# Modeling and Updating Uncertain Evidence within Belief Function Theory

Pierre Pomeret-Coquot

► To cite this version:

Pierre Pomeret-Coquot. Modeling and Updating Uncertain Evidence within Belief Function Theory. International Journal of Approximate Reasoning, 2025, 182 (July 2025), pp.109428. 10.1016/j.ijar.2025.109428. hal-04975896v2

HAL Id: hal-04975896

https://hal.science/hal-04975896v2

Submitted on 14 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

Check for
updates

# Modeling and updating uncertain evidence within belief function theory

Pierre Pomeret-Coquot a,b, ∗

ARTICLE INFO

Keywords:

Belief functions

Interpretation

Evidence

Credal set

Conditioning

Dilation

Convexity

A B S T R A C T

We propose a framework that enhances the expressiveness of the evidential and credal interpre-
tations of Belief Function Theory while remaining within its scope. It allows uncertain evidence
to be represented “as is” by associating meaningful intervals of N or R to focal elements, pro-
viding an intrinsic justification for belief values. This improves the modeling and manipulation
of knowledge. From a credal perspective, the framework enables the accurate representation of
non-maximal credal sets, when their extrema are belief and plausibility functions.

We introduce three update operations that extend Dempster's, geometric, and Bayesian condition-
ing to uncertain evidence. These updates are expressed in terms of transfer of evidence, ensuring
linear complexity relative to the number of focal elements. This approach provides clear eviden-
tial semantics to Bayesian conditioning, resolves several of its anomalies by making it tractable
and commutative, and explains its apparent dilation effect. Most importantly, it accurately yields
the updated credal set, rather than merely providing its bounds.

1. Introduction

Belief Function Theory enables the expression of imprecise knowledge [7,37]. Two interpretations are usually contrasted: the evidential interpretation, where the knowledge is made of several potentially imprecise and contradictory pieces of evidence [37,42]; and the credal interpretation, where the knowledge about a partially known probability measure is constituted by the set of possible probabilities or by its extreme values [44]. Section 2 illustrates this distinction with two simple examples. Section 3 recalls their common mathematical foundations.

Section 4 introduces the framework we propose to model uncertain evidence and reason with it. It extends the expressiveness of evidential models to express uncertainty on the evidence itself while remaining within the framework of Belief Function Theory. Its simple semantics, which rely neither on randomness nor on external decision process or canonical examples to set the values [38,42], facilitates an easy bridge to the credal interpretation (Section 5) in terms of uncertain random sets arising, e.g., from uncertain multi-valued mappings. Doing so, it allows delimiting non-maximal credal sets whose extrema are belief and plausibility functions. It hence allows for distinguishing distinct pieces of credal knowledge having the same lower and upper bounds.

In Section 6, we formally define the notion of update-by-transfer, then we define three update-by-transfer operations respectively generalizing Dempster's conditioning [7], geometric conditioning [43,41], and Bayesian conditioning [24,26]. They all achieve linear complexity relative to the number of focal elements. Notably, anomalies of the latter are resolved: the proposed Bayesian update

<footnote>ª IRIT, Université de Toulouse, CNRS, Toulouse INP, UT3, UT2, Toulouse, France</footnote>

<footnote>ᵇ ANITI, B612, 3 rue Tarfaya, Toulouse, France</footnote>

<footnote>* Correspondence to: IRIT, Université de Toulouse, CNRS, Toulouse INP, UT3, UT2, Toulouse, France.</footnote>

<footnote>E-mail address: pierre.pomeret@irit.fr.</footnote>
accurately represents the conditioned credal set (instead of its lower and upper bounds only), making the operation commutative. It hence properly captures focusing (or hypothetical conditioning) [13,14,11] in the Belief Function Theory and enables sequential updates. It also makes it clear it acts as a “no-assumption” update, explaining the dilation effect.

This work opens up several research perspectives which conclude the paper in Section 7.

**Preliminary remark: chance vs. probability**

The expression $Pr(x) = \frac{1}{2}$ can be read as “one chance in two” or “a probability of 0.5.” The phrase “one chance in two” is clearer, distinguishing it from, for example, “three chances in six,” even though both correspond to a probability of 0.5. This distinction matters because chances represent a countable ratio—favorable outcomes relative to total possible outcomes—which is lost when only the probability value is given.

Although this article focuses on Belief Function Theory, the contribution primarily relies on the simple observation that beliefs, like chances, are fundamentally countable and their normalization depends on the total number of possible outcomes.

**2. Introductory examples**

**Example 1 (The Murder of Mr. Jones—inspired from [42]).** Mr Jones was murdered at home; the only suspects are Rose, Bryan and Walter. Three witnesses saw someone enter Mr Jones’ building: one saw a woman, the two other saw a man.

This example illustrates evidential knowledge by considering both contradictory and imprecise testimonies: one implicates Rose (the only woman) while the two others implicate either William or Bryan (the men) without specifying which one of them specifically.

Modeling starts with the expression of the *frame of discernment* $\Omega$, made of atomic and mutually exclusive propositions: only one $\omega \in \Omega$ is true, but we don’t know which. Here $\Omega = \{r, b, w\}$, where $r, b$ and $w$ denote the propositions stating Rose, Bryan and William as the murderer, respectively. Subsets of $\Omega$ form compound propositions (e.g., $\{r, b\}$ means “either Rose or Bryan is the murderer”).

Each testimony states a proposition $B \subseteq \Omega$ exactly: one states $\{r\}$, the two others state $\{b, w\}$. Testimony weights are determined externally, for example by comparison with canonical examples or from a decision process; they are expressed on the quantitative scale [0, 1]—depending on the author these values are degrees of probability, of belief, etc.—denoting confidence levels [38,39,42]. Here, all testimonies have equal weight: $1/3$.

The degree at which the evidence supports propositions is deduced from the weights of the compatible testimonies. For some propositions, the degree of support is known (e.g. $\{b, w\}$ is supported by exactly two testimonies, hence a support degree of 2/3); for others, it can be bounded only (e.g. $\{r, w\}$ is supported by $\{r\}$ and maybe by $\{b, w\}$, hence a support degree between 1/3 and 1).

**Example 2 (The Three-Ball Urn—adapted from [17]).** Three balls have been placed in a opaque urn: one has been taken from BAG I, which contains only red balls; the two others have been taken from BAG II, which contains black and white balls. What are the probabilities of randomly drawing a red, black or white ball from the urn?

This example illustrates credal knowledge by considering several possible contents, i.e., several possible probability measures.

Modeling uses a *universe* $\Omega$ of possible atomic outcomes. Here, $\Omega = \{r, b, w\}$ where $r, b$ and $w$ represent the draw of a red, black or white ball respectively. Subsets of $\Omega$ are events denoting for compound outcomes (e.g., $\{r, b\}$ means “drawing a red or black ball”).

The urn’s content is only partially known: the possible content configurations are *rbb*, *rbw* and *rww*, each configuration leading to a particular probability measure. For some events, the probability is known (e.g. the probability of $\{b, w\}$ is necessarily 2/3); for others, it can be bounded only (e.g. the probability of $\{r, w\}$ is between 1/3 and 1).

