A Review of Bayesian Uncertainty Quantification in Deep Probabilistic Image Segmentation
Authors: Amaan Valiuddin
Ruud Van Sloun
Christiaan Viviers
Peter H.N. de With
Fons van der Sommen
Venue: Accepted by TMLR ()

Abstract: Advances in architectural design, data availability, and compute have driven remarkable progress in semantic segmentation. Yet, these models often rely on relaxed Bayesian assumptions, omitting critical uncertainty information needed for robust decision-making. Despite growing interest in probabilistic segmentation to address point-estimate limitations, the research landscape remains fragmented. In response, this review synthesizes foundational concepts in uncertainty modeling, analyzing how feature- and parameter-distribution modeling impact four key segmentation tasks: Observer Variability, Active Learning, Model Introspection, and Model Generalization. Our work establishes a common framework by standardizing theory, notation, and terminology, thereby bridging the gap between method developers, task specialists, and applied researchers. We then discuss critical challenges, including the nuanced distinction between uncertainty types, strong assumptions in spatial aggregation, the lack of standardized benchmarks, and pitfalls in current quantification methods. We identify promising avenues for future research, such as uncertainty-aware active learning, data-driven benchmarks, transformer-based models, and novel techniques to move from simple segmentation problems to uncertainty in holistic scene understanding. Based on our analysis, we offer practical guidelines for researchers on method selection, evaluation, reproducibility, and meaningful uncertainty estimation. Ultimately, our goal is to facilitate the development of more reliable, efficient, and interpretable segmentation models that can be confidently deployed in real-world applications.

Published in Transactions on Machine Learning Research (12/2025)
A Review of Bayesian Uncertainty Quantification in Deep Probabilistic Image Segmentation
Amaan Valiuddin
m.m.a.valkuddin&ttr nl r-j-g.v. f.g.v.sloun@ttin @uue.nl e.g. a. viviers@lue.. p.h.n. de. with nl fudsommen &tue.
Ruud van Sloun Christiaan Viviers
Peter de With
Fons van der Sommen
Department of Electrical Engineering Eindhoven University of Technology Eindhoven, The Netherlands
Reviewed OpenReview: https: /ptp::///penreveeu.. net/ forum? id= Yaf4anluao
Abstract
Advances in architectural design, data availability, and compute have driven remarkable progress in semantic segmentation. Yet, these models often rely on relaxed Bayesian assump- tions, omitting critical uncertainty information needed for robust decision-making. Despite growing interest in probabilistic segmentation to address point-estimate limitations, the research landscape remains fragmented. In response, this review synthesizes foundational concepts in uncertainty modeling, analyzing how feature- and paraameeerdiitrribttion mod- cling impact four key segmentation tasks: Observer Variability, Active Learning, Model rospecccton,, and Model Generalization. Our work establishes common framework by standardizing theory, notation, and terminology, thereby bridging the gap between method developers, task specialists, and applied researchers. We then discuss critical challenges, including the nuanced distinction between uncertainty types, strong assumptions in spa- tial aggregation, the lack of standardized benchmarks, and pitfalls in current quantification methods. We identify promising avenues for future research, such uncertainty-awaat tive learning, data-driven benchmarks, transformer- based models, and novel techniques to move from simple segmentation problems to uncertainty in holistic scene understanding. Based on our analysis, we offer practical guidelines for rescarchers on method selection, evaluation, reproducibility, and meaningful uncertainty estimation. Ultimately, our goal facilitate the development of more reliable, efficient, and interpretable segmentation models that be confidently deployed in real-world applications.
1 Introduction
Image segmentation entails pixel-wise classification of data, effectively delineating objects and regions of interest (Szeliski 2010). The advent of convolutional neural networks (CNNs) has led to major breakthroughs
al. 2015). obtaining impressive scores with largo-scale segmentation datasets (Lin al. 2014; Cordtt et al. 2016b; Richter et al. 2016). However, these models typically rely on strong assunptions and significant relaxations of the Bayesian learning paradigm, neglecting the uncertainty associated within their predictions. This lack of uncertainty modeling reduces both the reliability and interpretability of these predictions. In high-stake applications, such as autonomous driving or medical diagnosis, this can have severe consequences. For instance, misclassifying objects autonomous driving or overlooking uncertainty in lesion classification can both lead to critical decision- makinn errors.
in this domain (Ronneberger al. 2015; Shelhamer et. al.] 2014; Badrinarayanan
Fortunately, the merits of uncertainty quantification have been well-recognized in the field of CNN-basod segmentation, especially interpretability and reliability have become central in data-driven applications.

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Hence, extensive efforts have been made to align neural network optimization with Bayesian machine learning, such as learning parameter distributions rather than point estimates or to explicitly model the likelihood distribution of model features (Blundell et al. 2015 Guo et al. 2017 Kingma & Welling 2013 Kendall & Gal 2017p. At the same time, uncertainty in image segmentation often treated as an auxiliary tool to enhance downstream performance, rather than as a primary modeling objective. Consequently, many studies lack a rigorous theoretical foundation, prioritizing empirical gains over principled formulations and clear definitions of uncertainty. This narrow focus frequently results in methods tailored to specific datasets or modalities, which limits their generalizability. Due to this application-driven nature, most existing surveys adopt
medical perspective (Jungo & Reyes 2019: Kwon et al. (020)) often focused on specific modalities (MeCrindle lt al.
2021 Jungo et al. 2020; Ng et al. 2022 Roshanzamir et al. 2023). Additionally, the study by [ahh et al. segmentation, but focuses primarily on empirical evaluation within specific tasks. Moreover, theoretical contributions often originate from adjacent fields like Bayesian deep learning or information theory, and can be insufficiently contextualized within the domain of segmentation. This disconnect has resulted in a fragmented body of work, with inconsistent terminology, evaluation metrics, and assumptions. The resulting abundance of literature can be overwhelming, even for experienced researchers.
2024 introduces a valuable framework for benchmarking uncertainty disentanglement in semantic
Given the fragmented nature of the existing literature and the renewed interest in uncertainty (Papamarkou et al. 2024 Kirchhof et al., 2025), we note that a comprehensive overview in the field remains limited. Our central contribution is to unify core uncertainty modeling concepts across methods, tasks, and applications, thereby offering substantial theoretical insights and demonstrating their relevance across diverse domains. We unify and contextualize the underlying theory, and standardize terminology and notation. We systematically compare methodologies, link them to specific tasks, and connect those tasks to practical applications. Hence, this review is not only a conventional 'zero-to-hero' survey; rather, it additionally serves a cross-domain synthesis that bridges method developers, task specialists, and application- ieplicatioo-focueed researchers. Our aim is to provide a cohesive framework for navigating and integrating the complex uncertainty modeling landscape. This broader perspective helps clarify conceptual foundations, highlight cross-cutting insights and provide a comprehensive grasp of the key challenges and open research directions in the field. In the absence of broad quantitative consensus, we adopt a robust, evidence-based methodology for making recommendations. By leveraging deep qualitative analysis of the literature, we derive reliable method comparisons and guidelines that offer essential and practical guidance. However, this reliance on qualitative synthesis rather than quantitative evidence is an important limitation and our recommendations should be interpreted accordingly.
We begin by establishing the background on image segmentation in Section 2 Then, Section 3 introduces the notation and general theoretical principles of probabilistic image segmentation. Subsequently, we organize the review based on the source of introduced stochasticity: the feature level (Soction 4 and the parameter level (Section 5). This structure aligns with the underlying theoretical decomposition of uncertainty and the utility of the modeled uncertainty for the downstream tasks and application domains discussed in Section G The review concludes by moving to the discussion (Section where we highlight important pitfalls, challenges and opportunities in the field. This is followed by recommendation and guidelines for rescarchers (Section 8 before concluding in Section g The core structure and organizational flow of this paper is visualized in
Figure !
which serves as a tep-level roadmap for the reader.
2 Background
A common approach image understanding involves labeling pixels according to semantic categories, which is the core of semantic segmentation. This technique is particularly well-suited for amorphous or uncount- able subjects. In contrast, instance segmentation not only assigns class labels, but also distinguishes and delineates individual object occurrences. This makes it more appropriate for scenarios involving countable entities. A third variant, panoptic segmentation, unifics both semantic and instance-level labeling, offering a comprehensive view of scene composition. As summarized by Minaee el al. (2020), semantic segmentation has been performed using methods such as thresholding (Otsu, 1979), histogram- bastogram-based bundling, region- growing (Dhanachandra et al. 2015), k-means clustering (Nock & Nielson 2004), watershedding (Najman & Schmitt 1994), to more advanced algorithms such as active contours (Kass et al. 2004). graph cuts (Boykov
et al. 2001 conditional and Markov random fields (Plath et. al. 2009), and sparsity- methods (Starck
2

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Deep Probabilistic Image Segmentation
METHODS
TASKS
DISCUSSION
RECOMMENDATIONS
Feature space
Observer Variability
Disentanglement
7.1
Method selection
Pixel-level
Active Learuing
6.2
Spatial coherence
1.2
Evaluation
Latent-level
1.2
Model Intrespection
Standardization
Limitations
Parameter space
Model Gemeralization
Active Learning
'Goud' uncertainly
8.4
Varlational Inference
Application domains
6.5
Method limitations
dependency
Luplace Appr.
Backbonrs
7.7
Test-time
Contemporary methods
Scene understanding
7.9
Figure 1: Overview the sections. The leafs of the presented hierarchical tree are related to subsections of the article.
al. 2005 Minace & Wang 2017 While the literature on segmentation is vast and rapidly evolving, selection of backbone architectures has been particularly influential in shaping current probabilistic seg- mentation models. The focus here is not exhaustiveness, but on those models most relevant to the development of uncertainty-awarr approaches.
In particular, following the successful application of CNNs LoCun et al. 1998). image segmentation ex- perienced rapid progress driven by increasingly powerful and specialized deep architectures. Notably, the Fully Convolutional Network (FCN) (Shelhamer al. 2014) adapted the AlexNet (Krizhevsky et al. 2012) VGG16 (Simouyan & Zisserman 2014) and GoogLeNet (Szegedy al. 2014] architectures to enable end-to to end semantic segmentation. Furthermore, other CNN architectures such as DeepLabv3 (Chen al. 2017 and the MobileNetv3 (Howard et al.] (019g) have also been commonly used. As the research progressed. increasing success has been observed with encoder-decoder models (Noh et al. 2015 [Bdrinarayanan et al. 2015 Yuan et al. 2019; Ronneberger et al.) 2015). Initially developed for the medical applications, Ron- neberger et al. (2015) introduced the U-Net, which successfully relies on residual connections between the encoding-decoding path, to preserve high-frequency details in the encoded feature maps. To this day, the C-Net still often utilized the default backbone model for many semantie segmentation and even general image generation architectures (Ho et al., 2020 Song et al. 2020 particularly in the medical domain. In
fact, reports of recent research indicate that the relatively simple U-Net [[sensee et al. 2020) still outperform
more contemporary and complex models |Eiscnmann et al. 2023 Isensee et al. 2024
3 Probabilistic Image Segmentation
Assuming random-variable pairs (Y,X) Pr.x that take values E ZkxHxw and X E RCCH*W respectively, then instance y can be considered as the ground-truth of K-class segmentation task and instance the query image. The variables H, W and C correspond to the image height, width and channel depth, respectively. Conforming to the principle of maximum entropy, the optimal parameters given the data (i.e. posterior) subject to the chosen intermediate distributions can be inferred through Bayes Theorem as
p(y|x, 0)p(O) p(y]x)
r0 x)
assuming independence of x from 0, and where p(0) represents the prior belief on the parameter distribution and p(y' x) the conditional data likelihood (also commonly referred to the evidence). After obtaining
3

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posterior with dataset D yx containing N images, the predictive distribution for new datapoint x" can be denoted as
Model
p(Yx''
p (( do.
(2)
Data
As evident, both the variability in the empirical data and the inferred parameters of the model influence the predictive distribution. A straightforward approach quantify the total uncertainty is achieved by obtaining the predietive entropy, H H D], i.c the entropy of the predictive distribution p oY XYxx", ;
epistemic
H Y x" D) 1 0x' DJ ( H ,
(3)
sleatoric
where I represents the mutual information. The first term represents the expected information gain about the model paramcters 0 given the true label Y. A high value indicates the model is very uncertain about its prediction, meaning observing the true label would provide large information gain. The second term represents the inherent randomness in the data, found by averaging the predictive uncertainty across the posterior of plausible models from Equation ( This reflects the noise that cannot be reduced, regardless of how much more training data observed. This distinction allows us to determine if a model's uncertainty stems from its own ignorance or from ambiguity inherent. the data. Therefore, uncertainty often split into two types: epistemic uncertainty, which reflects the model's limited knowledge of its parameters, and aleatoric uncortainty, which captures the irreducible statistical randomness in the data itself. Nonctheless, determining the nature of uncertainty is rarely straightforward. For example, Hällermeier & Waegeman (2021) stated that "by allowing the learner to change the setting, the distinction between these two types of uncertainly will be somewhat blurred'. This sentiment is also shared by Der Kiureghian & Ditlevsen (2009) noting that "in one model an addressed uncertainty may aleatory, another model it may be epistemic" Sharing similar views, we highlight the necessity of careful analyses and possible subjective interpretation regarding the topic as we treat the realm of quantifying spatially correlated uncertainty. Especially with increasingly complex methodologies, treating the uncertainties separate concepts is mostly theoretical (often even more philosophical) and highly non-trivial in practice.
3.1 Conventional segmentation
Regardless of the elegantly formulated Bayesian posterior, most practical approaches make use of so-called "doterministic" segmentation networks. Such models are trained by Maximum Likelihood Estimation (MLE), specified as
MLEE arg max log (lyl(( 0),
(4)
0
which simplifies the training procedure by taking a point estimate of the posterior. This approximation improves as the training data increases and the model parameter variances approach zero. As such, MLE
pY 0)
fo
C1
C2
ea
C2
X
a
Σg eak
;
CK
Ck.
C1
1
K
Figure/ 2: Uncertainty modeling by treating pixel-level confidence values as parameters of Probability Mass Function.
4

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does not include any prior knowledge the structure of the parameter distribution. This, however, can be achieved through Maximum A Posteriori (MAP) estimation with
MAP arg max log ply x. 0) + log p(0).
(5)
For example, assuming Gaussian or Laplacian priors leads to regularizing the L2 norm (also known as ridge reyression or weight decay) or L1 norm of 0, respectively (Figueiredo, 2001 Kaban 2007). model ply|x, 0), we make use of function fo RCxD kKxD that infers the parameters of a Probability Mass Function (PMF). For instance, consider spatial image dimensions D and CNN with fo(x). Then, can write
eaa
11xx,, P(Y = x 0)
(6)
Σ. E
with channel-wise indexing over the denominator, which is commonly known the SoftMax activation (Figure 2). Hence, this approach is probabilistic modeling in the technical sense, although it is not referred to such in commno nomenclature. As such, the approximated distribution can represent and localize uncertain regions. However, the implicit pixel-independence assumption
KXD
X = II PY])),,
omits information on structural variation in the segmentation masks. In conventional classification, factoriz- ing the categorical distribution is lypically regarded logical simplification. In probabilist ic segmentation however, this assumption has caused the emergence of distinct research direction (See Figure 3
Key takeaways: Conventional models yield independent uncertainty estimates for each pixel, which omits spatial coherence. This failure to model spatial correlation precludes realistic inference and inhibits discerning common underlying causes for uncertainty in neighboring regions.
pY||,, 0)
arg max p (Y|X, 0)
y p(| 0)
(a) Likelihoods
(b) Thresholding
(c) Sampling
Figure 3: Ilussrationn of the likelihood function in segmentation models. Color intensities reflect the nor- malized confidence values that can be interpreted as probabilities. (a) The continuous output results in a horizontal gradient. (b) Maximum likelihood thresholding can be applied. (c) However, the coherence of the segmentation suffers when sampling because information on neighboring pixels is disregarded.
3.2 Review organization and structure
The body of this review is structured around three interconnected concepts that serve as the fundamental organizing framework for the field of uncertainty in image segmentation. This structure is intended to facilitate cross-domain synthesis and enable rescarchers from diverse backgrounds to navigate the literature effectively. Consistent with the decomposition in Equation 2 we structure the methods based on feature- level (Section 4 and parameter-level (Section uncertainty modeling. We then detail the utility of these

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uncertainties for four key downstream tasks in Section linking them to methods via Tables where the connection to practical application domains is summarized in Table Following this structured overview, we provide a detailed discussion (Section]) 7 to syuthesize key insights, culminating in practical recommendations for the field (Section
Core concepts:
Methods: How uncertainty is mathematically modeled and quantified.
Tasks: What specific goals accomplished using these uncertainty estimates.
Applications: Where these methods and tasks are applied in real-world domains.
4 Feature modeling
Following Equation 2 we focus in this section on approaches that place uncertainty at the level of features. In most practical scenarios, methods assume a paramcterized likelihood function p(Y[X, 0), where parameters are point estimates. Accordingly, the central design decision where to introduce auxiliary stochastic variables. Hence, categorize models based the location of these latent variables: those introduced near the output are discussed in Section 4.1 while lower-dimensional or latents embedded deeper within the architecture are covered in Section 4.2 Although their theoretical formulations are similar, the practical consequences can differ substantially.
4.1 Pixel-level sampling
Uncertainty in segmentation masks can be modeled directly al the pixel level. These approaches can be further categorized into those that assume independence between pixels (Section 4.1.1 and those that explicitly model spatial correlations (Section 4.1.2). In the former case, accurate uncertainty estimates rely on well-calibrated models, which is an assumption that often fails in practice without additional tuning on separate calibration set. Furthermore, such methods do not address the challenge of spatial coherence. In contrast, the latter approach introduces a stochastic variable to capture dependencies between neighboring pixels.
4.1.1 Independence
While it is well established that assuming independence across pixels is problematic, it standard approach in much of the early literature. As discussed demonstrated in Equation (6), neural network predictions are often normalized with SoftMax activation in order to interpret the confidence values as parameters of a probability mass distribution. However, these values rarely reflect the true probabilities modern neural networks, hypothesized due to the use negative log- ligelikelihood as the training objective, along with regularization ion techniques such as batch normalization, weight decay, and others (Guo et al. 2017) Thus, interpreting the confidences as true probabilities is often only justified after proper validation, which is referred to model calibration.
