# Evidential Deep Learning is not Evidential Learning: A
Clear Distinction

Arthur Hoarau

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# Evidential Deep Learning is not Evidential
Learning: A Clear Distinction

Arthur Hoarau¹²

¹ CentraleSupélec, Université Paris-Saclay, Metz, France

² Loria, CNRS, Université de Lorraine, Nancy, France

**Abstract.** A large portion of the recent literature on uncertainty quantification has focused on second-order risk minimization, renaming it Evidential Deep Learning. We argue that this naming is misleading, as the naming of the evidential framework has already been established by the Mathematical Theory of Evidence, which has been applied to numerous problems for decades and deep evidential learning was defined long before second-order risk minimization. However, this naming overlap has given rise to particularly interesting ideas for (evidential learning in the sense of) second-order risk minimization, which is applied here both theoretically and empirically to (evidential learning in the sense of) Dempster-Shafer Theory, establishing a rule of monotonicity for epistemic uncertainty as essential to evidential learning. This rule is proven to be true in this paper for Dempster-Shafer Theory, whereas it has been proven impossible for second-order risk minimization. As a result, our work clarifies the distinctions between these approaches while contributing new insights to the broader discussion on uncertainty quantification.

**Keywords:** Machine Learning · Uncertainty Quantification

## 1 Introduction

The intersection between machine learning and uncertainty quantification has been gaining momentum for some time. Being able to assign a reliability (or an uncertainty) to a model's prediction is indeed a valuable objective. Two types of uncertainties are widely distinguished in the literature [18]: aleatoric and epistemic. Aleatoric uncertainty stems from the stochastic nature of the event (i.e., the data-generating process) and is therefore irreducible within the same learning boundaries, whereas epistemic uncertainty corresponds to a lack of knowledge and can be reduced by acquiring more observations. This distinction has profound implications for the design of models and methods in machine learning, as it dictates how uncertainty is quantified and addressed in practical applications.

In 2018, a new method is introduced by [30], which allows for the estimation of epistemic uncertainty in the context of deep learning, addressing the issue of the model's ability to represent its ignorance rather than experiencing wrong confidence. The authors elaborate on the Mathematical Theory of Evidence, from which they draw inspiration, to propose a second-order risk minimization strategy. This approach aims to minimize both the error of the model
and its epistemic uncertainty, which is represented by a second-order probability. This work was subsequently widely adopted by several authors [35, 1, 28, 6, 25, 2], and the name of this application domain became recognized as Evidential Deep Learning.

Two decades earlier, the evidential framework [9,31], also known as Dempster-Shafer theory or the theory of belief functions, was applied to machine learning [11] using a K-Nearest Neighbors architecture based on Dempster's combination rule. This approach treated observations as sources of evidence and was applied to a wide range of real-world applications [15,27,7]. Five years later, the same framework was extended to neural networks [10], eventually leading to a proliferation of evidential and deep evidential models based on Dempster-Shafer theory [17,14,8,36,5,26,37]. It should also be noted that the application of the evidential framework to the field of artificial intelligence dates back to before 1995 [32], as retraced by the author of the theory himself, 40 years later.

However, the issue of overlapping nomenclature between these two domains is not solely due to its anachronism. The confusion was initially induced by the fact that a belief function can be interpreted as a Dirichlet distribution [19]. Therefore, second-order risk minimization can theoretically lead to a belief function. However, unlike credal sets, evidential framework, second-order probabilities (which include second-order risk minimization), and possibility theory, all of which are frameworks for representing uncertainty, second-order risk minimization is a method (and not a theory) for quantifying a second-order probability, fundamentally antagonistic to the theory of evidence. This frequentist approach does not align with Shafer's view that "the idea of chance and the idea of belief have been brought together under the same term of probability" [31] proposing the belief function to exclusively represent epistemic uncertainty in a more Bayesian than frequentist interpretation. Moreover, this method does not utilize any rule of combination such as Dempster's rule [9], which is at the core of the evidential framework.

This specific naming has led survey papers to use the term "Evidential Deep Learning", which can give the impression (especially to newcomers) that evidential and deep evidential learning did not exist before 2018. Our contribution includes highlighting the risks of such a naming convention, and therefore, we advocate for renaming the so-called approach (when epistemic regularization is incorporated directly into the loss) simply to *second-order risk minimization*, following the formalism of [20], as this is what it truly represents.

