Epistemic Deep Learning

1 Introduction

• Predictions can be brittle when test data differ from training data, while traditional machine learning is described as unable to address this non-incrementally.The introduction gives heavy rain and other drivers’ styles as examples of changed conditions.Source: §1 · p. 1
• Random sets assign probabilities directly to outcome sets, and belief functions provide their finite-target form for classification and a Dempster-Shafer framework for combining evidence.Source: §1 · p. 2

1.1 Contributions

• The paper contributes an epistemic deep-learning framework, a baseline epistemic CNN for multi-label image classification, and loss functions generalizing KL divergence and entropy.Source: §1.1 · p. 2
• It also proposes an approach to assessing epistemic predictions that includes Bayesian and traditional deep learning as special cases.Source: §1.1 · p. 2

1.2 Paper outline

• The paper reviews uncertainty modelling, develops epistemic deep learning and random-set CNNs, derives losses and evaluation methods, then reports experiments and conclusions.Source: §1.2 · pp. 2–3

2 State of the art

• The review groups uncertainty-quantification techniques into set-wise learning, Bayesian inference, evidential learning, and conformal prediction.Source: §2 · p. 3

2.1 Set-wise learning

• Set-wise approaches include credal classifiers, set-valued predictions, and minimax domain adaptation; related work also spans evidential classifiers, trees, forests, ensembles, and regression.Source: §2.1 · pp. 3–4

2.2 Bayesian deep learning

• Bayesian deep learning estimates predictive uncertainty through posterior inference, approximated with methods such as MCMC, variational inference, dropout, or ensembles.Source: §2.2 · p. 4

2.3 Evidential approaches

• The review describes evidential approaches including Dempster-Shafer CNNs that assign utilities to mass functions and classifiers using a Dirichlet representation.Source: §2.3 · p. 4

2.4 Conformal prediction

• Conformal prediction forms prediction sets by thresholding error, but the review says these uncertainty-theory efforts have had limited impact, possibly for lack of a solid underlying principle.Source: §2.4 · p. 5

3 Epistemic deep learning

• Epistemic deep learning uses random sets and belief functions to model uncertainty associated with limited training data by producing outcomes for selected target subsets.Source: §3 · p. 5

3.1.1 Epistemic artificial intelligence

• Unlike traditional learning from limited evidence, epistemic AI begins from near-ignorance and tempers it with evidence, seeking compatible sets of hypotheses rather than individual models.A set of models can provide robust prediction sets from which a cautious prediction may be chosen.Source: §3.1.1 · pp. 5–6, Figure 1 · p. 6

3.1.2 Target-level vs parameter-level representation

• Target-level models output uncertainty measures on the target space with deterministic weights, while parameter-level models represent uncertainty over the weights.The paper focuses on target-level representations while describing parameter-level modelling as the framework’s full generality.Source: §3.1.2 · p. 6

3.1.3 Training objectives

• Target-level training compares uncertainty-valued predictions with uncertainty-valued ground truth using a loss that depends on a deterministic weight assignment.Parameter-level training instead seeks an uncertainty measure over weights or hypotheses; its output for a given weight configuration can resemble a classical network’s scores.Source: §3.1.3 · pp. 6–7

4 Random-set convolutional neural networks

• The proposed random-set neural network outputs scores for sets of outcomes to encode epistemic uncertainty, with this initial work focused on target-level representations.The authors describe the architecture and training objectives as a novel class of models.Source: §4 · p. 7, §4.1 · p. 8

4.1 Random sets and belief functions

• A belief function assigns belief to a decision-space event from source outcomes whose compatible outcome sets fall within that event.A multivalued mapping Γ from the source space to subsets of the decision space, together with a source probability measure, induces the belief function.Source: §4.1 · p. 7
• Classical probability measures are a special case of belief functions that assign mass only to singletons.A basic probability assignment distributes mass across subsets of the frame of discernment; subsets with nonzero mass are focal elements.Source: §4.1 · pp. 7–8

