Epistemic Deep Learning: Enabling Machine Learning Models to Know When They Do Not Know

1.1 Motivation

• The thesis identifies principled uncertainty estimation as important for reliability, especially in safety-critical settings where uncertainty can inform fallback strategies or human intervention.It also describes existing approaches as limited in capturing the full spectrum of uncertainty and efficient deployment.Source: §1.1 · pp. 24–25

1.2 Research Questions

• The research questions cover feasible set-based quantification, behavior on unseen or adversarial inputs, and unified comparison of methods with different output structures.The questions specifically connect the second aim to out-of-distribution detection and robustness, and the third to fair evaluation.Source: §1.2 · p. 26

1.3 Thesis Structure

• RS-NN outperforms state-of-the-art Bayesian and ensemble methods in accuracy, uncertainty estimation, and out-of-distribution detection across multiple benchmarks.The thesis also reports scaling to large architectures, adversarial robustness, and statistical guarantees in a conformal setting.Source: §1.3 · p. 27
• RS-NNs predict belief functions over class labels, using the associated credal-set width to represent epistemic uncertainty from limited or unrepresentative training data.The thesis frames this as learning from ignorance through second-order uncertainty measures.Source: §1.3 · p. 27
• The thesis extends RS-NNs to text generation as RS-LLMs and applies them to weather classification for autonomous racing.These case studies are presented as practical applications in real-world scenarios.Source: §1.3 · p. 27, §1.4 · p. 28

1.4 Contributions

• A budgeting mechanism selects a representative subset of focal sets instead of handling the full power set, managing set complexity for practical random-set models.The contribution passage describes this as supporting strong performance and reliable uncertainty quantification without processing all possible combinations.Source: §1.4 · p. 28
• The thesis proposes a model-agnostic evaluation framework for heterogeneous epistemic predictions, with metrics for accuracy and imprecision and guidance for method selection.The framework includes experimental analysis and ablation studies.Source: §1.3 · p. 27, §1.4 · p. 28

2.1 Sources of Uncertainty: Aleatoric Vs. Epistemic

• Aleatoric uncertainty reflects irreducible randomness, whereas epistemic uncertainty reflects lack of knowledge that additional information can reduce, given sufficient data coverage and model capacity.In high-capacity models, the two sources may interact and may not be cleanly separable (de Jong et al., 2024).Source: §2.1 · pp. 32–33
• Insufficient or unrepresentative training data is a principal source of reducible epistemic uncertainty, and this thesis models it at the target level using random sets.The thesis leaves random-set modeling at the parameter level for future work.Source: §2.1 · p. 33

2.3. Why Uncertainty Quantification Matters

• The thesis proposes random sets over probability distributions and a unified metric space for comparing predictive entropy and epistemic uncertainty across models.The framework addresses the difficulty of direct comparison when uncertainty-aware models define epistemic uncertainty differently.Source: §2.3 · p. 34
• Uncertainty quantification matters because overconfident predictions on unfamiliar or perturbed inputs can undermine reliability and safety in real-world applications.The section identifies adversarial robustness, domain adaptation, confidence calibration, and safer sequential decision-making as motivations.Source: §2 · p. 32, §2.3 · p. 34

2.3.1 Adversarial Robustness

• 86.9% and 94.95% are the Traditional model’s confidence scores for two ImageNet-A misclassifications, versus 32.93% and 3.96% for the Epistemic model.Both ResNet50-based models were trained on ImageNet; the uncertainty-aware model is an RS-NN and assigns lower confidence to misclassifications.Source: Figure 2.1 · p. 34, §2.3.1 · p. 35

2.3.3 Out-of-distribution (OoD) Detection

• Out-of-distribution detection helps models recognize unfamiliar inputs and express caution; AUROC and AUPRC evaluate separation of in-distribution and OoD samples.AUPRC is particularly informative when OoD instances are rare.Source: §2.3.3 · p. 35

2.3.4 Calibration; 2.3.5 Sequential Decision-making

• In sequential decisions, uncertainty can compound errors and create unsafe outcomes, while calibration addresses whether confidence estimates match empirical correctness.The section notes that confidence may be miscalibrated and that existing calibration methods can overlook deeper issues in uncertainty representation.Source: §2.3.5 · pp. 36–37, §2.3.4 · p. 36

3.1 Traditional and Deterministic Methods; 3.2 Bayesian Methods

• Traditional networks output point probabilities without epistemic mechanisms, while Bayesian methods model parameter distributions but face computational, prior-specification, architecture, and real-time challenges.Bayesian methods predict distributions over distributions, often approximating posterior integration through sampling.Source: §3.1 · p. 38, §3.2 · p. 40

3.2 Bayesian Methods

• Bayesian model averaging can suppress informative posterior samples and produce incorrect point predictions, while mutual-information estimates are difficult to compare across uncertainty methods.The examples show BMA selecting automobile despite an airplane sample assigning it approximately 0.34 probability, and selecting truck when automobile has substantial sample support; the thesis therefore analyzes prediction samples before averaging.Source: §3.2 · pp. 41–42, Figure 3.2 · p. 42

3.4 Evidential Methods

• Evidential methods represent class predictions with Dirichlet distributions, but some losses fail to make epistemic uncertainty diminish with more data, and some approaches require unavailable OoD training data.EDL interprets network outputs as class evidence and infers a posterior Dirichlet, whereas Prior Networks parameterize the Dirichlet prior directly.Source: §3.4 · p. 44

3.5 Conformal Prediction

• Conformal prediction creates prediction sets with marginal coverage guarantees under exchangeability, but standard methods do not model uncertainty about the model and may lose coverage under distribution shift.The thesis treats conformal prediction as compatible with, rather than directly comparable to, its uncertainty methods; its model can serve as a conformal predictor’s underlying model.Source: §3.5 · p. 45