Nevertheless, this overlap in nomenclature has led to a plethora of criticisms regarding second-order risk minimization. Valuable insights can be drawn from these critiques and applied to Dempster-Shafer Theory. As part of this critique, the authors in [3] propose to formalize appropriate properties for second-order risk minimization with the following two rules when obtained through loss minimization: (i) The epistemic uncertainty of a second-order distribution should be monotonic with respect to an increase in sample size. (ii) The second-order distribution should converge to a Dirac delta function as the sample size tends to infinity. Several works have proven that this expected behavior is impossible
to achieve [4,3,20,29], with the general conclusion being that second-order risk minimization is unsuitable for epistemic uncertainty quantification<fnref n="3" />.

The mistake to avoid is generalizing these results to (evidential learning in the sense of) Dempster-Shafer Theory (DST). We therefore propose in this paper to generalize these two particularly relevant properties to DST and to demonstrate, both theoretically and experimentally, that they are indeed respected. More importantly, it would be relevant in machine learning and uncertainty quantification to define a proper evidential model as one that adheres to these rules. We will also show that there are several models that satisfy these properties and are already used in practice, highlighting a framework that truly allows for properly modeling epistemic uncertainty. It should be noted that this paper is not a critique of second-order risk minimization, but rather aims to highlight that some of the criticisms directed at this family are not (theoretically and practically) applicable to DST.

The paper is structured as follows: Section 2 introduces DST and presents the difference with second-order risk minimization, while also listing the risks associated with improper nomenclature concerning these two topics. Section 3 theoretically extends the expected rules to the current DST-evidential framework through a proof of the monotonicity of epistemic uncertainty. Section 4 experimentally demonstrates that there are already existing models in use that comply with these rules; we show application cases in both machine learning and deep learning. Finally, Section 5 concludes the article.

# 2 Evidential learning and Deep Evidential learning

## 2.1 Evidential framework - Dempster-Shafer theory

The theory of belief functions, also known as Dempster-Shafer theory, was introduced by Dempster [9] then was further developed and renamed as the “Theory of Evidence” by Shafer [31]. This theory extends the theory of probabilities by focusing on the notion of belief rather than chance.

The frame of discernment, denoted by $\Omega$, is the set of C exclusive hypotheses such that $\Omega = \{\omega_1, \dots, \omega_C\}$. The power set of $\Omega$, denoted by $2^\Omega$, represents the set of all subsets of $\Omega$: $2^\Omega = \{\emptyset, \{\omega_1\}, \{\omega_2\}, \{\omega_1, \omega_2\}, \dots, \Omega\}$. A source s can then assign a belief to any element of $2^\Omega$.

The function $m: 2^\Omega \to [0, 1]$, which assigns belief (or mass) to each element of $2^\Omega$, is called the mass function and always satisfies the following normalization condition: $\sum_{A \in 2^\Omega} m(A) = 1$. Non-zero mass elements are called focal elements, and the union of focal elements forms the core. A vacuous mass function assigns a mass of 1 to the frame $\Omega$, representing complete ignorance. A mass function with only one focal element is referred to as a categorical mass function. A mass function with all its focal elements concentrated on singletons is referred to as a

<footnote n="3">³ Recent work [34] has proposed addressing some of these issues within the framework of second-order risk minimization. However, this lies outside the scope of our paper, which focuses on evidential learning in the sense of DST.</footnote>
Bayesian mass function (special case of a probability function). A quasi-Bayesian mass function, on the other hand, has its focal elements on singletons as well as on the frame $\Omega$ (i.e., partial ignorance).

**Example 1.** Consider the toss of a coin that may or not be fair. The two events $h$: “the coin lands on heads” and $t$: “the coin lands on tails” form a partition of the frame $\Omega = \{h, t\}$. Two sources $s_1$ and $s_2$ express their beliefs about this coin. Source $s_1$ states “I don’t know whether the coin is biased or not” with $m_1$: $m_1(\{h\}) = 0, m_1(\{t\}) = 0, m_1(\{h, t\}) = 1$, while source $s_2$ expresses “I know that the coin is not biased” with $m_2$: $m_2(\{h\}) = 0.5, m_2(\{t\}) = 0.5, m_2(\{h, t\}) = 0$.

