exhibits high uncertainty across all classes, indicating difficulties in this language, whereas S SigLIP and SigLIP-2 remain as confident as in English. In Swahili, all models struggle and display high uncertainty across all classes; notably, birs bird and a aplanne share the same word in Swahili, complicating the prediction. These examples align well with models' performances across the different languages (see Table 3).
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FAL
DIATE
CLP
KaLP
SIGITS
OHIP
SIgIPS
(c)
FGL
Figure 11: Comparison of credal sets for CLIP, SigLIP, and SigLIP-2 on observations with high uncertainty with CLIP. Observation (a) shows swimming deer, where the MLE is ship. High uncertainty is spread across ship, two sky-related classes (airplamd bird), the amphibi- frog, and the correct class deer. Both Si gLIP models exhibit similar patterns with greater uncertainty. Observation (b) depicts dark image of a frog misclassified as ship, with high uncertainty again on that class; both SigLIP models additionally assign probability cat. Observation (c) is challenging deer image, where annotators themselves showed high disagree- ment. CLIP is confident it is either deer or horse, while SigLIP favors bird or frog, and SigLIP-2 remains certain it is an animal but not which. Observation (d) illustrates case where human annotators are nearly evenly split between cat and dog, and the uncertainties of three models capture this ambiguity.
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BlomadCLIP
CLP
DETTPPIIST shi retivicyte
(a)
BlomedCLIP
returne
(b)
meiinone
Figure 12: Comparison credal sets for BiomedCLIP and CLIP on DERMAMNIST for melanocytic nevi (a) and melanoma (b). While Bi BiomeCCLIP demonstrates higher overall per- formance than CLTP (see Table 3), it misclassifies the melanoma with high confidence and low uncertainty, which could be dangerous if applied in medical contexts. Interestingly, CLIP classifies the melanoma correctly, albeit with greater uncertainty. Both models predict the melanocytic nevi correctly, though omedCLIP shows increased uncertainty toward the related melanoma class, suggesting a more challenging case.
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F ABLATIONS
This section contains additional ablation experiments.
F.1 NUMBER OF ENSEMBLE MEMBERS IN OUT-OF-DISTRIBUTION DETECTION
We provide an additional ablation study on the impact of the ensemble size on out-of-distribution performance. Table 7 and Figure 13 demonstrate once more the efficiency of our approach: it requires only a single trained model. In contrast, ensemble-based baselines typically rely on at least five members and benefit from larger ensembles to improve performance.
Table 7: Ablation of different numbers of trained ensemble members for tection.
De-
Method
Members
SVHN
Places365
CIFAR- 100
FMNIST
ImageNet
EffCree.95
1
0.885+0.003
0.862+0.005
0.854+0.003
0.907
0.826+0.004
CreRLo.9e
5
0.917:0.012
0.894
0.885 0022
0.928
0.863
CreWra
5
0.943
0.004
0.905
0.039
0.879=0.001
CreEnsnoo
5
0.938:10.009
0.898110.00
0.900
0.929:10.001
0.874:0.001
CreBNN
5
0.843
0.829
0,831
0,851
0.809=0.007
CreNet
5
0.938 003
0.908
0.900 001
0.941 003
0.871+0.002
CreRL
10
0.92110.010
0.90510.002
0.896110.001
0.940.1..002
0.87210.002
CreWra
10
0.953+0.000 004
0.911 0.00
0.912+0.0m 000
0.948 001
0.886+0.001
CreEnSo.o
10
0.949 001
0.907=0.001
0.909+0.000 002
00.41
0.883+0.001
CreBNN
10
0.880
0.856
0.85941.0m .002
0.886
10.01
CreNet
10
0.944++0.007
0.915:1.000
0.90811..00
0.949:1.0001
0.881
CreRL 0.95
20
0.917=0.013
0.910=0.001
0.901 10.000
0.945
0.878=0.002
CreWra
20
0.957 003
0.916#0.001
0.916 .000
0.952 0000
0.890=0..01
CreEnso.0
20
0.955=20.001
0.913
0.914+0.000
0.949
0.008
CreBNN
20
0.907
0.885
1.88001..002
0.935
0.85910.002
CreNet
20
0.94310.003
0.918
0.91210.000
0.951 10.002
0.884
F.2 a- VALUES FOR ACTIVE IN-CONTEXT LEARNING
We provide an additional ablation on the effect that the a-value has on the performance of our method in active in-context learning. We evaluate runs for values a E {0.2, 0.4, 0.6, 0.8, 0.9, 0.95). In Figure 14, we provide the results for the TabArena datasets with OpenML. (Bischl et al., 2025) id 46941, 46963, and 46930. For the sake of legibility, we only consider the zero-one-loss-bate epistemic uncertainty measure (6).