3.6 Imprecise Models

• Imprecise-probability approaches represent uncertainty with sets of distributions, including credal sets and belief functions, while interval neural networks extend interval representations to uncertainty-aware prediction.Some interval networks use ensembles to quantify uncertainty in prediction bounds, while others model weights and biases as intervals to capture aleatoric and epistemic uncertainty and handle interval-valued inputs; set-valued decisions can support risk-aware classification.Source: §3.6 · pp. 45–46

3.7 Discussion On Existing Methods and Motivating This Thesis

• Existing uncertainty methods have shortcomings in faithful epistemic-ignorance modeling, including computational costs, poor scalability, unreliable heuristics, and sensitivity to data or design choices.The discussion cites BNN prior sensitivity and costs, ensemble dependence on diversity, evidential loss sensitivity, and conformal dependence on exchangeability.Source: §3.7 · p. 47

3.3 Ensemble Methods

• Deep ensembles aggregate predictions from multiple independently trained models and estimate epistemic uncertainty through predictive variance, but their costs grow with ensemble size and unreliable diversity can weaken estimates.The variance is described as the difference between entropy of the averaged prediction and the average entropy of individual predictions.Source: §3.3 · p. 43, §3.7 · p. 47

3.7. Discussion on existing methods and motivating this thesis

• The thesis advocates Epistemic AI and imprecise models based on random sets, belief functions, and credal sets to represent second-order uncertainty while preserving computational tractability.It argues that substantial performance improvements need not require the full expressive power of second-order measures; suitably expressive model structures can capture key aspects of that uncertainty.Source: §3.7 · p. 48

4.1 Concept

• Epistemic learning maps inputs to second-order uncertainty measures, rather than a single model’s probability distribution over the target space.The uncertainty measure can be a credal set on the target or parameter space and may be updated as new data arrive.Source: Figure 4.1 · p. 51, §4.1 · p. 51

4.2 Why Epistemic AI Is Essential

• The chapter argues that Bayesian methods can conflate ignorance with weak evidence, constrain set-valued evidence, smooth uncertainty through averaging, and incur high inference costs.It also notes prior sensitivity and arbitrary prior choices, particularly with scarce data.Source: §4.2 · p. 52

4.3.1 Theories of Uncertainty: Beyond Bayesian Probabilities

• Uncertainty theory models uncertainty about the probabilistic process generating data, with uncertainty refined as new evidence arrives and operators supporting reasoning.Some approaches use credal sets; random sets and belief functions assign mass directly to sets of outcomes.Source: §4.3.1 · p. 53, §4.3 · p. 54

4.3. Modeling A Lack Of Knowledge

• Uncertainty theories form clusters by rationale, ranging from robustified probabilities and generalized sets to generalized measures, random sets, and scoring-function frameworks.Figure 4.2 uses arrows to indicate relative generality; it cautions that approaches vary in quality and rigour.Source: Figure 4.2 · p. 53, §4.3 · p. 54

4.3.1.1 Random Sets

• Random sets model observations that may be sets: when die faces 1 and 2 are cloaked, either outcome maps to {1, 2}.Their non-additive structure provides additional degrees of freedom for expressing uncertainty about probability values.Source: §4.3.1.1 · pp. 54–55, Figure 4.3 · p. 54

4.3.1.4 A Unified Example

• Belief functions assign mass to subsets, and wider associated credal sets indicate greater epistemic uncertainty; a three-class example computes beliefs from masses.The figure describes the equivalent credal set’s boundaries as determined by lower probability bounds.Source: §4.3.1.2 · p. 55, §4.3.1.4 · p. 56, Figure 4.4 · p. 56

4.4 Classical Probabilities Vs. Random Sets & Belief Functions

• Softmax probabilities do not capture insufficient evidence under distribution shift; belief functions can retain support for a class set without forcing singleton probabilities.This distinction matters when the evidence cannot justify how support for a set should be divided among its members.Source: §4.4 · p. 57, §4.5 · p. 58

4.5 Leveraging Epistemic AI

• The thesis proposes RS-NN and notes collaborative credal-set models, then compares epistemic models with Bayesian and ensemble baselines across performance and runtime measures.The stated comparison includes test accuracy, out-of-distribution detection, expected calibration error, and training and inference times.Source: §4.5 · p. 58

5.2 Methodology

• At inference, RS-NN maps each input to a belief function over class sets, whose credal set supports a pignistic probability for point prediction.The network predicts belief values for a budget of focal sets, rather than a conventional softmax vector.Source: §5 · p. 59, §5.2 · p. 62

5.1 Key Contributions

• RS-NN predicts belief values for class sets rather than individual classes, serving as a wrapper for baseline networks and enabling conformal guarantees on pignistic probabilities.In the FGSM MNIST example, CNN confidence scores were 99.01% and 96.74%, versus 32.59% and 48.25% for RS-NN.Source: §5 · pp. 59–61, Figure 5.2 · p. 60, §5.1 · p. 61
• Across from-scratch experiments, RS-NN outperforms state-of-the-art Bayesian and ensemble methods on test accuracy, inference time, OoD benchmarks, uncertainty quantification, and scalability.The experiments include large-scale architectures and datasets, including ImageNet.Source: §5.1 · p. 61

5.2.1 Ground-truth Representation

• Ground truth is encoded as belief values for focal sets: a set receives value 1 when it contains the true class and 0 otherwise.The output layers can be added to baseline representation layers, allowing the approach to use pretrained networks while fine-tuning the final decision layers.Source: §5.2.1 · p. 62

5.2.2 Budgeting

• To avoid the powerset’s exponential output size, budgeting selects relevant non-singleton class sets by clustering class features and ranking their overlap.The selected sets are combined with singleton classes; experiments report that limited representative budgets can outperform the full powerset and improve estimation efficiency.Source: §5.1 · p. 61, §5.2.2 · pp. 63–64