These examples share similar conclusions: the value associated with $\{b, w\}$ is 2/3, while the one associated with $\{r, w\}$ is between 1/3 and 1. However, their interpretations differ. **Example 1** relies on the evidential interpretation: the identity of the murderer (a single $\omega \in \Omega$) is unknown, and we deduce the degrees of compatibility and non-contradiction of propositions with the testimonies. Conversely, **Example 2** relies on credal understanding: the proportion of balls (a probability over $\Omega$) is unknown, and we deduce the minimum and maximum probabilities of events.

Note that even different, the evidential and credal interpretations are not opposite. One may indeed see probability values as degrees of support, events as propositions, and possible probability measures as possible states of evidence, for the several way testimonies can be made precise.

**3. Belief function theory**

**3.1. Basic notions and definitions**

Belief Function Theory, pioneered by Dempster [7] and Shafer [37], is a mathematical framework that generalizes both probability theory and quantitative possibility theory [12,15,16], and is further generalized by the theory of imprecise probabilities [29,28,44]. There are several ways to understand it, which all accord to either the evidential or the credal interpretation, making it a powerful tool for reasoning under uncertainty in various domains.<fnref n="1" />

<footnote n="1">¹ For the several semantics of Belief Function Theory, see, e.g., [3, Part I, Chapter 3].</footnote>
<table><caption>**Table 1**
BPA *m*, belief function Bel, plausibility function Pl, and the credal set (Pr<sub>x</sub>)<sub>0≤x≤2/3</sub> delimited by Bel and Pl, for Examples 1 and 2.</caption><thead><tr><th>A</th><th>θ</th><th>{r}</th><th>{b}</th><th>{w}</th><th>{r, b}</th><th>{r, w}</th><th>{b, w}</th><th>{r, b, w}</th></tr></thead><tbody><tr><td>m(A)</td><td></td><td>1/3</td><td></td><td></td><td></td><td>2/3</td><td></td><td></td></tr><tr><td>Bel(A)</td><td>0</td><td>1/3</td><td>0</td><td>0</td><td>1/3</td><td>1/3</td><td>2/3</td><td>1</td></tr><tr><td>Pl(A)</td><td>0</td><td>1/3</td><td>2/3</td><td>2/3</td><td>1</td><td>1</td><td>2/3</td><td>1</td></tr><tr><td>Pr<sub>x</sub>(A)</td><td>0</td><td>1/3</td><td>x</td><td>2/3 - x</td><td>1/3 + x</td><td>1 - x</td><td>2/3</td><td>1</td></tr></tbody></table>

Let $\Omega = \{\omega_1, \dots, \omega_n\}$ be a finite set.

**Definition 1 (Basic Probability Assignment—BPA).** A BPA on $\Omega$ is a set function $m: 2^{\Omega} \to \mathbb{R}$ satisfying $m(\emptyset) = 0$, $\sum_{B \subseteq \Omega} m(B) = 1$, and $m(B) \ge 0$ for all subsets $B$ of $\Omega$.

A focal element is any subset $B$ such that $m(B) \neq 0$.

Evidential models use BPAs to encode the raw knowledge. In this paper, we reserve the term 'mass function' for evidential BPAs. In Example 1, $m(r) = 1/3$ and $m(b, w) = 2/3$ encapsulate the knowledge by describing the weights of the testimonies (i.e. their mass). Conversely, BPAs are not always considered in credal models. However, because BPAs are probability distributions on non-empty subsets of $\Omega$, they may be thought as random sets [27,30,31]. Each $m(\{\omega\})$ is then the amount of probability precisely assigned to $\omega$, and for $|B| > 2$, $m(B)$ is the amount of probability one does not know how to allocate to the $\omega$s in $B$. In Example 2, $m$ follows from the multi-valued mapping $\text{BAG I} \mapsto \{r\}$, $\text{BAG II} \mapsto \{b, w\}$ and from the known probability $\text{Pr(BAG I)} = 1/3$, $\text{Pr(BAG II)} = 2/3$. That is, $m(r) = 1/3$ is the probability precisely assigned to $r$ and $m(b, w) = 2/3$ is the amount of probability we do not know how to allocate to $b$ and to $w$ [7].

Two other set functions, Bel and Pl, known as the belief and plausibility functions, are considered.

**Definition 2 (Dual Belief and Plausibility Functions).** Set functions Bel: $2^{\Omega} \to \mathbb{R}$ and Pl: $2^{\Omega} \to \mathbb{R}$ are dual belief and plausibility functions iff there exists a BPA $m: 2^{\Omega} \to \mathbb{R}$ such that for all $A \subseteq \Omega$:

$$ \text{Bel}(A) = \sum_{B \subseteq A} m(B) \quad \text{and} \quad \text{Pl}(A) = \sum_{B \cap A \neq \emptyset} m(B). $$

These functions are dual: $\text{Bel}(A) + \text{Pl}(A^c) = 1$ for all $A$, where $A^c = \Omega \setminus A$ is the complement of $A$ in $\Omega$.

Belief functions can be characterized independently of their BPAs, however it is not clear how to interpret the required properties: Bel is a belief function if and only if $\text{Bel}(\emptyset) = 0 \le \text{Bel}(A) \le \text{Bel}(\Omega) = 1$ and $\text{Bel}(A_1 \cup \dots \cup A_k) \ge \sum_{\emptyset \ne I \subseteq \{1, \dots, k\}} (-1)^{|I|+1} \text{Bel}(\cup_{j \in I} A_j)$ for all subsets $A$ and $A_i$ of $\Omega$. The BPA of a belief function can always be retrieved using the inverse Möbius transform ([37], probably from [34]): $m(B) = \sum_{A \subseteq B} (-1)^{|B \setminus A|} \text{Bel}(A)$, hence Definition 2 is proper. For $m$, Bel and Pl can be retrieved from one another, they may be thought as three different views of the same information. However, the meaning of BPAs retrieved using the Möbius inverse is not always clear: some BPAs cannot be interpreted, neither in terms of mass functions, nor in terms of random sets (as for example the BPA $m(\cdot | \Omega, C)$ in Table 5 to come). When the initial knowledge is stated as a belief function instead of a BPA, it makes the requirement of Definition 2 hard to interpret too.

In evidential interpretations, $\text{Pl}(A)$ denotes the degree at which the proposition $A$ is compatible with the evidence, while $\text{Bel}(A)$ denotes the degree at which the evidence implicates $A$ (see Table 1), generalizing notions of possibility and necessity, respectively. If focal elements are nested (i.e., for all focal $B$ and $B'$, either $B \subseteq B'$ or $B' \subseteq B$), then they are mutually non-contradictory (the evidence is "logical consistent") and Bel and Pl are dual necessity and possibility measures, respectively [47,12].

In credal interpretations, Bel and Pl denote the minimum and maximum probabilities of events according to a set of possible probability measures, called *credal set* [29,20] (also see Table 1). When focal elements all are singletons, the knowledge is precise and Bel = Pl is the unique probability measure in the credal set.

Let $\text{Prob}(\Omega)$ denote the set of all probability measures over $\Omega$. Starting from an arbitrary credal set $F \subseteq \text{Prob}(\Omega)$, one can consider its extreme probabilities for each event, known as the *lower and upper probabilities*.