Calibration Different methods can be used to quantify whether the empirical accuracy of a model approx- imately equals the provided class confidence for class k, iee. P(Y = k (x) = cı. A fairly straightforward metric, the Expected Calibration Error (ECE), determines the normalized distance between accuracy (acc) and confidence (conf) bins as
B FECE Σ
nh acc((b) - conf(b)),
with ny the number of samples bin b and being the total sample size across bins. The ECE is prone to skew representations some bins are significantly more populated with samples due over- cunder-confience. Furthermore, the Maximum Calibration Error (MCE) more appropriate for high- risk applications, where only the worst bin is considered. Additionally, when background pixels have

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1.0
0.8
0.8
Accuracy
Accuracy
0.6-
0.6
0.4
0.4
0.2
0.2
0.0
0.8
1.0
Confidence
Confidence
(a) Calibrated
(b) Miscalibrated
Figure 4: Visualization of model calibration with reliability diagram, which demonstrates whether the predicted model confidences align with the observed empirical accuracies.
predominant influence each bin can be weighted equally using the Average Calibration Error (ACE) to mitigate this imbalance (Jungo et al. 2020 Neumann et al. 2018). Calibration is typically visualized with a reliability diagram, where miscalibration and underr/over-confddenee can be assessed by inspecting the deviation from the graph diagonal (Figure
Methods Most of the calibration techniques are post-hoc, i.e. applied after training, they necessitate separate validation set. For example. over-fitting has often been considered be the cause of over- confidence (Szegedy et al.] 2016 Percyra et al. 2017) and erroncous pixels therefore be penalized through regularizing low-entropy outputs (Larrazabal et al. 2021a). Another prominent method is Temperature Scaling (Guo et al.; 2017). which has been successfully applied in pixel-wise manner to segmentation problems (Ding et al.
2021 The standard softmax probability in Equation (6) is modified as
e
YY=
Σg
eau/T
with some scalar T optimized validation set. For overconfident models, T > 1 softens the estimated distribution, pushing output probabilities away from the extremes of 0 and 1 toward more moderale values. In contrast, label Smoothing (Silva & Oliveira, 2021 Liu et al. 2022a) require only modifying the training data. Instead of one-hot encoding Y in to [0, IJ, a small positive random variable is introduced, which converts the targets into [e, 1 - c). discouraging the model from producing overly confident logit outputs. The Focal Loss (Mukhoti al., 2020) adds modulating factor to down-weight the loss from well-classified examples. This forces the model to focus more on hard-to-classify pixels, which can implicitly improve calibration by preventing it from becoming overly certain about easy examples.
Ultimately, relying on raw model outputs probabilities typically yields sample-level spatial incoherence. Because modern neural networks are often uncalibrated by default, they require post-hoc calibration. Cru- cially, resolving spatial incoherence does not preclude good calibration, i.e., spatially correlated models can still well calibrated. Conversely, calibration alone cannot impose spatial structure. Well-calibrated models may remain spatially incohcrent. This asymmetry motivates placing greater emphasis on modcling spatial coherence rather than calibration alone.

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X
R
Y
ree
Y
PΦ
L X 2
1 x
Pz
Figure 5: Illustration of the PixelCNN-based PixelSeg (Zhang et al. 2022c Parameter "C" indicates the concatenation module, 'R' the resampling module and a the softmax activation. Dotted elements appear during test-time sampling.
4.1.2 Spatial correlation
Introducing spatial correlation can be achieved through an autoregressive approach. For instance, we can rephrase Equation 7 to
KXD
p(] = II pYY.).,..... X,,,,
(10)
where pixel Y, is predicted based the preceding pixels. A popular implementation of this formulation is known the PixeICNN (Van Den Oord et al. 2016). For dense predictions, Zhang al (2022c) propose PixelSeg, which predicts a downsampled segmentation mask with P (f(x,, and fuse this with a conventional CNN to predict the full-resolution mask with pe (yl x). The two masks are fused through resampling module, containing a series of specific transformations to improve quality and diversity of the samples (Figure 5). Notably, the recursive sampling process also enables completion/inaanntten of user-given inputs.
Monteiro et al. (2020) propose the Stochastic Segmentation Network (SSN), which models the output logits multivariate normal distríbution, parameterized by the neural networks f and fg. Given the output features fx(x) of deterministic model, we can denote the logits distribution as
p(a x 0) (aa: 1 = 1/' (x), E 10 (x)):
(11)
where the covariance matrix has a low-rank structure Σ PP" +A, with P having dimensionality ((Kx (K)D)> R), with R being a hyperparameter that controls the parameterization rank and A representing diagonal matrix. The low-rank assumption results in more structured distribution, while retaining reasonable efficiency. Monte Carlo sampling is used to generate predictions, which are then mapped to categorical values using the SoftMax activation. SSNs can theoretically be augmented to any pretrained CNNs as an additional layer (see Figure Gh.
R
P
a~ N( =
P
Σ =
A
X
ea
A
Y
Lyec
V
Σ, ak
Figure 6: Depiction of a Stochastic Segmentation Network (Monteiro et al. 2020). Here, the covariance of the likelihood distribution explicitly modeled through low-rank approximation.

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Key takeaways: Calibrating standard segmentation models can produce valid confidence values. However, it still preserves the incorrect pixel-independenne assumption and requires a separate valida- tion set. A possible solution is model the correlation directly in pixel space through auloregressive modeling or low-rank approximation of the output distribution.
4.2 Latent-level sampling
The limitation of modeling intricate, complex distributions directly the output level can be mitigated by using generative models, which often rely on simpler, lower-dimensional latent variables Z pz with Z E Rd, to instead learn the approximation through
/ po (z)x)))
Pe, v (Y =
(12)
with parameters 2b. As such, the spatial correlation is induced through mapping the latent variables to segmentation masks. On a related note, an unconditional prior p(z) can also be employed. However, con- ditioning the latent density, i.c. P (2 is usually preferred for smooth optimization trajectories (Zheng et al., 2022). The key overarching contribution of latent-level sampling methods lies in the conditioning of generative models on the input image. This section briefly introduces Generative Adversarial Networks (Section 4.2.1 Variational Autoencoders (Section 4.2.2), and Denoising Diffusion Probabilistic Models (Sec-
tion 1.33 and outlines how these architectures commonly adapted for probabilistic segmentation.
4.2.1 Generative Adversarial Networks (GANs)
A straightforward approach is simply learn the marginalization Equation 12 through sampling from an unconditional prior density, pz, and mapping this to segmentation Y through generator G X - V. Goodfellow et al. (2014) show that this approach can be notably enhanced through the incorporation of discriminative function (the discriminator), denoted as D RCxD [0, 1]. In this way, G learns to reconstruct realistic images using the discriminative capabilitics of Dy, making sufficient guidance from D& to G imperative. We can denole the cost of Gø in the GAN as the negative cost of Dxp as
JG, -J = -Ep Do (y)) Epz Epo log((1 - (G, x)))].
(13)
Conditional GANs have been used for semantic segmentation in early literature (Isola et al. 2017 However, Kassapis et al. (0021) explicitly contextualized the architecture within uncertainty quantification, using their proposed Calibrated Adversarial Refinement. (CAR) network (see Figure 7 The calibration network, Fo RCxD + IR KDD, initially provides SoftMax activated prediction Fx(x) c, with (eross-entropy) reconstruction loss
Cree -Ep [log pe(cxx)].
(14)
Then, the conditional refinement network Go uses e together with input image x and latent samples z; injected at multiple decomposition scales i, to predict various segmentation maps. Furthermore, the refinement network is subject to the adversarial objective
Ladv = EEpo (log D (G)
(15)
which is argued to elicit superior structural qualities compared to relying solely on cross-cntropy loss. At the same time, the discriminator opposes the optimization with
LD EPp Eppl log Du x)) Epp [log (v(y)).
(16)
Finally, the average of the N segmentation maps generated from Go are compared against the initial pre- diction of Fg through the calibration loss, which is the analytical KL-divergence between the two categorical densities, denoted by
Leal Epp KT[po(ylc, x) po([)))
(17)

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Lal
C
P
X
Y
X
D4
X
- Pz
4p
[0,1]
Ladv
Figure 7: Diagram of the Calibrated Adversarial Refinement network (Kassapis et al. 2021 1, based on the
Generative Adversarial Network (Goodfellow et al. 2014) with additional mechanisms and loss terms.
In this way, the generator loss can be defined as
La = Ladv + A Leal,
(18)
with hyperparameter X 2 The purpose of the calibration network argued to be threefold. Namely, it (1) sets calibration target for Leal, (2) provides an alternate representation of X to Gø. and (3) allows for sample-free uncertainty quantification. The refinement network can be seen modeling the spatial dependency across the pixels, which enables sampling coherent segmentation maps.
4.2.2 Variational Autoencoders (VAEs)
Techniques such as GANs rely implicit distributions and are void of any notion of data likelihoods. An allernative approach estimates the Bayesian posterior w.r.t. the latent variables, p(Z X), with an approximation qo Y X), obtained by maximizing the conditional Evidence Lower Bound (ELBO)
logp( (Y)) = log / (x) p(z, Y[X)
[z) Y, X) dz
pzz, XX)
40 (z) Y. log
dz
Qa (z) Y, X)
-E log Y zz X)
(19)
where Jensen's inequality justifies moving the logarithm inside the integral. The first. term in Equation 19 represents the reconstruction cost of the decoder, subject to the latent code Z and input image X. The second term is the Kullback-Leibler (KL) divergence between the approximate posterior and prior density. As consequence of the mean-field approximation, all involved densities are modeled by axis- gaussian densities and amortized through neural networks, parameterized by Φ, 0 and sb. The predictive distribution after observing dataset D is then obtained as
xYYx')== ff Po (Y (xxx""")
(20)
The implementation of the conditional ELBO in Equation 19 can be achieved through the well known VAE architecture (Kingma Welling, 2013). Furthermore, some additional design choices lead to the segmentatioonsspeefic Probabilistic U-Net (PU-Net) (Kohl et al. 2019b). Firstly, the latent variable is only introduced at the final layers of a U-Net conditioned on the images. The latent vector up-scaled through tiling and then concatenated with the feature maps of the penultimate layer, which is followed by sequence of 1x1 convolution for classification. When involving conditional latent variables in this manner, it is expected that most of the semantic feature extraction and delincation are performed in the U-Net, while the variability in the latent samples is almost exclusively related to the segmentation variability. Therefore, relatively smaller values of are feasible than what commonly used in conventional image generation tasks. Similar related rescarch on the VAE (Zhao al. 2017 Bousquet et al., 2017 Higgins et al.) 2017 Van Den Oord et al. 2017 Rezende & Mohamed 2015). much work has been dedicated improving the PU-Net.
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46
4y
Y
KL.
X
X
Lapez
1xl convolution
Samole upscale
X
Y
Postenor pa parameters
Prior parameters
Figure 8: Depiction of the Probabilistic U-Net Autoencoder (Kingma & Welling 2013). The latent samples are inserted at the final stages of U-Net through a tiling operation.
(Kohl et al. 2019bb based on conditional Variational
Density reparameterization Augmenting a Normalizing Flow (NF) to the posterior density a VAE is a commonly used tactic to improve its expressiveness RRzznnd & Mohamed, 2015). This phenomenon has also been successfully applied to VAE-like models such as the PU-Net (Valiuddin et al. 2021 Selvan et al. 2020 NN's are a class of generative models that utilize & consecutive bijective transformations fu RD + R f fK .... fk o... fi, to express exact log-likelihoods of arbitrarily complex approximations of the posterior p(z[x), where symbol indicates functional composition. The approximated posteriors in turn induce an approximate of p(x|z) = p(x|x)) piz)
p(x) and are often denoted as logr(x) for simplicity. The fractional
term accounts for the change in probability space and corresponds to the log-Jacobian determinant, as given
by the Change of Variables theorem. Thus, we can denote
Σ log del dfx(2x=1)
logp(x) logrz(Zo)
(21)
dz--1
where z and Z are variables from intermediate densities and Zo f- fx). Equation 21 can be substituted in the conditional ELBO objective in Equation 19 to obtain
p x 2 (x) log Y X)
log det (f
-KL[ ( X)] Eg YY,
(23)
dzz-
where the objective consists of a reconstruction term, sample-based KT KT-divergence and a likelihood correction term for the change in probability density induced by the NF. Bhat et al. (2023 2022a) compare this
approach with other parameterizations of the latent space, including a mixture of Gaussians and low-rank approximation the full covariance matrix. Valiuddin et al. (2024b) show that the latent space can converge contain non-informative latent dimensions, undermining generally referred to as mode posterior collapse (Chen al.) 2016 considers the alternative formulation of the ELBO, specified as
capabilities of the latent-variable approach,
Zhao al. 2017 Their proposition
logp(Y(X) 2 Eyp (Z) [l P (Y 2)] (Z(ga(z|XX) To [X] ((X,
(23
The proposed novel objective maximizes the contribution of the (expected) mutual information between latent and output variables, i.e. the aleatoric uncertainty term of the predictive entropy defined later in Equation
42 This enables the introduction of the updated objective
C = - EE ( / xxy x//g. XYX,
(24)
+ a KI. (g (z Y, X) || p (2) + B (e(gg0z(X)) (2x
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with Se being the Sinkhorn divergence (Cuturi 2013 and a, 3 denoting hyperparameters. This adaptation results in a more uniform latent space leading to increased model performance. Also, modeling the ELBO of the joint density has been explored (Zhang et al., 2022b). This formulation results in an additional roconstruction term and forces the latent variables to be more congruent with the data. Furthermore, constraining the latent space to be discrete has resulted in some improvements, where it is hypothesized that this partially addresses the model collapse phenomenon (Qiu & Lui 2020
Multi-scale approach Learning latent features over several hierarchical scales can provide expressive den- sities and interpretable features acToss various abstraction levels (Sønderby et al. 2016 Kingma et al. 2016 Klushyn et al. 2019 Gregor et al. 2015 Ranganath et al., 2016). Such models are commonly categorized under hierarchical VAE (HVAE) modeling. Often, an additional Markov assumption of length T is placed
on the posterior as
T 90 [2 Y, Y) II
9o (Z Y, X)
do (Z 11 X).
(25)
t=2
Consequently, the conditional ELBO is denoted as
(Y 2 E. oz x. log (2,
(26)
T KL Y, X, Z14-1) (11-11)llgglee (a XX]]oo/a1
This objective is implemented in the Hierarchical PU-Net (Kohl et al. 2019ah (HPU-Net, depicted in Fig- ure g which learns a latent representation at multiple decomposition levels of the U-Net. Residual connec- tions in the convolutional layers are necessary to prevent degeneracy of uninformative latent variables with the KL divergence rapidly approaching zero. For similar reasons, the Generalized ELBO with Constrained Optimization (GECO) objective is employed, which extends Equation (26) to
( LGECO A. I 12 ( log (Y X)z X)] -kl12
(27)
EL KL (2 Y, X, 21=-1) || (z [1)=11] (2) gp (X)
Hyperparameter À is the Lagrange multiplier update through the Exponential Moving Average of the recon- struction, which is constrained to reach target value K, empirically set beforehand to appropriate value. Finally, online negative hard mining is used to only backpropagate 2% of the worst performing pixels, which are stochastically picked with the Gumbel-SoftMax trick (Jang et al. 2016 Huijben et al. 2022). Further- more, PHiSeg (Baungartner et al. 2019) lakes a similar approach to the HPU-Net. However, PHiSeg places residual connections between latent vectors across decompositions rather than in the convolutional layers. Furthermore, deep supervision placed each decomposition scale to enforce disentanglement between the latent variables.
Extension to 3D Early methods for uncertainty quantification in medical imaging primarily utilized 2D slices from three-dimensional (3D) datasets, leading to potential loss of critical spatial information and subtle nuances often necessary for accurate delineation. This limitation has spurred research into 3D probabilistic segmentation techniques with CLBO--asedd models, aiming to preserve the integrity of entire 3D structures. Initial works (Chotzoglou Kainz 2019 Long et al. 2021a] demonstrate that the PU-Net can be adapted by replacing 2D operations with their 3D variants. Crucially, the fusion of the latent sample with 3D extracted features is achieved through a 3D tiling operation. Viviers et al. (2023c) additionally augment Normalizing Flow to the posterior density for improved expressivences. Further cnhancements to the architecture include the implementation of the 3D HPU-Net (Saha et al. 2020 an updated feature network incorporating the attention mechanisms, nested decoder, and different reconstruction loss components tailored to specific applications (Saha et al. 2021a
Conditioning on annotator It can be relevant to model the annotators independently in cases with consistent annotator-sogmeetation pairs in the datasct. This can be achieved by conditioning the learned densities on the annotator itself (Gao et al. 2022 Schmidt al. 2023). For example, features of U- Net. be combined with samples from annotator-speciff gaussian-distributel posteriors (Schmidt et al.
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X
Lrec
Y
P+
KL
KL
KL
convolution
40
Upscale
upscale
Posterior parameters
X
Prior parameters
Figure 9: Hierarchical Probabilistic U-Net
Kohl et al. 2019a) based hierarchical Variational Autoen-
coder [Kinmma & Welling 2013 Sønderby et al. 2016
Klushyn et al. 2019 Instead of a single latent code,
multiple decomposition scales encode the segmentation variability.
2023 Considering the approach from Gao et al. 2022), generating segmentation mask is achieved by first sampling annotator from categorical prior distribution (xTx(x)), governed by the image conditional parameters (x) for the k-th annotator. Then, samples are taken from its corresponding prior density as 24 ~ >P((z)) to reconstruct. a segmentation through an image-conditional decoder = F(x,2x). The parameters T& (2x) can also be used to weigh the corresponding predictions to express the uncertainty in the prediction ensemble. Additionally, consistency between the model and ground-truth distribution enforced through an optimal transport loss between the of predictions and labels.
