Possibilistic Predictive Uncertainty for Deep Learning
Yao Ni 1 Jeremie Houssineau 1 Yew Soon Ong 1 2 Piotr Koniusz 3 4
arXiv:2605.00600v1 [cs.LG] 1 May 2026
Abstract
tection (Wang et al., 2024), semantic segmentation (Kirillov et al., 2023) and image generation (Rombach et al., 2022).
Deep neural networks achieve impressive results across diverse applications, yet their overconfidence on unseen inputs necessitates reliable epistemic uncertainty modelling. Existing methods for uncertainty modelling face a fundamental dilemma: Bayesian approaches provide principled estimates but remain computationally prohibitive, while efficient second-order predictors lack rigorous derivations connecting their specific objectives to epistemic uncertainty quantification. To resolve this dilemma, we introduce Dirichletapproximated possibilistic posterior predictions (DAPPr), a principled framework leveraging possibility theory. We define a possibilistic posterior over parameters, projects this posterior to the prediction space via supremum operators, and approximates the projected posterior using learnable Dirichlet possibility functions. This projection-and-approximation strategy yields a simple training objective with closedform solutions. Extensive experiments across diverse benchmarks demonstrate that our approach achieves competitive or superior uncertainty quantification performance compared to state-of-theart evidential deep learning methods while maintaining both principled derivation and computational efficiency. Code will be available at https://github.com/MaxwellYaoNi/DAPPr.
Despite their impressive success, deep models are often prone to overconfidence (Nguyen et al., 2015; Mehrtash et al., 2020; Zhang et al., 2025), assigning high predictive confidence to incorrect outputs, particularly on inputs beyond their training distribution. Such overconfidence becomes critical in high-stake applications such as autonomous driving (Wang et al., 2021) and medical diagnosis (Mehrtash et al., 2020), where incorrect predictions can be catastrophic. To ensure safe deployment, models must recognize when knowledge is insufficient to make reliable predictions, which requires explicitly representing epistemic uncertainty arising from lack of knowledge, a capability absent in standard deep models. A principled approach to representing epistemic uncertainty is Bayesian learning, which characterizes uncertainty through the disagreement among multiple models that explain the observed data well. Formalizing this disagreement requires marginalizing over the posterior distribution of parameters, which is computationally intractable and necessitates approximations such as Bayesian neural networks (Blundell et al., 2015; Krueger et al., 2017; Jospin et al., 2022; Rudner et al., 2022), MC Dropout (Gal & Ghahramani, 2016; Ovadia et al., 2019; Mobiny et al., 2021), and ensemble methods (Lakshminarayanan et al., 2017; Wen et al., 2020; Löhr et al., 2025). While theoretically appealing, these approximations are computationally expensive and challenging to scale to large networks and datasets (Liu et al., 2020; He et al., 2020; Mukhoti et al., 2023). To avoid the computational burden of Bayesian marginalization, an alternative line of work models distributions over output predictions rather than over parameters. By representing predictions as distributions rather than point estimates, these methods form second-order representations that capture epistemic uncertainty through distributional properties such as concentration or dispersion. Such representations are realized using different distributional families, including Dirichlet distributions over class probabilities (Sensoy et al., 2018; Malinin & Gales, 2018; Charpentier et al., 2020), exponential-family predictive models (Charpentier et al., 2022), and Gaussian distributions over logits (Ghosh et al., 2016; Tagasovska & Lopez-Paz, 2019). Despite their effi-
1. Introduction Recent breakthroughs in deep neural networks (He et al., 2016; Dosovitskiy et al., 2021) are primarily driven by training expressive models (Kaplan et al., 2020) on massive datasets (Deng et al., 2009; Kuznetsova et al., 2020; Schuhmann et al., 2022). This data-driven paradigm has achieved remarkable success across diverse computer vision tasks, including image classification (He et al., 2016), object de1
Nanyang Technological University 2 A*STAR 3 The University of New South Wales 4 Data61 CSIRO. Correspondence to: Jeremie Houssineau <[email protected]>. Preprint. May 4, 2026.
1
Possibilistic Predictive Uncertainty for Deep Learning
ii. Under a high-capacity assumption for the considered neural networks, we derive a tractable implementation with closed-form solutions, achieving computational efficiency while remaining theoretically grounded. iii. Our method offers competitive performance against state-of-the-art methods across diverse benchmarks.
Pytorch-style pseudo code for DAPPr loss import torch.nn.functional as F def DAPPr loss(logits, labels, lamb, eps=1e-8): alpha = F.softplus(logits) + 1 y = F.one hot(labels, logits.shape[-1]).float() a star = alpha - y + eps # avoid log instability p star=(a star / a star.sum(dim=1,keepdim=True)).detach() a 0 = alpha.sum(1) loss=a 0*a 0.log()+(alpha*(p star/alpha).log()).sum(dim=1) reg = (alpha * (1 - y)).square().sum(dim=1) return loss.mean() + lamb * reg.mean()
2. Related work
for x, labels in train loader: # Training Loop logits = model(x) - loss = F.cross entropy(logits, labels) + loss = DAPPr loss(logits, labels, lamb) ...
Bayesian Deep Learning. Bayesian deep learning provides a principled framework for epistemic uncertainty by marginalizing predictions over the posterior distribution of parameters (Neal, 2012). This captures disagreement among models that explain the observed data, directly quantifying epistemic uncertainty. However, exact inference is computationally intractable for modern networks (Wilson & Izmailov, 2020), necessitating approximations such as variational inference (Blundell et al., 2015; Graves, 2011), Monte Carlo dropout (Gal & Ghahramani, 2016; Ovadia et al., 2019; Mobiny et al., 2021), and deep ensembles (Lakshminarayanan et al., 2017; Wen et al., 2020; Löhr et al., 2025). While effective, these methods require multiple models or extensive sampling, making them expensive and challenging to scale (Liu et al., 2020; He et al., 2020; Mukhoti et al., 2023). Despite recent efficiency efforts (Rudner et al., 2022; Wilson & Izmailov, 2020), the tension between theoretical rigor and computational tractability persists.
Figure 1. PyTorch-style pseudocode for the DAPPr loss (∼10 lines of code) and its usage by simply replacing the cross-entropy loss.
ciency, these approaches adopt heuristic objectives without rigorous justification for uncertainty quantification. These limitations reveal a dilemma for modeling epistemic uncertainty: Bayesian methods offer theoretical rigor but remain computationally intractable, while second-order predictors achieve efficiency but lack principled connections between their specific objectives and epistemic uncertainty quantification. To resolve this dilemma, we turn to possibility theory (Zadeh, 1978; Dubois & Prade, 1988), an alternative framework for epistemic uncertainty. This framework replaces probabilistic additivity and sum-normalization with supremum-based operators and max-normalization, enabling tractable uncertainty quantification without integrating over high-dimensional parameter spaces. Despite these advantages and its introduction decades ago, possibility theory remains largely unexplored in deep learning.