**Definition 3 (Lower and Upper Probabilities).** Let $F \subseteq \text{Prob}(\Omega)$ be a set of probability measures over $\Omega$. The lower and upper probability measures are the dual functions $P_*$ and $P^*$ defined by:

$$ P_*(A) = \inf_{\text{Pr} \in F} \text{Pr}(A) \quad \text{and} \quad P^*(A) = \sup_{\text{Pr} \in F} \text{Pr}(A). $$

The framework of imprecise probability is more general than that of belief functions: $P_*$ and $P^*$ may not be proper belief and plausibility functions. They are so if their Möbius inverse is a proper BPA, otherwise it yields some negative values. (In this paper, we only consider credal sets whose lower and upper probabilities are belief and plausibility functions.)
<figure><img src="image_4.png" /><figcaption>Fig. 1. Geometric representation of the 3 possible probability distributions of Example 2 (red dots) and of the maximal credal set $\mathcal{F}_{\text{Bel}}^{\text{max}}$ (red line). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)</figcaption></figure>

<table><caption>**Table 2**
The BPA $m'$ and its belief and plausibility functions $\text{Bel}'$ and $\text{Pl}'$.</caption><thead><tr><th>A</th><th>Ø</th><th>{r}</th><th>{b}</th><th>{w}</th><th>{r, b}</th><th>{r, w}</th><th>{b, w}</th><th>{r, b, w}</th></tr></thead><tbody><tr><td>$m'(A)$</td><td></td><td>1/3</td><td>1/3</td><td></td><td></td><td></td><td>1/6</td><td>1/6</td></tr><tr><td>$\text{Bel}'(A)$</td><td>0</td><td>1/3</td><td>1/3</td><td>0</td><td>2/3</td><td>1/3</td><td>1/2</td><td>1</td></tr><tr><td>$\text{Pl}'(A)$</td><td>0</td><td>1/2</td><td>2/3</td><td>1/3</td><td>1</td><td>2/3</td><td>2/3</td><td>1</td></tr></tbody></table>

Conversely, one may start from dual belief and plausibility functions $\text{Bel}$ and $\text{Pl}$ and consider the set of all probability measures $\text{Pr}$ such that $\text{Bel}(A) \le \text{Pr}(A) \le \text{Pl}(A)$, $\forall A$. This credal set is maximal in the sense that any compatible probability measure belongs to it: it is the largest credal set for which $\text{Bel}$ and $\text{Pl}$ are lower and upper probabilities [29,28]. (The term ‘convex’ is also used to characterize such sets. In this paper, we will rather stick to its conventional meaning: a set is convex if and only if it contains every combination of its elements.)

**Definition 4 (Maximal credal set).** Let $\text{Bel}$ and $\text{Pl}$ be dual belief and plausibility functions over $\Omega$. The maximal credal set they define is<fnref n="2" />:

$$ \mathcal{F}_{\text{Bel}}^{\text{max}} = \{ \text{Pr} \in \text{Prob}(\Omega) \mid \forall A \subseteq \Omega, \text{Bel}(A) \le \text{Pr}(A) \le \text{Pl}(A) \}. $$

When seen as a random set, a BPA $m: 2^{\Omega} \to [0, 1]$ expresses several possible probability distributions, hence directly induces a credal set. More precisely, $m(B)$ is the amount of probability which can be allocated to the $\omega$s in $B$; hence the BPA $m$ induces the credal set containing the several probability distributions corresponding to the several possible allocations. Formally, an allocation $(\gamma^B)_{B \subseteq \Omega}$ is such that each $\gamma^B: B \to [0, 1]$ satisfies $\sum_{\omega \in B} \gamma^B(\omega) = m(B)$; then $\gamma_B(\omega)$ is the amount of probability allocated from $B$ to $\omega$. Such an allocation defines the probability distribution $p: \Omega \to [0, 1]$ such that $p(\omega) = \sum_B \gamma^B(\omega)$. It is easy to check that the set of possible distributions is the maximal credal set of the belief function derived from $m$.

In general, there are many different credal sets admitting the same belief and plausibility functions as extrema. Because maximal credal sets are necessarily convex, the maximal credal set $\mathcal{F}_{\text{Bel}}^{\text{max}} = \{\text{Pr}_x \mid 0 \le x \le 2/3\}$ defined by the belief function of Table 1 obviously differs from the discrete credal set $\mathcal{F} = \{\text{Pr}_x \mid x \in \{0, 1/3, 2/3\}\}$ containing the three possible probability measures of Example 2 (Fig. 1).<fnref n="3" />

Moreover, there are credal sets that are convex but not maximal. In general, a belief function is the lower bound of many convex credal sets, only one of which is maximal. As for example, consider the two credal sets illustrated in Fig. 2. $\mathcal{F}_{\text{Bel}}^{\text{max}}$ is the maximal credal set derived from the BPA $m'$ (Table 2); it contains the probability distributions $(\frac{2+x_r}{6}, \frac{2+x_b+y_b}{6}, \frac{x_w+y_w}{6})$ for non-negative $x_i$ and $y_i$ such that $x_r + x_b + x_w = y_b + y_w = 1$. Conversely, $\mathcal{F}'$ is convex but not maximal; it contains the probability distributions $(\frac{1}{2+x+y}, \frac{1+x}{2+x+y}, \frac{y}{2+x+y})$ for non-negative $x$ and $y$ such that $x+y \le 1$. Yet $\mathcal{F}_{\text{Bel}}^{\text{max}}$ and $\mathcal{F}'$ have the same lower and upper probabilities.

<footnote n="2">² Since $\text{Bel}$ and $\text{Pl}$ are dual, it holds that $\forall A. \text{Bel}(A) \le \text{Pr}(A) \iff \forall A. \text{Pr}(A) \le \text{Pl}(A)$: one of the two inequalities is sufficient to state the definition. However, keeping both make the definition even clearer and allows for an easy parallel with Definitions 19 and 21.</footnote>

<footnote n="3">³ Geometric representations rely on the analogy between probability distributions over $\Omega = \{\omega_1, \dots, \omega_n\}$ and $n$-tuples $(p_1, \dots, p_n) \in \mathbb{R}^n$. Since $\forall i, p_i \ge 0$ and $\sum_i p_i = 1$, they all belong to the convex hyper-plan of $\mathbb{R}^n$ delimited by its base, called the standard $n$-simplex. In particular, probability distributions over $\{\omega_1, \omega_2, \dots, \omega_n\}$ belong to the segment whose ends are $(1, 0)$ and $(0, 1)$; the ones over $\{\omega_1, \omega_2, \omega_3\}$ belong to the triangle whose vertices are $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$. Credal sets on $\Omega$ hence are analogous to subsets of the standard $|\Omega|$-simplex. Geometrically, a credal set is maximal if and only if its frontiers are parallel to the edges and facets of the standard $n$-simplex [3].</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 2. Geometric representation of $\mathcal{F}_{\text{Bel}'}^{\text{max}}$ (at left) and $\mathcal{F}'$ (at right).</figcaption></figure>

## 3.2. Complexity of belief functions and their algorithms

Since BPAs are usually sparse, algorithms devoted to the Belief Function Theory often rely on BPAs. It is indeed sufficient to encode the focal elements and their corresponding mass only. Although some dedicated data structures can be considered, it may be sufficient to think about BPAs as lists of pairs $(B, m(B))$ for each focal element $B$ (such that $m(B) \neq 0$). It is then easy to retrieve belief and plausibility values, as for other computations, using a single loop on the list.