5.2.3 Training RS-NN: Loss Function

• Training uses binary cross-entropy for focal-set belief values, with mass regularization and a mass-sum term to encourage valid belief functions.When these constraints are insufficient, negative masses are set to zero and the universal set receives the remaining mass.Source: §5.2.3 · p. 65

5.2.4 Accuracy & Uncertainty Estimation

• Pignistic entropy measures predictive uncertainty, while credal-set width—based on probability bounds for the most likely class—measures epistemic uncertainty.The section reports that pignistic entropy better separates in-distribution from out-of-distribution samples than Bayesian entropy, while credal width is less correlated with confidence.Source: §5 · pp. 59–60, §5.2.4 · pp. 66–67

5.2.5 RS-NN Learning Mechanism

• The section explains the rationale for RS-NN’s ground-truth representation, loss, and training, and how it learns sets of outcomes.Source: §5.2.5 · p. 67
• RS-NN encodes each class subset as a separate binary target, trains with sigmoid binary cross-entropy, and derives mass functions and pignistic predictions from learned beliefs.The belief target marks every subset containing the true class as 1; the resulting pignistic predictions redistribute masses to original classes and are reported as more accurate than standard CNN predictions.Source: §5.2.5 · pp. 67–68

5.3 Experiments

• Experiments span multi-class image classification, OoD detection, calibration, uncertainty estimation, scalability, transfer learning, robustness, statistical guarantees, and parameter and focal-set ablations.Reported ablations find best accuracy at α = β = 1e-3 and K = 20 on CIFAR-10.Source: §5.3 · p. 68

5.3.1.4 Sample Prediction

• Two CIFAR-10 examples show a confident horse prediction at 99.9% confidence and entropy 0.0017, versus an uncertain cat prediction at 33.3% confidence and entropy 2.6955.Source: Table 5.2 · p. 70, §5.3.1.4 · p. 71

5.3.2 Performance: Accuracy & Expected Calibration Error

• RS-NN outperforms Bayesian and ensemble methods in test accuracy across datasets and shares the fastest CIFAR-10 inference time with CNN.The comparison uses a standardized setup; the authors note that FSVI’s performance may differ because it was optimized for ResNet-18, while their comparison uses ResNet-50 and less extensive augmentation.Source: §5.3.2 · pp. 71–73, Table 5.3 · p. 72

5.3.3 Out-of-distribution Detection

• RS-NN greatly outperforms Bayesian, ensemble, and CNN baselines on OoD detection across evaluated dataset pairs, and exhibits the lowest ECE.Deep Ensembles show comparable AUROC on some CIFAR-10 and MNIST pairs but lower AUPRC than RS-NN.Source: §5.3.3 · pp. 73–74, Table 5.5 · p. 73

5.3.4.1 Entropy of Pignistic Predictions

• RS-NN has the strongest iD/OoD entropy distinction, with lower entropy on familiar data and higher entropy on OoD data; its entropy also rises for challenging rotations.On rotated MNIST, all models misclassify at higher rotation angles; LB-BNN does not show a significant entropy increase for those errors.Source: §5.3.4.1 · pp. 74–75, Figure 5.8 · p. 74, Figure 5.10 · p. 75

5.3.4.2 Credal Set Width

• Incorrect CIFAR-10 predictions generally have larger, more dispersed credal-set widths than correct predictions, and OoD datasets show larger widths than iD datasets.The ImageNet/ImageNet-O widths are less distinct, and credal-set width is not directly comparable to entropy, variance, or mutual information.Source: §5.3.4.2 · p. 76, Figure 5.13 · p. 76, Table 5.6 · p. 77

5.3.4.3 Entropy/Credal Set Width Vs. Confidence Score

• For CIFAR-10 and OoD predictions, iD samples concentrate at high confidence and low entropy or credal width, while OoD samples are more dispersed.Source: §5.3.4.3 · p. 77
• Credal-set width better reflects the epistemic uncertainty across plausible outcomes than entropy, and is less dependent on confidence scores in the shown in-distribution and out-of-distribution comparisons.Entropy is described as correlated with confidence, whereas credal-set width varies less dependently on confidence for SVHN and Intel Image predictions.Source: §5.3.4.3 · p. 77, Figure 5.16 · p. 78, Figure 5.15 · p. 78

5.3.5 Scalability To Large-Scale Architectures

• RS-NN outperforms standard CNNs across the tested large-scale architectures, and ImageNet-pretrained ResNet50 improves its CIFAR-10 accuracy over the same architecture without pretraining.The tested architectures include WRN-28-10, VGG16, InceptionV3, EfficientNetB2, and ViT-Base-16.Source: §5.3.5 · p. 78, Table 5.7 · p. 79

5.3.6 Overcoming the Overconfidence Problem in CNNs

• On a noisy rotated MNIST example, CNN predicts ‘6’ incorrectly at 99.95% confidence, while RS-NN predicts the true class ‘8’ at 49.7% confidence with pignistic entropy 2.1626 and credal-set width 0.482.Across incorrectly classified samples, CNN has higher confidence than RS-NN; the example illustrates RS-NN’s use of confidence and entropy together.Source: Table 5.8 · p. 79, §5.3.6 · pp. 79–80, Figure 5.17 · p. 79

5.3.7 Robustness To Noisy and Rotated Data

• For randomly rotated MNIST, RS-NN reaches 47.71% test accuracy versus 45.86% for CNN, and it predicts noisy or rotated examples correctly where CNN can be highly confident but wrong.For a noisy ‘3’, CNN predicts ‘8’ at 96.9% confidence while RS-NN predicts {‘3’} at 42.7%; for a rotated ‘9’, CNN predicts ‘5’ at 98.8% while RS-NN predicts {‘9’} at 69.9%.Source: Table 5.9 · p. 80, Table 5.11 · p. 81, §5.3.7 · p. 81