Second-Order Predictive Modeling. An alternative paradigm bypasses parameter-space inference by modeling distributions over predictions. These methods learn predictive distributions whose properties, such as concentration or dispersion, serve as proxies for epistemic uncertainty. Representative approaches include prior networks trained with KL divergence (KL-PN) (Malinin & Gales, 2018) and reverse KL divergence (RKL-PN) (Malinin & Gales, 2019), DUQ (Van Amersfoort et al., 2020) which estimates uncertainty via feature distances, PostNet (Charpentier et al., 2020) which employ normalizing flows for Dirichlet modeling, Natural Posterior Networks (Charpentier et al., 2022) which extend to exponential families, and RS-NN (Manchingal et al., 2025) which predicts set-valued outputs. Despite their efficiency, these methods lack principled foundations, relying on heuristic interpretations of epistemic uncertainty.
To bridge this gap, we introduce Dirichlet-approximated possibilistic posterior predictions (DAPPr), a principled framework that leverages possibility theory for secondorder predictors, achieving both theoretical rigor and computational efficiency. Our approach defines a possibilistic posterior over parameters, projects it to the prediction space via supremum operators, and approximates it using learned Dirichlet possibility functions. This projection-andapproximation strategy yields tractable closed-form solutions under the widely-adopted cross-entropy loss, resulting in a simple training objective with minimal regularization (as shown in Fig. 1). Despite this simplicity, extensive experiments demonstrate that our method achieves competitive or superior uncertainty quantification performance compared to state-of-the-art evidential deep learning methods across diverse benchmarks, including standard datasets, long-tailed distributions, distribution shift detection, and fine-grained classification tasks.
Evidential Deep Learning. Evidential deep learning (EDL) (Sensoy et al., 2018) has emerged as a leading second-order approach by interpreting network outputs as Dirichlet distribution parameters over class probabilities. Deriving its objective from subjective logic (Jøsang, 2016) and DempsterShafer theory (Dempster, 1968; Shafer, 1976), EDL models both aleatoric and epistemic uncertainty through the Dirichlet’s total evidence (concentration parameter), with higher evidence indicating lower epistemic uncertainty. However, this connection lacks rigorous justification (Bengs et al.,
Our contributions are as follows: i. We introduce Possibilistic Uncertainty Modeling, bridging possibility theory and second-order predictors for principled epistemic uncertainty modelling. 2
Possibilistic Predictive Uncertainty for Deep Learning
2022), and EDL can exhibit pathological behaviors such as increasing uncertainty with more data (Bengs et al., 2022; Yoon & Kim, 2025). Subsequent variants address specific issues: I-EDL (Deng et al., 2023) introduces Fisher information regularization, R-EDL (Chen et al., 2024) relaxes training constraints, DA-EDL (Yoon & Kim, 2024) incorporates density-aware mechanisms, and F-EDL (Yoon & Kim, 2025) employs flexible Dirichlet modeling. Despite these improvements, EDL and its variants fundamentally lack principled probabilistic derivations rigorously connecting their objectives to epistemic uncertainty quantification.
f : Θ → [0, 1] satisfying supθ∈Θ f (θ) = 1. Given this representation, the possibility of an event B ⊆ Θ is defined as . Π(B) = supθ∈B f (θ). Accordingly, an event is impossible if Π(B) = 0, while Π(B) = 1 indicates that the event cannot be excluded given the available information. When such non-exclusion holds simultaneously for an event B and its . complement B c = Θ \ B, i.e. Π(B) = 1 and Π(B c ) = 1, the representation provides no information to exclude either event. When this non-exclusion holds for all events, the representation corresponds to total ignorance, where f (θ) = 1 for all θ ∈ Θ and the possibility function is denoted by 1.
Possibility Theory. Possibility theory (Zadeh, 1978; Dubois & Prade, 1988) provides an alternative framework for epistemic uncertainty from insufficient knowledge. Unlike probability theory’s additive measures summing to one, possibility theory uses supremum-based operators normalizing to a maximum of one (Dubois, 2006), naturally accommodating imprecise information (Dubois & Prade, 2015). Possibilistic inference uses max-based rules rather than integration, offering computational advantages (Gebhardt & Kruse, 1995). Applications include point processes (Houssineau, 2021), control problems (Chen et al., 2021), and deep ensembles (Löhr et al., 2025). Despite its appeal for epistemic uncertainty, possibility theory remains underexplored in deep learning for second-order predictive modelling. Our work addresses this gap by connecting possibility theory with efficient second-order uncertainty quantification.
Dirichlet possibility function. When the parameter space Θ is a probability simplex ∆K−1 = {p ∈ [0, 1]K : PK k=1 pk = 1}, a natural structured possibility function on this simplex is given by the Dirichlet possibilityPfunction K Dir(·; α) with α ∈ [0, ∞)K and α0 = ∥α∥1 = k=1 αk , defined as: αk K Y pk Dir(p; α) = α0α0 , (1) αk
3. Preliminary for Possibility Theory
Possibilistic Bayes Posterior. Having introduced possibility functions for representing epistemic uncertainty, we now introduce the general form of the Bayesian posterior in possibilistic setting. Let the relationship between a parameter θ and a data point y be characterised by a loss ℓ(θ, y), and let prior on θ be represented by a possibility function f (θ). The corresponding possibilistic posterior is defined as
k=1
with the convention that αkαk = 1 when αk = 0. This formulation constitutes not only a conjugate prior for the multinomial likelihood, mirroring the Dirichlet distribution, but also a valid possibility function satisfying supp Dir(p; α) = 1. This supremum is uniquely attained at p = α/α0 when α0 > 0, while spanning the entire simplex when α0 = 0 to represent total ignorance Dir(·; α) = 1.
Probability theory modelsRuncertainty through additive measures normalized to one ( p(x)dx = 1), thereby enforcing a complete allocation of probability mass. This complete allocation is appropriate to modeling aleatoric uncertainty from intrinsic data randomness, provided sufficient knowledge is available. However, when information is limited, complete probability mass allocation becomes ill-suited for representing epistemic uncertainty arising from ignorance.
f (θ | y) =
exp(−ℓ(θ, y))f (θ) . supθ′ ∈Θ exp(−ℓ(θ′ , y))f (θ′ )
(2)
Importantly, if the loss admits a probabilistic interpretation, i.e., ℓ(θ, y) = − log p(y | θ) for some distribution p(· | θ), the posterior f (·|y) is consistent with the standard Bayesian posterior and inherits familiar properties such as conjugacy and Bernstein–von Mises behaviour (Hieu et al., 2025).