However, in general, there may be up to $2^{|\Omega|} - 1$ focal elements, which leads to an exponential space representation and exponential time algorithms. Some authors propose to approximate such BPAs with low complexity ones, by restricting the number or the size of focal elements [1,23].

**Definition 5 (k-Additivity).** Let $m: 2^{|\Omega|} \to [0, 1]$ be a BPA.<fnref n="4" />

$$m \text{ is } k\text{-additive} \iff (\forall B, |B| > k \implies m(B) = 0).$$

A $k$-additive BPA has fewer than $\sum_{i=1}^k (|\Omega|^i) \in \mathcal{O}(|\Omega|^k)$ focal elements.<fnref n="5" />

## 3.3. Updating knowledge in the belief function theory

When new information becomes available, the knowledge must be updated. Let $C \subseteq \Omega$ denote the new information “only the $\omega \in C$ are possible.” Obviously, the frame of discernment should be restricted to $C$. But how to update the knowledge about the still possible states $\omega \in C$?

Several updating rules have been proposed, depending on the nature of the information $C$ conveys [13,14] and, supposedly, to the adopted evidential or credal interpretation [24].<fnref n="6" /> They all transform the initial knowledge (the prior BPA, belief and plausibility functions) into a new, updated knowledge (the posterior ones), but in different ways. The three rules we present below all generalize the classic probabilistic conditioning, which is recovered when $\text{Bel} = \text{Pl}$ is a probability measure.

### 3.3.1. Three conditioning operations

Dempster's conditioning (denoted $|\Omega|$) is suited for knowledge revision. If the information $C$ conveys is about the knowledge itself, some conclusions made out of $C$ can be used to revise the prior. Formally:

**Definition 6 (Dempster Conditioning [7]).** Let $\text{Pl}: 2^{|\Omega|} \to [0, 1]$ be a plausibility function and $C \subseteq \Omega$ such that $\text{Pl}(C) > 0$. Its Dempster's conditioning given $C$ is the plausibility function defined as follows:

$$\text{Pl}(A|\Omega, C) = \frac{\text{Pl}(A \cap C)}{\text{Pl}(C)}.$$

Consider Example 1 and assume Walter was abroad the day Mr. Jones was murdered. Walter is not suspected anymore, hence $C = \{r, b\}$. Moreover, the witnesses that saw a man surely saw Bryan, hence the testimonies stating $\{b, w\}$ should be corrected to $\{b\}$ (Table 3).

Updating is easily achieved on the BPA by transferring the mass of $B$ to $B \cap C$, aggregating the transferred mass on the same sets and normalizing. It requires linear time (w.r.t. the number of focal elements, assuming set operations in $\mathcal{O}(1)$) and ensures $k$-additive priors lead to $k$-additive posteriors.

<footnote n="4">⁴ In contrast with [23]'s definition, we state $k$-additivity as a property of the BPA, not the belief function, and we do not assume that there actually exists a focal element with size $k$ exactly. Hence, if a BPA is $k$-additive, it is also $(k+1)$-additive.</footnote>

<footnote n="5">⁵ $O(|\Omega|^k)$ is reasonable for small values of $k$; but becomes quite loose for larger ones.</footnote>

<footnote n="6">⁶ Section 6 shows that the presented updates may be understood in both interpretations.</footnote>
<caption>**Table 3**</caption>

BPA, belief and plausibility functions after Dempster's conditioning given $C = \{r, b\}$ for Example 1.

<table><caption>**Table 4**</caption><thead><tr><th>A</th><th>Ø</th><th>{r}</th><th>{b}</th><th>{r, b}</th></tr></thead><tbody><tr><td>$m(A|_D C)$</td><td></td><td>1/3</td><td>2/3</td><td></td></tr><tr><td>$\text{Bel}(A|_D C) = \text{Pl}(A|_D C)$</td><td>0</td><td>1/3</td><td>2/3</td><td>1</td></tr></tbody></table>

BPA, belief and plausibility functions after geometric conditioning given $C = \{r, b\}$ for Example 1.

<table><thead><tr><th>A</th><th>Ø</th><th>{r}</th><th>{b}</th><th>{r, b}</th></tr></thead><tbody><tr><td>$m(A|_g C)$</td><td></td><td>1</td><td></td><td></td></tr><tr><td>$\text{Bel}(A|_g C) = \text{Pl}(A|_g C)$</td><td>0</td><td>1</td><td>0</td><td>1</td></tr></tbody></table>

**Property 1.** For any non-empty $B \subseteq C$:

$$m(B|_D C) = \frac{1}{\text{Pl}(C)} \sum_{B' \cap C=B} m(B').$$

Geometric conditioning (denoted $|_g$) achieves another kind of *knowledge revision*: conclusions made out of $C$ differ from those of Dempster's conditioning.

**Definition 7 (Geometric Conditioning [43,41]).** Let $\text{Bel}: 2^\Omega \to [0, 1]$ be a belief function and $C \subseteq \Omega$ such that $\text{Bel}(C) > 0$. Its geometric conditioning given $C$ is the belief function defined as follows:

$$\text{Bel}(A|_g C) = \frac{\text{Bel}(A \cap C)}{\text{Bel}(C)}$$

Consider Example 1 again, but with a different alibi: Walter lives in the same building as Mr. Jones. Although he entered the building, he is no longer suspected: $C = \{r, b\}$. Since the two witnesses surely saw him coming home, their testimonies should not be considered, as they saw an innocent people. Only the testimony $\{r\}$ remains (Table 4). Note that Bryan has no alibi and is still suspected ($b \in C$), even though no one saw him.

Updating is easily achieved on the BPA by retaining only the pieces of evidence $B$ such that $B \subseteq C$ and normalizing. It requires linear time and ensures $k$-additive priors lead to $k$-additive posteriors.

**Property 2.** For any non-empty $B \subseteq C$:

$$m(B|_g C) = \frac{1}{\text{Bel}(C)} m(B).$$

Bayesian conditioning (denoted $|_{\Omega_3}$) achieves *focusing*: it changes the point of view from $\Omega$ to $C$ without making extra assumptions.

**Definition 8 (Bayesian Conditioning [18]).** Let $\text{Bel}: 2^\Omega \to [0, 1]$ be a belief function, $\text{Pl}$ its dual plausibility function and $C \subseteq \Omega$ such that $\text{Bel}(C) > 0$. Bel's Bayesian conditioning given $C$ is the belief function defined as follows:

$$\text{Bel}(A|_{\Omega_3} C) = \frac{\text{Bel}(A \cap C)}{\text{Bel}(A \cap C) + \text{Pl}(A^c \cap C)}.$$

Consider that in Example 2, one drew a ball and states “this ball is not white”. It is either red or black, hence $C = \{r, b\}$. Here, we might consider all previously possible probability measures and update each one individually, to conclude, e.g., that $\text{Pr}(b | C)$ ranges from 0 (if all the balls in the urn are either red or white) to 2/3 (if they all are either red or black), leading to the knowledge expressed in Table 5.