4.2.3 Denoising Diffusion Probabilistic Models (DDPMs)
Recent developments in generative modeling have resulted in a family of models known as Denoising Diffusion Probabilistic Models (Ho et al. 2020 Song et al. 2022 2020 Such models are especially renowned for their expressive power by being able to encapsulate large and diverse datasets. While several perspectives can be used introduce the DDPMs, we build upon the earlier discussed HVAE (Section 4.2.2) to maintain cohesiveness with the overall manuscript. Specifically, we describe three additional modifications to the HVAE Luo 2022). Firstly, the latent dimensionality is set equal to the spatial data dimensions, i.e. d= D. As consequence, redundant notation of Z is removed and Y is instead subscripted with t E [1,..., T): indicating the encoding depth, where Yo is the initial segmentation mask. Secondly, the encoding process (or forward process) is predefined as linear Gaussian model such that
T ((y) = pY,)) II (Y.Y...)).
(28)
with q(y,/Y(-_a.. Nyy: Va, 1,-1, Σ = (1-a:) 1), and noise schedule a (ar)"= Then, the decoding reverse process be learned through Ps (Y. Y.. The ELBO for this objective is defined
p( PYY) > Lo
log p(yr)
+
(Yr
(29)
g(yr Yo)
As denoted, the regularization term is ignored, since it is assumed that a sufficient amount steps T are taken such that ((rrly)) is approximately normally distributed. With the reparameterization trick (Kingma Welling 2013). the forward process is governed by random variable ~ (O 1). As such, the KL divergence in the second term be interpreted predicting Yo, the source noise the score Vv, log g(ve) (gradient
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of the data log-likeiihood) from Y,, depending the parameterization. This is in almost all instances approximated with U-Net variants (Ronneberger et al. 2015 Song et al. 2020
The proposed methodologies for segmentation vary in the conditioning of the reverse process on the input image. For instance, Wolleb et al. (2021) concatenate the input image with the noised segmentation mask. Wu et al. (2023b) insert encoded image features to the U-Net bottleneck. Additionally, information on predictions Y, at time step t is provided in the intermediate layers of the conditioning encoder. This is performed by applying the Fast Fourier Transform (FFT) the U-Net encoding, followed by learnable attention map and the inverse FFT. The procedure of applying attention
the spectral domain of the U-Net encoding has also been implemented with transformers in follow-up work (Wu et al. 2023a Amit et al. (021)) also encode both current time step and input image, but combine the extracted features by simple summation prior to inferring through the U-Net. It has also been proposed to model Bernoulli
instead of Gaussian noise (Chen 2023a Zbinden al. 2023 Rahman al. 2023
Bogensperger al.
2023
(0%))%)
Yo
Y1
Y
Yt
YT
(V./V,)
(V,--/VV))
KL
KL
Frec
Pe XX)
P4 -% (V,.)
YO
O
P₂
Y
Pz
X
X
Figure 10: Illustration of conditional Diffusion Probabilistic Models (Ho et al. 2020 remove noise that has been gradually added the input image.
The model learns to
Key takeaways: There is large focus improving amortized inference in VAE-based models (richer posterions, hierarchies, refined objectives). The GAN-based CAR model has received limited follow-up. More contemporary work incline towards using DDPMs, where innovations largely center on input-conditioning strategics and employing categorical distributions in the noising process.
5 Parameter modeling
A significant advantage of this parameter-sampling approach, when compared to modeling the output distribution directly, is that it naturally induces spatial coherence across the output segmentations. As a result, existing methods are often simple extensions of uncertainty modeling in conventional classification, rather than leveraging techniques tailored for segmentation tasks. As established in Section
3 the posterior
distribution from Equation (1p used to generate a predictive distribution via Equation (2), suggesting
parameter distribution modeling as an alternative to the modeling in feature space. We refer to any neural network that approximates the Bayesian posterior over model parameters as Bayesian Neural Network (BNN). However, evaluating the true Bayesian posterior is impeded by the intractability of the data likelihood in the denominator. This section discusses the methods used in image segmentation for approximating the parameter distribution. The most commonly used approach is Variational Inference (VT) over model parameters (Section 5.1). Laplace Approximations are examined in Section 5.2
Test-Time Augmentation
(TTA) is covered in Section 5.3 [we justify this non-trivial choice later in Scction 7.1
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5.1 Variational Inference
Consider a simpler, Iractable densily ((O)n), parameterized by 7, to approxinate posterior (00ly,x), Vari- ational Inference (VT) w.r.t. to the parameters can be employed, by minimizing the Kullback-Leibler (KL) divergence between the truc and approximated Bayesian posterior as
v = arg min KL (9(0)) [1((0))))10000
[[]))
arg min / ((()))) log p(Y 0)p(x) 0) d0
((0
arg min KL (g(0 p( -
0)),
(30)
where the parameter-inddeeeddent terms are constant and therefore excluded from notation. In the case of a deterministic encoder, i.e. y(0(n) 8(0 0*), the formulation collapses the MAP estimate in Equa- tion (5). A popular choice for the approximated variational posterior is the gaussian distribution, i.e., a mean I and covariance parameter for cach element of the convolutional kernel, usually with zero-mean unitary gaussian prior densities. However, the priors can be also learned through Empirical Bayes (Bishopl 1995). Furthermore, backpropagation is possible with the reparameterization trick (Kingma & Welling 2013 and
within this context, the procedure is referred to as Bayes by Backprop (BBB) (Blundell et al. 2015 and
has later been improved with the Local Reparameterization trick (Kingma et al., 2015). In addition to approximating the posterior, sampling from yields multiple parameter permutations, effectively enriching the model's hypothesis space, a phenomenon also referred to as model combination (Minka) 2000) Clarke 2003 Lakshminarayanan 2016). In most cases, VI is not performed explicitly. Instead, simpler techniques are employed, with the resulting parameter permutations hypothesized to serve proxy. We discuss two such methods: Monte Carlo Dropout (Section 5.1.1 and Ensembling (Section 5.1.2p. p. See Figure 11 for an illustration how each method approximates the convolutional kernel distribution. All of these methods have been widely applied to segmentation. However, unlike feature-level distribution modeling, their implementa- tions are often more generic and not specifically tailored to segmentation tasks. Therefore, we provide only brief overview here, followed by a more in-depth discussion later sections related to various applications.
5.1.1 Monte Carlo Dropout
Dropout is a common technique used to regularize neural networks by randomly "switching off" nodes of the neural network. Using Dropout can also be interpreted as a first-order equivalent L2 regularization with additionally transforming the input with the inverse diagonal Fisher information matrix (Wager el al., 2013). Furthermore, with Monte- Corlo Dropout (MC Dropout), the random node switching is continued during test time, effectively sampling new sets of parameters, mimicking samples from implicit parameter distribution 0 p), defined
Bernoullipp),
(31a)
n,
(31b)
7 o3
(011))
n ~ Bernoulli(p)
0 0 n
0x (0,, 0g. ..., 0x)
(a) Variational Inference
(b) Monte Carlo Dropout
(c) Ensembling
Figure 11: Three techniques for sampling convolutional kernels, with grayscale values indicating kernel weight magnitudes. These include: (a) Sampling an gaussian approximation parameterized by mean and
standard deviation kernel. (b) Applying dropout at inference time. (c) Training independently trained models, providing a discrete set of parameter configurations.
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with probability p and vector operating clement-wise on the parameters. It has been shown that MC Dropout can be interpreted as approximate VT in Deep Gaussian Process [Gal & Ghahramani 2016). In this manner, such method is able to provide multimodal estimates of the uncertainty. The dropout rate is usually optimized through grid search or simply set to = 0.5. The relationship between p and the magnitude of the model weights can also be exploited to probabilistically prune neural nelworks Fronzlez-CCrrabnn et al. 2022). As noted by Gal al. (2017a), the variance in the model output primarily governed by the magnitudes of the weights rather than the dropout rate p. Hence, they propose to additionally learn p using wradient-based methods, known Concrete Dropout, to increase the influence of p. As the name suggests, continuous approximation to the discrete distribution is used, known as the Concrete distribution (Maddison et al.
2016 Jang et al. 2016), to enable path-wise derivatives through Also, generalization of MC dropout has been proposed, where the weights of the network, rather than its hidden output, are set lo zero Mobiny et al. 2021
5.1.2 Ensembling
As mentioned earlier, MC dropout effectively optimizes over a set of sparse neural networks. From this perspective, explicitly ensembling multiple models can also be considered as an approximation of VI (Lkk- shminarayanan al. (207)). To this end, we define the set of functions f = (fØ Im=1, with M representing the number of models in the ensemble. Then, it relatively simple to obtain @ {0m]m which can be interpreted samples from an approximale posterior. Ensembling in only the latter parts of neural network (typically the decoder) is referred to M-heads, i.e. the network has multiple outputs. Often, the M obtained paramcters are from M separate training sessions. However, it has also been proven effective to ensemble from single training session, by saving the parameters at multiple stages or training with different weight initializations (Dahal et al. 2020 Xie et al.] 2013 Huang et al. 2017). A closely related concept to ensembling is known as Mixture of Experts (MoE), where each model in the ensemble (an 'expert') is 1991). In such settings, however, gating mechanism is usually applied after combining the expert hypotheses.
trained on specific subsets of the data (Jacobs
5.2 Laplace approximation
It can be shown that the intractable posterior in Equation (2 can be estimated as
p(0lD) (det H) = exp (-1 -MMP) 'H (0- (MAP)
(2T))
(32)
NO E
with OMAP being the Maximum Posteriori solution and Hessian matrix H -V log i.c. the second derivative of the loss function evaluated at OMAP. This approach, known the Laplace Approxima- tion (LA) [aplace 1774), has the merit that it can be performed post-hoc on pretrained neural networks. However, estimating the Hessian can become infeasible, as computation scales quadratically with the model parameters. This is usually circumvented by treating neural networks partly probabilistic and/or approx- imating the Hessian with a more simplified matrix structure (Martens & Grosse 2015 Daxberger et al. 2021).
Botev al., 2017
5.3 Test-time augmentation
An image X can be understood as a one-of-many visual representations of the object of interest. For example, systematic noise, translation rolation result. in many realistic variations that approximately retain image semantics. Hence, data, augmentation has been used at test time (explaining the name test- time augmentation, of the likelihood function Ayyhn & Berens, 2022). By randomly augmenting input images with invertible transformation x = (x),, with transformation parameters Ç. a prediction obtained with ý fo()) and can then be inverted through y = (y). Repeatedly performing this procedure results a set of segmentation masks.
TTA), argued to obtain uncortainty estimates by efficiently exploring the locality
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Key takeaways: Compared with earlier methods, parameter distribution modeling is largely model agnostic and rarely incorporates segmentation-ssoccie architectures. In practice, many implemen- lations reduce to approximate Variational Inference techniques, such as MC Dropout or Ensembling, followed by methods like Test-Time Augmentation and Laplace Approximations. Echoing their origins in classical classification, these concepts apply broadly to other domains.
6 Tasks
This section reviews literature on the downstream utilization of uncertainty in segmentation models, classi- fying the corresponding tasks into four main groups. Furthermore, Table 5 in Section 6.5 explicitly connects the downstream tasks to the specific application domains where they have been successfully employed.
Tasks:
Estimating Observer Variability, the range of plansible segmentation masks (Section 6.1
Reducing labeling costs using Active Learning (Section 6.2
Giving the model the ability to self-assess with Model Introspection (Section 6.3
Improving Model Generalization (Section 6.4
6.1 Observer variability
After observing sufficient data, the variability the predictive distribution is oflen considered be negligible and is therefore omitted. Nevertheless, this assumption becomes excessively strong ambiguous modalities, where its consequence is often apparent with multiple varying, yet plausible annotations for a single image. Additionally, such annotations can also vary due to differences in expertise and experience of annotators. The phenomenon of inconsistent. labels across annotators is known as the ånter-observer variability, while variations from a single annotator are referred to as the infra-observer variability (see Figure 12
For example, in critical applications like radiation therapy planning, the precise delineation of tumor contours is essential. However, inherent ambiguity the image acquisition process often leads to significant inter- observer variability. Multiple expert radiologists might annotate the same tumor with different, yet equally plausíble, boundaries. In this context, no single annotation can be considered definitively correct, and this irreducible uncertainty in the ground truth must be accounted for to mitigate potentially serious impacts patient care. Similarly, the perception of an autonomous driving system can suffer from label-level uncertainty. Consider an object partially obscured at a distance. The sensor data may be insufficient to distinguish whether it is a pedestrian about to cross the road or stationary object like a mailbox due to a genuine ambiguity in the scene itself. The decision makiinnn--aingggg process must handle this uncertainty, as the consequences of misclassifying a vulnerable road user are severe.
Annotators are models To contextualize this phenomenon within the framework of uncortainty quan- tification, annotators can be regarded models themselves. For example, consider K separate annotators modeled through parameters Φ with k = 1,2. K. For simple segmentation task, it be expected that Var [[r[p(x)]]- 0. In other words, each annotator is consistent in their delineation and the intra-observer variability is low. For cases with consensus across experts, i.c. yielding negligible inter-observer variability, the marginal approaches to Var[p(()] + 0. Asserting these two assupptions, it is valid to simply consider a point estimate of the posterior. Yet, this rarely the case in many real-life life applications and, as such, explicitly modeling the involved distributions becomes necessary.
Evaluation For evaluation, a commonly used metric minimizes the squared distance between arbitrary mean embeddings of the ground truth and predicted annotations using the kernel trick SShawe-Taylor 2004). This metric is known Maximum Mean Discrepancy (MMD) the Generalized Energy Dis-
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E
+
YE R
E D
Inter Mrleeno
Intra-observer
2
variability
3
4
Visualization of the intra-observer variability in parameter space (left) and the inter-observer variability in data space (right).
Figure 12:
lance (GED) Gretton et al. 2012 denoted as
GED(Py: Pv) = Py [k(y,y)) By.y [k( - Py [kyy, 3)1,
(33)
with marginals Py and py: representing the true and predictive segmentation distribution, and distance represented by kernel k ) x>x - R, usually the (1-IoU) or (1-Dice) score (squared by the GED). However, solely relying on the GED has been criticized due to undesirable inductive biases (Kohl et al., 2019a Zepf let al. 2.24). Hence, Kohl et al. (2019a) introduce an alternative metric known as Hungarian Matching (HM), which compares the predictions against the ground-truth labels through a cost matrix. This can be also formally denoted as finding the permutation matrix P, subject to the objective
1 HM(),)) N2 min Tr(PM).
(34)
where the elements of M are Mij k(),))))) and N2 represents the number of elements within the
matrix. Subsequently, the unique optimal coupling between the two sets that minimize the average cost is determined through a combinatorial optimization algorithm. Hu et al. (2022) use the Normalized Cross Correlation (NCC) to measure the similarity between the true and predictive set, which can formally defined as
NCC = nao (0:--M) (b.
(35)
with being the number of pixels, and μt, and f., the mean and standard deviation of the true and predicted segmentations set, respectively. Variables and 6 represent the uncertainty map calculated through the pixel-wise variance across the true and predicted segmentations, respectively.
Approaches The most straightforward approach directly model the empirical stochasticity in the an- notations with stochasticity in the features. Given that annotators have an intrinsically associated expertise, it also be possible model the annotator distribution in the parameters. However, relying the model to infer ambiguity in the parameters by observing the data, can become quite burdensome. This observation is reflected in literature, where we see that most approaches use conditional generative models to encapsulate observer variability (See Table D. While conditional generative models have been successfully applied for the task at hand, it has been shown that such models, without explicit conditioning, do not encapsulate more subtle variations such as distinct labeling styles of annotators (Zepf et al., 2023a). VAE-based Probabilistic U-Net is the most popular approached followed by its hierarchical variants (HPU-Net and PHI-Seg) and SSNs. More recently, the growing popularity of DDPMs is apparent in the field.
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Table 1: Methods used for the application of encapsulating Observer Variability.
Method
Literature
PixelCNN
Zhang al. (202cc
SSN
Monteiro et al.
2020
Kahl et al. (2024] Zepf et al.
2024
2023a.
Philps et al.
2024
GAN
Kassapis ct al.
2021
Gao et al. 2022
Kohl et al. [2019b
Valiuddin et al. 2021
Selvan
2020 Bhat
et al.
20223 2022a
Valiuddin et al.
2024bb
Quu & Lui [2020
Valiuddin et al.
2024a
VAE
Viviers et al.
2023c
Schmidt et al.
20033
Long et al.
(2021) 1b
Zepf et al.
(2023a
2024
Hu et al.
2022 Viviers et al. (2023b
Savadikar et al.
2021
Philps et al.
22024
Zhang et al.
2022b); b): Rafael-Palon et al.
2021
Kohl et al.
2019a
Baumgartner et al.
HVAE
2019
Gantenbein et al. (2020)
DDPM
Chen et al.
2023a
Zbinden et al.
2023
Rahman et al.
(2023
MC dropout
Kahl et al.
2024
Philps et al.
2024
Ensembling
Zepf et al.
2024
Hu
2023
TTA
Kahl et al.
2024
6.2 Active Learning
The field of Active Learning aims to reduce the costly annotation procedure by careful selection of the most informative samples for training (Settles) 2009: Ren al.] 2021). Instead of randomly labeling data, the model directs expert time toward the most challenging examples, leading more efficient learning. Active Learning is an extensively researched topic in both traditional (Cohn et al. 1996) and deep Machine Learning (Gal et al. 2017b). including uncertainty-based approaches applied to segge-segmentation pairs. In such methods, it is assumed that predictions with high uncertainty provide the most informative samples for training. Moreover, this core idea extends to active perception, where an autonomous agent, like a robot or a drone, physically acts to resolve its own environmental uncertainty. For instance, instead of guessing what an obscured object is, it might move its camera to get a better view. In both cases, the system actively socks the most valuable information, either from human annotator or by interacting with the world, improve its understanding and performance.
Evaluation Selecting the most informative samples for training contains two elements. Firstly, the un- certainty needs to be quantified. This is usually the predictive entropy (Kasarla et al.] 2019 Shen et al. 2021 Xie et al. 2022 variance (Ozdemir al., 2021 but in some instances the uncertainty is estimated through Bayesian Active Learning by Disagreement (BALD) (Houlsby et al. 2011 Ma et al. 2024b Shen et al. 2021). Simply stated, the reduction in posterior entropy indicates the informativeness of a data sam- ple. In turn, this can also be formulated by the reduction in predictive entropy in the dual formulation. We can formally denote this as
BALD H(ODD] - B D) H x", D]},
H D] Hyy x DD.