We observe that higher a values consistently improve performance across all three datasets until the performance converges at a = 0.8. particular, lower a values result in larger predicted sets with high epistemic uncertainty, which reduces the meaningful separation between instances. Consequently, the optimal order for selecting instances during active learning is lost when a is small, explaining the drop performance.
F.3 ACCURACY AND EXPECTED CALIBRATION SCORE EVALUATION FOR SINGLE MODELS
If we have to commit to precise probabilistic prediction, a natural choice is to use the maximum likelihood estimate, which theoretically well-established approach. If, additionally, a class- wise prediction is sought, the argmax class can be predicted. To provide sense of the quality of the underlying models trained for our experiments, we report standard supervised-learning metrics, namely accuracy and expected calibration error, for each individual model in Table 8 based on the original CIFAR-16 10 test set. This serves as a sanity check to ensure fair comparison with the baselines. For experiments with TabPFN and CLIP models the pre- rre-trained models.
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0.95
0.95
4.
AUROC
AUROC
0.80
Cxmr
0.00
10 12 1920 Number of Ensemble Members
1611 121114 1916 19 Number of Ensemble Members
(a) Tashion-MNIST
(b) CIFAR-100
1.00
1.00
Drffrr Orke
AUROC
0.90
Cole
AUROC
0.90
0.85
0.85
0.75
0.75
10 121314 16 1920
10 Number of Ensemble Members
Number Ensemble Members
(c) Places365
ImageNet
Figure 13: Out-of-distribttion detection performance (based AUROC score) function of ensemble size. CIFAR- 10 the in-distribution data while various datasets are used as OOD data.
Active Context 469411
Active Context
Active
0.55
0.50
0.85
0.50
Accumery
5.75
0.80
2.05
0.35
0.50
enCre (a- 3.21
EMCre
enCr
0.15
effrr
0.05
efcr
100
200
300
392
520
700
0
200
400
600
500
250
500
750
1000 1250 1500 1750
Number of Instancees
Number Instances
Number of Instancee noes
(a)
(b)
(c)
Figure 14: Active In-Context Learning with TabPFN. Performance on TabArena datasets 46941, 46963, 46930 for different values of a.
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Method
Model
CIEAR10
ChaosNLI
QualityMRI
ECE
Ace
ECE
Acc
ECE
Acc
EffCre
0.04 + 0.00
0.93 + 0.00
0.04 0.01