In contrast to the other two updates, the BPA expression is complex and cannot be understood in term of mass: new focal elements may appear (potentially more numerous and with larger cardinality than those of the prior),<fnref n="7" /> and the normalization is not constant for a given $C$. As for today, there only exist exponential-time algorithms implementing Bayesian conditioning [35], as $k$-additive priors may lead to $k'$-additive posteriors with $k' > k$. The exact characterization of posterior focal elements can be found in [26].

<footnote n="7">⁷ Let, e.g., $\Omega = \{\omega_1, \dots, \omega_n\}$ and $m$ a prior such that each $\{\omega_i\}$ and each $\{\omega_i, \omega_j\}$ is focal. $m$ is 2-additive. Then, given any $C$, all non-empty subsets of $C$ are focal elements of the posterior, which is hence $|C|$-additive.</footnote>
<table><caption>**Table 5**
BPA, belief and plausibility functions after Bayesian conditioning given $C = \{r, b\}$ for Example 2.</caption><thead><tr><th>A</th><th>θ</th><th>{r}</th><th>{b}</th><th>{r, b}</th></tr></thead><tbody><tr><td>m(A |<sub>23</sub> C)</td><td></td><td>1/3</td><td></td><td>2/3</td></tr><tr><td>Bel(A |<sub>23</sub> C)</td><td>0</td><td>1/3</td><td>0</td><td>1</td></tr><tr><td>Pl(A |<sub>23</sub> C)</td><td>0</td><td>1</td><td>2/3</td><td>1</td></tr></tbody></table>

**Property 3 (Bayesian Conditioning).** For any non-empty $B \subseteq C$:

$$m(B |_{23} C) = \sum_{\substack{(B_0, \dots, B_k) \subseteq (2^\Omega)^\infty \\ C \cap \bigcup_{i=0}^k B_i = B \\ B_0 \subseteq C \\ B_i \cap C \neq \emptyset, i \ge 1 \\ B_i \cap C^c \neq \emptyset, i \ge 1 \\ B_i \neq B_j, i \neq j}} \frac{\prod_{j=0}^k m(B_j)}{\prod_{i=0}^k (\text{Pl}(C) - \sum_{j=1}^i m(B_j))}.$$

Credally, Bel and Pl are the lower and upper probabilities of a credal set. Conditioning a credal set amounts to conditioning individually each probability measure it contains.

**Definition 9 (Conditioning of a Credal set).** Let $\mathcal{F}$ be a credal set containing probability measures on $\Omega$ and $C \subseteq \Omega$ such that $\forall \text{Pr} \in \mathcal{F}$, $\text{Pr}(C) > 0$. $\mathcal{F}$'s conditioning given $C$ is the credal set $\mathcal{F} | C$ defined as follows:

$$\mathcal{F} | C = \{ \text{Pr}(\cdot | C) | \text{Pr} \in \mathcal{F} \}.$$

The conditioned credal set is generally not maximal, even though the prior is so. If the prior is maximal and delimited by a belief function, Bayesian conditioning yields the lower and upper probabilities of the posterior.

**Theorem 1 ([18]).** Let $\text{Bel}: 2^\Omega \to [0, 1]$ be a belief function and $\mathcal{F}_{\text{Bel}}^{\text{max}}$ be the maximal credal set it delimits. Then for all $A \subseteq \Omega$,

$$\inf_{\text{Pr} \in \mathcal{F}_{\text{Bel}}^{\text{max}}} \text{Pr}(A | C) = \inf_{\text{Pr} \in \mathcal{F}_{\text{Bel}}^{\text{max}} | C} \text{Pr}(A) = \text{Bel}(A |_{23} C).$$

Let us insist that in the statement of Theorem 1, the prior Bel delimits a maximal credal set but the posterior Bel($\cdot |_{23} C$) does not. In general indeed, conditioned credal sets are not maximal: there may be probability measures Pr such that $\text{Bel}(A |_{23} C) \le \text{Pr}(A) \le \text{Pl}(A |_{23} C)$ and $\text{Pr} \notin \mathcal{F}_{\text{Bel}}^{\text{max}} | C$.

Note that Dempster's and geometric conditioning also have clear credal interpretations in terms of likelihood, as they amount to condition only a subset of the delimited credal set. Dempster's conditioning conditions only the most likely probability measures of the credal set, i.e. the ones maximizing $\text{Pr}(C)$, while geometric conditioning works on the less likely ones [33].

### 3.3.2. Commutativity of conditioning methods

Commutativity is an essential property that any conditioning method should satisfy to consider sequential knowledge updating. It corresponds to the natural assumption that acquiring several new pieces of information leads to the same updated knowledge whatever the order they are acquired.

**Definition 10 (Conditioning's Commutativity).** A conditioning $|_f$ commutes iff for any belief function $\text{Bel}: 2^\Omega \to [0, 1]$ and any subsets $C_1$ and $C_2$ of $\Omega$ such that $C_1 \cap C_2 \neq \emptyset$, it holds that for all $A$:

$$\text{Bel}(A |_f C_1 |_f C_2) = \text{Bel}(A |_f C_2 |_f C_1) = \text{Bel}(A |_f C_1 \cap C_2).$$

When the conditioning commutes, the updated knowledge does not depend on $C_1$ is acquired before, after or at the same time $C_2$ is acquired. From their BPA expressions, it is clear that Dempster's conditioning and geometric conditioning do commute, hence they seem well-defined and are suited for sequential conditioning. Credal set conditioning also commutes, since the probabilistic conditioning does. On the contrary, Bayesian conditioning fails to satisfy this property. This follows from the inability to represent non-maximal credal sets using their lower and upper probabilities only [18].<fnref n="8" />

<footnote n="8">⁸ For example, let $\Omega = \{r, b, w, y\}$, $m_5$ such that $m(\{r\}) = m(\{b\}) = m(\{b, w, y\}) = 1/3$, $C_1 = \{r, b, w\}$ and $C_2 = \{r, b, y\}$. It holds that $\text{Bel}_5(\{b\} |_{23} C_1 |_{23} C_2) = 2/5$ but $\text{Bel}_5(\{b\} |_{23} C_1 \cap C_2) = 1/2$, the latter being the intended result (as it is the lower probability in $\mathcal{T}_{\text{Bel}_5} | C_1 | C_2 = \mathcal{T}_{\text{Bel}_5}^{\text{max}} | C_1 \cap C_2$). This example is discussed in detail in Subsection 6.4 to illustrate how and why uncertain evidence resolves this issue.</footnote>
### 3.3.3. Dilation

When the knowledge is imprecise, a surprising conditioning effect known as dilation may be observed. Consider the following credal example.

**Example 3 (The fighters and the coin flip [21]).** A wrestler and a boxer are scheduled to fight. Let $w$ (resp. $b$) denote the win of the wrestler (resp. the boxer). Also, let $h$ (resp. $t$) if a toss of a fair coin yields head (resp. tail).