Hyy|x", D] - E, H H 0|| = I(y. 0|x", DD).
(36)
Note the similarity between the mutual information / obtained from the predictive entropy decomposition in Equation 3 Secondly, to evaluate whether the active lcarning strategy is beneficial, the performance is mostly measured subject to increasingly stringent. budget requirements (i.e. a percenlage of the original dataset) Yang et al.)
2017 Sourati et al.) 2018 Kasarla et al.) 2019 Mahapatra et al. 2018 Sinha et al.
2019 Siddiqui et al. 2020
Li & Alstrom 2020 Kim et al., 2021 Shen et al.) [021 Burmeister et al. 2022; Ma et al.) 2024b). Kahl et al. (2024) argue that Active Learning should be evaluated on image-level humans do not annotate on pixel-level. Assuming saturated performance in-distribution data (training
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Table 2: Methods used for the application of Active Learning.
Method
Literature
Kasarla et al. 2001
García Rodriguez et al. 2020
Xie et al.
(0022
Wu al. 2021
SoftMax
Burmeister et al.
2022
Gaillochet et al. 2023
SSN
Kahl et al.
2024
Adversarial
Mahapatra et al.
2018
Kim et al.
2021
Sinha et al.
2019
VAE / GAN
Ensembling
Yang et al.
2017
Nath et al.
2020
Khalili et al. 2004
Kahl et al.
2024
Gorriz et al.
2017 7 Ozdemir et al.
(0021] ; Hiasa et al. 2019 Li &
MC dropout
Alstrøm 2020
Shen et al.
2021
Ma ct al.
(2024b)
Gaillochet et al. 2023 Siddiqui
ct al.
22020) Sadaf et al. 20199
TTA
Gaillochet et al.
2023
cycle t1), the author propose to measure the improved relative performance, C, subject to a second training sample (12) from selected samples
Crotal = Cmethod - Crandom:
(37)
with
C= Ptz Ps.
(38)
Pt.
where Cmethod denotes the relative improvement achieved by the employed method, and Crandom is cor- rection term accounting for gains due to random sampling, and P suggested to be the Dice-score, but can theoretically be any segmentation metric.
Approaches As can be seen in Table 2 most of the literature use MC dropout or ensembles. Further- more, using the plain SoftMax activation has also been commonly used. Often, solely relying on the most uncertain samples can result in samples with limited diversity and therefore samples selected on the repre-
sentativeness (Ozdemir et al. 2021 Shen et al. 2021 Wu et al., 2021 Mahapatra et al. 2018 Sinha et al. 2019). Another crucial modeling element pertains the aggregation of the pixel-level uncertainties. Therefore, closeness to boundary (Gorriz al. 2017 Kasarla al.) 2019 Ma et al. (2024b) or specific regions (Gail- additionally incorporated when combining the uncertainty values across pixels. Siddiqqi al. (2020) determine the entropy through estimating the entropy across viewpoints in multi-view datasets. Gaillochet et al. (2023 argue the superiority of sclecting samples based on batch-level uncertainty.
lochet al.) 2023 Wu et al. 2021 |
Kasarla et al. 2019)
6.3 Model introspection
The provided uncertainty can be used to gauge prediction quality of model. For example, an agricultural drone's segmentation model provides clear example. When distinguishing 'crop" vs. 'weed", it will segment healthy fields with low uncertainty, signaling reliable output. However, for anomalous patch of stressed plants, it will produce segmeniation with high uncertainly. This high score directly flags the prediction as low-quality and likely erroneous, preventing a costly automated action like misapplying herbicide. As evident, the described problem can also be considered as Out-of-Distribution (OOD) detection (Lambert
et al. 2022 Holder & Shafique 2021).
The relationship between uncertainty and model accuracy has been formalized by Mukholi & Gal (2018) through the Patch Accuracy vs Patch Uncertainty (PAvPU) metric. Firstly, the accuracy given certain prediction, p(A(C), and socondly, the uncertainty given an inaccurate prediction PUU Given threshold ur that distinguishes certain from uncertain pixels or patches, we can define pixels that accurate and certain, accurate and uncertain, inaccurate and certain, inaccurate and uncertain.
Evaluation
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Table 3: Methods used for the application of Model Introspection.
Method
Literature
VI
Ng et al.
2022
LaBonte al.
2019
LA
Zepf et al.
2003b
Kahl et al.
2024]
Wang et al.
2019b
Whitbread & Jenkinson
2022
Roshanzamir et al.
TTA
(2023]
Dahal et al.
2020)
Jungo et al.
2020
Mchrtash et al.
2020
Czolbe et al.
2021
Holder & Shafique (2021
Ensembling
Jungo & Reyes (2019) Ng et al. (2022
Hann et al. (2021
Jungo & Reyes 2019
Linmans
al.
2020
Pavlitskaya al.
2020
Valada al.
22017
Jungo et al.
[2020];
Kahl et al.
(004)
Whitbread & Jenkinson 2022
Kendall et al. 2016
Kampfineyer al.
12016
Dechesne et al.
(2021
Morrison et al. 2209
;
Qi et al. 20023
Baton-Rosen et al.
[2018
Jungo et al.
201
Roy et al. 20 80l8al b
Mehrtash et al.
(2020)
Nair et al.
2020
Sander et al.
2019
Hasan & Linte (0022b
Camarasa et al.
MC Dropout
(2021]
Roshanzamir
al.
(2023 ; Sedai et al.
(0018) Seeböck
al.
(019 ; DeVries &
Taylor
2018]; [Zoobe [ et al.
[2021] Hocbel
al.
[019 Bhat al. 2021
Dahal ct al.
[2020
Antico et al. (2020)
Lambert et al.
2022
Tasan & Linte 2022a
Jungo & Reyes
2019
Ng et al.
2022
Twamoto et al. 2021
Hoebel et al. 2020
SSN
Kahl et al. 2024)] Ng et al.
(022
VAE
Chotzoglou & Kainz
2019
Viviers et al.
2023a Czolbe et al.
2021
Bian al. 2020
denoted by Mac, Vaus they this respectively. Consequently, the authors combine p(A = naa/ (nee + nau) and P(U[T) = //s///// + niu) to obtain the Patch Accuracy vs Patch Uncertainty (PAvPU) metric, defined as
Dac + Rau
PAvPU = Nac + Nau Nie 1luu
(39)
It can be noted that uncertainty is usually only obtained on pixel basis, while crucial information can be present in structural statistics. Hence, [ooy et al. (2018b) propose to use the Coefficient of Variation (CoV), which addresses this by measuring structural uncertainty through dividing the volume variance over the mean for all samples. The authors also evaluate structural uncertaintics by assuming predictions with the pair-wise average overlap between all respective samples.
= Epyix- (Dice(((r))) Vi
(40)
Approaches Considering Table 3 it can be observed that parameter modeling methods such as MC
Dropout and Ensembles are by far the most popular choice of uncertainty encapsulation method. However,
Concrete dropout (Rumberger al. 2020b Mukhoti & Gal 2018), M-heads (auxiliary networks) Jungo
& Reyes 2019 Linmans et al. 2020 MoE (Pavlitskaya al. 2020 Laplace Approximation (Zepf et al.
2023b), Variational Inference in 3D LaBonte et al.) 2019), and test-time augmentation (Dahal et al.) 2020
have also been used for this application. Moreover, including information on localized uncertainly 1o the
training objective has shown to improve generalization capabilities (Ozdemir et al. 2017 Bian et al.
2020:
Twamoto et al. 2021 Li ct al. 2021
6.4 Model generalization
As mentioned carlier in Scction 5.1 sampling new parameter permutations often improves segmentation performance, due to the model combining eflect. In contrast. to other downstream tasks, evaluation does not require specialized metrics besides quantifying the relative change in model performance, For instance, dropout layers the deeper dccomposition levels of the SegNet (Badrinarayanan et al. 2015) has shown improve model performance [Kendall al. (2016). MC Dropout, and specifically Concrete Dropout (Mukhoti & Gal 2018 Rumberger al. 2020b)). has also led to improved performance. The improved generalivation
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Table 4: Methods used for the application of improved Model Generalization.
Method
Literature
Kasarla al. [2019]
García Rodriguez et al.
2020
Xie et al. (002
Wu al. 2021
SoftMax
Burmeister et al.
2022
Gaillchhet et al. 2023
SSN
Zepf et al. 2023a
Ng et al. 2022
VAE
Vivicrs et al.
2023a
Zepf ct al. (2023a
Hu ct al.
2022
Wolleb et al.
2021
Wu et al. (2023b
Amit et al. (2021)
Zbinden et al.
2023
Bo-
DDPM
gensperger et al.
(0023]
Hoogeboom et al.
(2021
Mukhoti & Gal
2018
Rumberger et al.
2020b)
Saha et al.
2021a
Zhang et al. 2022a
MC dropout
Whitbread & Jen kinsooo
2022
Huang et al.
2018
Mukhoti et al.
2020 Ng et al.
2022
Wickstrøm et al.
2020
Ng et al. (2022
Larrazabal et al.
2021b
Kamnitsas et al.
2018 Saha et al.
2021b
Hu
Ensembling
et al. [2023
et al.
[2021]
TTA
Wang et al.
2019a Rakic et al. (0024) Whitbread & Jenkinson 2022
from ensembling has also shown produce more calibrated outputs (Mchrtash et al. 2020 Ng et al. (2022) found that ensembling results in the best performance improvement, while BBB is more robust to noise distortions. In other work, orthogonality within and across convolutional filters of the ensemble is enforced through minimizing their cosine similarity, which reaped similar merits (Larrazabal et al. 2021b).
Ensembling has also been performed using M-Hcads (Hu et al. 2023 Jungo & Reyes 2019). Nonetheless, individual models in conventional ensembles receive data in an unstructured manner. Hence, specific subsets of the data can also be assigned to particular models with MoEs (Pavlitskaya et al.) 2020 Ji et al. 2021 Gao
ct al. 2022 Generative models, though typically used for other applications, have also led to performance
improvements across wide range of studies (Saha et al. 2021a; Wolleb et al. 2021
Wu et al. 2023b Amit
et al.
2021
Bogensperger et al. 2023 Zbinden et al. 2023 Viviers et. al. 2023a
Zepl
al. 2023a
Hu al.
2022
Key takeaways: Parameter distribution methods are commonly utilized for Active Learning, Model Introspection, and Model Generalization, while feature distribution modeling is employed for Observer Variability. This observation suggests the irreducibility of uncertainty lied to parameter-level modeling, while reducibility aligns with feature-level modeling.
6.5 Application domains
In 3 we link the discussed methodologics and tasks to their respective application domains. It is evident that the medical domain, followed by the automotive field, is the most prominent for probabilistic segmen- tation tasks. The high-stakes nature of both domains contributes to their prominence, as uncertainty-aware models are especially vital in safety-critical applications. Several benchmark datasets central to these applications. For instance, the LIDC-IDRI dataset (Armato) 2011), with its multiple expert delineations of lung nodules in CT scans, is a key benchmark for quantifying observer variability. In the autonomous driving domain, outdoor scene understanding applications widely utilize the Cityscapes dataset. Similarly, MRI-based research is often advanced by datasets like BraTS or challenges such as QUBIQ (Menze el al. 2021
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Table 5: Overview that links literature to various domains and applications.
Observer Variability
Active Learning
Model Introspection
Model Generalization
Paclitskaya
2020
Sinha
2019
Hasan & Lante
2022b
Ng
2022
Huang
Kahl et al.
2024
Sander
al. 2019 Ng
2018
Zbinden
2023
Kahl
2024
Kohl al.
al [2022 Hassan & Linte
Outdoor scenes
12019b
Kassa pis et (0221)
2022
Kim al.
(2021
2019
Xie
(2022
20220
Kendall
Mehrtash et al
Minchoti
al 2021 Hoogeboom of
2011
ct ml. 2022
Wn
Kahl of al. 2024
Holdes
202
Valada nt al. 2017
2021
Shalique
2022
ml
Rakie
(024
Siddiqui al.
2020
scones
Wu et al. 2021
2023
Kendall et al 2016 Qi et al
Various objects
Philpe et al. 2024
2023 ] Morrinon
Dechesme
Gareia Rodrigues et 2020
Remote sonsing
Kamphimeyrr ot mb
2016
Decheene al.
2021
Kohl et al.
2019B
Zepf et al
Khalili et & Alsrron
2024
Microscopy
Schmldt 20233
2120
Linmans
2020
Wamoooo
Bogensserggr
al.
2023
Yang et al. 2017 et al.
2021
Amit et al. [2121
al.
Mehrtashi 2020 Jungo & Reyes
(2018b
Wolleb et al. (202 | Wi et al
et sl.
Philpe et al. 22044 Bhat et
Zepf et al
2023b
Ma
20019
Eaton- Rosen et al.
Wang et al.
20233 5 Clenn
(2015s
Kamant
Brain MRI
2023
sl.
2021 Savadikn al.
Hu al.
et 6l
2021
20246
Shee
Jungo el
Wanng
(2020 g Whitbread
(201
Eas
al. 2121b
al
12018
Wang et al.
Larrazabal
Zhang et.
Jenkinson [2222 Monseiro
2019h
Whtbbrr & Jenkin-
2020
Lembert al
2022
Jungo et al.
20185
Baungartner ct al. 2019
In
Burmeister
n6 Gallloche
Mchrtash et ml.
2020
Zopf
Iu ct al.
2022
Saha et
Prostate MRT
2022
2023
Zepí
2022
a
Ra kil
al. 20233
2024
ct al.
2023
Saha ct al
2021b
Ozdemir et al. 2021
al. 2010
Bhat et
2021
Camaracs
2722 Burmess
Various MRI
Rahmaa et al.
2023
Ser cf
al
2172 Ma
Roshanzamir
al (222
Nair al
2023
Ji ct ml.
2021
Ng ct al.
2022
Hann al. 2021
ct m1
2720
Gaillo
catll x.
2024
OCT imaging
Selvan et al. 2020
Seeböck et
2019
Bian
Rakic et
2024
2020
(Sedal
2018
2021
Wu
2024a
Zepf al
2023b
Jungo &
Dermoscopy
Corriz al.
Alstrom
2020
Reyes
(2010
DeVries & Tay-
2023n
2023a
Zepf
2018
Coolbe
2021
Rafael-Palou
2021 Gao et al. (202) : Valuddii
ab
2024b CCee es
2023x
Zhang
(0222b
Monteizo
et sl. (2020
Quu
(2220)
2019h
1hm et al.
2122
Zhang
et al. 22/22 Gam--
tembein
m..
(2020
Viviers
Lung CT
al.
20233
22oindon
3] Valiudaii
Hoabel ct al.
Kahi
al.
2020
2024
Kabl et al
2024 Chozzoglon
Coolbe et al
Hu
20/4
Zhang et a..
2022
Rakic al.
20/22
2023
Zep! sl. (0121)
2019
Kahl et al. (022 [Bhat
Long
Valuddin
et al. (2121 Raahe
maa
(2023) Kohn
20199
Hu al. 2028 BBat
ct al.
20223 Kassapis et
2022
Loug el al. a2
Sclvan
es
ml 2020)
Baunngart
peer et
2019
Various CT
Valiudiı et
2024b
Hiasa el al
2019
Hoebel
2019
Bonte
Viviers et al
2023a
21123
2019
Antico
2020 Dahal
Wa. al. berger ot ml.
2023b
Rum-
Rahman al.
Yang
st. al.
2020 Ban at. al.
2021
2020b
Others
Valiuddio al
2021
Mahapatra
et
Wickstrem
2020
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7 Discussion
This section builds on the reviewed theory and literalure guide researchers selecting methods suited their goals. Furthermore, it highlights key gaps and limitations, offering insights address current challenges and identify promising directions for future rescarch.
Main discussion points:
The interpretation of epistemic and aleatoric uncertainty remain complex and warrant careful
reconsideration, despile the literature often suggesting a clear distinction (Section 7.1
Challenges related to spatial coherence remain unaddressed (Section 7.2
The application-driven field lacks standardization across the downstream tasks, hindcring benchmarking and further progress (Section 7.3
Uncertainty-based Active Learning is promising and has many open challenges (Section 7.4
There exist various limitations and trade-offs across methods (Section 7.5)
Understanding the critical influence of data characteristies on uncertainty models remains a relatively under-explored area of research (Section 7.6
The field of segmentation could advance significantly by treating the backbone critical modeling choice and embracing contemporary architectures (Section 7.7
The segmentation community should actively explore and integrate novel uncertainty quan- tification methodologies (Section 7.8)
Uncertainty modeling should address the growing importance of multiclass, instance, and
panoptic segmentation problems (Section 7.9
7.1 Disentangling uncertainties
Proper disentanglement of the uncertainties allows to accredit high-entropy outputs to either model ignorance about the data (epistemic), or inherent noise in the data generation process (alcatoric). As Equation (2) implies, this can be achieved by introducing stochasticity into either the model's parameters or its features. This fundamental distinction not only provides the organizing principle for our subsequent review but also aligns with the approaches empirically observed in the literature. Seemingly straightforward, sufficient literature and discussion in the scientific community exist related to the applicability of this taxonomy practical cases. As paraphrased from Kirchhol et. al. (2025). "defimitions of uncertainly resemble the shapes of clouds clearly defined from distance, but losing clarity during approach by dissolving in one another". We further explore this, highlighting key nuances in uncertainty modeling and contextualize this to segmentation problems. To begin, we first establish the original definitions of these concepts to better understand why these concepts have lead to incorrect definitions, claims and general confusion.
Aleatoric uncertainty reconsiders the non-deterministic relationship between æ E and y E V, which implies that
p( P(Y,X) X) p(X)
+ 5 F(X)).
(41)
with Dirac-delta function 6, and mapping F X + y. This relationship is characterized by the ambiguity in X and is inherently probabilistic, due to various reasons such as noise in the data (occlusions, sensor noise, insulficient resolution, etc.) or variability within a class (e.g. not all cats have (ails). Hence, the observance of substantial aleatoric uncertainty can in some cases be inevitable, but may also signal the need for higher-quality data acquisition or shifting to another modality. The possible input dependency of the uncertainty develops into further categorization of either heteroscedasttcc (input dependent) or homoscedastic (input independent) aleatoric uncertainty.