4.2 Architecture

• Set-valued classification maps inputs to subsets of classes, and the architecture is presented as a random-set convolutional neural network.Figure 2 is identified as the random-set convolutional neural network.Source: §4.2 · p. 8, Figure 2 · p. 8
• Training encodes each example’s ground truth with either mass values or belief values for subsets of the class set.The naive output has 2^N scores for N classes, one per subset; the corresponding target vector encodes masses or beliefs.Source: §4.2 · pp. 8–9

4.3 Training

• Belief-value targets sum the masses of subsets, giving the network set structure to learn; mass-value targets resemble padded traditional targets and allow standard cross-entropy.Set-valued ground truths are intended to help the model learn set structure and estimate uncertainty.Source: §4.3 · p. 9
• Because the network predicts scores for sets of outcomes, training requires a loss that measures the difference between set-valued predictions and ground truth.The section frames selecting suitable loss functions as the training question for epistemic neural networks.Source: §4.3 · p. 9

• Traditional loss functions such as cross-entropy cannot serve directly as objectives for networks that output scores for sets of outcomes.The predictions and ground truth are set-valued.Source: §5 · p. 9
• One route to generalizing cross-entropy is to use suitable entropy measures for belief functions.Source: §5 · p. 9
• A second route is to generalize Kullback-Leibler divergence to belief functions.Source: §5 · p. 9

5.1.1 KL divergence in the random set setting

• The proposed extension treats a belief function as induced by a source probability through a multivalued mapping from the source space to subsets of the decision space.The true and predicted mass functions are represented as images of source probabilities P and Q under Γ.Source: §5.1.1 · p. 10, Figure 3 · p. 10

5.1.2 Approximating the KL divergence of the source probabilities

• Mass-function KL divergence equals source-probability KL divergence when each decision-space subset is the image of a unique source point.The authors describe mass-function KL divergence as a reasonable general approximation to source-probability KL divergence.Source: §5.1.2 · pp. 10–11
• The relationship for belief-function KL divergence is more complicated because belief values have a combinatorial structure.The authors are working toward theoretical support for candidate extensions and analytical optimization solutions under additional constraints.Source: §5.1.2 · p. 11

5.2 Generalised entropy measures

• A suitable belief-function entropy candidate should reduce to Shannon entropy on probability measures, and the section considers several generalized measures.Source: §5.2 · p. 11

5.2.1 Nguyen’s entropy

• Nguyen’s entropy replaces probability values with mass values, and minimizing it produces satisfactory training-loss metrics in random-set CNNs.The measure is defined over the focal elements of the mass function.Source: §5.2.1 · p. 11

5.2.2 Hohle’s entropy

• Hohle’s measure of confusion replaces predictive probabilities with belief values.Source: §5.2.2 · p. 11

5.2.3 Yager’s entropy

• Yager’s entropy replaces predictive probabilities with plausibility values.Source: §5.2.3 · p. 11

5.2.4 Klir and Ramer’s global uncertainty measure

• Klir and Ramer’s global uncertainty measure combines non-specificity with discord, a measure of conflict.Its first component measures conflict and its second measures non-specificity.Source: §5.2.4 · p. 12

5.2.5 Desired properties of entropy for belief functions

• Nguyen’s, Hohle’s, and Yager’s measures capture only conflict and therefore fail non-negativity, maximum entropy, and monotonicity; Klir and Ramer’s measure violates maximum entropy but satisfies the other listed properties.The other listed properties are consistency with Dempster-Shafer theory semantics, probability consistency, and additivity.Source: §5.2.5 · pp. 12–13

5.3 Distance measures

• The distance-loss approach uses a belief-function distance rather than generalizing KL divergence.The paper considers Jousselme’s distance as one such approach.Source: §5.3 · p. 13

5.3.1 Jousselme’s distance

• Jousselme’s distance measures two basic probability assignments on the same frame using similarities among subsets.Its associated matrix is called the similarity measurement matrix.Source: §5.3.1 · p. 13