The concepts of upper and lower probabilities, referred to as plausibility and belief respectively, serve to bound the belief of a source in an event. The belief function, denoted as Bel, represents the minimal belief of the source and is calculated as follows for any $A \subseteq \Omega$: $Bel(A) = \sum_{B \subseteq A} m(B)$. It is the sum of the elements of information that imply $A$ and represents the total belief in $A$.

The plausibility function $Pl$ is the maximum belief that could be attributed to $A$, defined as: $Pl(A) = \sum_{A \cap B \neq \emptyset} m(B)$. It is the sum of elements of information that are non-contradictory with $A$. One can relate the plausibility of an element $A$ to the belief of its complement $\bar{A}$ with $Pl(A) = 1 - Bel(\bar{A})$.

**Example 2.** Consider example 1, the belief of $s_1$ that $h$: “the coin lands on heads” can be bounded by the functions Bel and Pl, where $Bel_{s_1}(\{h\}) = 0$ and $Pl_{s_1}(\{h\}) = 1$. Therefore, its belief in the event is bounded by [0, 1]. Similarly, the belief of $s_2$ is bounded by [0.5, 0.5], since this source knows that the coin is fair, $Bel_{s_2}(\{h\}) = Pl_{s_2}(\{h\}) = 0.5$.

To capture epistemic uncertainty within the evidential framework, the Hartley entropy was extended to the evidential framework by Dubois and Prade [13]. The Non-specificity NS of a mass function $m$ is defined as:

$$NS(m) = \sum_{A \subseteq \Omega} m(A)\log_2(|A|), \qquad (1)$$

with $|A|$ the cardinality of the set $A$.

**Example 3.** Consider example 1, the Non-specificity of $m_1$ is given by $NS(m_1) = 1\log_2(2) = 1$. This represents maximal epistemic uncertainty, as $s_1$ is in a state of complete ignorance. The Non-specificity of $m_2$ is $NS(m_2) = 0.5\log_2(1) + 0.5\log_2(1) = 0$, meaning the epistemic uncertainty is zero, since $m_2$ corresponds to a purely aleatoric event, of a fair coin toss.

When multiple sources express themselves, it is possible to combine two or more mass functions defined on the same frame of discernment. Several methods exist, each with its own advantages and disadvantages depending on the fusion context (cf. [24]). The conjunctive combination (Dempster’s rule) of two mass functions $m_1$ and $m_2$, defined on the same frame of discernment $\Omega$, is denoted by $m_1 \oplus m_2$ and is defined as follows:

$$m_1 \oplus m_2(A) = \frac{1}{1-\kappa} \sum_{B \cap C=A} m_1(B)m_2(C), \quad \forall A \subseteq \Omega, \qquad (2)$$
where $κ$ is the degree of conflict defined as:

$$κ = \sum_{B \cap C = \emptyset} m_1(B)m_2(C), \qquad (3)$$

where $B$ and $C$ are subsets of $Ω$. This rule combines the information from the two mass functions by aggregating the contributions of all intersecting subsets $B$ and $C$ that form $A$. In the case of more than two masses, the combination of multiple mass functions $m_i \in M$ is denoted by $\bigoplus_{m_i \in M_j} m_i(A)$.

*Example 4.* Let's consider three suspects: Alice, Bob, and Eve, forming the frame of discernment $Ω = \{Alice, Bob, Eve\}$. During a trial, two reliable sources, $s_1$ and $s_2$, express the following beliefs: “the culprit’s name has three letters”, represented by the mass function $m_1 : m_1(\{Bob, Eve\}) = 1$, and “the culprit is a woman” represented by the mass function $m_2 : m_2(\{Alice, Eve\}) = 1$. These two pieces of evidence can be combined into $m$ such that $m = m_1 \oplus m_2$, resulting in $m(\{Eve\}) = 1$, meaning “the culprit is Eve”. It is also noteworthy that the Non-specificity of $m_1$ and $m_2$ are equivalent $NS(m_1) = NS(m_2) = 1$, while the Non-specificity of $m$ is zero. Epistemic uncertainty has been reduced to zero.

By adding a few assumptions and using the multiple mathematical tools of the evidential framework, it is possible to learn from data, resulting in evidential learning, and, in the case of deep architectures, in deep evidential learning. The following section demonstrates that it would be problematic to have two frameworks coexisting under the same name and suggests drawing inspiration from second-order risk minimization approaches to make evidential learning even more robust.