5.3.8 Robustness To Adversarial Attacks

• On CIFAR-10, RS-NN outperforms other models under FGSM as perturbations increase; under PGD, it leads at all tested CIFAR-10 perturbations and performs better at higher MNIST perturbations.The authors attribute this robustness to RS-NN’s precise predictions having zero loss gradients, leaving predictions unaffected by the attacks.Source: Table 5.12 · p. 83, §5.3.8 · pp. 83–84, Table 5.13 · p. 83

5.3.9 Application To Text Classification

• On BBC text classification with fine-tuned BERT, RS-NN BERT achieves higher AUROC and AUPRC than CNN BERT, alongside higher entropy on both in-distribution and out-of-distribution data.The authors interpret the higher entropy on both as indicating greater predictive uncertainty on this small dataset.Source: Table 5.14 · p. 85, §5.3.9 · p. 85

5.3.10.1 ImageNet Vs. ImageNet-O

• RS-NN distinguishes CIFAR-10 from CIFAR-100 well, but ImageNet versus ImageNet-O is difficult; ViT-B-16 RS-NN shows better separation than baseline measures, and the difficulty is specific to this dataset pairing.The passages attribute the ImageNet difficulty to ImageNet-O’s construction and suggest ImageNet class imbalance and uncertainty across many classes as possible factors.Source: §5.3.10.1 · pp. 85–86, Table 5.16 · p. 86, Table 5.15 · p. 86

5.3.10.2 Ablation Study On Hyperparameters α and β

• On CIFAR-10 with K = 20, accuracy is highest when α and β equal 1e-3, then drops as either regularization weight increases beyond that point.The regularizers promote valid belief functions, but excessive weighting constrains them and harms classification accuracy.Source: §5.3.10.2 · pp. 86–87, Figure 5.21 · p. 87
• 92.47% accuracy with softmax-normalized masses was close to 93.53% with mass regularization; softmax enforces nonnegative unit-sum masses, whose reconstructed beliefs are trained against the original beliefs.Source: §5.3.10.2 · pp. 87–88

5.3.10.3 Ablation Study On the Number of Focal Sets

• On CIFAR-10 with ViT-Base-16, accuracy peaked at K=20; a small K near the class count outperformed K=0, while larger K increased complexity and could overfit.The passage also reports that set prediction with the budgeting algorithm helps induce uncertainty quantification and improve performance.Source: §5.3.10.3 · p. 88, Figure 5.22 · p. 88
• Using 30 budgeted sets (K=20) rather than 1,024 full sets improved CIFAR-10 RS-NN performance, especially iD-versus-OoD entropy, AUROC, and AUPRC; excessive set degrees of freedom may hinder learning and generalization.Source: §5.3.10.3 · p. 89, Table 5.17 · p. 89, Table 5.18 · p. 89

5.3.10.4 On the Feasibility of Budgeting of Sets

• Replacing t-SNE with UMAP yielded comparable test accuracy, OoD detection, and uncertainty estimates at lower computational cost; t-SNE remained the main choice for preserving local-neighbor structure.Both methods embed in 3D; UMAP uses five nearest neighbors and a minimum distance of 0.9.Source: §5.3.10.4 · p. 89, Table 5.19 · p. 90, Table 5.20 · p. 90

5.4 Discussion and Implications

• RS-NN predicts belief functions over outcome sets, derives pignistic probabilities, and uses budgeted subsets to scale across architectures and tasks; the thesis reports gains in accuracy, calibration, OoD detection, and adversarial robustness over state-of-the-art uncertainty models and standard CNNs.Applications include Random-Set Large Language Models for text generation and weather classification for autonomous racing.Source: §5.4 · p. 90

6 Credal Set Models

• A credal set is a convex set of plausible class-probability distributions, elicited from lower and upper probability bounds for each class.Its extremal-point representation can vary in size with the number of classes and the complexity of the network prediction.Source: §6 · p. 91
• The chapter introduces Cre-INN and CreDE as distinct mechanisms for constructing and using credal sets in deep learning.The models are summarized here and compared with other approaches in the thesis’s later evaluation.Source: §6 · p. 91

6.1 Credal Interval Neural Networks (Cre-INN)

• CreINNs use deterministic intervals for weights and biases and predict upper and lower class-probability bounds that define a credal set.Their Interval SoftMax produces valid, ordered probability intervals that are differentiable for backpropagation; reachable-bound refinement removes redundant intervals, and interval batch normalization helps prevent interval explosion.Source: §6.1 · pp. 92–93, Figure 6.1 · p. 92
• CreINNs achieve superior or comparable uncertainty-estimation quality to variational BNNs and Deep Ensembles, with significantly lower variational-BNN inference complexity.The experiments cover standard multiclass and binary tasks, and also verify uncertainty quantification with interval-valued inputs.Source: §6.1 · p. 93

6.2 Credal Deep Ensembles (CreDE)

• CreDEs aggregate credal-set predictions from CreNets, whose lower and upper class-probability bounds are trained with a DRO-inspired loss.The loss encourages interval widths to reflect potential dataset shift by simulating divergences between training and test distributions.Source: §6.2 · p. 94, Figure 6.2 · p. 94
• Across OoD benchmarks and ResNet50, VGG16, and ViT Base, CreDEs outperform traditional ensembles and variational BNNs on ECE, AUROC, and AUPRC.The evaluations include CIFAR10/100 versus SVHN or Tiny-ImageNet and CIFAR10 versus CIFAR10-C.Source: §6.2 · p. 94