6.1.1 Traditional deep learning

• Traditional classification evaluation extracts a maximum-likelihood class, then computes accuracy and average precision from point predictions.A suitable evaluation strategy is needed for fair comparison with traditional and Bayesian methods.Source: §6.1.1 · p. 13

6.1.2 Bayesian deep learning

• Bayesian networks form predictive distributions through weighted model averaging and are commonly evaluated for predictive performance and calibration.The paper argues that current Bayesian metrics do not test the true nature of epistemic predictions.Source: §6.1.2 · p. 14

6.2 Performance evaluation in epistemic deep learning

• Epistemic performance is assessed by distance between a credal-set prediction and one-hot ground truth in the probability simplex.Prediction size also matters for balancing accuracy and precision.Source: §6.2 · p. 14, Figure 4 · p. 14

6.2.1 Vertices of a random set prediction

• For belief-function predictions, evaluation first requires computing the vertices of the credal set consistent with the predicted belief function.A belief function identifies probabilities satisfying P(A) ≥ Bel(A); its associated credal-set vertices have a permutation-based form.Source: §6.2.1 · p. 15
• Equation 12 derives the finite set of vertices, allowing distances between ground truth and a random-set prediction to be computed from those vertices.Because a convex set can be handled using its vertices, the distance is a function of them.Source: §6.2.1 · p. 15

Pima Indian Diabetes

• 72.9% test accuracy for the random-set network with mass-value ground truth compared with 73.4% for the traditional network and 67.7% for the Bayesian network.The Pima Indians Diabetes experiment used set-valued classes for binary classification.Source: Pima Indian Diabetes · pp. 15–16

MNIST

• 99.95% test accuracy for the random-set CNN on MNIST, compared with 99.87% for the traditional CNN and 98.28% for the Bayesian CNN.The experiment trained on four of the ten MNIST classes using mass values for subsets as labels.Source: MNIST · p. 16

Intel Image Classification

• 87.63% test accuracy for the random-set CNN on Intel Image Classification, compared with 84.70% for the traditional CNN and 73.62% for the Bayesian CNN.The experiment used four of the dataset’s six classes.Source: Intel Image Classification · p. 16

7.3 Results

• Random-set CNNs produced better test accuracies and test losses than traditional and Bayesian networks in most reported datasets.Table 1 compares test accuracies for traditional, Bayesian, and epistemic CNNs.Source: Table 1 · p. 17, §7.3 · p. 18

CIFAR10

• 85.90% test accuracy for the random-set CNN on CIFAR-10, compared with 84.95% for the traditional CNN and 69.74% for the Bayesian CNN.The experiment considered four of CIFAR-10’s ten classes.Source: CIFAR10 · p. 17

7.2.1 Kullback-Leibler divergence

• 99.92% test accuracy with KL-divergence loss for the random-set CNN, compared with 99.87% for a traditional CNN on MNIST.The reported result is attributed to the mass-function approximation of KL divergence.Source: §7.2.1 · p. 17, §7.2 · p. 17

7.2.2 Entropy measures

• Nguyen’s entropy, described as a generalization of Shannon entropy, yielded 99.85% test accuracy and 0.0017% test loss on MNIST.The authors note that mass-function encoding resembles the original target vector with zeros for non-singleton sets.Source: §7.2.2 · p. 17

7.3 Results

• The authors report that the power-set representation increases network complexity exponentially and limit these experiments to a few classes.Experiments with Hohle’s measure, Yager’s entropy, and Jousselme’s distance were ongoing.Source: §7.3 · p. 18, §7.2.2 · p. 18

8 Conclusions

• The method produces better classification test accuracies than current state-of-the-art uncertainty-estimation neural networks and is comparable to traditional networks.Source: §8 · p. 18
• The work discusses only target-level random-set CNN representations; parameter-level representations and detection tasks are ongoing extensions.Detection also requires suitable localisation losses consistent with the epistemic uncertainty principle.Source: §8 · p. 18
• Classical single-class ground truths make learning random sets and their 2^N subset structure challenging, while random-set networks require distinct performance metrics and a generalized evaluation framework.Source: §8 · p. 18