## 2.2 An overview of the issue

A second-order risk minimization technique was introduced by [30]. It has been further extensively revisited by the community [35,1,28,6,25,2], and adopted under the name of Evidential Deep Learning. The actual Mathematical Theory of Evidence, nearly half a century older [9,31], at its heart is more about what we model than how we get to that representation. The evidential and deep evidential learning (in the sense of DST) partially address this second point, using various hypotheses. The authors of [30] clearly phrase that it could be useful if a model could simply say “I do not know”. Evidential and deep evidential models have inherently and transparently addressed this assertion (as shown in section 2.1). However, it seems that these recent second-order risk minimization approaches do not inherently respond to it. Therefore, the two nomenclatures struggle to coexist.

The second problem is that the conclusions drawn regarding second-order risk minimization and epistemic uncertainty are disappointing [3,20,4,29,34] and sometimes mistakenly categorize evidential deep learning as inadequate. This risks tarnishing the reputation of DST, which satisfactorily meet the same criteria (as shown theoretically in section 3 and empirically in section 4). Therefore, it is important to be careful to critique second-order risk minimization and not generalize to evidential and deep evidential learning.
The last but most relevant problem concerns nomenclature. It pertains to the acceptance of this second-order risk minimization method as evidential deep learning by the scientific community. This is not evidential learning; any of the combinations in the evidence theory such as Dempster's rule are even utilized in these methods. It is scientifically questionable to use the name of Shannon's entropy from its Mathematical Theory of Communication [33] to refer to a different object. We advocate renaming these methods simply as "Second-Order Risk Minimization" (a term proposed by [20]) instead of "Evidential Deep Learning" for greater transparency and scientific integrity. This would avoid misleading conclusions such as "evidential deep learning methods do not represent epistemic uncertainty in a faithful manner", where the authors really mean "second-order risk minimization methods do not represent epistemic uncertainty in a faithful manner" supported by their strong theoretical claims and remarkable work.

There remains a positive aspect related to this overlap in nomenclature. The flourishing literature on second-order risk minimization has led to very relevant conclusions that can be applied to any evidential and deep evidential learning.

## 2.3 Second-order expected properties

Two particularly interesting properties from the literature on second-order risk minimization, as defined by [3], are presented below. We believe that these properties should also apply to DST's evidential and deep evidential learning. Notably, they have been proven to be impossible [20,4] within second-order risk minimization framework, and we will demonstrate in this paper that these properties are validated both theoretically (section 3) and experimentally (section 4) in the framework of evidential and deep evidential learning.

*Property 1 (Monotonicity).* The epistemic uncertainty of a second-order probability distribution estimate should be monotonic as the sample size increases.

*Property 2 (Convergence).* A second-order probability distribution estimate should converge to a Dirac delta function when the sample size grows to infinity.

# 3 Extension of the monotonicity rule

In this section, the monotonicity rule is extended to the evidential framework in the case of quasi-Bayesian mass functions (see Section 2.1). Only this first rule will be studied in detail here and in the experiments, as it appears to be the most important and should be respected by any evidential and deep evidential model. Regarding the second rule, it is obvious that when available evidence becomes infinite, its combination will result in a categorical mass function, which is the discrete equivalent of a Dirac delta function (convergence rule).

Non-specificity has been advocated [21] as the best measure of DST for capturing epistemic uncertainty. Therefore, this measure will be used in this document to prove its monotonicity when combining multiple sources of information. For this demonstration, we consider the case of quasi-Bayesian mass functions
(a common application scenario, such as in the widely used evidential learning model EK-NN [11]).

**Lemma 1.** The Non-specificity (epistemic uncertainty) of two quasi-Bayesian pieces of evidence $m_1$ and $m_2$ is monotonic with regard to their combination.

Given the Non-Specificity and the conjunctive combination defined respectively by equations (1) and (2), after combining the two mass functions $m_1$ and $m_2$, the Non-Specificity NS of the resulting mass $m_1 \oplus m_2$ is given as follows:

$$NS(m_1 \oplus m_2) = \sum_{A \subseteq \Omega} m_1 \oplus m_2(A) \log_2(|A|). \quad (4)$$

This can be trivially addressed in the general case of null conflict $\kappa = 0$, but in practical applications, there is often a positive conflict $\kappa \ge 0$ (e.g., the conflict is null in the example 4, but if we combine two pieces of evidence, such as “I think the culprit is Bob” and “I think the culprit is Alice”, conflict may arise).