The epistemic uncertainty, induced by lack of knowledge, concerns which hypotheses cannot be excluded given limited information. This emphasis on non-exclusion is naturally formalised in possibility theory (Zadeh, 1978; Dubois, 2006), which represents uncertainty through plausibility measures and, at a fundamental level, decouples epistemic uncertainty from aleatoric uncertainty (Hieu et al., 2025).
Possibilistic change of variable. In inference, we often need to characterize uncertainty about a transformed quantity ψ = T (θ) rather than the original parameter θ. Characterizing this uncertainty requires propagating uncertainty from θ through the mapping T , ensuring the plausibility of ψ reflects the plausibility of θ that generates it. In probability theory, such propagation uses a Jacobian determinant since densities change under transformation. However, possibility functions are not densities. Therefore, possibilistic transformation cannot simply replace integration with supremum
In this section, we introduce key concepts from possibility theory. First, we define possibility functions, with Dirichlet possibility functions as an important special case. We then present essential operations: possibilistic Bayesian inference, change of variables, and divergence measures. Possibility function. In possibility theory, uncertainty about a parameter θ ∈ Θ is represented by a possibility function 3
Possibilistic Predictive Uncertainty for Deep Learning
while retaining the Jacobian determinant. Instead, (Baudrit et al., 2008) define the possibilistic change of variable as: f (ψ) =
sup
f (θ).
predictions are plausible. Since predictions in classification lie on the probability simplex, the Dirichlet possibility function in Eq. (1) naturally represents such distributions. We thus instantiate this predictor as a neural network Φ′ψ parameterised by ψ, that maps x to Dirichlet parameters α, defining a Dirichlet possibility function gψ (· | x) to represent epistemic uncertainty over predictions.
(3)
θ∈T −1 (ψ)
where T −1 (ψ) ⊆ Θ is the set-valued pre-image of ψ (with supθ∈∅ f (θ) = 0). This formulation assigns to ψ the highest plausibility among all its pre-image.
Ideally, if Dirichlet parameters α or uncertainty labels were available for each input, training would be straightforward. However, such supervision is unavailable in practice. To find such a principle without explicit supervision, we examine where epistemic uncertainty about a prediction comes from. The prediction for an input x deterministically given by p = Φθ (x): once θ is known, the prediction follows without ambiguity. This means epistemic uncertainty about a prediction can only arise from uncertainty about the parameters that produced it. In other words, how much we trust a prediction comes from how plausible the model is. Epistemic uncertainty about a prediction is thus entirely governed by the plausibility of the model parameters.
Maxitive pseudo-divergence. Building on the concepts above, we now explain how to compare possibility functions. The comparison is based on a natural partial order: for two possibility functions f and g on Θ, we write f ⪯ g if f (θ) ≤ g(θ) for all θ ∈ Θ. Under this relation, every possibility function satisfies f ⪯ 1. This ordering further enables a pointwise comparison of possibility functions via the maxitive pseudo-divergence (Singh et al., 2025): Dmax (f ∥g) = max log θ∈Θ
f (θ) ≥ 0. g(θ)
(4)
The non-negativity of Dmax follows from the fact that both f and g have a supremum equal to 1, so that f (θ) < g(θ) cannot hold everywhere on Θ, and f ⪯ g requires that f (θ) = g(θ) when θ is in arg maxθ f (θ). It follows that Dmax (f ∥g) = 0 when f ⪯ g, rather than only when f = g.
This motivates a path to learning the second-order predictor: first quantify plausibility over parameters and project it to prediction space, then train the predictor to match. 4.2. Defining the objective To quantify the plausibility of model parameters θ, we relate it to the empirical loss L(θ; D) defined in Eq. (5). The intuition is simple: parameters with lower loss better fit the observed data and are thus more plausible. This intuition can be formalised using the possibilistic posterior in Eq. (2). Under a uniform prior possibility function 1, which assumes no prior preference for any particular parameters, the possibilistic Bayesian posterior becomes:
4. Methodology 4.1. Motivation A standard neural network trained for classification maps an input x ∈ X to a probability vector p ∈ ∆K−1 over K classes. This mapping is realised by a predictive model Φθ : X → ∆K−1 , parametrised by θ ∈ Θ and trained on a dataset D ⊆ {X × Y} by minimizing an empirical risk: X L(θ; D) = ℓ(Φθ (x), y), (5)
π(θ | D) =
(x,y)∈D
exp(−L(θ; D)) . supθ′ ∈Θ exp(−L(θ ′ ; D))
(6)
This posterior assigns higher plausibility to parameters with lower empirical loss. The normalisation by the supremum ensures supθ π(θ | D) = 1, satisfying the requirement of a valid possibility function. Particularly, when the loss is cross-entropy loss, the resulting posterior reduces to the relative likelihood, a well-established prior-free approach in statistical inference (Birnbaum, 1962; Wasserman, 1990; Walley & Moral, 1999; Giang & Shenoy, 2012), with axiomatic justification based on the likelihood principle, compatibility with Bayes’ rule, and the minimal commitment principle (Denoeux, 2014; Löhr et al., 2025).
where ℓ(·, ·) is a standard classification loss, such as the PK widely-adopted cross-entropy ℓ(p, y) = − k=1 yk log pk . After training, the learned θ produces a point prediction ptest = Φθ (xtest ) for a test input xtest . While this prediction provides class probabilities, it does not capture epistemic uncertainty and thus cannot indicate prediction reliability, especially when xtest is weakly supported by D. A principled way to quantify this missing epistemic uncertainty is Bayesian inference, which places a posterior distribution over parameters θ. However, performing this in the high-dimensional parameter space of modern neural networks is computationally expensive. A more efficient alternative is a second-order predictor that directly models epistemic uncertainty at the prediction level, mapping an input x to a distribution over predictions p to indicate which
With the posterior π(· | D) quantifying plausibility over parameters, the next step is transferring it to the prediction space to determine which predictions are plausible. For a given input x, we project the posterior onto the simplex ∆K−1 via the mapping θ 7→ Φθ (x). By the possibilistic 4
Possibilistic Predictive Uncertainty for Deep Learning
4.3. Approximating the objective
change-of-variable rule in Eq. (3), the projection yields: gx∗ (p | D) = sup{π(θ | D) : θ ∈ Θ, Φθ (x) = p}.