0.61 = 0.01
0.30 + 0.05
0.49 + 0.05
0.04 0.00
0.00
0.09 0.03
0.60 0.02
0.33
0.48 + 0.04
2
0.06 +.00
0.87 0.01
111++000
0.57 + 0.01
0.50 + 0.05
3
0.06
0.00
0.10
0.56
0.37
0.51 + 0.02
4
0.00
0.00
0.09
0.58
0.49 0.30
5
0.06
0.00
0.12
0.00
0.08
0.48
CreRL-o.s
0.000
0.00
0.10
0.07
0.00
0.09
0.06
0.01
0.09 0.01
0.09
0.01
0.51 + 0.04
0.00
0.00
0.11
0.60+000
0.33
0.49 +49+0.00 0.04
9
0.00
0.09 0.01
0.59 1 0.00
0.35
0.47 + 0.03
10
0.06 0.0000000
0.91 0.00
0.09 0.01
0.39 + 0.01
0.38 + 0.09
0.49 + 0.04
0.94
0.12 - 0.02
0.60 0.01
0.39 0.02
0.48 0.08
0.03
0.11 1 0.01
0.60 0.01
0.39
0.51 + 0.04
3
0.03 0.00
0.00
0.12 0.02
0.59 +.01
0.02
0.48 + 0.03
4
0.03
0.91+
0.15 01.01
0.57 + 0.02
0.01
0.49 + 0.02
5
0.00
0.94
0.03
0.55
0.07
0.46 1 0.01
CreWra
6
0.03 0.00
0.10 0.00
0.38
0.48 + 0.04
7
0.00
0.00
0.01
0.09
0.09
0.46 + 0.05
8
0.03
0.00
0.13 0.01
0.59
0.01
0.51 0.06
0.00
0.00
1.120000
0.00
0.02
0.03 0.00
+.90
0.10 = 0.01
0.58 - 0.02
0.34 0.04
0.47 + 0.04
0.03 0.00
0.94 (1.0
0.12 + 0..02
0.60 0.02
0.34
0.47 + 0.04
0.04
2
0.03 0.00
0.00
0.10 0.02
0.60
0
0.48 0.02
0.00
0.00
0.11 0.02
0.00
0.47 0.04
0.03
0.14 0.00
0.02
0.08
0.48
5
0.03
0.00
0.07 = 0.03
0.59
0.02
0.46
CreEis
0.00
0.00
0.12 0.03
0.02
+
7
0.03
0.00
0.09 = 01.03
0.59 0.01
0.01
0.49 + 0.02 0.02
8
0.00
0.12 0.00
0.59 0.01
0.39
0.51 + 0.04
9
0.00
0.00
0.10 + (1.01
0.59 0.0T
0.37
0.47 + 0.03
10
0.03
0.00
0.10 = 0.04
0.59 - 0.01
0.38 0.04
0.48 0.04
0.04 0.01
0.00
0.11 + 0.02
0.59 - 0.01
0.41 + 0.02 0.02
0.47 + 0.02
2
0.00
0.90
0.11 111=0.00 0.02
0 0.01
0.39
0.50 + 0.04
3
0.02
0.01
0.11 + 0.02
0.59
0.38
0.48 + 0.03
4
0.03
0.00
0.05
0.02
0.49 0.34
0.03 0.00
0.00
0.13+ (L00
0.55
0.07
0.46 0.01
CreNet
0.00
0.00
0.00
0.02
0.04
0.00
0.01
0,12
0.00
0.47 0.02
8
0.02
0.00
0.01
0.59
0.04
0.06
9
0.00
0.02
0.12 0.02
0.58
0.33
0.48 + 0.04
10
0.00
0.94
0.10 0.01
0.58 - 0.02
0.34 0.04
0.48 + 0.03
0.87 0.04
0.15
0.54 0.11
0.0T
0.87
0.09 + 0.03
0.49
0.14
0.54 0.11
3
0.00
0.01
0.10 0.05
0.49
0.15
0.14
4
0.01
0.88
0.07
0.44
0.17
0.54
5
0.00
0.00
0.00
0.44
0.54 0.24
CreBNN
6
0.00
0.81
0.13
0.08
0.18
0.46
7
0.00
0.00
0.03
0.06
0.14
0.49
0.01
0.86 0.02
0.03
0.07
0.53 0.19
0.63
0.04
0.02
0.00
0.19
0.11
0.63 + 0.00
0.86 0.00
0.12 + 0.04
0.07
+17+00112
0.52 + 0.12
Table 8: Comparison of ECE and accuracy of single models per method across datasets.