*Proof.* We therefore address the case of non-null conflict in the combination of quasi-Bayesian mass functions. Let a quasi-Bayesian mass function be defined as follows:

$$ \begin{align} m(\{\omega_i\}) &= \alpha_i, \quad \forall \omega_i \in \Omega, \nonumber \\ m(A) &= 0, \quad \forall A \subset \Omega, |A| \neq 1, \tag{5} \\ m(\Omega) &= 1 - \sum_{i=1}^{|\Omega|} \alpha_i, \nonumber \end{align} $$

with $\alpha_i \in [0, 1]$. Consider two quasi-Bayesian mass functions $m_1$ and $m_2$ such that:

$$ \begin{align} m_1(\{\omega_i\}) &= \alpha_i, \quad m_1(\Omega) = 1 - \sum_{i=1}^{|\Omega|} \alpha_i, \nonumber \\ m_2(\{\omega_i\}) &= \beta_i, \quad m_2(\Omega) = 1 - \sum_{i=1}^{|\Omega|} \beta_i. \tag{6} \end{align} $$

The Non-specificity of $m_1$, given by equation (1), is:

$$NS(m_1) = (1 - \sum_{i=1}^{|\Omega|} \alpha_i) \log_2(|\Omega|). \quad (7)$$

By combination of $m_1$ and $m_2$, the conflict $\kappa$ given by equation (3), is:

$$\kappa = \sum_{i=1}^{|\Omega|} \alpha_i \sum_{i=1}^{|\Omega|} \beta_i - \sum_{i=1}^{|\Omega|} \alpha_i \beta_i. \quad (8)$$

During the combination, the mass on the ignorance in then adjusted by the conflict $\kappa$ such that:

$$ \begin{align} m_1 \oplus m_2(\Omega) &= \frac{m_1(\Omega)m_2(\Omega)}{1-\kappa}, \nonumber \\ m_1 \oplus m_2(\Omega) &= \frac{(1-\sum_{i=1}^{|\Omega|} \alpha_i)(1-\sum_{i=1}^{|\Omega|} \beta_i)}{1-\sum_{i=1}^{|\Omega|} \alpha_i \sum_{i=1}^{|\Omega|} \beta_i + \sum_{i=1}^{|\Omega|} \alpha_i \beta_i}. \tag{9} \end{align} $$
The Non-specificity of $m_1 \oplus m_2$ is given as follows:

$$NS(m_1 \oplus m_2) = \frac{(1 - \sum_{i=1}^{|\Omega|} \alpha_i)(1 - \sum_{i=1}^{|\Omega|} \beta_i)}{1 - \sum_{i=1}^{|\Omega|} \alpha_i \sum_{i=1}^{|\Omega|} \beta_i + \sum_{i=1}^{|\Omega|} \alpha_i \beta_i} \log_2(|\Omega|). \quad (10)$$

We can now compare left-hand terms of $NS(m_1)$ and $NS(m_1 \oplus m_2)$. For simplicity, we pose $\Gamma_\alpha = \sum_{i=1}^{|\Omega|} \alpha_i$ and $\Gamma_\beta = \sum_{i=1}^{|\Omega|} \beta_i$. Using arithmetic we can prove the following statement by assuming $\Gamma_\beta(1 - \Gamma_\alpha)^2 + (1 - \Gamma_\alpha) \sum_{i=1}^{|\Omega|} \alpha_i \beta_i \ge 0$, since all elements are positive:

$$\begin{align}
& \Gamma_\beta(1 - \Gamma_\alpha)^2 + (1 - \Gamma_\alpha) \sum_{i=1}^{|\Omega|} \alpha_i \beta_i \ge 0, \nonumber \\
& \Gamma_\beta - 2\Gamma_\alpha \Gamma_\beta + \Gamma_\alpha^2 \Gamma_\beta + (1 - \Gamma_\alpha) \sum_{i=1}^{|\Omega|} \alpha_i \beta_i \ge 0, \nonumber \\
& -\Gamma_\alpha \Gamma_\beta + \Gamma_\alpha^2 \Gamma_\beta + (1 - \Gamma_\alpha) \sum_{i=1}^{|\Omega|} \alpha_i \beta_i \ge -\Gamma_\beta + \Gamma_\alpha \Gamma_\beta, \nonumber \\
& 1 - \Gamma_\alpha \Gamma_\beta - \Gamma_\alpha + \Gamma_\alpha^2 \Gamma_\beta + (1 - \Gamma_\alpha) \sum_{i=1}^{|\Omega|} \alpha_i \beta_i \ge 1 - \Gamma_\alpha - \Gamma_\beta + \Gamma_\alpha \Gamma_\beta, \tag{11} \\
& (1 - \Gamma_\alpha)(1 - \Gamma_\alpha \Gamma_\beta + \sum_{i=1}^{|\Omega|} \alpha_i \beta_i) \ge (1 - \Gamma_\alpha)(1 - \Gamma_\beta), \nonumber \\
& (1 - \Gamma_\alpha) \ge \frac{(1 - \Gamma_\alpha)(1 - \Gamma_\beta)}{1 - \Gamma_\alpha \Gamma_\beta + \sum_{i=1}^{|\Omega|} \alpha_i \beta_i}. \nonumber
\end{align}$$