7.2 Classes of Epistemic Predictions

• Prediction forms differ across model families: Bayesian models use posterior predictive distributions, ensembles aggregate model predictions, evidential models use Dirichlet parameters, and deterministic networks output softmax point estimates.Imprecise-probabilistic models instead produce credal or random sets.Source: §7 · p. 97

7 Evaluation under Uncertainty

• No common evaluation setting covers the diverse prediction types, making consistent evaluation or ranking of uncertainty models by predictive performance difficult.The chapter reviews evaluation metrics and the forms of predictions produced by uncertainty-aware models.Source: §7 · p. 97

7.1 Current Metrics For Evaluation

• Uncertainty metrics such as Bayesian-model mutual information and ensemble predictive variance are not directly comparable because they use different ranges and model-specific definitions.The cited discussion distinguishes aleatoric uncertainty from epistemic uncertainty arising from limited knowledge of the data distribution or model parameters.Source: §7.1 · p. 97

7.2. Classes of Epistemic Predictions

• The proposed model-selection approach evaluates a trade-off between accuracy and imprecision rather than relying solely on test accuracy.The passage describes this as an approach for selecting uncertainty-aware models.Source: §7.2 · p. 98

7.2 Classes of Epistemic Predictions

• Bayesian model averaging approximates a predictive distribution by averaging samples, but can smooth individual predictions; the proposed evaluation also uses prediction samples before averaging.The passage says this smoothing can dilute uncertainty present in individual models.Source: §7.2 · pp. 98–99
• EDL predicts Dirichlet-distribution parameters and derives a pointwise prediction, while DDU detects in- versus out-of-distribution inputs but outputs softmax probabilities.The proposed evaluation considers individual EDL prediction samples because averaging may not be optimal.Source: §7.2 · p. 99
• Credal models encode class-probability lower and upper bounds, while belief-function models use the belief function’s pignistic probability as their predictive distribution.Credal sets are represented by extremal points; their number can vary with class-set size and prediction complexity.Source: §7.2 · p. 99
• The unified evaluation framework maps the predictive distributions of uncertainty-aware classifiers to a credal set.The chapter depicts the considered prediction types in a probability simplex.Source: §7.2 · pp. 98–99, Figure 7.1 · p. 98

8.1 Key Contributions

• The framework ranks uncertainty-aware predictions by jointly assessing distance from the ground truth and prediction imprecision, supporting model selection across model types.Its metric reduces to the standard KL divergence for point-wise predictions.Source: §8.1 · p. 100, §8.3 · p. 104

8.2 Methodology: Mapping Predictions To Credal Sets

• Bayesian, ensemble, and evidential predictions are mapped to credal sets using lower probabilities; multiple model outputs are used before averaging to retain prediction information.Credal and belief-function models output credal sets directly, while traditional networks and Bayesian BMA yield singleton sets.Source: §8.2 · p. 101, §8.2.2 · p. 102

8.2.2 Computing Lower Probabilities From a Sample of Probability Vectors

• For large label spaces, the method computes lower probabilities only for a budgeted set of relevant class subsets rather than all 2^N subsets.Source: §8.2.2 · p. 102

8.2.3 Computing Masses By Moebius Inverse

• The method obtains mass functions by Möbius inversion, setting negative masses to zero and assigning the remaining mass to the universal label set to ensure normalization.Source: §8.2.3 · pp. 102–103

8.2.4 Computing the Credal Set of Predictions

• To approximate the credal set, the method places each class first and last in permutations, producing 2 · N vertices instead of computing all permutations.The approximation is proposed as an effective alternative for datasets with many classes.Source: §8.2.4 · p. 103

8.3.1 The Metric

• The evaluation metric combines distance to the closest credal-set vertex with λ-weighted non-specificity; for precise point-wise predictions, it reduces to classical KL.The closest-vertex distance is a tractable proxy for distance to the credal-set boundary, and λ controls the relative importance of imprecision.Source: §8.3.1 · p. 104, §8.3.1.1 · p. 105, §8.3.1.2 · p. 106

8.3.1.2 Non-Specificity

• Non-specificity measures imprecision and epistemic uncertainty: it increases with the size of focal sets and the predicted credal set.Source: §8.3.1.2 · p. 106

8.3.3 Model Selection and Scenarios

• Model selection minimizes E, with λ chosen to match application needs, such as allowing abstention for manual review or requiring decisive predictions for autonomous driving.The framework is intended to balance predictive accuracy and precision according to task requirements.Source: §8 · p. 100, §8.3.3 · p. 107
• Models are selected by minimizing E = KL(y||ŷK) + λ·NS[m], balancing predictive fit against prediction non-specificity.The procedure computes non-specificity on the test set and selects the model with the lowest E.Source: §8.3.3 · p. 107
• Low λ favors KL divergence when crop-disease classification permits abstention, whereas higher λ penalizes non-specificity when autonomous driving requires decisive predictions.The approach is presented as tailoring reliability to the decision context rather than selecting solely by accuracy.Source: §8.3.3 · pp. 107–108

8.3.4 Practical utility of the Evaluation metric

• Practitioners choose λ by considering whether abstention is allowed, whether precision or accuracy matters more, and the risks of incorrect predictions or abstention.The section contrasts crop-disease classification, where abstention is allowed, with autonomous driving, where it is not.Source: §8.3.4 · p. 108
• A toy example compares evaluation metric E for models A, B, C, and D under different λ values.The provided passage identifies the table's scope but does not include its values.Source: Table 8.1 · p. 108, §8.3.4 · p. 108

• When abstention is allowed, low λ gives little weight to non-specificity and favors low-KL predictions closer to the ground truth.Source: §1 · p. 109
• The metric is used only for model selection; selected models retain their inherent uncertainty predictions without λ-based rescaling.Source: §1 · p. 109
• High λ penalizes uncertainty, leading to selection of more precise predictions and limiting the benefit of abstention.Source: §1 · p. 109