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Epistemic uncertainty arises from model ignorance (i.e. related to the parameters). Consequently, epistemic uncertainty should not only be quantified but ideally also minimized. Epistemic uncertainty can be further categorized into two distinct types (Hüllermeier & Waegeman 2021). The first type pertains uncertainty related to the capacity of the model. For example, palle--pramcterized models or approximate model posteriors, that can become loo stringent to appropriately resemble the true posterior. The ambiguity on the best parameters given the limited capacity induces uncertainty in the learning process, which also known as model uncertainly. Nevertheless, given the complexity of contemporary parameteer-ntenssve CNNs, the model uncertainty is often assumed to be negligible. A more significant contribution to the epistemic uncertainty is due to the limited data availability, known as approzimation uncertainly, and can often be reduced by collecting more data. Additionally, we have found it to be common to interchangeably use the total epistemic and solely model uncertainly, especially in the context of Monte Carlo Dropout methods (Kendall et al. 2016 Roy et al., 20188 DeVries & Taylor| 2018 Mehrtash et al. 2020 Burmeister et al. 2022 [wamoto et al. 2021 To prevent confusion in already established terminology, the term model uncertainty should be exclusivel used for the induced ambiguity due to limited model complexity, rather than to denote general or the entire epistemic uncertainty.
Task dependency It is essential to consider that interpretation of the uncertainty source often differs depending on the modeling context. For example, epistemic uncertainty to be considered as the number of required models to fit the data (Wimmer et al. 2023 evident in ensembling and M-head approaches for tasks related to model introspection and generalizen lizaroiizaiion (Sections 6.3 and 6.4. Another perspective pertains model disagreement (Houlsby et al.) 2011 Gal et al., 2017b Kirsch 2024), relating to the source of observer variability (Section 6.1] and methods that employ mixturo-of-cxpert approaches (Section 5.1.2) Also, it has been simply considered to be the remainder after subtracting the aleatoric from the total uncertainty (Depeweg et al.] 2018)) such as commonly done in Active Learning (Section 6.2). Aleatoric uncertainty also knows several definitions such as being the Bayes-optimal residual risk (Apostolakis 1990; Helton 1997), but remains consistent in related literature as the inherent, irreducible noise in the data inducing variance in the ground-truth distribution (Section 6.1 To avoid confusion, this task-dependency should be considered when interpreting uncertainty.
Model dependency Moreover, past literature as well as our overview indicate the influence of specific design choices on the nature of the modeled uncertainty (Der Kiureghian & Ditlevsen 2009). For instance, the expected entropy w.r.l. a stochastic parameter variable is often assumed to represent. the epistemic uncertainty as shown in Equation (3). However, when employing generative models, this behavior changes. As argued by Kahl et al. (2024). the predictive uncertainty can be decomposod as
epistemic
H / 0 +
I Z.
(42)
aleatoric
In contrast to Equation 3). the mutual information term here encapsulates the alcatoric uncertainty, while the expected entropy reflects the epistemic uncertainty. This is because generative models operate under the assumption that the latent variable Z captures the inherent data ambiguity, thus representing the aleatoric uncertainty. The mutual information describes how much about this aleatoric uncertainty can be learned by observing the output Y. If one were to know the optimal latent code Z, the mutual information would be zero. In this case, all stochasticity in the output Y is perfectly represented by Z, meaning the alcatoric uncertainty has been fully captured. Under this view, if all data uncertainty can be fully encapsulated by the latent variable Z, there is no need to model this irreducible stochasticity in the predictive model p(Y]Z, 0), implying it corresponds to the epistemic uncertainty. This is consistent with the principle that epistemic uncertainty is the residual component isolated by subtracting the alcatoric uncertainty from the total predictive entropy. Hence, the calibration network of the GAN-based CAR model of [assapis et al.] (202I) (discussed in Section 4.2.1] is actually more inclined to model the epistemic rather than aleatoric uncortainty. This dependency is also demonstrated by the conflicting idcas on uncortainty encapsulated with TTA. In some works, it is argued that this approach models the alcatoric uncertainty in a better
way Wang et al. 2019a Ayhan & Berens 2022 Zhang et al. 2022a Wang et al. 2019b Rakic et al.] 2024
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Whitbread & Jenkinson, 2022 Roshanzamir et al. 2023 However, literature opposing this also exist and
even suggests that TTA enables the modeling of epistemic uncertainty (Hu et al. 2022). Kahl et al. (2024) draw parallels with BNNs and demonstrate that
aleatoric
HY 0) = -| + E
(43)
eplstemic
where are sampled augmentations from the input space T. A well-trained model should be invariant data augmentations. Consequently, predictive variance across augmentations is treated as irredueible aleatoric uncertainty, which is quantified by non-zero expected entropy over these predictions. Non-zero mutual information indicates the model has not yet learned invariance to the augmentations and can be reduced with further training, deeming it to be epistemic. This decomposition is analogous to the one for conventional BNNs (Equation 3), while it contrasts with the decomposition used for generative models (Equation 42) The key message thus far is that, after considering the type of uncertainty to model, is equally important to carefully choose the appropriate method for its representation, as confusion and misunderstandings can cause deviation from the targeted uncertainty.
Intertwinedness Despite the seemingly rigorous and well-defined distinction, it can be seen that imple- mentation rapidly blurs this notion. A systematic study has found that the ability to separate the uncer- tainties can strongly depend the data. (Kahl et al. 2024). These findings are also supported by rescarch presenting a strong correlation between epistemic and aleatoric uncertainty quantification methods (de Jong et al. 2024 Mucsányi et al. 2024), indicating that they represent uncertainty of similar kind. [Keddall & Gal (017 have also observed that explicitly modeling a single uncertainty tends to compensate for the lack of the other. In probabilistic models, we can also find that the parameter modeling methods, tradi- tionally used for epistemic uncertainty, have been used to model observer variability, by for example using MC-Dropout or ensembling (see Table []. Furthermore, using a combination of distribution and paramter modeling techniques can also beneficial (Gao et al., 2022). These findings further highlight the notion of the uncertainties being intertwined. On slight side note, it is notable that popular BNN methods such the Local Reparameterization trick use stochasticity in the output features of intermediate layers (usually considered to encapsulate aleatoric uncertainty) as a proxy to model the weight distributions (Kingma et al. 2015), further supporting this argument.
Modeling vs. quantifying A common misconception pertains that modeling a specific uncertainty also implies that the predicted output distribution represents that same uncertainty. This undermines the entropy decomposition in Equation (3) distinguishing between encapsulating, i.e. targeting an intended uncertainty type, versus quantifying, i.c. measuring the modeled uncertainty in isolation. For example, the estimation of mutual information is required to express the epistemic uncertainty in isolation. Nonetheless it is common for papers to quantify the uncertainty by inspecting the output distribution subject through repetitive sampling from the parameters densities, thereby approximating the predictive entropy encapsulating both uncertainties (Camarasa et al.] 2021 Seeböck et al. 2019 Roshanzamir et al. 2023 Roy et al. 2018a
Rumberger et al.] 2020b Hoebel et al. 2022 Seeböck et al., 2019 Wundram et al. 2024b; Rumberger
et al. 2020a Sedai et al. 2018). This similarly holds for literature emphasizing aleatoric uncertainty when
estimating the predictive variance (Zhang al.) 2022c Monteiro al. 2020 Valiuddin et al. 2024b Gao
et al. 2022). Since predictive variance can encompass both uncertainties, a crucial distinction must be made.
Targeting specific uncertainty for task through either paramcter or feature modcling is diffcrent from truly quantifying lype in isolation, which may require separate validation.
7.2 Spatial coherence
The inter-pixel dependency of segmentation masks often introduces additional complexity in proper un- certainty quantification, questioning the commonly repurposed techniques from conventional image-level classification. In this section, we further dive into important details and challenges arising due to the added dimension(s) in both 2D and 3D data.
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(a)
states, entropy
(b)
states, entropy
(c) states,
entropy
loga (2)
bit, factorized
(2) x 2.32 bits, factorized
loga
bita, factorized
entropy P: Σ lo 112
entropy pt Σ log2 Pt Rs bita
entropy pc Σ
logz pa 190
bits
bits
Figure 13: This figure illustrates why factorized entropy is a poor proxy for the true uncertainty in structured images. The common method to estimate this entropy by averaging multiple sampled binary segmentation masks, where the resulting pixel intensity indicates the average confidence value. The shapes shown possess only translational uncertainty. The factorized model fails to capture the object's spatial coherence and leads to significant overestimation of the true entropy.
Aggregation methods The mutual information is often utilized to express the reducible, epistemic uncer- tainty. This requires subtracting the expected entropy from the total entropy of the predictive distribution (see equations
3 and (36] In such cases, the entropy is usually estimated via mean or summation over
all pixel-level uncertainty values (Mukhoti & Gal 2018 Camarasa et al. 2021 Bhat et al. 2022b Mukhoti
2024b [Shen et al.) 2021 Li Alstrom 2020] Dechesne et al.j 2021 Huang et al. 2018 Nair ct al. 2020 Zepf al.) 2023b). However, this assorts the assumption of factorized categorical dist distribution, i.e. uncorrelated pixels, which is strong assumption that ignores the core rationale for the field's existence. Hence, spatial coherence is fundamental consideration and is frequently neglected in the uncertainty estimation frameworks used for Active Learning.
et al. 2020 Ma et al.
The problem with factorized entropy can be demonstrated with a simple example: a binary segmentation of circle on a 32x32 grid. If we assume that the 'latent' uncertainty is purely translational, Figure [3 visualizes the example distributions possible segmentation maps. The true entropy of the generating source scales with the number of possible states N (NV). However, using factorized approach severely overestimates the latent, true uncertainty. This behavior is direct consequence of the subadditivity property of entropy, which states that
D X2, Xpl & Σ H[]]].
(44)
Importantly, this fallacy could explain failure cases of entropy- based uncertainty evaluation methods in tasks such as Active Learning where sample informativeness//ivee diieesstyy is crucial. Elements such as object class dependency, size, shape, proximity, contextual information etc. can all influence the decision-making of the aggregation method. Notably, Kahl et al. (2024) argue that pixel aggregation should be considered separate, distinct modeling choice in probabilistic segmentation. The authors demonstrate that naive summation-based aggregation techniques (Wang et al. 2019a Jungo et al. 2020 Zhang et al.) 2022a Czolbo 2021 Camarasa el. al. 2021) can cause the foreground object size to correlate with the uncertainty score. Some empirical studies have explored other types of aggregation (Camarasa et al. 2021 Roy et al., 2018b Jungo al. 2020 Kasarla et al. 2019 Xie al. 2022 Wu et al. (222). However, we advocate for deoper investigation into well-informed imago-level aggregation strategies and their implications for uncertainty
le al.
quantification.
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Volumetric uncertainty modeling Processing volumetric clinical data knows two dominant strategies. The first, 2D slice processing, analyzes the volume as a stack of individual images, preserving full context within each slice but severing spatial continuity along the third dimension. The second, 3D patch processing, divides the volume into smaller ones, which maintains local 3D information but loscs the global anatomical structure. As compute power increases, end-to-end 3D segmenlation models are gaining traction because they can analyze increasingly more volume, preserving spatial continuity and reducing the inconsistencies inherent in these slice- or patch- based methods (Liu et al.) 2025] Burmeister et al. 2022 Viviers et al. 2023b; Monteiro et al. 2020). As we have seen the implications of the additional dimension in 2D uncertainty, we similarly argue that 3D segmentation introduces or exacerbates previously mentioned issues related to entropy estimation and aggregation of pixels. Probabilistic segmentation has been employed on 3D data
but often are straight.orwars extensions of patch-wise segmenlation architeclures (Viviers et al. 2023b;
Chotzoglou & Kainz 2019; Long et al. 2021b Saha et al., 2021a 2020; Hoebel et al.) 2019). Limited
diversity of incertainty-based sample selection Active Learning requires more focus, since 2D slices of 3D volumes are oflen highly redundant (Shen al., 2021; [Burmeister et al. 2022). Furthermoree given the high computational cost of many architectures, we emphasize the need for efficiency with higher-dimensional input tensors. Moreover, methodologies need to be developed in order to appropriately compare 2D to 3D models. For example, valid volumetric GED cannot be estimated with 2D models. Hence, general guidelines related to translating 2D segmentation models to 3D can be of great benefit for practitioners considering to phase out patch-based approaches for volumetric data.
7.3 Standardization
The widespread success of contemporary machine learning models can be largely attributed to open-source initiatives and rigorous benchmarking practices. The field of probabilistic segmentation falls short on this. A key factor contributing to these diverging standards is that the literature is predominantly ipplication-driven, uncertainty often considered merely an auxiliary tool, leading to a fragmented landscape where studics rely on varied datasets and methodologies without a shared evaluation protocol. This problem has been partly addressed by contemporary research attempting to standardize the field through a unified framework after identifying key pitfalls (Kahl et al. 2024). Our manuscript further contributes by introducing coherence and a unifying synthesis across the distinct disciplines of applications, tasks, and methodological developments, with these concepts being further detailed in our recommendations in Section g
Despite the aforementioned points, efforts to quantify observer variability show some implicit consensus. Notably, some of the recommendations of Kahl et al. (2024) diverge from these practices, such as how to employ the widely used LIDC-IDRI dataset (Armato| 2011) or the Cityscapes Cordts al. (2016a) dataset, suggesting a more rigorous examination of this application. Furthermore, the evaluation metrics used across various studies also diverge substantially. While GED and Hungarian Matching are commonly applicd to assess model performance, key details, such as the choice of distance kernel, the number of predictive samples, and the handling of correct empty segmentations (which cause NaNs), vary widely. This inconsistency is particularly evident in the work of Zbinden et al. (2023), who attempt to identify common ground across benchmarks but find over 10 (!) different evaluation methods. Additionally, many reported results are not taken from original papers but from follow-up studies using them baselines. See Table D for comparison of benchmarks on the LIDC-IDRI datasct.
7.4 Future of Active Learning
Among the applications considered, uncertainty quantification emerges as a promising approach Active Learning, showing true potential of potentially reducing annotation costs while preserving even improving model generalization. Active Learning directly addresses the labeling bottleneck through uncertainty-drive selection, aligning with model introspection and fostering human-Al collaboration in specialized domains. Furthermore, when combined with federated learning, this field can accelerate privacy-centric collaboration by enabling efficient, human-in in in-the-loop improvement of safety-critical models across decentralized data silos. Similarly, these principles can also translate tp Reinforcement Learning, where uncertainty guides an agent 's exploration policy to accelerate learning in complex environments. However, key limitations remain and require further research.
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Table 6: Comparison of test evaluations on two versions of the LIDC-IDRI dataset. Table extended from (Zbinden ct a 2023). "Indicates evaluation results not obtained from the original work.
LIDCv1
LIDCv2
Method
GED16 16
1M-1oU16
GED16
HM-IoU16
PU-Net (Kohl et al.
2019b
0.310 = 0.010
0.552 = 0.000
0.320 = 0.030
0.500 = 0,030
HPU-Net (Kohl et al.,
2019a
0.270 + 0.010
0.530 = 0.010"
0.270 + 0.010
0.530 + 0.010
PhiSeg (Baungartner et al.
2019
0.262 0.000
0.586 = 0.000
SSN (Monteiro al.]
2020
0.259 + 0.000
0.558 + 0.000
CAR Kassapis et al.
2021
0.264 = 0.002
0.592 + 0.005
JProb. U-Net (Zhang et al.
2022d
0.262 1 0.000
0.585 + 0.000
PixelSeg (Zhang et al.
2022c
0.243 = 0.010
0.614 0.000
0.260 = 0.000
0.587 = 0.010
MoSE (Gao et al. 2022
0.218 - 0.001
0.624 0.004
AB (Chen et al. 2022
0.213 = 0.001
0.614 0.001*
CIMD Rahrnan et al.
2023
0.234 + 0.005
0.587 + 0.001
CCDM Zbinden et 2023
0.212 + 0.002
0.623 + 0.002
0.239 + 0.003
0.598 + 0.001
Biff (Chen et al. 2023a
0.238 + 0.010
0.596 0.000
MedSeg Dill
(Wu el al..
2023b
0.420 + 0.030'
0.413 + 0.030'
SPU-Net Valiuddin et al.
2024b
0.327 = 0.003
0.560 = 0.005
Model sensitivity and baselines Active Learning requires a well-generalized model trained on limited data, rendering performance highly sensitive to budget constraints, model architecture, hyperparameters, and regularization (Mittal al., 2019 Kirsch 2024 Munjal et al.] 2022 Kirsch et al. 2019 Gaillochet al.
2023). Furthermore, random sampling, particularly when combined with MC Dropout, often proves be a
surprisingly strong baseline that is difficult to outperform (Kahl et al. 2024 Mittal et al. 2019 Lüth et al.
2023 Burmeister et al. 2022 Siddiqui et al. 20200; Kim et al. 2021 Sinha et al. 2019 Nath et al. 2020).
especially with highly imbalanced data (Ma et al. 2024ah. This observation can partly be attributed to the fact that MC Dropout likely fails to capture true Bayesian uncertainly (see Section of other techniques.
7.5
inviting exploration
Sample informativeness A set of highly uncertain samples may share similar semantic content, leading the model to overfit a narrow and homogeneous region of the data distribution during early training. Hence, sample selection should be based on both uncertainty and diversity OOdemir et al. 2021 Wu et. al.] 2021: Jensen et al. 2019: Burmeister et al.] 2022). Ozdemir et al. (2021) leverage the latent space of a VAE to select representative samples, whereas other studies use feature vectors from prediction models (Wu et al., 2021 Burmeister et al. 2022). Furthermore, the sample diversity offered by such models can be effectively combined with adversarial training (Mahapatra et al. 2018 Sinha al. 2019).
Uncertainty calibration and aggregation As discussed in Section 4.1 using the predictive entropy of MostMex uncertainty requires proper calibration. Unfortunately, this is rarely verified in prac- tice Kasarla et al., 2019 García Rodríguez et al., 2020 Xie et al., 2022 Wu et al. 2021 Burmeister et al.