37


Reviews:
[{"review_id": "EVMW24vyjI", "reviewer": "Reviewer_7AnL", "summary": "This paper proposes a novel, efficient, and model-agnostic method for credal prediction — a framework for representing epistemic uncertainty via credal sets (i.e., convex sets of probability distributions). The proposed method, termed decalibration, allows the construction of credal predictions without retraining or ensembling, which has historically been a major computational bottleneck in credal learning.\n\nCore Idea\nInstead of relying on Bayesian ensembles or multiple retrained models, decalibration works post hoc on a trained classifier. It perturbs the classifier’s logits within a relative-likelihood budget (α), thereby generating class-wise probability intervals that define a credal set.\n\t•\tWhen α = 1, predictions coincide with the maximum likelihood estimator (MLE).\n\t•\tAs α decreases, the set expands, capturing more epistemic uncertainty.\n\nThis approach maintains a clear likelihood-based interpretation: the resulting predictions represent all distributions “reachable without sacrificing more than an α-fraction of training likelihood.”\n\nTheoretical Contributions\n1.\tConvex feasibility and optimization properties:\n\t•\tThe set of permissible logit perturbations under a likelihood constraint is shown to be convex and compact on an identifiability hyperplane.\n\t•\tThe upper bounds of class probabilities correspond to the solution of a convex optimization problem, while lower bounds are attained at the boundary of this convex region.\n2.\tAnalytical results for 1D (class-specific) logit shifts:\n\t•\tEach class-wise bound can be efficiently computed through small convex programs or simple 1D searches.\n\t•\tThe resulting credal sets are nested and monotonic in α.\n\nEmpirical Contributions\n\t•\tExtensive experiments show the method’s competitive performance on coverage–efficiency trade-offs and out-of-distribution (OOD) detection, outperforming or matching credal baselines while being orders of magnitude faster.\n\t•\tThe approach is scalable to large architectures such as TabPFN and CLIP, where traditional ensemble-based credal methods are computationally infeasible.\n\t•\tVisual tools like credal spider plots are introduced to illustrate uncertainty across multi-class predictions.\n\nSignificance\nThis paper advances epistemic uncertainty quantification by providing a principled yet practical alternative to computationally intensive Bayesian or ensemble-based credal methods. Its theoretical soundness, post-hoc simplicity, and broad applicability to modern foundation models make it a potentially impactful contribution to the ICLR community.", "questions": "1.\tClarification on the Likelihood Decay Parameter (α):\n\t•\tHow should practitioners choose or interpret α in practice?\n\t•\tIs there a connection between α and known uncertainty measures such as expected calibration error (ECE) or Bayesian posterior variance?\n\t•\tCould α be automatically tuned using a validation objective (e.g., coverage vs. set size trade-off)?\n2.\tRelation to Distributionally Robust Optimization (DRO):\n\t•\tThe likelihood-based constraint defining the credal set seems conceptually close to DRO formulations (e.g., χ²-divergence or f-divergence balls).\n\t•\tCan the authors clarify whether decalibration is theoretically equivalent to or inspired by DRO methods?\n\t•\tIf not equivalent, how does its uncertainty behavior differ under covariate shift or adversarial perturbations?\n3.\tDistinction from Prior Post-hoc Calibration Methods:\n\t•\tDecalibration operates directly on logits, similar to temperature scaling, confidence calibration, and logit perturbation techniques.\n\t•\tCould the authors explicitly explain how their formulation mathematically differs from those methods and why it better captures epistemic (not aleatoric) uncertainty?\n4.\tComputational Complexity Claims:\n\t•\tThe paper claims “orders of magnitude” improvement in efficiency over ensemble-based methods.\n\t•\tCould the authors provide quantitative runtime comparisons (e.g., seconds per image or FLOPs) for fair assessment?\n\t•\tIs the optimization step fully parallelizable, and how does performance scale with the number of classes?\n5.\tGeneralization to Regression or Structured Outputs:\n\t•\tThe current framework appears classification-specific.\n\t•\tIs there a theoretical extension of decalibration to continuous outputs (e.g., regression) or structured prediction (e.g., segmentation, detection)?\n\t•\tIf so, how would the likelihood constraints translate?\n6.\tCredal Set Visualization and Intuition:\n\t•\tThe “credal spider plots” are compelling but may lack clear interpretability for practitioners.\n\t•\tCould the authors provide an example of how such visualization could inform human decision-making (e.g., in safety-critical applications)?\n7.\tUncertainty Decomposition:\n\t•\tDoes decalibration allow separating epistemic vs. aleatoric uncertainty components?\n\t•\tIf not, could an ensemble of decalibrated models or Bayesian prior over α achieve that?\n8.