2. Autonomous driving

• In autonomous driving, where abstention can cause accidents, high λ prioritizes low non-specificity: Model D ranks first at λ = 2, while Model B ranks first at λ = 0.1.This prioritizes decisive predictions when a decision must be made, rather than selecting an imprecise model.Source: §2 · p. 109

8.3.5 Evaluation Metric or Uncertainty Measure?

• KL divergence reflects prediction accuracy rather than uncertainty, whereas non-specificity tracks epistemic uncertainty through credal-set size.Together with λ, the terms provide a holistic assessment of predictions.Source: §8.3.5 · pp. 109–110

8.4.2 Analysis of the Evaluation Metric

• DE leads test accuracy on CIFAR-10 and CIFAR-100, while RS-NN has the lowest E on CIFAR-100 and highest MNIST accuracy; RS-NN and LB-BNN Avg tie for lowest MNIST E.The comparison uses trade-off λ = 1.Source: §8.4.2 · p. 111, Table 8.2 · p. 112

8.4.3 Evaluation of the Trade-off Parameter

• As λ grows, E increases for LB-BNN, DE, and CreINN, while EDL and E-CNN remain highly imprecise; CIFAR-100 also shows increasing E across models.On MNIST, most models instead show relatively consistent E across λ values.Source: §8.4.3 · p. 115

8.4.4 Model Selection Using the Evaluation Metric

• DE ranks first for most λ values, but RS-NN ranks first at λ = 1; higher λ favors RS-NN and LB-BNN, while lower λ favors DE and CreINN.The ranking reflects the task-dependent trade-off between tolerating imprecision and penalizing it.Source: Table 8.4 · p. 116, §8.4.4 · pp. 116–117

8.4.5.1 Metric Behavior For Correct (CC) Vs. Incorrect Classifications (ICC)

• Incorrect predictions generally have higher KL, non-specificity, and E than correct predictions, but E-CNN has high non-specificity for both.E alone can obscure whether a low score reflects confident error or uncertainty, so it should be read alongside KL and non-specificity.Source: §8.4.5.1 · p. 118, Table 8.7 · p. 118

8.4.5.2 The Effect of Approximating Credal Set Vertices

• For DE, the approximated credal set using 8 vertices can represent the metric better than the naive set using all 24 vertices.The approximation retains the most relevant permutations.Source: §8.4.5.2 · p. 119, Table 8.8 · p. 119
• E-CNN produces the widest credal sets across predictions, indicating imprecision whether its predictions are correct or incorrect.Larger credal sets indicate greater uncertainty; smaller sets align with lower non-specificity and greater confidence.Source: §8.4.5.2 · pp. 119–120, Figure 8.7 · p. 120, Figure 8.8 · p. 120

8.4.5.3 Ablation Study On Different Distance Measures

• KL divergence for incorrect predictions often spans a wider range than for correct ones, while JS is bounded and operates on a smaller scale.KL’s greater sensitivity to mismatches supports its selection for distinguishing correct and incorrect predictions.Source: §8.4.5.3 · pp. 121, 123, 124, Figure 8.9 · p. 121, Figure 8.10 · p. 122
• Excluding point-prediction models, KL and JS both rank RS-NN and DE among the top two; RS-NN often leads under JS, while DE Avg replaces DE as λ increases under KL.Rankings vary with λ because KL and JS have different ranges, so the JS contribution should be scaled accordingly.Source: §8.4.5.3 · pp. 121, 123, Table 8.9 · p. 122

8.4.5.4 Ablation Study On Different Non-specificity Measures

• Non-specificity generally distinguishes correct from incorrect predictions better than credal uncertainty, which remains elevated in both cases; E-CNN is an exception to the broader trend.Without point-prediction models, DE and RS-NN rank best under non-specificity, while DE and E-CNN rank best under credal uncertainty.Source: §8.4.5.4 · pp. 124, 126, Table 8.10 · p. 126

8.4.5.5 Distance Measure Vs. Confidence Score

• Distance-based evaluation captures prediction spread and overconfidence in wrong predictions, whereas entropy closely tracks confidence and does not isolate epistemic uncertainty.Non-specificity shows little correlation with confidence, unlike entropy.Source: §8.4.5.5 · pp. 126–128, Figure 8.14 · p. 127

8.4.5.6 Credal Set Size Vs. Non-specificity

• Credal set size and non-specificity are directly correlated across CIFAR-10, MNIST, and CIFAR-100, supporting non-specificity as a measure of prediction uncertainty.E-CNN also produces very wide credal sets across MNIST samples.Source: Figure 8.15 · p. 128, Figure 8.16 · p. 128, §8.4.5.6 · p. 129

8.4.5.7 Ablation On Number of Prediction Samples

• Increasing posterior samples for LB-BNN or ensemble members for DE increases evaluation metric E, with a stronger effect for DE.More samples make predictive distributions more expressive, broadening credal sets and increasing non-specificity.Source: §8.4.5.7 · p. 129, Figure 8.18 · p. 130

8.5 Discussion and Implications

• The evaluation framework uses prediction information beyond accuracy, jointly assessing ground-truth distance and credal-set extent to compare uncertainty and reliability across models.Both measures operate within the probability simplex and preserve distinctions between models with similar accuracy but different uncertainty.Source: §8.5 · p. 130

9.1 Methodology

• RS-LLMs train token-by-token with teacher forcing and generate sequentially; mass on a joint token set signals uncertainty, unlike equal mass on individual plausible tokens.Pignistic probabilities are computed from the generated mass function to sample the next token.Source: §9.1 · pp. 133–134, Figure 9.1 · p. 134
• Hierarchical clustering over token embeddings budgets focal sets, reducing the vocabulary’s combinatorial subset space to K + T sets, including singleton tokens.K controls cluster granularity; the method selects clusters of similar tokens as focal sets.Source: §9.1 · p. 134, Figure 9.2 · p. 135