2022 Gaillochet ct al. 2023). Furhtermore, most works rely on houristics to aggregate the pixel-level in-
formation such as closeness to edges, boundaries or the use of super-pixels (Gorriz et al. 2017 Kasarla
et al. 2019 Ma et al.) 2024b
Gaillochet al.) 2023 Wu et al.] [2021] Kasarla et al.) 2019
Bengar et al.
2021 h. Moreover, the mutual information is often quantified by plain sunmation (i.e. assuming pixel inde-
pendence) (Shen et al.) as mentioned in Section
2021
Ma et al. 2024b). We hypothesize that these ad-hoc aggregation techniques,
7.2 are key limiting factor to uncertainty-based Active Learning.
The Cold-Start Problem Finally, the "cold start" problem (Nath et al. 2022 Houlsby et al. 2014 Yuan et al.] 2020a), the notion that uncortainty methods work poor on the initial samples, is rarely addressod. Most approaches assume a well-trained model is already available, yet methods like MC Dropout require warm-up epochs to yield meaningful uncertainty. Addressing this gap, post-hoc techniques such as that of
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Sourati et al. 2018)) which enable uncertainty estimation from pretrained deterministic networks, represent valuable and underexplored direction for early-stage sample selection.
7.5 Method suboptimality
To provide necessary contrast to the benefits of many models, this section discusses key methodological limitations. These limitations will serve guidance for readers selecting the most optimal methods and will be utilized when we present our recommendations in Section X
Invertible covariance of SSNs Stochastic Segmentation Networks (SSNs), operate by placing a multi- variate gaussian over the classification head of an existing model. However, they can suffer from training instability primarily due to the requirement. that the covariance matrix remain invertible. To address this critical limitation, the authors propose two distinct remodial strategies: masking out the background to prevent exploding variances and employing uncorrelated gaussians when the covariance matrix becomes singular, which may impact model performance and theoretical purity.
GAN training The Calibrated Adversarial Refinement. network (CAR, as treated in Section 4.2.1 on the other hand, is less commonly used method in literature. This can perhaps be accredited due to the well-known instability of GAN-based models (Arjovsky & Bottou 2017). GANs oflen require additional heuristic terms in the loss function to ensure stable training. CAR is no exception to this, its objective function combines four separate losses. The multiplicity of individual losses is critical to the model's success. Nonetheless, this complex objective function suggests the training process may pose substantial fine-tuning challenges, potentially explaining the model's limited practical adoption.
Mode collapse with ELBOs Known for its flexibility to various datasets, fast sampling time and the interpretable latent space, Variational Autoencoders (VAEs) seem to be the most popular choice for the task of encapsulating Observer Variability. (See Table I in Section 6.1p. Nonetheless, the shortcomings of VAËs are well known. For example, such models suffer from inference suboptimality related to ELBO optimization (Cremer 2018 Zhao et al. 2017) and literature on the VAE-basod PU-Net often describe behavior similar to the well-known phenomena of mode collopse (Valiuddin et al. 2024a by Qiu & Lui 2020 This has been hypothesized to be caused by excessively strong decoders {Chen et al. 2016) and especially apparent when dealing with complex hierarchical decoding structures, where additional modifications such as the GECO objective (Kohl et al., 2019a], residual connections (Kohl et al. 2019a
Baumgartner et al.
2019), or deep supervision |Baumgartner et al. 2019) are required for generalization
Limitation of sequential modeling The adoption of sequential sampling models DDPMs or PixelSeg is rather limited compared to the VAE-based models. This is unexpected, as both models outperforms the latter. This likely due their tedious sequential inference procedure (Song al., 2020 Zheng et al., 2023 Meng et al., 2023 Zhang et al., 2022c), a crucial limitation that is exacerbated in supervised settings, which often require validation through sampling on separate data split. Despite these shortcomings, the strengths of DDPMs should not be overlooked. Their iteralive sampling allows for highly flexible modeling and preserves figh-frequency details often lost in latent-variable approaches like VAEs, which can produce blurry reconstructions.
Discrete vs. continuous It can be noted in Table 6 that the dest-performing Denoising Diffusion Prob- abilistic Models (DDPMs) are discrete in nature (Chen et al., 2023a Zbinden et al. 2023). While it is debatable whether shifting to categorical distributions is required for complex image generation CChn et al. 2022). This can visually be alrcady quite apparent for the multi-class casc, where the transition to noise is visually more gradual in the discrete transition (see Figure 14p. However, Table G shows AB (Chen) lt al. 2022) performs almost identically to CCDM (Zbinden et al., 2023), suggesting that straightforward thresholding of continuous models have little-to-no performance sacrifice compared with explicitly modeling discrete distributions. These observations suggests that the modeling intuition of categorical latent space, despite its fundamental soundness, lacks demonstrable merit.
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Catagorical Goussian
Catagorical Gaussian
M
(a) LIDC-IDRI (Armato 2011
(b) CityScapes Cordds et al. 2016a
Figure 14: Continuous vs. categorical forward diffusion process with cosine noise scheduling (Nichol &
Dhariwal 2021). Note the use of categorical diffussin that results in more gradual transition for the multi- class case.
Criticism of MC Dropout The majority of literature on parameter distribution modeling appears to rely on approximations variational inference (VI). Among these, MC Dropout is the most commonly used due its simplicity, low computational cost, and ease of implementation. However, MC Dropout has faced substantial criticism (Folgoc ct al.) 2021 Osband 2016 Kingma et al., 2015 Gal et al.
2017a). For instance,
it can assign zero probability to the true posterior introduce erroneous multi-modalityy [Folgoc et al. 2021
It is also sensitive to model size and dropout rate rather than the observed data (Osband 2016 Kingma et al. 2015). To address these issues, some alternatives attempt to learn the dropout rate (Kingma et al. 2015 Gal et al., 2017a). In cases of improved Model Generalization (Section (6.4). it is also debatable whether the performance gains stem from better uncertainty modeling or from other factors. For instance, both ensembling and MC Dropout introduce a model combination effect, effectively averaging multiple networks. MC Dropout also resembles placing L2 penalty weights (Gal & Ghahramani] 2016). While these effects may improve generalization, similar gains might be achieved with simpler, more efficient regularizers. This suggests that perhaps it is the inherent stochasticity of the methods that is the key beneficial factor. Still, if achieving peak performance is the primary goal, uncertainty modeling is likely not the most efficient route due to its added computational cost. Instead, the improved performance may simply be an unintended side effect of techniques chosen for different purpose.
7.6 Data dependency
In contrast to many other domains within computer vision, this field exhibits notably greater absence of single state-of-the-art method. Performance is heavily contingent upon multiple factors such as mod- eling context and application. However, data dependency often remains unexamined. Especially in cases where parameter distributions are modeled. For instance, MC dropout was found to perform better than ensembblig (Hoebel et al. 2020 Roy et al. 2018b), while in other works ensembling excels (Mehrtash et al., 2020 Ng al. 2022). Also, there is convincing evidence to prefer Concrete Dropout rather than MC
Dropout Mukkhoti & Gal 2018 All things considered, the preference for particular methodology scems to carry strong da data-
(Dahal et al. 2020: Jungo & Reyes 2019 Burmeister et al. 2022 Ng
et al. 2022 Kahl et al. 2024). For feature modeling, some models in Table may appear to be overall best performers. However, their validation is often limited to binary, mulli-annotaate data, thereby compromising any claims of genoralization to other data types, such as multi-class or single-annotated scenarios. Current research often focuses on incorporating uncertainty into already high-performing systems rather than com- paring distinct methods. This trend necessitates more rigorous, data-driven perspective to validate and understand the optimal uncertainty modeling strategy. Hence, valuable benchmark study should compre- hensively compare all available methods (including both feature and parameter modeling approaches) across all relevant task with special focus on the data dependencies.
7.7 Segmentation backbone
Due to emphasis on the mothod, task and application, the backbone fcature extractor often an overlooked element in uncertainty modeling. For example, CNN-based encoder-dccoder models such as the U-Net, remain the preferred backbone architecture (Fisenmann et al. 2023). Nevertheless, Vision Transformers
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(ViTs) Dosovitskiy et al., 2020 such SegFormer. Swin U-Net, Mask2Former already outperform CNNS in general segmentation problems (Xie et al. 2021 Zhang & Yang, 2021; Cheng et al. 2022). With the exception of experiments in Kahl et al. (2024) (combining SSN (Monteiro et al. 2 with HRNet (Yuan ct al.] 2020b ), all employed models use a CNN backbone. The limited adoption of ViTs likely stems from CNNs' advantageous inductive biases, whereas Transformers lypically require extensive pretraining on large datasets (Caron et al. 2021). Nonetheless, CNNs have benefited from the recent developments in Transformers. For instance, CowNNert takes inspiration from contemporary Transformers to modernize existing ResNet-based CNNs, retaining the inductive biases of convolutional filters and achieving significant performance gains (Liu et al. 2022c). These same inductive biases have been added to ViTs through CNN adapters to also enhance performance (Chen et al, 2023b). Building upon these observations, we conclude that future research should actively explore the integration of state-of-the-arl Vision Transformer (ViT) and backbones to advance the field.
hybrid
7.8 Contemporary uncertainty
While current efforts frequently leverage established uncertainty modcling tochniques for segmentation, the practical inlegration involves siguificant challenges related to ensuring architectural and task compatibility, To accelerate progress, the segmentation community should actively explore and integrate newly developed uncertainty quantification methodologies, rather than relying solely on standard, established practices.
VT alternatives Approaches besides VI, such as Markov Chain Monte Carlo (MCMC) or Laplace Ap- proximations, are also viable options to approximate the Bayesian posterior. Especially the Laplace Ap- proximation can be very beneficial, as it is easily applicable to pretrained networks. Notably, both Laplace Approximation and VT are biased and operate in the neighborhood of a single mode, while MCMC methods are a more attractive option when expecting to fit multi-modal parameter distributions. Zepf al. (2023b) explored the impact of the Laplace Approximation on segmentation networks. However, beyond this work, the method has received limited attention in the field.
Sampling- free uncertainty The multiple forward passes required in many Bayesian uncertainty quan- tification can incur cumbersome additional costs. Hence, considerable efforts have been made towards
sampling-free free uncertainty models (Mukhoti et al. 2023 Liu et al.] 2020 Van Amersfoort et al. 2020 Postels
et al. 2019), only depending on a single forward pass. Mukhoti et al. (2023) show that Gaussian Discriminant
Analysis after training with a SoftMax predictive distribution can in some instances surpass methods such as MC Dropout and ensembling. Along similar lines, Evidential Deep Learning also possesses the advantage of quantifying both uncertainties with a single forward pass. This framework is based on a generalization of Bayes theorem, known as the Dempster-Shafer Theory of Evidence (DST) (Dempster 1967]- While com- mon in Bayesian probability, DST does not require prior probabilities and bases subjective probabilities belief masses assigned on frame of discernment, i.e. the set of all possible outcomes. The use of Evidential Deep Learning has seen success in conventional classification problems (Sensoy et al. (018),, and (nchaa et al. 2024h recently applied this concept to segmentation attempt to decouple alcatoric and opistemic uncortainty within a single model. However, follow-up research in this direction remains limited to date.
Reliable uncertainty estimates A distribution-fre framework known as Conformal Prediction (CP) produces prediction sets that guarantee inclusion of the ground truth with a user-specified probability. By using an additional calibration set, CP transforms heuristic model confidence scores into rigorous uncertainty estimates. This technique is particularly renowned for being model-agnostic, simple, and highly flexible (An- gelopoulos & Bates 2021). Recently, CP has been applied to segmentation tasks (Wieslander et al.) 2020: Brunekreef et al. 2024 Mossina et al. 2024 Wundram et al.) 2024a), benefiting from these advantages and reflecting the growing interest in the framework for structured prediction. However, the validity of conformal prediction requires exchangeable draws for train, calibration, and lest data. In practical settings test data often comes with distribution shift, which can coverage to fall below the nominal level and predictions can become overconfident. Such overconfidence yields prediction sets that are too tight, suppressing deferrals and increasing missed or undersegmentedd regions which can be espocially detrimental in safety-critical tasks. We encourage further research to explore novel applications and benchmark CP against existing architectures.
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Generative models Given the established time-lag in the adoption of SOTA techniques from the broader generative modeling field, there remains persistent opportunity for research to effectively bridge the gap by translating the latest architectural and methodological improvements from image generation into more performant and robust segmentation solutions. For example, DDPMs were initially proposed for general image generation (Ho et al. 2020p, but was quickly adopted for stochastic segmentation. Furthermore, flow- based models also enjoy many benefits yet to be explored in this field, such as exact likelihood modeling and faster sampling compared DDPMs (Lipman et al., 2022
7.9 Towards complex scene understanding
A clear trend shows that most uncertainty applications remain narrowly focused on binary semantic segmen-- lation. While limited work exists in probabilistie instance segmentation (Kohl et. al. 2019a Morrison et al. 2019), there is notable absence of research addressing uncertainty in this domain, well in panoptic segmentation. Furthermore, work involving uncertainty across multiple classes is also scarce. This represents a critical gap, given that contemporary segmentation ion research is increasingly centered on multiclass problems across all segmentation types (Kerssies et al., 2025), often even incorporating class hierarchies (Atigh et al. 2022). Consequently, the significance of research focused solely on binary-class semantic segmentation is diminishing.
8 Guidelines and recommendations
This section synthesizes the preceding discussions into practical guidelines and recommendations. We will first critically examine specific methods and offer criteria for selecting the optimal approach for different tasks. Subsoquently, we will discuss the essential topic of evaluating uncertainty and the potential biases that can arise. The section concludes with a more philosophical look at what constitutes truly meaningful uncertainty in machine learning.
8.1 Method selection
The proposed framework is designed to simplify the complex landscape of available techniques by decon- structing the decision into a series of manageable questions. By systematically addressing these questions, practitioners can navigate the available options more effectively and clearly justify their final choice of method.
Decision framework:
1. Baseline: Does the vanilla backbone reach sufficient performance? (Section 8.1.1
2. Reducible: is the target uncertainty reducible? (Section 8.1.2
3. Purposc: Is the uncertainty auxiliary crucial to the main task? (Section 8.1.3
4. Data regime: Are labels multi-annotated multi-class? (Section 8.1.4
5. Budget: Are there budgetary constraints that limit compute? (Table
For cach element, evaluate potential method with criteria:
Prioritizing methods with stronger evidence in literature.
Considering the thcorctical soundness.
Favoring architecture--gnostic and post-hoc designs.
Evaluating computational efficiency, prioritizing methods that offer fast sampling time.
The final two points address the modern paradigm of using large, pro-trained models. These models are typically fine-tuned once for specific task but are then deployed for continuous use. While the initial training significant one-time cost, deployment involves ongoing inference. Therefore, optimizing for
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train
Tune parameters, collect data, im- prove backbone architecture
Baseline performance?
insufficient
sufficient
Uncertainty reducible?
Purpose?
SSN
Volumetric?
yes
no
insufficient
MCD
V1
Multiple labels labels/classees classes?
VAE
auxiliary critical
yes
yes/no
no/no
no/yes
ycs/yes
DDPM
(H)
DDPM
(H) VAE
Figure 15: Flowchart illustrating the complete framework for selecting the appropriate uncertainty modeling method.
inference efficiency is paramount and will lead to substantial long-term savings in computational resources. A comprehensive comparison of available methods is provided in Table which will used as key reference when making vocommendations. The complete selection process for our proposed framework outlined in the flowchart in Figure 15 We now proceed lo detail each component. of this framework in the subsequent sections.
8.1.1 Baseline
Probabilistic segmentation models perform best when they are built on a robust, capable backbone. Hence, it is crucial establish a strong deterministic baseline first. If the underlying model underfits the data or fails to learn the task, adding probabilistic layer will not solve this issuc. Only when the baseline is solid can probabilistic modeling effectively provide calibrated uncertainty and support decision-making, Besides model backbone, improvements through data collection, parameter tuning, and different training schemes can also be explored.
8.1.2 Reducibility of uncertainty
Unless it is the core scientific contribution, argue that disentangling uncertainty sources is often unnec- essary for dense prediction applications. The primary benchmark for success is improved performance a downstream task. If a model's total uncertainty score achieves this, its practical value is already established, regardless of our ability to precisely attribute that uncertainty to its underlying sources. This pragmatic view avoids the need for complex validation methods designed solely to prove disentanglement. Instead, the distinction between alcatoric and epistemic uncortainty is most valuable as conceptual guide to com- municate whether the uncertainty reducible and, consequently, select. the most appropriate modeling strategy, noted in prior work (Der Kiureghian & Ditlevsen 2009 Faber 2005).
Disentanglement shouldn't be mistaken for the primary objective, but might be useful diagnostic tool. For example, it makes sense to assume disentanglement of the uncertainty to choose the appropriate method. In a theoretical view, a model that perfectly captures all learnable structure would have its remaining prediction error driven solely by irreducible noise. This suggests that the fitted parameters can be treated as ncar-point estimates, and consequently, most residual uncertainty would be expected arise from the predictive dis- tribution hat than from parameter variability. In contrast, if were to model the reducible noise within
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the parameter distribution by markedly narrowing the predictive distribution while the parameter variance remains materially above zero, would be suggesting that a single sample from the parameter distribution could yield predictions with near-zero variance. This seems in tension with the premise of inherent data noise, since some uncertainty should still appear in predictions when the data-generating process is noisy. Thus, although the disentanglement of uncertainty is a theoretical debate (see Section 7.1). 1, both our analyses and literature overview conclude that it remains a useful tool for method selection. Specifically, by mod- eling parameter distributions for reducible uncertainty and feature distributions for irreducible uncertainty. While a conventional neural network, by contrast, should theoretically be able to model both uncertainties within the feature-baser output distribution, the empirical failure of this simple approach to provide robust uncertainty quantification is, in fact, the central driving force behind the entire field of study.
8.1.3 Purpose of modeling
Once decision has been made the uncertainty reducibility, the next step is to determine its purpose. For cases where the uncertainty is irreducible, literature overview has identified single use case: estimating Observer Variability (Section 6.1 For reducible uncertainty, this overview identified three key tasks. For the task of Model Generalization (Section 6.4), uncertainty is not the primary objective, but a beneficial byproduct of the modeling approach. In contrast, for Active Learning and Model Introspection (Section 6.2 and 6.3 respectively), uncertainty quantification is the core objective and must therefore be critically sound and theoretically justified.