Learning the second-order predictor would be straightforward if the conceptually well-defined gx∗ (·|D) in Eq. (9) were available. However, evaluating this projected posterior involves constrained optimisation over the high-dimensional parameter space Θ, which renders direct computation intractable. To overcome this intractability, we derive in this subsection a tractable surrogate for gx∗ that enables efficient optimisation of the inner maximisation in Eq. (9).
(7)
The function assigns to each prediction p the maximum plausibility among all parameters that yields p for input x. We denote this assignment compactly by supΦθ (x)=p π(θ | D). This supremum-based projection plays a similar role to marginalisation in Bayesian inference, where uncertainty over predictions is obtained by integrating the parameter posterior over the high-dimensional Θ and applying a Jacobian correction to define a density on ∆K−1 . Such marginalisation is generally intractable, requiring approximation such as ensembles. In contrast, Eq. (7) replaces integration with constrained optimisation. While non-trivial, this admits tractable approximations without ensembles.
We first consider a training sample (x, y) ∈ D, the negative log-possibility−log gx∗ (p|D) in Eq. (9) can be expressed as − log gx∗ (p | D) = − log =ℓ(p, y) +
inf Φθ (x)=p
We realise this approximation by learning a second-order predictor Φ′ψ whose output is a possibility function gψ (·|x) over p. This output is matched to the projected posterior gx∗ (· | D) using the pseudo-divergence Dmax in (4): L(ψ; D) = Ex∼X Dmax gψ (· | x)∥gx∗ (· | D) =Ex∼X max log gψ (p|x) − log gx∗ (p|D) . (8)
sup Φθ (x)=p
π(θ | D)
L(θ; D \ {(x, y)}) + c.
(10)
Here c = log supθ′ exp(−L(θ ′ ; D)) is a constant independent of p and x, and therefore can be safely ignored. Once c is ignored, evaluating Eq. (10) reduces to analysing how the empirical risk behaves under the constraint Φθ (x) = p. This constraint fixes the contribution of the sample (x, y) to the empirical risk L(θ; D) as ℓ(p, y), reducing the problem to the infimum over the leave-one-out loss L(θ; D \ {(x, y)}).
p∈∆
This objective enforces the ordering gψ (· | x) ⪯ gx∗ (· | D) by penalising the maximal pointwise ratio between the learned possibility function and the projected posterior. With an uninformative initialisation of gψ (· | x), optimisation reduces gψ only where this ordering is violated. Once the ordering holds, the divergence vanishes and cannot be further reduced, as discussed in Sec. 3.
The key observation is that this infimum depends on p only through the constraint Φθ (x) = p. A standard assumption in the overparameterised regime is that networks have sufficient capacity to fit any individual sample without affecting predictions on other samples (Hornik et al., 1989; Zhang et al., 2017; Koh & Liang, 2017). Under this sufficient capacity condition, the constraint can be satisfied for any p without affecting the infimum of the leave-one-out loss. This infimum is thus approximately independent of p:
Minimizing the loss L(ψ; D) in Eq. (8) yields a min-max problem, with an inner maximisation over p ∈ ∆ and an outer minimisation over ψ. Directly optimising it is challenging because the inner the maximiser depends on ψ. To address this, we apply Danskin’s theorem (Danskin, 1967), which computes the gradient of the outer objective by differentiating at the inner maximum, assuming the maximiser is unique; otherwise, the update is interpreted as a subgradient. Accordingly, the inner maximizer p∗ for an input x is: p∗ = arg max log gψ (p | x) − log gx∗ (p | D) . (9)
inf Φθ (x)=p
L(θ; D \ {(x, y)}) ≈ cx ,
(11)
where cx does not depend on p. We empirically verify this approximation in §C. Under this approximation, the projected posterior simplifies to the tractable surrogate: gx∗ (p | D) ∝ exp − ℓ(p, y) , (12)
p∈∆
which in turn yields a practical approximation of the maximiser p∗ defined in Eq. (9) by
∗
This shows that p is the point at which gψ (· | x) most overestimates gx∗ (· | D), i.e., where their ratio is maximal.
p̃∗ = arg max log gψ (p | x) + ℓ(p, y). p∈∆
With the inner maximiser obtained at each iteration, we can then minimise the loss L by stochastic gradient descent: 1 X ψ ←ψ−ρ ∇ψ log gψ (p | x)|p=p∗ , |B|
(13)
The approximation in Eq. (11) may be violated in the rare case where two samples share the same input x but have conflicting labels y, y ′ . Here, the constraint induces coupled effects, and a faithful surrogate replaces Eq. (12) with: gx∗ (p | D) ∝ exp − ℓ(p, y) − ℓ(p, y ′ ) ,
x∈B
where ρ is the learning rate, B ⊆ X is a mini-batch of inputs. This update pushes ψ to decreases gψ (p | x) at the point of maximal overestimation p = p∗ (ψ).
reflecting the interaction induced by Φθ at x. 5
Possibilistic Predictive Uncertainty for Deep Learning Table 1. AUPR (↑) results on MNIST. Conf.: confidence estimation using aleatoric uncertainty. OOD detection uses epistemic uncertainty against KMNIST and FMNIST. Baseline results from (Chen et al., 2024) where available; others are our reproduction.
4.4. Parametrising the Dirichlet possibility function With the approximate maximiser p̃∗ in Eq. (13) establised, training the second-order predictor Φ′ψ requires evaluating the surrogate objective associated with the base prediction loss ℓ. In classification, this base loss is standardly taken to be the cross-entropy, which we therefore adopt in the sequel. Under the cross-entropy loss, evaluating the surrogate objective in Eq. (13) necessitates an explicit functional form for the learned possibility function gψ . We therefore realise gψ (· | x) as a Dirichlet possibility function whose parameters are produced by Φ′ψ . According to Eq. (1), the corresponding log-possibility for a given input x and p is: log gψ (p | x) = α0 log α0 +
K X
αk log
k=1
pk , αk
OOD Detection KMNIST FMNIST MC Dropout 99.26±0.0 99.98±0.0 94.00±0.1 96.56±0.3 DUQ 98.65±0.1 99.97±0.0 98.52±0.1 97.92±0.6 KL-PN 99.01±0.0 99.92±0.0 93.39±1.0 98.16±0.0 RKL-PN 99.21±0.0 99.67±0.0 53.76±3.4 72.18±3.6 PostNet 99.34±0.0 99.98±0.0 94.59±0.3 97.24±0.3 EDL 98.22±0.3 99.99±0.0 96.31±2.0 98.08±0.4 I-EDL 99.21±0.0 99.98±0.0 98.33±0.2 98.86±0.2 R-EDL 99.33±0.0 99.99±0.0 98.69±0.2 99.29±0.1 F-EDL 99.30±0.2 99.93±0.0 98.74±0.3 99.31±0.2 DAPPr 99.26±0.1 99.99±0.0 98.81±0.2 99.55±0.1 Method
5. Experiments We conduct comprehensive experiments to evaluate our uncertainty modeling method across diverse settings and compare it with state-of-the-art EDL methods.