Since we don't know the odds for the wrestler or the boxer to win, $\text{Bel}(w) = \text{Bel}(b) = 0$ and $\text{Pl}(w) = \text{Pl}(b) = 1$. On the contrary, the coin is known to be fair: $\text{Bel}(h) = \text{Pl}(h) = \text{Bel}(t) = \text{Pl}(t) = 1/2$. Formally, the frame of discernment is the cartesian product $\Omega = \{w, b\} \times \{h, t\}$, and the knowledge is captured by the BPA $m$ defined as follows:

$$m(\{wh, bh\}) = m(\{wt, bt\}) = \frac{1}{2} \quad \text{and} \quad m(B) = 0 \text{ otherwise.}$$

Suppose the wrestler wins: $C = \{wh, wt\}$. It follows that $\text{Bel}(h|_{\Omega} C) = \text{Bel}(t|_{\Omega} C) = 0$ and $\text{Pl}(h|_{\Omega} C) = \text{Pl}(t|_{\Omega} C) = 1$. Conversely, suppose the boxer wins. It also follows that $\text{Bel}(h|_{\Omega} C^c) = \text{Bel}(t|_{\Omega} C^c) = 0$ and $\text{Pl}(h|_{\Omega} C^c) = \text{Pl}(t|_{\Omega} C^c) = 1$. Whatever the winner, it seems that we forget the fairness of the coin. This surprising effect is known as dilation [36].

**Definition 11 (Dilation).** Let $\text{Bel}$ and $\text{Pl}$ be dual belief and plausibility functions over $\Omega$, $|_f$ a conditioning method, $A \subseteq \Omega$ and $C$ a partition of $\Omega$. $C$ dilates $A$ iff:

$$\forall C \in C, [\text{Bel}(A), \text{Pl}(A)] \subseteq [\text{Bel}(A|_f C), \text{Pl}(A|_f C)].$$

Whatever the observation $C \in C$, the posterior belief on $A$ is more imprecise that the prior belief. Generally, Bayesian conditioning dilates more than Dempster's conditioning: $\text{Bel}(A|_{\Omega} C) \le \text{Bel}(A|_{\Omega} C) \le \text{Pl}(A|_{\Omega} C) \le \text{Pl}(A|_{\Omega} C)$; and than geometric conditioning: $\text{Bel}(A|_{\Omega} C) \le \text{Bel}(A|_{\Omega} C) \le \text{Pl}(A|_{\Omega} C) \le \text{Pl}(A|_{\Omega} C)$ [22]. In Section 6 we show the dilation effect of Bayesian conditioning is only apparent: posterior beliefs are actually as precise as the prior ones (Theorem 11). This result hence also covers Dempster's and geometric conditioning.

## 4. Uncertain evidence

Belief Function Theory allows us to represent a partial knowledge. As illustrated by Example 1, where some testimonies are imprecise, and by Example 2, where the color of some balls is not precisely known, such knowledge is partial but obtained from reliable sources: we never doubt the validity of the provided information. The evidence may be weak and imprecise, but it is certain it has to be taken in account. BPAs, seen as mass functions or as random sets, directly capture such reliable but imprecise information, by associating reliable values to the subsets of $\Omega$. However, partial knowledge may also arise because of the information is doubtful: one does not know whether it should be taken in account.

Some approaches allowing for uncertainty on the evidence itself can be found in the literature. They are mainly of two types.

After Shafer [37, Chapter 4], several authors have considered simple support functions to represent doubtful evidence. A simple support function supports a single proposition $A \subseteq \Omega$ to a given extent $\sigma$; its BPA is such that $m(A) = \sigma$, $m(\Omega) = 1 - \sigma$ and $m(B) = 0$ otherwise. By combining several simple support functions, one can express complex evidence that is only partially reliable. However, this is not our focus. Support functions are used to express evidence that is *certainly* partially reliable (i.e., from $m(A) = \sigma$, we know that the evidence supporting $A$ must be considered with degree $\sigma$). In contrast, our goal is to model evidence where it is *uncertain* whether it is reliable or not (i.e., we do not know how much the evidence supporting $A$ should be taken in account, and $\sigma$ is not known).

Other authors have proposed considering interval values for the BPA, which allows for uncertainty about the evidence itself (e.g., [8,46,45]). This approach is closer to our objective, although not fully aligned. It involves considering a set of possible BPAs rather than a single representation of the uncertain evidence. Additionally, the upper and lower probabilities of such sets are generally not belief and plausibility functions, as with probability intervals [6], where only singletons are mapped to non-zero intervals.

### 4.1. The uncertain evidence framework

The framework proposed in this paper allows for expressing questionable evidence using unnormalized intervals that express neither mass nor probability. Importantly, the knowledge is described by a single set function that encodes for a single belief function; hence, the framework enhances the expressiveness of Belief Function Theory while it remains within its scope. It is first presented for the evidential interpretation, where intervals denote possible numbers of pieces of evidence. Its credal understanding, where intervals denote possible counts of individuals (e.g., colored balls in urns), will be addressed in Section 5. Belief values are set without referring neither to randomness nor to external decision processes or scales of canonical examples, making it easy to understand and to apply.

**Example 4 (Are the Testimonies Reliable?).** Mr. Jack has been murdered and only two people are suspected: Rose and Bryan. A reliable witness saw Rose enter Mr. Jack's building, but it is unknown whether the two other witnesses, who claim to have seen Bryan enter,
are reliable. Given that they were drunk at that time, it is not sure whether their testimonies should be taken in account: they may be reliable, but we do not know so.

As in classical evidential models, modeling starts with the definition of a finite frame of discernment $\Omega = \{\omega_1, \dots, \omega_n\}$ consisting of mutually exclusive atomic propositions; the subsets of $\Omega$ represent compound propositions.

The knowledge comes from the testimonies: each of them is a piece of evidence that states for an atomic or compound proposition $B \subseteq \Omega$, as in Example 1. However, no external justification is needed to set the values: pieces of evidence are simply not weighted. To enforce a particular one, it is simply repeated. (Note that it is equivalent to consider relative weights: pieces of evidence are compared, not to an external scale, but to one another.)

Moreover, we need to express both *certain* and *uncertain* pieces of evidence because some testimonies are known to be reliable, but for some others, it is unclear whether they should be considered (they also may be reliable, but we do not know whether they are). In Example 4, the testimony stating $\{r\}$ is reliable but the two testimonies stating $\{b\}$ are questionable. Therefore, the modeling of knowledge merely consists of a pool of certain and/or uncertain pieces of evidence.

**Definition 12 (Piece of Evidence).** A piece of evidence over $\Omega$ is a pair $(B, x) \in \mathfrak{P}(\Omega) = (2^{\Omega} \setminus \{\emptyset\}) \times \{\text{!}, \text{?}\}$.

For the sake of readability, any $(B, x) \in \mathfrak{P}(\Omega)$ is denoted $Bx$. Pieces of evidence $B$! represent reliable testimonies stating the proposition $B$; they are said to be *certain*. Conversely, pieces of evidence $B$? represent questionable testimonies stating $B$; they are said to be *uncertain*. Piece of evidence cannot state the impossible proposition, that is, neither $\emptyset! \notin \mathfrak{P}(\Omega)$ nor $\emptyset? \notin \mathfrak{P}(\Omega)$.

**Definition 13 (Pool of Evidence).** A pool of evidence $E$ over $\Omega$ is a multiset whose elements are pieces of evidence over $\Omega$, that is, $E \in \mathbb{N}^{\mathfrak{P}(\Omega)}$.