Observer Variability As demonstrated in Table D the lop-performing methods for this task are SSN and DDPM-based approaches. Given its model-agnostic nature, fast sampling capabilities, compatibility with 3D data, and strong recent benchmarks (Kahl et al. 2024 Monteiro et al. 2020), we recommend prioritizing SSNs. Should the performance prove insufficient, further exploration based on specific data characteristics and computational constraints is warranted. As discussed in Section 7.6. the existing literature on the data dependency of these specific methods is limited. However, we will provide an overview and discussion in the following scction basod on the available studics.
Model Generalization Though their theoretical grounding are limited (see discussion in Section 7.5 pragmatic first step is to cither MC Dropout or TTA. Both methods simple, incxpensive, and have been shown to be elfective. Their limited theoretical support is not major concern here, the uncertainty is auxilaary to the main task of improving generalization. Nevertheless, MC dropout has more evidence in literature to perform well (see Table 4 and is generally better understood (i.e. the connection to variational inference made in Section 5.1 than TTA (its connection to Bayesian learning is under discussion as mentioned in Section 7.1
Active Learning & Model Introspection In contrast to Model Generalization, the reducible uncer- tainty is crucial factor for both tasks. MC Dropout remains overwhelmingly the most popular uncertainty estimation method (see Tables 2 and 3) due primarily to its minimal implementation overhead, making it an casy baseline for numerous papers. Despite this prevalence, studies have reported that the simple random sampling baseline in Active Learning remains a highly competitive approach (Sinha et al.) 2019). Recent benchmarks consistently show that ensemble methods are suporior (Kahl et al. 2024). and Variational Infer- ence (VT) closely trails ensembling while clearly outperforming MC Dropout [g et al. 2022). Since VI offers a more theoretically sound Bayesian approach that is also more cost-efficient than full ensembling, we rec- ommend researchers adopt rigorous methods like VT for uncertainty characterization. Furthermore, caution is warranted concerning the aggregation methods detailed in Scction 7.2 Namely, entropy calculation, often used to quantify uncertainty severily, is frequently severely overestinated. Future research must therefore pursue fundamentally sound methodologies that avoid these aggregation pitfalls.
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8.1.4 Nature of the data
Our analysis in Section 7.6 indicates that uncertainty modeling is strongly data-dependent and remains largely inconclusive for modeling parameter distributions. Nonetheless, limited but practical ordering can be established for modeling irreducible uncortainty via feature distribution methods, using criteria like class count, labels, and data dimensions.
Multi-label, binary class The LIDC-IDRI benchmark heavily influences current. literature, leading to focus on (H))AE-based methods. However, shown in Table 6 DDPMs achieve superior scores this key dataset. While (H)VAB approaches remain more popular for other datasets, the lack of robust bench- marking makes broader conclusions difficult. We therefore recommend DDPMs for superior performance, but recognize that (H) remain a viable alternative when constrained by computational limits.
Singlo-label, binary class When evaluating single-label problems, accounting for ground-truth uncer- lainty (i.e., observer variability) is critical. The only rigorous verification method involves ablation studies multi-annotated datasets where the model is trained only on single mask. Since these crucial studies are performed almost exclusively
(AE-bbed approaches (e.g., (Zhang et al. 2022d Gao et al. 2022; Zepf et al.) 2023a Selvan et al. 2020), plus one instance with PixelSeg (Zhang et al. 2022cl), our recom- mendation favors (H)VAF models due to this strong accumulated evidence. However, given their superior mode coverage (Xiao et al. 2021). there is no indication that DDPMs would perform poorly in this specific ablation.
Single label, multi-class Crucially, we have not found a multi-annotaled, multi-class dataset used for ablation by training only single mask. In the absence of this rigorous evidence, we must draw inferences from related applications, such as Model Gencralization. Within this context, Table 4 suggests that DDPMs are a popular choice.
Multi-label, multi-class In the rare instances multi-class, multi-annotated problems, such as the Cityscapes datasot when combined with artificial labels, (H)VAEs currently have the most supporting evi- dence (Kohl et al. 2019b Gao et al., 2022
Volumetric data For problems involving 3D data, the primary evidence strongly supports the use of VLE-bbadd methods (Viviers et al. 2023c Long et al. 2021a). This body of work encompasses both multi-class single-annotated and binary-class multi-annotated datasets. Given that this is the most substan- tial lead, outside of SSNs, we recommend the adoption of the (H)VAE model architecture for 3D uncertainty estimation.
8.2 Evaluation and reproducibility
Based on our literature overview and in-depth discussions, this section outlines general and task-specific recommendations aimed at improving evaluation and reproducibility. Compliance with these guidelines is essential for validating experimental findings and ensuring comparability across the broader research landscape.
8.2.1 General
Reporting Achieving robust reproducibility and generalizability hinges detailed reporting, area where current rescarch practice often falls short, as discussed in Scction 7.3 Their inconsistent application and reporting inhibit the validation and critical assessment of published methods, eroding confidence in the field's reported progress. Furthermore, the routine omission of source code and model checkpoints exacerbates this issue, making result verification impractical. This practice stands in sharp contradiction to the standards of modern machine learning, which mandates standardized data splits, clear evaluation metries, and the highly encouraged release of code and model weights.
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Table 7: Comparison between methods. N represents number of parallel or sequential models in ensembles and DDPM denoising, respectively. Params. indicate the number of parameters required for feature based methods. For parameter-based modeling, this indicates how the parameter count scales with the uncertainty method. Sumples indicate how many evaluations are needed for single forward for feature-based modeling, For varameter-basee modeling, this indicates if the uncertainty evaluation quality scales with the number of forward passes of parameter-base approaches.
Method
Advantages
Limitations
Params. Samples
Likelihoods
Extra notes
FEATURE MODELING
Occasional diag-
onal covariance during training. prior, likelihonds impractical.
Model agnostic,
explicit, appr.
SSN
possibly unstable
411
1
fast sampling
High sample qual- ity
Sequential pling, weak global structure
sam-
Explicit,
PixelCNN
33.4
x
Unstable training, poorly defined objective, extensive tuning
Fast sampling, flex- ible
Additional losses for segmentation
CAR
1
implicit
Fast sampling, flex- ible, interpretable latent space
Mode/posteris collapse, amortiza- tion gap
deep supervision, GECO objective
additional
(H)VAE
~ (19)5M
appr. bound
lower
Compute can be re- duced with DDIMs cost. of sample quality
DDPM
Flexible, expressive
Soquential pling
SaIn-
~00
appr.
lower
bound
PARAMETER MODELING
Model agnostic, no extra parameters
Local approx., posterior infeasible
Explicit act
exc Usually on layer
LA
x1
No
Full weight posto- rior; principled un- certainty
Explicit VI
slow convergence
No
appr. bound
lower
Usmally on selected layers
simple, paralleliza- tion
expensive, member correlation risk
variants: M-Heads, MoE
Ensemble
x
Yes
Implicit
Minimal
varisnts:
concrete, heiphweights, the- oretical grounded- ness be disputed
poor uncertainly, principled
MC dropout
change; regularizes; model-sgnostic
x1
Yes
Implicit
Zero
training boosts
boundary aggregation sensi- tivity
errors,
TTA
changes;
Implicit
robustness
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Structural splitting Furthermore, to prevent information leakage and ensure truly generalizable model, data must be split strategically based on its underlying structure. For instance, splitting by patient, time point, or acquisition condition (e.g,, device, institution, or environment). A notable failure to enforce the critical step of pationt wise splitting is frequently observed (sce, for example, [Baumgartner et al. 2019 Valiuddin el. al. 2024b; Monteiro et al. 2020)).
Inference details To ensure fair and accurate comparisons, the number of forward passes used to generate
uncertainty maps and evaluation scores must. be explicitly reported for every method. This is a prerequisite this hyperparameter (see Table 7
because the quality of the uncertainty estimate is often dependent Such transparency, exemplified by the standardization work of Zbinden et al. (2023), allows the community to determine optimal validation strategies. Similarly, the specific parameters for any data augmentation must be detailed, particularly when using TTA (Section 5.3 For latent variable models, it is crucial to specify the dimensionality of the latent space and detail the precise mechanism for incorporating these variables into the network architecture (e.g., through tiling in Kohl et al. (2019b) or by using scale and translation parameters as done in Kassapis al. (2021)).
Quantifying uncertainty types As argued by Kahl et al. (024)) claiming to quantify an uncertainty type would ideally require validation across all relevant applications. For example, validation of epistemic uncertainty requires a clear distribution shift in the data (domain, semantic, covariate etc.), but it is often found in literature that either no distribution shift or the required metrics are used (Zhang el. al.) 2022a Wang
iet al. 2019a; Postels et al.) 2019 Mukhoti & Gal 2018 Mobiny et al. 2021 Whitbread Jenkinson 2022)
As discussed in Section 6.1 alcatoric uncertainty should be validated with multiple labels and appropriate
metrics, but can often be found to be evaluated on a single annotator or without the use of distribution-level
statistics (Wang al. 2019a; Whitbread & Jenkinson 2022 Kendall & Gal 2017 Liu et al. 2022b Schmidt
et al. 2023; Savadikar et al. 2021). Hence, simply avoiding any serious claims related to either type alleviates this burden, and allows one to focus solely on the task at hand. In contrast to the work of Kahl et al. (2024), however, we do not consider calibration of models itself as a downstream task, but rather as a tool to gauge model reliability to be later used for other tasks. As a consequence, the pixel-wise uncertainties can be later used for possible downstream applications.
Aggregation The aggregation method should be considered as a standalone design choice (discussed Section 7.2) and researchers must explicitly state how per-pixel probabilities are combined. Ideally, multiple aggregation techniques should be evaluated to rule out potential biases, such as those related to object size. When working with volumetric data, it is crucial to evaluate both 3D volumetric and 2D slice-based approaches. Furthermore, entropy must be calculated in one of two ways: either correctly, using the true likelihood under the model or by using a factorized assumption. If the latter approach is taken, this significant methodological limitation must be made explicit.
Feature extractors The feature extracting backbone key component in uncertainty modeling (Sec- tion 7.7). Thus, any proposed method or improvement must be ablated across diverse feature extractors to confirm its general functionality. For instance, latent variable methods should demonstrate performance stability when interchanging CNN or ViT encoders, and parameter modeling approaches should similarly be tested across various architectures.
Dataset Data dependency is a crucial element in uncertainty modeling (Section 7.6 For new methodolog- ical contributions, ablation across diverse datasets is essential. These datasets should encompass variations in: number of classes, modality, input channels, dimensionality (2D vs. 3D), and annotation types (single vs. multi- .nntteennotated). This scope should also extend to complex tasks beyond simple semantic segmentation, notably instance and panoptic segmentation.
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8.2.2 Task-specific guidelines
Observer Variability We recommend the standard of 16 predictive samples for evaluating the GED and HM-IoU on the LIDC-IDRI dataset, a convention established in literature (Table 6 Another commonly used dataset is the CityScapes benchmark. for this, advisable to use the label-switching parameters proposod by Kohl al. (2019b). Other multi-annotated datasets, such as those from the QUBIQ Challenge (Menze et al. 2021), include MRI and CT scans of various organs (Valiuddin et al.) 2024b Ji et al. 2021). although they follow less standardized preprocessing. The RIGA dataset for retinal fundus images (Almazroa et al. 2018 can be used as in Wu et al. (2022). Furthermore, it is crucial to explicitly report the strategy for handling correct emply segmenlation maps; we recommend assigning a maximum score (Valiuddin et al., 2024b) rather than removing NaNs (Kohl et al.) 2019a).. Finally, evaluations should include an ablation study on the number of annotations to fully assess the generalization capabilitics of model.
Active Learning Researchers should report the relative change in performance (corrected for random sampling as shown in Equation (37) and evaluate sample diversity using metrics like FID. Furthermore, an ablation study should be conducted where varying percentages of the data are withheld from the initial training pool to assess performance under different data availability scenarios.
Model Introspection It essential to report PAvPU and CoV (see Section 6.3 based on rigorous uncer- fainty calculations. Furthermore, studies must critically analyze the relationship between the characteristics of the ground truth segmentation mask and the model's predicted error. This analysis is crucial for ensuring that the model's uncertainty estimates are reliable and not biased by specific data attributes (i.e. size, complexity, location, contrast).
Model Generalization Report the relative change in performance as discussed in Section 6.4
8.3 Limitations in the recommendations
Selection bias A limitation of this review stems from the prevalence of specific methods in the current literature such as VAE- models. While dominance of such methods may reflect the technique's genuine efficacy or versatility, it may also be attributed to factors such as case of implementation, readily available codebases, or simplicity of initial adoption, thereby potentially introducing a selection bias into the overall analysis. Furthermore, their extensive application may not inherently confirm their superiority. In fact, it could simply reflect the community momentum and historical popularity the technique.
Literature gaps The heterogeneity and lack of rigor in the current literature necessitated a degree of gener- alization in method classification, which is a key constraint on our comparative analysis. To maintain clarity, we primarily refer to methods by their underlying conceptual class (e.g., VAE, DDPM) rather than specific implementations, as many variants (including fine-graing -ee-grained hierarchical or categorical versions) offer only incremental changes. Exccptions are made solely for highly task-specific mothods, such as SSNs and GAN- based CAR, which lack a broader architectural category. This constraint was amplisied by the widespread absence of standardized benchmarks and comprehensive failure analyses in literature. Consequently, to en- sure analytical coherence across all surveyed tasks, we sometimes infer the properties of a specific method from its parent class rely qualitative observations based on general architectural principles, prioritizing broad insight across fragmented literature over impleenentation-speeiii empirical precision.
Soundness vs. prevalence Finally, notable tension emerged between theoretical soundness and em- pirical prevalence. Methods built upon stronger theoretical justifications, such as those utilizing princi- pled Bayesian inference, are often less prevalent in the literature than more hcuristic, theoretically weaker, approaches such as MC dropout or ensembling. This disparily suggests a practical barrier to adoption (e.g., complexity or computational cost). In several instances, our analytical preference for theoretical rigor superseded the volume of available literature on more widcly adopted alternatives. This choice risks under-representing the pragmatic, empirically dominant approaches that currently drive much of the applied progress in the field.
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8.4 Criteria for good uncertainty
Following the review of the fundamental need and methodological approaches for uncertainty modeling, this analysis culminates in the essential assessment of utility. This section transcends evaluations based on algorithricc complexity, theoretical soundness, and standard metrics to articulate the defining characteristics of 'good' uncertainty. We will therefore establish the practical criteria required to ensure the resulting estimates possess genuine relevance in real-world scenarios.
8.4.1 Reliable
Trustworthy systems fundamentally require high accuracy in the point prediction, as being correct most of the time is the prerequisite for the model's uncertainty to be low (sharp). Secondly, and critically, the model must be well-calibrated. In cases of potential inaccuracy, the model must reliably report high uncertainty to maintain Jrustworthines: A miscalibrated model, often manifesting dangerous overconfidence, is funda- mentally unreliable. High confidence misleads users to assumne low risk, which is particularly problematic in safety-critical fields like medicine or autonomous driving, thus compromising safe docision-makiin maasinnn
8.4.2 Explainable
To be effective, uncertainty must be well-communicated and casily understood. For example, raw entropy map is often insufficient because provides only relative comparison between pixels while lacking absolute, actionable meaning. Therefore, better approach is to process this raw data into interpretable signal for the end-user. This can be achieved by displaying distinct, plausible alternative boundaries or by using bright color to highlight only the regions of highest uncertainty, which effectively guides an expert's attention where it needed most.
8.4.3 Actionable
The key insight that uncertainty should not be seen merely as a measure model doubt, but as a crucial actionable signal. In practical applications, this signal is used to trigger specific, risk-mitigating behaviors, such as altering a plan, halting procedure, or deferring control a human expert, ensure safety and prevent errors. For effective utility, users must be clearly informed about the reducibility of the incertaint. High uncertainty should signal whether there is a need to collect more training data refine the model or if it simply represents irreducible, inherent data noise. In this latter case of irreducible uncertainty, the system must still recommend a specific risk-miligating strategy based on the bounds of the prediction.
8.4.4 Unbiased
A fundamental tenet of robust and equitable modeling is that the system's predictions must remain unbiased with respect to the underlying data distribution. To address this core challenge, we will categorize and discuss several common type of biases, providing examples of how they manifest.
Distributional bias If the validation data does not reflect the real-world deployment environment, the model's uncertainty estimates will not be reliable in practice. This bias can stem from subtle shifts in acquisition conditions population characteristics. Therefore, to ensure robustness, datasets must represent balanced collection encompassing diverse acquisition and demographic variations. Examples include an object detection model validated exclusively on footage from sunny highways deployed in a snowy snowstorm. Or, a diagnostic model validated on MRI from one hospital and specific demographic deployed at a new institution with different scanners and a diverse patient population.
Annotation bias Relying on a single ground-truth mask introduces a significant source of bias, as the model learns to be certain about one annotator's idiosyneratic choices, even where boundaries are ambiguous other experts. This results a false sense of cortainty, where the model's confidence reflects its ability replicate single opinion rather than the inherent ambiguity of the task. Furthermore, the selection of annotators critical consideration. To mitigate biases from individual annotators and their tools,
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rescarchers should collect multiple annotations from diverse group of experts to build ground truth that more accurately captures the inherent ambiguity of the task. Examples of this can be delineations provided by highly trained medical experts often differing systematically from those of trainees or multiple experts, which may have their own valid interpretation of the boundary. Another example of this can be that a polygon tool for annolating encourages simplified masks with straight edges, while a freehand brush tool allows for more detailed curves but can introduce minor inconsistencies.
Modeling bias Different uncertainty quantification methods may have their intrinsie biases. For instance, Gaussian densities assume single-moded uncertainty well light-tail bias. Consider segment- ing brain tumor with ambiguous boundaries, single-moded Bayesian model produces only one average segmentation. This misrepresents the true, multimodal uncertainty to climician. For autonomous vehicles segmenting a road, the light-tail bias, often inherent in models relying on assumptions like the Gaussian distribution, causes the system to view critical safety hazards, such as massive potholes, as statistically near-impossible events because training focused primarily on smooth surfaces.