9.2.1 Implementation

• The evaluation uses CoQA and OBQA-Additional with Llama2-7B, Mistral-7B, and Phi-2, trained by supervised fine-tuning with 4-bit quantization and rank-64 LoRA adapters.The described datasets include 7,200 CoQA training samples and 4,957 OBQA training samples.Source: §9.2.1 · p. 135, Figure 9.3 · p. 135

9.2.2 Performance

• RS-LLMs consistently outperform standard LLMs in cosine similarity on CoQA and accuracy on OBQA under identical training conditions.The authors interpret this as support for the expressiveness of set-valued outputs.Source: §9.2.2 · p. 136, Table 9.1 · p. 136

9.2.3 Uncertainty Estimation

• Credal width correlates negatively with correctness and shows some separability on OBQA, while entropy has overlapping correct/incorrect distributions there and no obvious CoQA trend.The reported entropy patterns limit its usefulness for distinguishing correctness in these evaluations.Source: §9.2.3 · p. 136, Figure 9.4 · p. 137

9.2.4 Hallucination Evaluation

• With misleading contexts, both Llama2 models show increased uncertainty, while RS-Llama2 separates correct from incorrect contexts better via credal width.CoQA questions were swapped and OBQA answer choices randomized.Source: §9.2.4 · p. 136, Table 9.2 · p. 136

9.2.5 Budgeting Analysis

• The budgeted focal sets group semantically close tokens; centroid distances peak near zero, and sets of sizes 2, 3, and 4 are most frequent.Their reported counts are 2,142, 1,940, and 1,562, respectively; sets larger than 30 are excluded from the figure’s visualization.Source: §9.2.5 · p. 137, Figure 9.5 · p. 138

9.2.6 Ablation Studies

• Cosine similarity is best at α = β values 0.01 and 0.0001; the authors choose 0.01 for stronger validity enforcement, and report best CoQA results at K = 8000.They note that small budgets behave like classical LLMs, while large budgets add complexity.Source: §9.2.6 · pp. 137–138, Table 9.3 · p. 138, Table 9.4 · p. 138

10.1 Experimental Setup For Weather/cone Classification

• The chapter evaluates RS-NN, a standard CNN, and LB-BNN on weather and cone datasets; it states that this application requires no changes to RS-NN.The datasets include ROAD-WAYMO, OBR-A, and a cone-centric dataset; the dataset overview summarizes their sources, classes, and sizes.Source: §10 · p. 140, §10.1 · pp. 140–141, Figure 10.1 · p. 141

10.2 Cone Classification

• For one CONE sample, the pignistic probability of blue is 1.0 and entropy is 1.52 × 10^-18; a second sample has yellow and blue probabilities of 0.506 and 0.391, with entropy 1.3658.The chapter characterizes the first prediction as highly confident and the second as uncertain.Source: §10.2 · p. 141, Table 10.1 · p. 141

10.3 Weather Classification

• Across R-WAYMO, OBR-A, and CONE, RS-NN has the highest reported accuracy and favorable uncertainty metrics, while CNN has higher uncertainty on incorrect predictions.The chapter also reports that RS-NN outperforms its counterparts on OBR-A accuracy and uncertainty behavior.Source: §10.3 · p. 142, Table 10.2 · p. 142

10.3.1 Domain Adaptation Setting

• 75.51% test accuracy on shared R-Waymo classes was reported for RS-NN, versus 63.78% for CNN and 61.83% for LB-BNN after training on OBR-A.The chapter also reports that RS-NN had the best generalization in both directions across the OBR-A and R-WAYMO adaptation tasks.Source: §10.3.1 · p. 143, Table 10.3 · p. 144
• For unseen R-Waymo classes, RS-NN uncertainty remains high, whereas LB-BNN entropy is predominantly lower; the chapter reports RS-NN as stronger under domain shift.Training used 3,468 OBR-A images and testing used 39,987 R-Waymo images.Source: §10.3.1 · p. 143, Figure 10.2 · p. 143, §10.4 · p. 144

10.4 Discussion

• The chapter concludes that RS-NN improves predictive reliability in autonomous-driving perception tasks, including under uncertainty and domain shift.The conclusion summarizes results across cone and weather classification against CNN and LB-BNN baselines.Source: §10.4 · p. 144

11.1 Thesis Summary

• RS-NNs predict belief functions over class sets, while budgeting selects informative focal sets to reduce combinatorial complexity and support large-scale use.The unified framework maps diverse uncertainty outputs to credal sets and evaluates them using KL divergence and non-specificity, without declaring a universally superior model.Source: §11.1 · p. 145

11.2. Impact on the Epistemic AI paradigm

• RS-NNs outperformed Bayesian and ensemble baselines by up to 4% on CIFAR-10 and competitors by as much as 7% on ImageNet.The thesis also reports notable improvements in accuracy, robustness, uncertainty quantification, and OoD detection on large-scale benchmarks.Source: §11.2 · pp. 146–147
• RS-LLMs improved token-level prediction accuracy by up to 7%, while weather-classification RS-NNs achieved up to 40% better accuracy under domain shifts.For unseen R-WAYMO classes, RS-NNs maintained high entropy and wide credal widths, outperforming LB-BNN.Source: §11.2 · p. 146
• RS-NNs achieved up to 6% higher OoD detection performance, with a CIFAR-10–SVHN entropy gap of 0.45 and credal widths of 0.007 for iD versus 0.260 for OoD.The thesis also reports up to 20% better calibration than Bayesian models and ECE below 0.05 under dataset shifts.Source: §11.2 · p. 146
• RS-NNs maintained up to 50% higher accuracy than baselines under FGSM and PGD attacks, with inference at approximately 1.91 ms per sample.The reported inference times were 340.25 ms/sample for FSVI and 13,163.50 ms/sample for Deep Ensembles.Source: §11.2 · p. 146