The Non-specificity $NS(m_1 \oplus m_2)$ is therefore less than or equal to $NS(m_1)$. For any quasi-Bayesian mass functions $m_1$ and $m_2$, the monotonicity of the Non-specificity with respect to the combination of two pieces of evidence can thus be expressed as:

$$NS(m_1) \ge NS(m_1 \oplus m_2). \quad (12)$$

This demonstrates the monotonicity of Non-specificity for two quasi-Bayesian mass functions.

**Theorem 1 (Extended monotonicity rule).** The Non-specificity of any given quasi-Bayesian set of evidence and a larger set is monotonic with regard to their combination.

The combination of two quasi-Bayesian mass functions results in another quasi-Bayesian mass function, and since Dempster's rule (2) is both commutative and associative, the equation can be extended to any set of evidence. This establishes the extended rule of monotonicity. In other words, epistemic uncertainty decreases as the sample size increases. It may also be useful to note the following propositions derived from the properties of belief functions.

**Proposition 1 (Zero property).** Combining any mass function $m_1$ to a Bayesian mass function $m_{\alpha_i}$ (first order probability) such that $m_{\alpha_i}(\{\omega_i\}) = \alpha_i, \forall \omega_i \in \Omega$, and $\sum \alpha_i = 1$, gives a zero Non-specificity: $NS(m_1 \oplus m_{\alpha_i}) = 0$.

Simply put, adding specific knowledge (purely aleatoric) reduces epistemic uncertainty to zero (e.g., adding the information “the coin is not biased” removes all epistemic uncertainty about the toss and degenerates into the probability of getting heads = probability of getting tails = 0.5).
<figure><img src="image_3.png" /><figcaption>Fig. 1: Epistemic uncertainty vs. knowledge increase.</figcaption></figure>

**Proposition 2 (Identity property).** Combining any mass function $m_1$ to a vacuous mass function $m_\Omega$ such that $m_\Omega(\Omega) = 1$ and $m_\Omega(A) = 0, \forall A \in 2^\Omega \setminus \Omega$, gives a Non-specificity identical to that of $m_1$: $NS(m_1 \oplus m_\Omega) = NS(m_1)$.

In other words, adding completely ignorant information does not affect epistemic uncertainty (e.g., adding the information “I don’t know whether the coin is biased or not” does not change the prior epistemic uncertainty of the problem).

The monotonicity rule, particularly relevant in machine learning, has been extended to the evidential framework in this section. Although it has just been proven to be true, machine learning and deep learning models relying on evidence theory can make many assumptions to estimate such mass functions and can thus deviate from the theory. In the following section, we demonstrate that several models indeed adhere to this rule and, more importantly, that a proper evidential or deep evidential model must respect this rule.

# 4 Experiments

In this section, the extension of the monotonicity rule (clearly recalled as “the epistemic uncertainty of an evidential model should be monotonic when the sample size increases”) is empirically demonstrated for a case of machine learning and deep learning. Initially, this behavior is demonstrated on the most popular evidential machine learning model, the Evidential *K*-nearest neighbors [11]. Then, a deep evidential learning version is also applied to show the continuity of the results within the deep architecture.

To simplify reproducibility, the uncertainty quantification framework is available to the scientific community at https://github.com/ArthurHoa/deep-eknn.
The framework is compatible with pre-trained models and can be seamlessly integrated at the penultimate layer of a deep neural network.