Sampling bias system trained reduce uncertainty often foeuses its labeling budget on rare or complex outlier samples. This wastes resources by developing expertise in statistically insignificant cases, ultimately resulting in model that fails to reliably generalize or improve performance in the common, high-value scenarios it was designed to analyze. For example, in flawed active perception, soccer robot aiming to segment the ball focuses its attention where segmentation uncertainly is highest, i.e. the visually complex sidelines. This draws focus away from the simple green field, causing the robot to become an expert at tracking the ball the least important part of the game. Another example can be model used in satellite imaging may focus its uncertainty-driven budget on unique, complex buildings instead of common houses resulting in model expert in architectural oddities but failing reliably segment critical residential areas.
Validation bias Overemphasizing metric improvement can incentivize models to undermining the original goal. A textbook case of Goodhart's law: " Whhnn measure becomes larget, it ceases to be a good measure". Examples of this are optimization of metrics that minimize the discrepancy between the prediction and ground-truth distribution can incentivize the model to simply replicate the observed ground-truth masks rather than capture the full range of plausible, unseen segmentations.. Furthermore, a model can achieve perfect calibration (ECE score) while remaining dangerously useless for quantification; this occurs when the model only expresses binary confidence (100% or 0%). While technically calibrated, this failure mode is useless because the system fails to communicate the necessary nuanced doubt about ambiguous regions, Another example can be entropy-guided uncertainty can perversely incentivize models to suppress doubt and minimize uncertainty in genuinely ambiguous regions to achieve better metric scores.
Key takeaways: Good predictive uncertainty is defined by:
Reliability: The uncertainty should accurately reflect the model's true error rate.
Explainability: The source and meaning of the uncertainty should be clear to the end-user.
Actionability: The uncertainty estimate must be useful for downstream decision-making.
The estimates should not be systematically skewed by factors like subgroup characteristics.
Unbiasedness:
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9
Conclusion
This review synthesizes the fragmented literature on uncertainty in semantic segmentation, establishing a unified framework to bridge the gap between method developers, task specialists and applied researchers. Our analysis highlights critical challenges, including the nuanced distinction between uncertainty types, flawed assumptions spatial aggregation, and a lack of standardization. We identify active learning, data-driven benchmarks, and transformer-basee models as the most promising avenues for future research, particularly for exlending novel uncertainty meth to holistie tasks like instance seslance/panptie panoptic segmentation. Our conclusion provides clear method selection guide and places strong emphasis the need for future work to produce uncertainty that is reliable, explainable, actionable, and unbiased.
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Reviews:
[{"review_id": "9fjQk6ENy4", "reviewer": "Reviewer_2cbv", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors provide a very extensive overview of different approaches and uncertainty estimation techniques for semantic segmentation. They formalize how the different approaches model and disentangle uncertainty. They further analyze downstream applications for uncertianty quantification. The paper ends with a comprehensive discussion of the problems and future outlook for various parts of the field.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Comprehensive review covering the main approaches of the field.", "audience_yn": "Yes", "audience_explanation": "It is a very comprehensive overview of the topic and will be very helpful for everyone who seeks to start working on this topic.", "requested_changes": "- Some methods could be elaborated a bit more on how they are connected to the current topic: e.g. label smoothing in 4.1.1. is only mentioned after a long background on calibration\n- Explain the differences between the entropy decompositions. I really like the fact that this is done, but each time the authors could spend 2-3 sentences more to really explain what is shown and how it differs to the other equations and assumptions\n- The applications sections could benefit from a high-level example in the beginning each time: e.g. observer variablility could be made very clear given an example there\n\nNot necessarily requested, but I believe it would help the work: \n- Streamlining the paper. Some paragraphs explain very basic background, like GANs, Diffusion models, etc. I could see a more distilled main text while adding this information in the appendix. This could help with a clear focus on the work.\n\nTypos:\n- In 5.2 laplace approximation: „.e.g.“\n- In 5 parameter modeling last sentence in the overview has the incorrect order of the sections", "broader_impact_concerns": "No concerns", "additional_comments": "Question: \nDo the authors believe that the attempts to disentangle uncertainty sources during modeling is hindering progress of using UQ for downstream applications?", "extra_scores": {}, "extra_text": {}}, {"review_id": "R99Okc1Zh2", "reviewer": "Reviewer_HJet", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This review paper provides a survey of Bayesian uncertainty quantification in deep probabilistic segmentation, organizing a fragmented literature across domains like medical imaging and autonomous driving. It offers taxonomic organization of uncertainty modeling approaches and identifies practical challenges in the field.\n\n**Strengths:** Comprehensive literature coverage; identifies important practical challenges; valuable as reference work for the field\n\n**Weaknesses:** Perpetuates problematic epistemic/aleatoric framework despite acknowledging its limitations; lacks critical analysis of method effectiveness; insufficient empirical support for comparative claims; shallow treatment of validation approaches; limited practical guidance for method selection", "claims_and_evidence_yn": "No", "claims_explanation": "* The paper repeatedly discusses uncertainty \"disentanglement\" methods while acknowledging these categories are \"entangled or conflicting.\" No evidence is provided that existing methods can reliably achieve this disentanglement, making discussions of epistemic vs. aleatoric methods potentially misleading.\n\n* The paper suggests methods addressing spatial coherence solve\n  fundamental problems without providing evidence that these approaches\n  actually improve uncertainty calibration.\n\n* This evidence gap is particularly problematic given the paper's survey nature. Readers expect claims about method effectiveness to be supported by rigorous analysis of the literature, not assertions based on selective citation.", "audience_yn": "Yes", "audience_explanation": "Uncertainty quantification in segmentation is highly relevant to the TMLR community, particularly given applications in safety-critical domains like medical imaging and autonomous driving.", "requested_changes": "**Critical:**\n\n* Either defend the epistemic/aleatoric distinction with new evidence and arguments, or reorganize around empirically grounded categories (e.g., by computational approach, application domain, or uncertainty estimation strategy).\n* Replace anecdotal effectiveness claims with structured analysis using consistent evaluation criteria across methods. Include quantitative comparisons where possible rather than qualitative descriptions.\n* Develop frameworks for evaluating uncertainty quality in segmentation\n  contexts, especially for applications with ambiguous ground truth.\n  Cooment on fundamental questions about what constitutes good\n  uncertainty estimates. Poorly calibrated uncertainty estimates in medical imaging could lead to missed diagnoses or unnecessary procedures, while unreliable uncertainty in autonomous driving could affect collision avoidance systems. The authors should discuss validation strategies and potential biases in uncertainty estimates for high-stakes applications.\n* Identify when and why specific methods fail, rather than only describing successes. Analyze patterns across applications to provide clearer guidance on practical trade-offs.\n* Provide specific recommendations for method\n  selection based on data characteristics, computational constraints,\n  and application requirements rather than general observations.\n\n**Strengthening:**\n\n* Provide deeper analysis of spatial coherence implications: Section 7.2 identifies spatial aggregation as a key challenge, but the theoretical implications for uncertainty modeling frameworks deserve more thorough exploration.", "broader_impact_concerns": "N/A", "additional_comments": "Consider whether the goal is purely organizational (in which case acknowledge theoretical limitations more explicitly) or whether you aim to contribute to resolving conceptual issues (in which case provide deeper critical analysis and potential solutions).", "extra_scores": {}, "extra_text": {}}, {"review_id": "j1QYD8UBTV", "reviewer": "Reviewer_rS4H", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors review common approaches to probabilistic image segmentation, covering both aleatoric uncertainty modelling techniques (e.g. based on GANs, VAEs and diffusion models) and methods for estimating epistemic uncertainty (mainly covering MC-dropout and ensembling).\n***\n***\n***\n***\n\n\n\n\n\n\n\nStrengths:\n- The studied problem is interesting/important, a review of this topic could definitely be useful.\n- The paper is well-written overall, clean figures etc, solid work overall.\n***\n\n\n\n\nWeaknesses:\n- At least to me, the paper feels somewhat disjointed overall. The different sections and subsections don't quite tie together in an intuitive way. For example, the transition from Section 3 to Section 4 could be smoother, and it could be made more clear at the start of Section 4 what will be covered in this section, how it relates to the rest of the paper etc. In general, it could be made more clear what is covered in the different sections/subsections and _why_ this is covered (this is a long paper, and I quite often found myself reading and thinking roughly _\"yes, this is a solid description of method X, but why am I reading this? Will this be used later in the paper in some way? If so, how? And how did we get here? How does this relate to the previous sections and what the paper aims to do overall?\"_).\n- Personally, I didn't find this review overly interesting/useful, I don't quite feel like it provided me with a lot of new knowledge or important new insights. At the same time, I'm not entirely sure that someone who is less familiar with uncertainty estimation methods would find this to be a good introduction to the field, I think there are more pedagogical / digestible / easy-to-follow resources for that. In short, it's somewhat unclear to me what the main target audience for this paper is, or what type of person who would find this paper really helpful/useful/interesting.\n***\n\n\n\n\nQuestions/suggestions:\n- I don't quite understand Figure 11, don't quite see what this is intended to illustrate/explain?\n- In Section 7.6, _\"Table 1 shows AB (Chen et al., 2022) performs almost identically to CCDM (Zbinden et al., 2023)\"_: I don't quite understand how this is shown in Table 1? Or, what exactly do you mean here?\n- Section 7.7, \"Reliable uncertainty estimates\" paragraph: Should at least mention that the coverage guarantee of conformal prediction depends on the assumption of exchangeably drawn train/calibration and test data (true for i.i.d. data, for example)? At least in my opinion, this is a quite serious limitation of CP, it can produce overconfident predictions in many practical settings due to distribution shifts (when the exchangeability assumption generally does not hold) etc.\n***\n\n\n\n\n\nMinor things:\n- Section 2, \"Ronneberger Ronneberger et al. (2015) introduced the U-Net\": typo.\n- Section 3: \"the predictive distribution from a new datapoint\" --> \"the predictive distribution for a new datapoint\"?\n- Section 3, \"Hence, uncertainties stemming from the conditional likelihood distribution are classified as either aleatoric, implying from the statistical diversity in the data, or epistemic, which stems from the posterior, i.e. the variance of the model parameters\": perhaps just restructure a bit?\n- Section 3.1: \"does not include any prior knowledge on the structure of the parameter distribution but can be achieved\" --> \"does not include any prior knowledge on the structure of the parameter distribution, but this can be achieved\" or similar perhaps?\n- Section 3.1: \"we make use of function\" --> \"we make use of a function\"?\n- Section 4.1.1, \"true probabilities in modern neural networks Guo et al. (2017)\": citation formatting, \"Guo et al. (2017)\" --> \"(Guo et al., 2017)\".\n- 4.1.1, \"considered.Additionally\": typo.\n- 4.1.1, \"imbalance Jungo et al. (2020); Neumann et al. (2018)\": citation formatting.\n- 4.2: \"to instead learn the approximate through\" --> \"to instead learn the approximation through\"?\n- 4.2, \"optimization trajectories Zheng et al. (2022)\": citation formatting.\n- Fig 7 caption, \"network Kassapis et al. (2021)\": citation formatting.\n- 4.2.2, \"Sinkhorn divergence Cuturi (2013)\": citation formatting.\n- 5, \"Mean-Field Variational Inference Blundell et al. (2015), Markov Chain MonteCarlo (MCMC) Neal (2012), Monte-Carlo Dropout Gal & Ghahramani (2016); Kingma et al. (2015), ModelEnsembling Lakshminarayanan et al. (2017), Laplace approximations Mackay (1992), Stochastic GradientMCMCs Korattikara Balan et al. (2015); Springenberg et al. (2016); Welling & Teh (2011), assumed densityfiltering Hernández-Lobato & Adams (2015) and expectation propagation Hasenclever et al. (2017); Louizos& Welling (2016)\": citation formatting.\n- 5, \"Test-time augmentation, often mistakenly regarded as an aleatoric method, is addressed in Section 5.3. Finally, Laplace Approximations are examined in Section 5.2\": incorrect ordering.\n-  5.1.1: \"The relationship between p and the magnitude of the model weights has also be exploited\" --> \"The relationship between p and the magnitude of the model weights has also been exploited\" / \"The relationship between p and the magnitude of the model weights can also be exploited\"?\n- 5.1.1, \"prune neural networks Gonzalez-Carabarin et al. (2022)\": citation formatting.\n- 5.1.1: \"where the weights of the network, rather that its\" --> \"where the weights of the network, rather than its\"?\n- 5.2, \"post-hoc to .e.g.\": typo.\n- 5.2: \"by treating neural network partly probabilistic\" --> \"by treating the neural network partly probabilistic\" / \"by treating neural networks partly probabilistic\"?\n- 6.1: \"each annotator is consistent in his delineation\" --> \"each annotator is consistent in their delineation\"?\n- 6.1, \"inductive biases Kohl et al. (2019a); Zepf et al. (2024)\": citation.\n- 6.1: \"it can also be possible to modeling the annotator distribution\" --> \"it can also be possible to model the annotator distribution\"?\n- 6.1, \"labeling styles of annotators Zepf et al. (2023a)\": citation.\n- Missing period at the end of the caption for Table 2 and 3.\n- 6.2, \"predictive entropy Kasarla et al. (2019); Shen et al. (2021); Xie et al. (2022) or variance Ozdemir et al. (2021)\": citation.\n- 6.2, \"Yang et al. (2017); Sourati et al. (2018); Kasarla et al. (2019); Mahapatra et al. (2018); Sinha et al. (2019); Siddiqui et al. (2020); Li & Alstrøm (2020); Kim et al. (2021); Shen et al. (2021); Burmeister et al. (2022); Ma et al. (2024b)\": citation.\n- 6.2: \"literature use MC dropout of ensembles\" --> \"literature uses MC dropout or ensembles\"?\n- End of 6.2, \"Ozdemir et al. (2021); Shen et al. (2021); Wu et al. (2021); Mahapatra et al. (2018); Sinha et al. (2019)\" and \"Gaillochet et al. (2023); Wu et al. (2021); Kasarla et al. (2019)\": citation.\n- 6.3: \"Hence, Roy et al. (2018b) propose to use the Coefficient of Variation (CoV) addresses this by\" --> \"Hence, Roy et al. (2018b) propose to use the Coefficient of Variation (CoV) to address this by\"?\n- 6.4, \"Saha et al. (2021a); Wolleb et al. (2021); Wu et al. (2023b); Amit et al. (2021); Bogensperger et al. (2023); Zbinden et al. (2023); Viviers et al. (2023a); Zepf et al. (2023a); Hu et al. (2022)\": citation.\n- 7.1.1, \"total uncertainty Depeweg et al. (2018)\" and \"residual risk Apostolakis (1990); Helton (1997)\": citation.\n- 7.1.1: \"This is dependency is also demonstrated\" --> \"This dependency is also demonstrated\".\n- 7.1.1, \"the data Kahl et al. (2024)\", \"quantification methods de Jong et al. (2024); Mucsányi et al. (2024)\", \"distributions Kingma et al. (2015)\", \"Zhang et al. (2022c); Monteiro et al. (2020); Valiuddin et al. (2024b); Gao et al. (2022)\", \"Kendall et al. (2016); Roy et al. (2018b); DeVries & Taylor (2018); Mehrtash et al. (2020); Burmeister et al. (2022); Iwamoto et al. (2021)\": citation.\n- 7.1.2: \"but can often found to be evaluated on\" --> \"but can often be found to be evaluated on\"?\n- 7.1.2, \"Wang et al. (2019a); Whitbread & Jenkinson (2022); Kendall & Gal (2017); Liu et al. (2022b); Schmidt et al. (2023); Savadikar et al. (2021)\": citation.\n- 7.2: \"we further dive into import details and challenges\" --> \"we further dive into important details and challenges\"?\n- 7.2, \"Mukhoti & Gal (2018); Camarasa et al. (2021); Bhat et al. (2022b); Mukhoti et al. (2020); Ma et al. (2024b); Shen et al. (2021); Li & Alstrøm (2020); Dechesne et al. (2021); Huang et al. (2018); Nair et al. (2020); Zepf et al. (2023b)\", \"redundant Shen et al. (2021); Burmeister et al. (2022)\", \"direction Liu et al. (2025); Burmeister et al. (2022); Viviers et al. (2023b)\": citation.\n- 7.2: \"appropriately compare 2D- to 3D models\" --> \"appropriately compare 2D to 3D models\"?\n- 7.2: Looks like there's an extra space at the beginning of the Goodhart’s law quote.\n- 7.4: \"the notion that uncertainty methods work poor on the initial samples\" --> \"the notion that uncertainty methods work poorly on the initial samples\"?\n- 7.4: Multiple instances of incorrect citation formatting here as well. Same also for 7.6 (including in the Figure 13 caption). And for 7.7.\n- 7.5, \"enhance performance Chen et al. (2023b)\": citation.\n- 7.6: \"while classified PixelCNNs and SSNs are classified as pixel-level methods, both can also be phrased as a latent-variable model\" --> \"while PixelCNNs and SSNs are classified as pixel-level methods, both can also be phrased as latent-variable models\"?\n- 7.7: \"However, it’s debatable whether\" --> \"However, it is debatable whether\"?\n- 7.7: \"In particular, such study can\" --> \"In particular, such a study can\"?\n- 7.7: \"after training with SoftMax predictive distribution\" --> \"after training with a SoftMax predictive distribution\"?", "claims_and_evidence_yn": "No", "claims_explanation": "The authors claim that _\"By the end of this paper, readers will have a clear understanding of the various forms of uncertainty, their relevance to segmentation tasks, and a comprehensive grasp of the key challenges and open research directions in the field\"_. Overall, I'm not entirely sure that the paper fulfills this claim, see \"Weaknesses\" above.", "audience_yn": "Yes", "audience_explanation": "The studied problem is interesting/important, a review of this topic could definitely be useful.", "requested_changes": "This is a quite well-written and solid paper overall, and I think it could be relevant for the TMLR audience.\n\nHowever, I think the current version requires some clarifications and tweaks, see \"Weaknesses\" and \"Questions/suggestions\" above.", "broader_impact_concerns": "No concerns.", "additional_comments": "No additional comments.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)