Proposition 1. Let the loss be the cross-entropy loss and let gψ be parameterised as a Dirichlet possibility function with α = Φ′ψ (x). Then, for a labelled sample (x, y) with one-hot label vector y, the approximate maximiser is 1 (α − y). α0 − 1
5.1. Experimental Setup Datasets. Following (Deng et al., 2023; Chen et al., 2024; Yoon & Kim, 2025), we evaluate on standard image classification datasets: MNIST (LeCun, 1998), CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2009). We further include CIFAR-10-LT (Cui et al., 2019), an artificially imbalanced version of CIFAR-10 with imbalance ratio ρ and two finegrained datasets, CUB-200-2011 (Wah et al., 2011) and StanfordDogs (Khosla et al., 2011), which contain numerous categories and high-resolution images.
(15)
The proof is provided in Appendix A. Proposition 1 shows that, under a Dirichlet parametrisation and cross-entropy loss, the surrogate maximiser p̃∗k admits a closed-form expression. This expression is valid only if p̃∗k lies in the probability simplex, which requires αk > 1. In practice, we implement this as α = softplus(Φ′ψ (x)) + 1.
Evaluation Metrics. For each dataset, we assess classification accuracy, confidence estimation, and out-of-distribution (OOD) detection. Confidence estimation is evaluated using the area under the precision-recall curve (AUPR), with confidence defined as negative aleatoric uncertainty (labels: correct=1, incorrect=0). OOD detection is also evaluated via AUPR based on negative epistemic uncertainty (labels: ID=1, OOD=0). All AUPR scores are normalized to 0100, with higher values being better. For OOD detection, we use KMNIST (Clanuwat et al., 2018) and FashionMNIST (Xiao et al., 2017) for MNIST; SVHN (Netzer et al., 2011) and CIFAR-100 for CIFAR-10; SVHN and TinyImageNet (Deng et al., 2009) for CIFAR-100, and ImageNet-O (Hendrycks et al., 2021), DTD (Cimpoi et al., 2014) and Places365 (Zhou et al., 2017) for fine-grained datasets.
Under the surrogate objective in Eq. (13), each labelled sample (x, y) can be fitted essentially independently. This independence may drive the Dirichlet possibility function to become arbitrarily sharp, corresponding to large total evidence α0 around the true label. To prevent such unbounded precision, we penalise evidence assigned to incorrect classes via a spurious evidence regulariser: R(x) = ∥(1 − y) ⊙ α∥22 .
(16)
Combining the surrogate likelihood term with this regulariser yields the final per-sample training loss ℓψ (x) = log gψ (p̃∗ | x) + λR(x),
Conf.
(14)
PK where α = Φ′ψ (x) and α0 = ∥α∥1 = k=1 αk . Under this parametrisation, the surrogate maximisation in Eq. (13) admits a closed-form solution.
p̃∗ =
Test Acc.
(17)
Implementation. Following (Charpentier et al., 2020), we use ConvNet (3 convolutional + 3 dense layers) for MNIST, VGG16 (Simonyan & Zisserman, 2014) for CIFAR10/CIFAR-10-LT, ResNet-18 (He et al., 2016) for CIFAR100, and ResNet-50 for fine-grained datasets. We train for up to 100 epochs (200 for fine-grained datasets) with batch
where λ ≥ 0 controls the strength of regularisation. After training, aleatoric uncertainty is measured by 1 − maxk αk /α0 , while epistemic uncertainty is measured by K/α0 , which decreases with increasing total evidence. 6
Possibilistic Predictive Uncertainty for Deep Learning Table 2. AUPR (↑) on CIFAR-10/100. Conf.: confidence estimation using aleatoric uncertainty. OOD detection using epistemic uncertainty against SVHN and CIFAR-100 (for CIFAR-10) or TinyImageNet (for CIFAR-100). Baseline results are token from (Yoon & Kim, 2025). CIFAR-10 Method MC Dropout EDL I-EDL R-EDL DAEDL F -EDL DAPPr
Test Acc.
Conf.
82.84±0.1 83.55±0.6 89.20±0.3 90.09±0.3 91.11±0.2 91.19±0.2 92.00±0.2
97.15±0.0 97.86±0.2 98.72±0.1 98.98±0.1 99.08±0.0 99.10±0.0 99.23±0.0
CIFAR-100
OOD Detection SVHN CIFAR-100 51.39±0.1 45.57±1.0 79.12±3.7 84.18±0.7 82.96±2.2 84.84±0.6 85.00±1.2 87.73±0.3 85.54±1.4 88.19±0.1 91.20±1.3 88.37±0.3 91.72±1.2 89.39±0.3
Test Acc.
Conf.
65.94±0.6 45.91±5.6 66.38±0.5 63.53±0.5 66.01±2.6 69.40±0.2 70.85±0.2
92.00±0.3 91.28±0.8 92.84±0.1 92.69±0.2 86.00±0.3 94.01±0.1 94.39±0.1
OOD Detection SVHN TinyImageNet 71.83±2.0 74.93±0.6 56.21±3.1 70.13±2.0 67.51±2.9 75.86±0.3 61.80±3.4 69.78±1.3 72.07±4.1 77.40±1.6 75.35±2.3 80.58±0.2 73.32±3.3 79.11±0.1
Table 3. AUPR (↑) on CIFAR-10-LT (long-tailed) setting. Conf.: confidence estimation using aleatoric uncertainty. OOD detection using epistemic uncertainty against SVHN and CIFAR-100. Baseline results are obtained from (Yoon & Kim, 2025). Method MC Dropout EDL I-EDL R-EDL DAEDL F -EDL DAPPr
Test Acc. 39.22±3.1 42.62±2.7 57.88±1.3 63.36±1.0 63.36±1.4 63.73±1.4 68.81±0.9
CIFAR-10 (ρ = 0.01) OOD Detection Conf. SVHN CIFAR-100 63.62±2.7 33.33±1.7 54.17±1.1 82.63±1.7 51.99±3.8 66.86±0.9 84.10±1.3 52.85±6.8 69.19±1.3 78.34±1.0 48.71±7.1 64.20±1.4 82.15±1.0 51.03±5.6 65.31±1.2 85.99±1.7 62.56±2.8 70.18±2.0 87.01±1.3 64.80±3.6 71.77±1.3
Table 4. AUPR (↑) for detecting distribution shift from CIFAR-10 to CIFAR-10-C using aleatoric uncertainty. C ∈ {1, 2, 3, 4, 5} denotes corruption severity levels. Results are averaged over 19 corruption types. Baseline results are from (Yoon & Kim, 2025).