\tRobustness under Data Shift:\n\t•\tHave the authors tested decalibration under distributional shift scenarios (e.g., corrupted datasets, domain transfer)?\n\t•\tIf so, how does it compare to ensemble or conformal methods in terms of coverage stability?\n\nSummary of Key Questions for Rebuttal Focus:\n\t1.\tTheoretical connection to DRO and uncertainty calibration frameworks.\n\t2.\tJustification and interpretation of α.\n\t3.\tExplicit differentiation from existing logit-perturbation and calibration methods.\n\t4.\tQuantitative validation of efficiency claims.\n\t5.\tPotential extension beyond classification tasks.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "•\tOriginality:\nThe paper presents a novel post-hoc approach to credal prediction — decalibration — that eliminates the need for retraining or ensemble-based inference, which have been the dominant approaches in credal and epistemic uncertainty estimation. The idea of adjusting logits within a relative-likelihood constraint is both elegant and conceptually original, bridging Bayesian epistemic reasoning with practical optimization.\n\t•\tTechnical Quality:\nThe theoretical exposition is mathematically sound and internally consistent. The authors derive and justify convexity properties of the credal set under the proposed perturbation scheme, ensuring interpretability and computational tractability. Analytical insights into the monotonicity of the α-parameterized likelihood bounds reinforce the approach’s rigor.\n\t•\tPractical Relevance:\nThe method is computationally efficient and easily applicable to large-scale deep networks and foundation models (e.g., CLIP, TabPFN). This directly addresses a key bottleneck in existing credal learning methods, which often require retraining or expensive ensembles.\n\t•\tClarity and Presentation:\nThe paper is generally well written, with clear organization, intuitive explanations, and informative visualizations (e.g., credal spider plots). The connection between likelihood decay and epistemic expansion is articulated clearly and grounded in statistical reasoning.\n\t•\tExperimental Strength:\nThe experiments are broad and diverse, covering OOD detection, reliability under label noise, and uncertainty calibration across various architectures. The results show consistent improvements in efficiency–coverage trade-offs, supporting the method’s robustness.\n\t•\tSignificance:\nThe proposed framework provides a principled and scalable solution to epistemic uncertainty quantification in deep learning, which is an increasingly critical research area in reliable AI. Its post-hoc and model-agnostic nature make it particularly relevant for the ICLR community focused on trust, calibration, and interpretability.\n\nOverall Strength Summary:\nThe paper is a solid and meaningful contribution that balances theoretical insight with practical usability. It offers an innovative and efficient solution to credal prediction, addressing both the computational and conceptual limitations of prior approaches.", "weaknesses": "•\tLimited Theoretical Depth Beyond Convexity:\nWhile the convexity and boundedness of the credal sets are clearly demonstrated, the paper lacks deeper theoretical guarantees. For instance, there are no formal proofs of coverage calibration, robustness under data shift, or asymptotic optimality compared to Bayesian posteriors.\nSuggestion: Strengthen the theoretical contribution by connecting decalibration to known uncertainty frameworks such as PAC-Bayesian bounds, conformal coverage guarantees, or distributionally robust optimization.\n\t•\tPotential Overlap with Prior Work:\nThe approach resembles ideas from temperature scaling, likelihood perturbation, and distributional robustness via logit adjustment (e.g., Stutz et al., 2021; Ahuja et al., 2023). The conceptual novelty might appear incremental to readers unless clearer distinctions are drawn.\nSuggestion: Explicitly clarify how decalibration differs mathematically or conceptually from logit perturbation in confidence calibration or adversarial robustness literature.\n\t•\tEmpirical Evaluation Scope:\nThe experimental section, though diverse, is mostly limited to classification tasks. Since credal methods are general, it remains unclear how decalibration performs in structured or regression contexts, where uncertainty has different semantics.\nSuggestion: Add at least one structured prediction or regression experiment (e.g., depth estimation or tabular uncertainty).\n\t•\tInterpretability of α-Parameter:\nThe α hyperparameter controlling likelihood decay is intuitive but empirically opaque. Its practical selection and relationship to epistemic uncertainty remain heuristic.\nSuggestion: Provide either a principled selection rule (e.g., based on validation likelihood or calibration metrics) or a sensitivity analysis showing stable performance over α ranges.\n\t•\tComparative Baselines:\nWhile results are favorable, the baselines do not include recent strong probabilistic calibration models such as Dirichlet Prior Networks or Deep Ensembles with temperature tuning. Without these, the strength of decalibration over modern uncertainty quantifiers remains somewhat uncertain.