11.3 Limitations

• The evaluation metric’s ranking depends on its mapping, distance, non-specificity, and linear-combination choices; E-CNN training on CIFAR-100 proved infeasible.RS-NN budgeting can be time-consuming on very large datasets, although it is a one-time pre-training procedure for a given dataset.Source: §11.3 · pp. 148–149

11.4 Broader Challenges Within Epistemic Deep Learning

• The thesis focuses primarily on static training settings, and a unified framework linking continual learning with second-order epistemic uncertainty remains largely unexplored.It also identifies foundation-model scale as an open question for epistemic representations.Source: §11.4 · p. 149

11.4. Broader challenges within Epistemic Deep Learning

• Epistemic models do not yet provide frequentist statistical guarantees, while scaling random-set methods to foundation models will require new focal-set management and uncertainty-propagation methods.The thesis notes that RS-NNs can be combined with conformal prediction for distribution-free coverage guarantees.Source: §11.4 · p. 150

Advancing Random-Set Learning

• Future work proposes extending random-set learning to multimodal generation, including diffusion models that represent images as discrete tokens.High-resolution images remain a challenge for this direction.Source: §12 · p. 151
• RS-NNs could support regression, including object detection, by predicting Dirichlet distributions over Borel closed intervals for bounding-box coordinates.Class labels could remain set-valued, representing uncertainty in both spatial and categorical outputs.Source: §12 · p. 151

Towards a More Unified Evaluation Framework

• The evaluation framework could become a training objective and be extended to structured outputs, multilabel classification, and real-world decision tasks.Further analysis is proposed to assess model performance without overly coarse aggregations.Source: §12 · p. 151

Towards an Epistemic Generative AI

• RS-LLMs predict belief functions over vocabulary tokens, with hierarchical embeddings for focal sets and belief-based sampling for generation.The passage highlights potential usefulness for polysemous or synonym-rich languages such as Japanese or Arabic.Source: §12 · p. 152

Opportunities in Science and Engineering

• How epistemic uncertainty can be quantified in neural operators’ functional spaces remains an open question amid limited data and PDE misspecification.The passage also notes that conformal prediction depends on costly calibration data.Source: §12 · p. 152
• For climate forecasting, the thesis asks how models trained on insufficient, sparse data can quantify epistemic uncertainty to avoid forecasting errors.It links this question to dynamically evolving atmosphere, ocean, and land systems and the challenge of reliable long-term forecasts.Source: §12 · p. 152

Final Remarks

• The thesis argues uncertainty quantification should be embedded in model architecture and communicated in ways that support human trust and decision-making.It identifies interpretability, human-computer interaction, and ethical considerations as relevant to high-stakes use.Source: §12 · p. 153

Towards an Epistemic Generative AI

• Future work must examine how input properties, training dynamics, and architecture shape uncertainty predictions in LLMs and instruction-tuned vision-language models.The passage states that the transfer of epistemic AI to instruction-fine-tuned VLMs remains open.Source: §12 · p. 153

A.0.1 Algorithm For Budgeting

• Budgeting avoids generating all 2N class subsets by selecting K relevant focal sets from reduced features and choosing subsets with the highest overlap ratios.The procedure fits class-specific GMMs, constructs ellipsoids covering 95% of data, and uses early stopping when larger subset cardinalities no longer change the top-overlapping sets.Source: §A.0.1 · p. 156

A.0.2 Algorithm For ECE

• The appendix defines ECE by comparing average confidence and accuracy within each bin, weighting each difference by the bin’s share of instances, and summing the weighted differences.The steps are given as the ECE calculation procedure.Source: §A.0.2 · p. 156

22: Combine selected non-singleton focal sets with N singleton sets

• RS-NN exhibits the lowest ECE in Table 5.5.ECE weights the absolute confidence–accuracy difference in each bin by that bin’s share of instances, then sums across bins.Source: §A.0.2 · p. 156, §22 · p. 157

A.0.3 Algorithm For AUROC, AUPRC

• The AUROC and AUPRC algorithm takes uncertainty scores for in-distribution and OoD samples and outputs ROC and precision-recall metrics.The appendix identifies AUROC and AUPRC as the areas under their respective curves.Source: §A.0.3 · p. 158
• RS-NN’s curve outperforms the other models on AUROC and has the highest precision-recall curve in the cited comparisons.The appendix describes CIFAR-10 and MNIST plots and states that these results show discrimination between in-distribution and OoD samples.Source: §A.0.3 · p. 159, Figure A.1 · p. 159, Figure A.2 · p. 159

B Statistical guarantees

• Inductive conformal learning uses RS-NN’s predictive belief function to score candidate labels and returns labels whose calibration-score empirical CDF meets the chosen significance level.The proposed non-conformity score treats a label as non-conformal when its predicted plausibility is low.Source: B Statistical guarantees · p. 160
• A calibration threshold set at the 1 − α percentile, with α = 0.05, is used to target 95% coverage on CIFAR-10.The calibration split contains 1,000 of the 60,000 listed data points.Source: B Statistical guarantees · p. 160, Figure B.2 · p. 161
• Coverage averages across classes exceed the desired threshold, while average prediction-set sizes are considerably larger for classes 1, 2, 3, 4, and 6.The reported figures show per-class coverage and average set size; the text relates set-size variability to adapting to sample difficulty.Source: Figure B.3 · p. 161, Figure B.4 · p. 161, Appendix B · p. 162

Appendix B. Statistical guarantees 139

• The authors plan to investigate generalizing the empirical CDF used in conformal learning to belief-function or random-set representations while retaining statistical guarantees.They describe this work as ongoing in a separate paper.Source: Appendix B · p. 162