Test Acc. 70.87±3.0 79.09±0.4 84.86±0.1 85.35±0.2 84.95±0.4 85.46±0.2 86.37±0.3
CIFAR-10 (ρ = 0.1) OOD Detection Conf. SVHN CIFAR-100 89.82±2.3 37.37±1.4 61.18±1.3 95.36±0.1 72.18±2.1 80.09±0.7 97.31±0.2 79.83±3.9 83.50±0.4 94.35±0.2 60.58±5.0 69.53±1.6 95.22±0.4 69.40±4.5 74.56±1.7 97.60±0.1 85.36±1.5 83.64±0.7 97.69±0.1 85.63±2.0 84.32±0.5
F-EDL (Yoon & Kim, 2025) without requiring spectral normalisation (Miyato et al., 2018) or additional MLP layers. CIFAR-10-LT. Table 3 evaluates long-tail settings, better reflecting real-world scenarios. With the same simple formulation, our method achieves the best accuracy and AUPR scores at both ρ values, demonstrating superior robustness.
Method C = 1 C=2 C=3 C=4 C=5 CE 56.39±0.7 61.88±1.1 65.86±1.3 69.91±1.5 75.01±1.8 EDL 54.76±0.3 59.01±0.4 62.46±0.5 65.87±0.6 70.21±0.8 I-EDL 56.33±0.2 61.52±0.5 65.44±0.5 69.45±0.5 74.56±0.5 R-EDL 57.37±0.5 62.20±1.0 65.74±1.4 69.33±1.9 73.58±2.6 DAEDL 57.89±0.3 63.23±0.4 67.53±0.4 72.21±0.4 77.74±0.4 F -EDL 59.01±0.8 65.11±0.7 69.48±0.5 73.88±0.3 78.72±0.4 DAPPr 58.81±0.2 65.20±0.3 69.83±0.5 74.29±0.3 79.73±0.6
Distribution Shift. Table 4 shows that distribution shift detection improves as corruption severity increases. CE: cross-entropy using 1 − maxk pk for aleatoric uncertainty. Our method achieves the best AUPR at severity levels 2–5. Fine-Grained Datasets. Table 5 evaluates fine-grained datasets with many classes and high resolution. CE and RS-NN use 1 − maxk pk for aleatoric and entropy for epistemic uncertainty. Implementation details are in §B. Our method achieves best accuracy, confidence estimation, and OOD detection on ImageNet-O and Places365, outperforming all EDL-based methods without requiring additional hyperparameters or architectural modifications. It also surpasses RS-NN (Manchingal et al., 2025), which requires extra computation and pre-trained knowledge. Notably, our single-model method outperforms 10-model ensembles on accuracy, confidence estimation, and Places365 OOD detection. Combining with ensembling yields further gains.
size 64 (256 for fine-grained) using validation-based early stopping. We report mean ± std over 5 random seeds. See §B for details on datasets and implementation. 5.2. Comparison with state-of-the-arts MNIST. Despite using only a simple loss with a regulariser, our method achieves the best AUPR scores for confidence estimation and OOD detection, while remaining competitive in test accuracy (Table 1). These gains are achieved without the multiple hyperparameters or computational overhead required by DUQ, PostNet, and EDL-based methods. CIFAR-10/100. Table 2 shows that our method achieves the best accuracy and confidence estimation on both CIFAR-10 and CIFAR-100, with best or second-best OOD detection. It outperforms I-EDL, R-EDL, and DAEDL, which require Fisher information regularisation, extra hyperparameters, or post-hoc Gaussian fitting. It also matches or surpasses
5.3. Ablation studies and analysis Ablation of Regularisation Components. Table 6 evaluates our spurious evidence regulariser by comparing four variants: (1) DAPPr without regularisation, (2) DAPPr with 7
Possibilistic Predictive Uncertainty for Deep Learning Table 5. AUPR (↑) results on CUB-200-2011 and Stanford Dogs. Conf.: confidence estimation using aleatoric uncertainty. OOD detection using epistemic uncertainty against Image-O, DTD and Places365. Baseline results are reproduced by us. CUB-200-2011 Stanford Dogs OOD Detection OOD Detection Test Acc. Conf. Test Acc. Conf. ImageNet-O DTD Places365 ImageNet-O DTD Places365 CE 47.17±0.6 80.26±0.6 85.21±0.6 65.16±1.3 36.69±1.1 56.35±0.5 84.63±0.6 94.32±0.4 82.29±1.3 66.16±1.9 MC Dropout 40.43±0.7 73.61±0.8 81.40±0.6 60.02±1.3 31.33±1.0 49.84±1.0 79.99±0.8 90.07±0.6 73.26±1.0 51.51±2.0 RS-NN 46.21±0.1 79.96±0.5 85.05±0.3 61.56±0.7 45.10±0.8 57.83±0.4 85.05±0.3 93.31±0.2 81.33±0.4 67.92±0.5 EDL 44.01±0.7 82.47±0.3 85.02±1.0 59.98±2.3 46.33±2.0 53.06±1.0 83.97±0.2 93.87±0.2 77.14±0.6 74.28±0.5 I-EDL 44.05±1.2 82.17±0.8 84.99±1.2 58.87±2.7 46.09±2.5 53.22±0.4 84.20±0.3 93.92±0.3 77.42±1.8 74.32±0.4 DAEDL 38.25±0.9 77.28±1.0 85.87±0.8 67.30±1.8 45.73±2.2 53.34±0.9 80.38±0.5 94.62±0.2 82.10±1.5 77.79±0.8 F -EDL 42.54±0.7 77.97±1.2 85.26±0.5 67.67±1.3 37.75±1.1 53.00±0.6 82.99±0.2 95.14±0.3 82.59±1.4 70.69±1.6 R-EDL 51.83±0.3 86.25±0.2 89.37±0.7 67.83±1.9 55.74±2.5 53.69±0.8 83.40±0.4 95.08±0.3 82.03±0.7 78.71±0.6 DAPPr 55.95±0.5 88.22±0.4 89.53±0.7 67.77±2.5 59.39±2.1 61.59±0.8 87.89±0.3 95.47±0.3 82.37±1.1 78.76±1.4 Ensemble 52.57±0.2 85.12±0.1 88.98±0.1 71.30±0.3 47.18±0.2 61.11±0.4 87.83±0.1 96.79±0.1 85.75±0.2 75.89±0.4 +DAPPr 61.79±0.3 91.37±0.1 91.02±0.2 72.16±0.4 69.26±0.3 67.08±0.1 90.82±0.1 96.97±0.1 86.44±0.1 85.71±0.1 Method
Test Acc
71 68 65
OOD AUPR
OOD AUPR
Test Acc
EDL’s KL regularisation, (3) EDL with our regulariser, and (4) our full method. Removing the regulariser or replacing it with KL regularisation degrades all metrics, confirming its effectiveness. Applying our regulariser to EDL causes complete failure (near-zero accuracy), indicating that the regulariser is unique to our possibilistic method.