## 4.1 DST's Evidential Learning

An evidential version of *K*-NN is introduced by [11], this Evidential *K*-Nearest Neighbors (*EK*-NN) uses belief functions to assign a label to a new incoming instance. Given a machine learning scenario, predictions then can be made. Let an instance space $\mathcal{X} = \mathbb{R}^P$ be a collection of *P* random variables and an output
space $\Omega = \{\omega_1, \dots, \omega_C\}$ be a set of $C$ possible outcomes. Let $\mathbf{x}_i = \{x_i^1, \dots, x_i^P\} \in \mathcal{X}$ be an instance associated with a class $\omega \in \Omega$ and the distance $d_{i,j}$ be the $l_2$-norm between two instances $x_i$ and $x_j$. Given the $K$ nearest neighbors of $x_i$ according to $d$, one can assign a mass function to each neighbors $x_j, 1 \le j \le K$ of class $\omega^j \in \Omega$ discounted according to the distance from $x_i$. For further details, the reader is referred to [11] due to space constraints. Note that in this setup, we only consider quasi-Bayesian mass functions (c.f., Section 2.1).

According to the theoretical proof outlined in Section 3, the number of neighbors should ideally be set to the size of the training set. However, we demonstrate that such complexity is unnecessary and that simpler parameters can yield satisfactory monotonicity. Therefore, for this experiment, we arbitrarily selected 7 as the number $K$ of neighbors. The masses of all neighbors are then aggregated into the predicted belief function, enabling the Non-specificity (1) to be computed directly as the epistemic uncertainty.

Figures 1a and 1b illustrate the monotonicity of epistemic uncertainty in the context of machine learning. Two datasets from the UCI repository [12] are used, and the model's performance is shown as a function of the number of samples in the training set. The datasets are split into training and testing sets using a 70/30 ratio. This process is repeated 100 times, and the average model error is plotted. Along with the error, epistemic uncertainty (Non-specificity) is presented as an interval around the error. The results clearly show that Non-specificity decreases as the number of labeled instances increases. In other words, epistemic uncertainty decreases with regard to a sample size increase.

## 4.2 DST's Deep Evidential Learning

In this section, we use a Deep EK-NN version. This framework enables the quantification of epistemic uncertainty for most deep learning architectures and is compatible with pre-trained models. It can be seamlessly applied to the penultimate layer of a neural network (whether convolutional, residual, transformer, encoder, etc.) to perform uncertainty quantification. EK-NN [11] is used to predict the membership of new observations in the form of a belief function. For the experiments, observations are encoded using the penultimate layer of a neural network, and EK-NN is applied to this reduced feature space, generating a belief function. Epistemic uncertainty is computed according to Equation (1).

LeNet [23] is trained on the MNIST dataset with varying dataset sizes, and a ResNet18 [16] is pre-trained on CIFAR-10 [22]. The EK-NN framework is then applied to estimate epistemic uncertainty after each model's training. The model's error is plotted along with its epistemic uncertainty, showing that non-specificity decreases as the number of labeled instances increases. Figures 1c and 1d illustrate the monotonicity of epistemic uncertainty in deep learning, where a small value of $K = 10$ is sufficient for monotonicity. Again, the results show that non-specificity decreases with more labeled instances, confirming that epistemic uncertainty remains monotonic with respect to sample size in deep learning. This valuable property demonstrates that, both theoretically and empirically, the model becomes less ignorant as it gains access to more knowledge.
# 5 Conclusion

The main takeaway of this article is that second-order risk minimization is not Dempster-Shafer Theory but both are known under the same name of evidential (deep) learning. It has been showed that modeling a Dirichlet distribution is not sufficient for a model to be considered evidential. Furthermore, recent critiques of second-order risk minimization emphasize the difficulty in modeling epistemic uncertainty, whereas Dempster-Shafer Theory offers a built-in, reliable approach to modeling this type of uncertainty. A mathematical proof demonstrated the monotonicity of the learner's epistemic uncertainty in the application context of the evidential framework. Practical experiments also showed a satisfactory decreasing of epistemic uncertainty both in machine learning and deep learning, leveraging a proper framework to model the ignorance state of the learner. This property is particularly important, as it guarantees that the learner's ignorance decreases as it acquires more information. Although reliable and efficient in our *scenario*, evidential learning is a broad domain, and some models considered evidential do not adhere to this axiomatic behavior. Thus, the insights gained from second-order risk minimization allow us to extend valuable properties to the DST case, enabling the definition of an evidential model that properly estimates epistemic uncertainty, such as those presented in our experimental setup.

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