Multisets are denoted using double round brackets. Example 1's pool of evidence is $E_1 = \{\{r\}!, \{b, w\}!, \{b, w\}!\}$ and Example 4's pool of evidence is $E_2 = \{\{r\}!, \{b\}?, \{b\}?\}$. Since a multiset $E$ allows for multiple occurrences of the same element, the values of its indicator function $I_E: \mathfrak{P}(\Omega) \to \mathbb{N}$ are natural numbers, e.g., $I_{E_2}(\{b\}?) = 2$.

Note that the above definition allows for representing the absence of knowledge. The empty pool of evidence $\{\}$ represents the total absence of knowledge: there is no evidence at all. This should not be confused with the so-called 'total ignorance', which arises when the available evidence is not informative (w.r.t. the frame of discernment $\Omega$): $\{\Omega!\}$ expresses that the evidence may support all propositions. More generally, it is possible we only dispose pieces of uncertain evidence, e.g., $\{\mathfrak{B}?\}$ models that the evidence stating $B$ may be totally unreliable, leading to a possible absence of knowledge. Such situations may arise, in particular, after the evidence is updated to take in account new information (Section 6).

Proportional pools of evidence contain the same pieces of evidence, not in equal numbers but with equal relative proportions. They convey similar knowledge; in this work they'll always lead to identical conclusions.<fnref n="9" />

**Definition 14 (Proportional Pools of Evidence).** Let $E$ and $E'$ be two pools of evidence over $\Omega$. $E$ and $E'$ are proportional if there exists $k$ such that $\forall B \in \mathfrak{P}(\Omega)$, it holds that $I_{E'}(B) = k \times I_E(B)$.

In such a case, we'll denote $E' = kE$, e.g., $2E_2 = \{\{r\}!, \{r\}!, \{b\}?, \{b\}?, \{b\}?, \{b\}?\}$ is the pool in which $E_2$'s pieces of evidence are repeated twice.

A pool of evidence describes the raw knowledge. To simplify its use, we consider its *evidence function*, which provides the minimum and maximum number of pieces of evidence stating each proposition exactly.

**Definition 15 (Evidence Function).** Let $E$ be a pool of evidence over $\Omega$. Its evidence function $M_E: 2^\Omega \to \mathbb{N}^2$ is defined for every $B \subseteq \Omega$ as follows:

$$M_E(B) = [I_E(B!), I_E(B!) + I_E(B?)]$$

Let $M_E^-(B)$ and $M_E^+(B)$ denote the bounds of $M_E(B)$, that is, $M_E(B) = [M_E^-(B), M_E^+(B)]$. $M_E^-(B)$ is the number of pieces of certain evidence stating $B$ exactly, while $M_E^+(B)$ is the total number of pieces of evidence stating $B$, whether certain or uncertain. Of course, $M_{kE}(B) = [kM_E^-(B), kM_E^+(B)]$ for all $k$, $E$ and $B$. The evidence function for Example 4 is:

$$M_{E_2}(\{r\}) = [1, 1], \quad M_{E_2}(\{b\}) = [0, 2] \quad \text{and} \quad M_{E_2}(B) = [0, 0] \text{ otherwise.}$$

An evidence function resembles an unnormalized BPA yielding interval values. Apart from the fact that it allows for empty pools of evidence, the information an evidence function conveys is subtly different from the one an interval-valued BPA conveys. Specifically, since a BPA is normalized, its values depend on the total number of pieces of evidence: they are proportions without a unit (i.e.

<footnote n="9">⁹ Note that if $k > 1$ then $kE$ contains more pieces of evidence than $E$: one may argue that $kE$'s evidence is more robust than $E$'s one, since it has a higher statistical significance.</footnote>
<table><caption>**Table 6**
Support count function of Example 1.</caption><thead><tr><th>A</th><th>Ø</th><th>{r}</th><th>{b}</th><th>{w}</th><th>{r, b}</th><th>{r, w}</th><th>{b, w}</th><th>{r, b, w}</th></tr></thead><tbody><tr><td>#<sup>-</sup><sub>E<sub>1</sub></sub>(A)</td><td>0</td><td>1</td><td>0</td><td>0</td><td>1</td><td>1</td><td>2</td><td>3</td></tr><tr><td>#<sup>+</sup><sub>E<sub>1</sub></sub>(A)</td><td>0</td><td>1</td><td>2</td><td>2</td><td>3</td><td>3</td><td>2</td><td>3</td></tr></tbody></table>

<table><caption>**Table 7**
Support count function of Example 4.</caption><thead><tr><th>A</th><th>Ø</th><th>{r}</th><th>{b}</th><th>{r, b}</th></tr></thead><tbody><tr><td>#<sup>-</sup><sub>E<sub>1</sub></sub>(A)</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>#<sup>+</sup><sub>E<sub>2</sub></sub>(A)</td><td>0</td><td>1</td><td>2</td><td>3</td></tr></tbody></table>

“percentages”). On the contrary, evidence functions yield commensurable values that only depend on the considered propositions: $M_E(B)$ only depends on the pieces of evidence stating $B$.

As for example, the knowledge of Example 4 may be described using “mass intervals” (these are probability intervals [6]). The proportion of testimonies stating $\{r\}$ is between $1/3$ (if the two other testimonies are reliable) and $1$ (if they are not), hence the mass interval should be $m(\{r\}) = [\frac{1}{3}, 1]$. This value is imprecise. However, the imprecision does not stem from the piece of evidence stating $\{r\}$ but rather from the pieces of uncertain evidence stating other propositions. We know exactly the number of pieces of evidence stating $\{r\}$; this is accurately captured by $M_{E_2}(\{r\}) = [1, 1]$.

This observation applies when some pieces of evidence are uncertain. If the evidence is totally certain, then $M_E^- = M_E^+$: all values of $M_E$ are single-point intervals of the form $[x, x]$ and $M_E$ is analogous to an unnormalized BPA, for which all the evidence is reliable and taken for granted.

**Definition 16 (Pool of Certain Evidence).** A pool of evidence $E$ over $\Omega$ is said to be certain if and only if all its pieces of evidence are certain: $I_E(B?) = 0$, for all $B \subseteq \Omega$.

**Lemma 1.** Let $E$ be a non-empty pool of certain evidence. Then $M_E^- = M_E^+$ is an unnormalized BPA.

A piece of certain evidence stating the proposition $B$ exactly necessarily supports all the propositions $A$ such that $B \subseteq A$. It is also possible, but not sure, that it supports the propositions $A$ such that $B \cap A \neq \emptyset$. In contrast, a piece of uncertain evidence only possibly supports such propositions, since it is not sure whether it should be taken in account. The minimum and maximum numbers of supporting pieces of evidence are provided by the *support count function* (or simply *count function*) denoted $\#_E$.