76 74 72
λ = 0 1e-4 1e-3 2e-3 5e-3 0.01 0.02
temic uncertainty and accuracy on CIFAR-100. Ideally, epistemic uncertainty should decrease as more data is observed. Figure 3 reveals a flaw in EDL: its epistemic uncertainty increases with more data at certain points, contradicting expected behavior. EDL shows unstable training, with accuracy dropping at 5K datapoints. In contrast, our method consistently decreases epistemic uncertainty as data grows and maintains stable, higher accuracy throughout. These results confirm that our possibilistic formulation provides a principled foundation for uncertainty modeling, whereas EDL’s heuristic interpretation leads to unreliable estimates.
62 60 58 85 80
Figure 4 shows the distribution of normalised α0 for ID and OOD samples. EDL produces heavily overlapping distributions, making OOD detection difficult. DAPPr clearly separates ID (high α0 ) from OOD (low α0 ), enabling reliable detection of distribution shift.
75
λ = 0 1e-5 1e-4 2e-4 5e-4 1e-3 2e-3
1 (a) CIFAR-100.
1 (b) Stanford Dogs.
0.88 0.85 0.82 0.84 0.82 0.80
Acc EDL
DAPPr
0.5 1 2 3 4 5 6 8 10 Data Points (× 1000)
50 40 30 20 10 0
16 Density
Epistemic Uncertainty
Figure 2. Test accuracy and OOD AUPR (↑) for varying λ on CIFAR-100 and Stanford Dogs. OOD AUPR averaged over their corresponding OOD datasets. EDL DAPPr
ID (CIFAR-10) OOD (SVHN) OOD (CIFAR-100)
12 8
ID (CIFAR-10) OOD (SVHN) OOD (CIFAR-100)
4 0 0.0
0.2
0.4
0.6
0.8
1.0 0.0
0.2
0.4
0.6
0.8
1.0
1
(a) EDL (b) DAPPr (Ours) Figure 4. Distribution of normalised α0 on CIFAR-10. DAPPr achieves clear separation; EDL does not.
0.5 1 2 3 4 5 6 8 10 Data Points (× 1000)
1 (a) Epistemic1 Uncertainty. (b) Accuracy. Figure 3. Epistemic uncertainty and accuracy on CIFAR-100 with varying training data size. Our method (DAPPr) decreases epistemic uncertainty with more data; EDL does not.
6. Conclusion We introduced Dirichlet-approximated possibilistic posterior predictions (DAPPr), a principled framework leveraging possibility theory to achieve both theoretical rigor and computational efficiency for epistemic uncertainty quantification. By defining a possibilistic posterior over parameters and projecting it to the prediction space via supremum operators, our approach yields a simple training objective with closedform solutions. Extensive experiments demonstrate that DAPPr achieves competitive or superior performance com-
Ablation on λ. Figure 2 shows effect of regularisation strength λ on CIFAR-100 and Stanford Dogs. Without regularisation (λ = 0), OOD detection degrades significantly. Small values such as 2e-4 or 5e-3 achieve strong accuracy and OOD detection. Performance remains stable across this range, indicating low sensitivity to hyperparameter tuning. Analysis. To verify our method properly captures epistemic uncertainty, we vary training data size and measure epis8
Possibilistic Predictive Uncertainty for Deep Learning Table 6. AUPR (↑) results for ablation on CUB-200-2011 and Stanford Dogs. Conf.: confidence estimation using aleatoric uncertainty. OOD detection via epistemic uncertainty against Image-O, DTD, and Places365, with scores averaged across the three datasets. Method DAPPr w/o Regularisation DAPPr w/ KL Regularisation EDL + our Regularisation DAPPr
Test Acc. 49.60±0.8 49.84±1.2 0.78±0.0 55.95±0.5
CUB-200-2011 Conf. OOD 83.43±0.5 53.73±1.3 84.23±1.0 61.52±1.3 0.90±0.1 46.57±0.8 88.22±0.4 72.23±1.7
pared to SOTA EDL methods across diverse benchmarks while properly capturing epistemic uncertainty behavior.
Test Acc. 58.07±1.2 58.92±1.4 1.53±0.1 61.59±0.8
Stanford Dogs Conf. 86.14±0.9 85.45±0.7 1.84± 0.1 87.89± 0.3
OOD 72.95±1.0 75.40±1.1 55.79± 0.6 85.53±0.9
2024. URL https://openreview.net/forum? id=Si3YFA641c. Chen, Z., Ristic, B., Houssineau, J., and Kim, D. Y. Observer control for bearings-only tracking using possibility functions. Automatica, 133:109888, 2021.
Impact Statement This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Cimpoi, M., Maji, S., Kokkinos, I., Mohamed, S., , and Vedaldi, A. Describing textures in the wild. In Proceedings of the IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), 2014.
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Possibilistic Predictive Uncertainty for Deep Learning
A. Proof for Proposition 1 With the log-possibility log gψ (p | x) = α0 log α0 +
K X
αk log
k=1
pk , αk
and the cross-entropy loss ℓ(p, y) = −
K X
yk log pk ,
k=1
Eq. (13) is derived as follows: p̃∗ = arg max
n
p∈∆
o log gψ (p | x) + ℓ(p, y)
K K n o X pk X = arg max α0 log α0 + − αk log yk log pk p∈∆ αk k=1
= arg max
K nX
p∈∆
k=1
k=1
o (αk − yk ) log pk ,
(18)
where terms independent of p are omitted. PK Since p ∈ ∆ implies the constraint k=1 pk = 1, we introduce a Lagrange multiplier β and form J =
K X k=1
(αk − yk ) log pk + β
K X k=1
pk − 1 .