\nSuggestion: Include these baselines or discuss expected trade-offs to contextualize improvements.\n\t•\tTerminological Ambiguity (“Decalibration”):\nThe term decalibration may be confusing since in standard uncertainty literature, “calibration” typically denotes improving reliability, not relaxing likelihood constraints.\nSuggestion: Clarify this choice early in the paper and consider an alternative framing such as “likelihood-scaling credalization” or “post-hoc credal expansion.”\n\t•\tComputational Claims Need Quantitative Backing:\nThe paper asserts substantial efficiency gains (“orders of magnitude faster”) but provides limited runtime comparisons or profiling details.\nSuggestion: Include explicit runtime or FLOPs analysis versus ensemble-based credal methods to substantiate this claim.\n\nOverall Weakness Summary:\n\nThe paper is well-executed and conceptually clear, but its mathematical guarantees, empirical breadth, and comparative depth could be strengthened. Clarifying the novelty relative to prior calibration and robustness work, expanding evaluation beyond classification, and providing stronger empirical or theoretical justifications would elevate the paper’s impact and credibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "YchbXYgumN", "reviewer": "Reviewer_yG26", "summary": "This work deals with estimating epistemic uncertainty of predictive models through credal sets. Specifically, it deals with uncertainty estimation for large pretrained foundation models such as LLMs and VLMs, where retraining and finetuning can be exceedingly expensive. The authors address this problem by proposing a training-free method that estimates plausible intervals by modifying the inference procedure. They do so by defining a credal set over the base model’s predictions by adding a variable vector c to their logits; The credal set consists of all the values of this modified predictive distribution that are within a threshold of the maximum likelihood estimate. This credal set allows uncertainty estimation and cautious decision-making. The authors evaluate empirically on 9 domains, including CIFAR, SVHN, FMNIST, DermMNIST, ImageNet, Places365, ChaosNLI, QualityMRI, and TabArena. On these domains, they compare the proposed method with 5 credal uncertainty estimation methods. They evaluate these methods on standard credal classification metrics like coverage and efficiency, on out-of-distribution detection, on active in-context learning, and on zero-shot classification, with the latter two focusing on large pretrained models. They find that the proposed method performs on par with baselines without requiring training and that it provides informative uncertainty estimates for active learning and cautious zero-shot classification.", "questions": "- The proposed method shifts the logits for each class separately, and the conclusion distinguishes this from the more general coupled case. Can you give an example where the class-wise shift would be a bad approximation of the general case?\n- Table 4 compares the runtimes of the proposed method and credal baselines, showing that the baselines take 10x more time than the proposed method, but it is not clear how this runtime is defined. Does it include training time or is the 10x difference only in inference time?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The proposed method enables uncertainty estimation and cautious inference in large pretrained models like CLIP and TabFPN without expensive retraining & finetuning\n- The proposed method is straightforward to implement yet quite effective (as demonstrated by the empirical results)\n- The authors introduce credal spider plots to visualize credal sets represented as box intervals", "weaknesses": "- While the method does not require retraining, it does require the original training data or an appropriate surrogate to calculate the relative likelihood\n- The presentation is unclear in places, e.g., while the credal spider plots are quite informative, a full explanation about what they represent is presented in the appendix, which makes earlier references to them (e.g., figure 1) unclear. Section 4 (Empirical results) presents a lot of information without emphasizing key parts like research questions, datasets, metrics, and baselines; instead interleaves it with details about the setup for each experiment. This section could be made easier to read by explicitly listing the research questions datasets, metrics etc., before the subsections, which may focus on more specific details", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "QxVfD7UfMC", "reviewer": "Reviewer_XAJU", "summary": "The paper tackles the problem of efficiently computing credal sets over predictions of probabilistic classifiers. To this end, the authors propose perturbing the logits of the model to generate upper and lower bounds of plausible class-wise probabilities while adhering to a likelihood-ratio budget. Additionally, the authors propose an efficient method to compute class-specific credal sets. The proposed method is evaluated against relevant credal prediction baselines on coverage-efficiency trade-off and out-of-distribution detection tasks.", "questions": "Please refer to the weaknesses section. I am mainly unconvinced about the appeal of computational efficiency in so-called *safety-critical* domains where other credal prediction methods, as evident in fig. 3 on the OOD detection task, perform better.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "The scope of the problem considered - model-agnostic credal predictions - is sizeable and will be of interest to a wide community. Additionally, the post-hoc approach that does not require any retraining as proposed in this paper will encourage its adoption as an added post-training step that can be used to quantify model's epistemic uncertainty. The proposed approach itself, to the best of my knowledge, is sound and the theoretical results seem reasonable, if not unsurprising. The experimental results suggest that the proposed approach is at least on par with the considered credal prediction baselines on the coverage-efficiency trade-off task while allowing, by design, a wide range of coverage. Finally, the efficient computation aspect allows the proposed approach to scale to large models which was previously infeasible.", "weaknesses": "Credal predictions are particularly useful in data-scarce and safety-critical domains such as healthcare where the lack of data can lead to higher epistemic uncertainty and understanding the plausible range of model predictions can help avoid catastrophic decisions. In this regard, the motivation behind the need for computationally efficient credal predictions is not very compelling. In the same vein, while the authors note such safety-critical domains in their introduction, the experimental evaluation is primarily on large benchmark image datasets. Lastly, some of the baselines seem to perform slightly better on the OOD detection task, although requiring more computational time.\n\nOverall, I believe the strengths far outweigh the weaknesses.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "WDI9fz1e5K", "reviewer": "Reviewer_9AwT", "summary": "In this work, the authors present a post-hoc approach for generating the credal prediction by using class-wise plausible probability intervals. This is achieved by perturbing a trained model’s logits under a global likelihood-ratio budget, thereby exploring less-likely yet still plausible predictions without retraining. Multiple experiments from different perspectives are conducted.", "questions": "1. How would the authors place the work? A theoretical one or practical work? What would be the potential practical use case for the approach?\n\n2. Proposition 2.1 is a desired design principle for this method, right? It is a sufficient condition for controlling the trade-off between efficiency and coverage, not a necessary condition, am I correct?\n\n3. The EffCre significantly reduces the training complexity via a single model. What is the inference time complexity of EffCre?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper is relatively well-structured and easy to follow.\n\n2. Although the work is built on recent work--the likelihood-based notion of plausibility (Löhr et al., 2025), the motivation and technical routes are different. It is novel and interesting to me.\n\n3. Mathematical proofs for the relative propositions are provided.\n\n4. Multiple experiments are performed.", "weaknesses": "1. *Extensive experiments show that our method yields credal sets with strong coverage and efficiency and performs well on out-of-distribution detection tasks.* The main empirical claim seems a bit misleading. As from the OOD detection benchmarks, e.g., in Table 5, the EffEct only performs reasonably when \\alpha is close to 1, e.g., $\\alpha$ = 0.95 (still visibly lower than the other baselines). If we fix to use these values of $\\alpha$, how would one conclude that the EffCre has strong coverage and efficiency?\n\n2. The practicality of evaluating efficiency and coverage is limited. A key difficulty in supervised learning is the absence of ground truth for test instances. In addition, this work lacks theoretical guarantees for the coverage and does not provide a clear recipe for choosing the parameter $\\alpha$.\n\n3. In classification tasks, the prediction performance, e.g., test accuracy and calibration performance (expected calibration error), also matters. The performance of this approach in this matter remains unclear. As well as how $\\alpha$ will influnce the prediction performance.\n\n4. The paper highlights its good performance in epistemic uncertainty estimation for OOD detection using a single model. To support this claim, it would be valuable to include comparisons with other single-model-based epistemic uncertainty estimation methods, such as those from the evidential deep learning family or deterministic approaches. If EffCre continues to outperform these additional baselines, it would substantially strengthen the paper’s significance and credibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Poster)