(19)
Taking derivatives of Eq. (19) gives the stationarity condition
⇒ Enforcing
αk − yk ∂J = +β =0 ∂pk pk αk − yk . pk = − β
(20)
PK
k=1 pk = 1 yields
1=
K X k=1
⇒
K
pk = −
1X 1 (αk − yk ) = − (α0 − 1), β β k=1
β = −(α0 − 1).
Substituting Eq. (21) into Eq. (20) gives p̃∗k =
(21) αk − yk . α0 − 1
Thus, the closed-form expression for the approximate maximizer is p̃∗ =
α−y . α0 − 1
(22)
B. Datasets and Experiment Details B.1. Datasets MNIST (LeCun, 1998) consists of 60K training and 10K test examples. We use an 80/20 split for training and validation. KMNIST (Clanuwat et al., 2018) and FashionMNIST (Xiao et al., 2017) serve as OOD datasets. We train with a learning rate of 5e-4, batch size of 64, for 100 epochs. 13
Possibilistic Predictive Uncertainty for Deep Learning
CIFAR-10 and CIFAR-10-LT CIFAR-10 (Krizhevsky et al., 2009) contains 50K training and 10K test images across 10 classes, split 95/5 for training and validation. SVHN (Netzer et al., 2011) and CIFAR-100 serve as OOD datasets. We train VGG16 (Simonyan & Zisserman, 2014) with a learning rate of 5e-4, batch size of 64, for 100 epochs. CIFAR-10-LT is a long-tailed variant with artificially imbalanced class distributions, where the imbalance factor ρ, denotes the ratio of head-class to tail-class samples. We evaluate with ρ ∈ {0.01, 0.1}. CIFAR-100 (Krizhevsky et al., 2009) contains 50K training and 10K test images across 100 classes, split 95/5 for training and validation. SVHN and TinyImageNet (Deng et al., 2009) serve as OOD datasets. We train ResNet-18 (He et al., 2016) with a learning rate of 5e-4, batch size of 64, for 100 epochs. CUB-200-2011 (Wah et al., 2011) is a fine-grained bird classification dataset with 200 classes. Following (Jia et al., 2022; Ni et al., 2024), we use 5,394/600/5,794 images for training/validation/testing. ImageNet-O (Hendrycks et al., 2021), DTD (Cimpoi et al., 2014), and Places365 validation set (Zhou et al., 2017) serve as OOD datasets. We train ResNet-50 (He et al., 2016) with a learning rate of 2e-3, weight decay of 1e-4, batch size of 256, for 200 epochs. Stanford Dogs (Khosla et al., 2011) is a fine-grained dog classification dataset with 120 classes. Following (Jia et al., 2022; Ni et al., 2024), we use 10,800/1,200/8,580 images for training/validation/testing. We use the same OOD datasets and training configuration as CUB-200-2011. B.2. Hyperparameters Table 7 presents the regularization hyperparameter λ used across different datasets. Notably, λ remains consistently small across all datasets, requiring minimal tuning, which demonstrates the stability and ease of use of our method. Table 7. Hyperparameters used for our regularizer. Dataset λ
MNIST 1e-5
CIFAR-10 2e-3
CIFAR-10-LT 2e-3
CIFAR-100 5e-3
CUB-200-2011 2e-4
Stanford Dogs 2e-4
B.3. Baseline Implementations for Fine-Grained Datasets CE: We train the network with cross-entropy loss. For confidence estimation, we use 1 − maxk pk as aleatoric uncertainty. For OOD detection, we use the entropy over class probabilities as epistemic uncertainty. MC Dropout: After training with cross-entropy, we perform T P = 10 stochastic forward passes with dropout rate 0.5, T obtaining predictions {p(t) }Tt=1 . The averaged prediction is p̄ = T1 t=1 p(t) . We use 1 − maxk p̄k as aleatoric uncertainty PT 1 (t) for confidence estimation, and EU = H(p̄) − T t=1 H(p ) as epistemic uncertainty for OOD detection. PT Ensemble: We train T = 10 models from scratch with different random seeds. Predictions are averaged as p̄ = T1 t=1 p(t) . Aleatoric and epistemic uncertainties are computed identically to MC Dropout. Ensemble+DAPPr: We train T models using DAPPr loss with different random seeds. The averaged outputs serve as Dirichlet parameters for computing aleatoric and epistemic uncertainties. RS-NN: Following (Manchingal et al., 2025), we use a pretrained ResNet-50 for feature extraction and construct random sets as new classes. We search hyperparameters over {1e-6, 5e-6, 2e-5, 1e-4, 5e-4, 1e-3, 5e-3} and report the best results. EDL, I-EDL, DAEDL, F-EDL, R-EDL: For each EDL-based method, we tune the KL regularization coefficient along with method-specific hyperparameters (e.g., Fisher information regularizer for I-EDL, evidence offset for R-EDL, and MLP architecture for F-EDL). We report the best results from the hyperparameter sweep.
C. Additional Results To evaluate our approximation assumption, we conduct a leave-one-out analysis on CIFAR-100 using a pre-trained ResNet-18 model θ0 . For each sample (x, y) in the training set, we perform two fine-tuning experiments: 1. True label fine-tuning: Fine-tune θ0 for three epochs on D \ {(x, y)} while forcing the model to fit (x, y) by including it in every batch during training, yielding model θtrue and loss Ltrue = L(θtrue ; D \ {(x, y)}). 14
Possibilistic Predictive Uncertainty for Deep Learning 1.00
Sx
0.75 0.50 0.25 0.00
310
320 Ltrue
330
340
Figure 5. x axis:1Ltrue . y-axis: Sx
2. Perturbed label fine-tuning: Fine-tune θ0 identically, but replace the label of x with a randomly sampled soft label p ∈ ∆K−1 for x, yielding model θp and loss Lp = L(θp ; D \ {(x, y)}). For each sample, we compute the maximum loss deviation Sx = maxp |Lp − Ltrue | over multiple random perturbations. Figure 5 plots (Ltrue , Sx ) for all samples. The results show that Sx is negligibly small relative to Ltrue (e.g., Sx ≈ 0.25 versus Ltrue ≈ 320, validating that perturbing a single sample’s label has minimal impact on the overall loss, a key assumption underlying our approximation.
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