**Definition 17 (Support Count Function).** Let $E$ be a pool of evidence over $\Omega$. Its support count function $\#_E: 2^\Omega \to \mathbb{N}^2$ is defined for any $A \subseteq \Omega$ as follows:

$$ \#_E(A) = \left[ \sum_{B \subseteq A} M_E^-(B), \sum_{B \cap A \neq \emptyset} M_E^+(B) \right]. $$

Let $\#_E^-(A)$ and $\#_E^+(A)$ denote the bounds of $\#_E(A)$, that is, $\#_E(A) = [\#_E^-(A), \#_E^+(A)]$. The support count functions for Examples 1 and 4 are listed in Tables 6 and 7.

Since $E_1$ is certain, the support count of $\Omega$ is known: $\#_E^-(\Omega) = \#_E^+(\Omega)$. Conversely $E_2$ is uncertain and $\#_E^-(\Omega) < \#_E^+(\Omega)$.

**Lemma 2.** A pool of evidence over $\Omega$ is certain iff $\#_E^-(\Omega) = \#_E^+(\Omega)$.

It follows from Definition 17 that $\#_E^-$ is an unnormalized belief function with BPA $M_E^-$, and that $\#_E^+$ is an unnormalized plausibility function with BPA $M_E^+$. However, they are not dual (there exist no normalization constants $x$ and $y$ such that $\forall A, x \#_E^-(A) + y \#_E^+(A^c) = 1$) unless the evidence is precise.

**Lemma 3.** Let $E$ be a non-empty pool of certain evidence over $\Omega$. Then $\#_E^-$ and $\#_E^+$ are unnormalized dual belief and plausibility and their unnormalized BPA is $M_E^- = M_E^+$. The normalization factor is $\#_E^-(\Omega) = \#_E^+(\Omega)$.

$\#_E(A)$ provides the minimum and maximum support for the proposition $A$ in terms of number of supporting pieces of evidence. Finally, we wish to determine the proportion of the evidence supporting $A$. This proportion may not be precisely known but can be bounded by the extreme values of $\frac{\text{Evidence supporting } A}{\text{Total evidence}}$. In this fraction, it is crucial that the numerator and the denominator are consistent with each other, meaning that a piece of evidence considered as supporting $A$ on the one side must also support $A$ on the other side, and conversely.
<table><caption>**Table 8**
Support proportion function for Example 1.</caption><thead><tr><th>A</th><th>Ø</th><th>{r}</th><th>{b}</th><th>{w}</th><th>{r, b}</th><th>{r, w}</th><th>{b, w}</th><th>{r, b, w}</th></tr></thead><tbody><tr><td>P<sup>-</sup><sub>E<sub>1</sub></sub>(A)</td><td>0/3</td><td>1/3</td><td>0/3</td><td>0/3</td><td>1/3</td><td>1/3</td><td>2/3</td><td>3/3</td></tr><tr><td>P<sup>+</sup><sub>E<sub>1</sub></sub>(A)</td><td>0/3</td><td>1/3</td><td>2/3</td><td>2/3</td><td>3/3</td><td>3/3</td><td>2/3</td><td>3/3</td></tr></tbody></table>

<table><caption>**Table 9**
Support proportion function for Example 4.</caption><thead><tr><th>A</th><th>Ø</th><th>{r}</th><th>{b}</th><th>{r, b}</th></tr></thead><tbody><tr><td>P<sup>-</sup><sub>E<sub>1</sub></sub>(A)</td><td>0/3</td><td>1/3</td><td>0/1</td><td>1/1</td></tr><tr><td>P<sup>+</sup><sub>E<sub>1</sub></sub>(A)</td><td>0/1</td><td>1/1</td><td>2/3</td><td>3/3</td></tr></tbody></table>

**Theorem 2.** Let E be a pool of evidence. The proportion of evidence supporting A in E is bounded by:

$$P_{E}^{-}(A) = \frac{\#_{E}^{-}(A)}{\#_{E}^{-}(A) + \#_{E}^{+}(A^{c})} \quad \text{and} \quad P_{E}^{+}(A) = \frac{\#_{E}^{+}(A)}{\#_{E}^{+}(A) + \#_{E}^{-}(A^{c})}.$$

We define $P_E(A) = [P_E^-(A), P_E^+(A)]$ and refer to $P_E$ as the support proportion function of E, or simply as its proportion function. Those for Examples 1 and 4 are listed in Tables 8 and 9. The unsimplified fractions highlight that these proportions depend on the total number of pieces of evidence, which is known for $E_1$ but uncertain for $E_2$. In particular, note that $P_{E_2}({\color{red} \{r\}})$ ranges from 1/3 to 1/1 because of the uncertainty in the denominator only.

Since we do not prohibit empty pools of evidence, the expression above may result in the value $\frac{0}{0}$. This does not pose a problem in this paper, where this value signifies both “No piece of evidence supports A” (since $\frac{0}{x} = 0$) and “All pieces of evidence support A” (since $\frac{x}{x} = 1$), which is logically true when considering the empty set. $\frac{0}{0}$ is then identified with both 0 and 1. To avoid this behavior for all A, it is necessary to include at least one piece of certain evidence in the pool.

$P_E^-(A)$ and $P_E^+(A)$ are the minimum and maximum proportions of evidence supporting A in E. For all $E$, $P_E^-$ and $P_E^+$ are dual measures.

**Lemma 4.** Let E be a pool of evidence. Then for all $A$, $P_E^-(A) + P_E^+(A^c) = 1$.

More precisely, $P_E^-$ and $P_E^+$ are dual belief and plausibility functions. When the evidence is certain, it directly follows from $M_E^- = M_E^+$ is an unnormalized BPA (Lemma 1).

**Lemma 5.** Let E be a non-empty pool of certain evidence over $\Omega$. Then $P_E^-$ and $P_E^+$ are dual belief and plausibility functions; their BPA m is defined by $m(B) = M_E^-(B)/\#_E^-(\Omega)$.

In general, $P_E^-$ and $P_E^+$ are dual belief and plausibility functions, even if the evidence is uncertain. The general theorem relies on [18] and on Theorem 7 (Section 6). Stating it now makes the reading easier.

**Theorem 3.** Let E be a non-empty pool of evidence. Then $P_E^-$ and $P_E^+$ are dual belief and plausibility functions.

The presented framework allows for modeling uncertainty on the evidence itself in a straightforward manner. As Example 4 shows, pools of uncertain evidence, or equivalently interval-valued evidence functions, make the framework intuitive and easy to use. Moreover, minimum and maximum proportion functions are dual belief and plausibility functions, even if the evidence is uncertain. This framework hence extends the expressiveness of evidential models while remaining within the scope of the Belief Function Theory!

## 4.2. Complexity of evidence functions and their algorithms

As for BPAs, evidence functions are usually sparse. They provide a succinct way to represent the knowledge when the evidence is uncertain. Algorithms devoted to uncertain evidence would benefit from relying on evidence functions rather than on pools of evidence or count functions. Here, we consider them as lists of pairs $(B, M_E(B))$ where B is a focal element and $M_E(B)$ is the interval associated with it, for all $B \subseteq \Omega$ such that $M_E(B) \neq [0, 0]$. Computing support counts $\#_E(A)$ and proportions $P_E(A)$ would then be linear-time operations (assuming set-operations in $O(1)$), since it requires only iterating over the list. We can extend the notion of $k$-additivity (Definition 5) to evidence functions.

**Definition 18 (k-additivity).** Let E be a pool of evidence over $\Omega$.

$$M_E \text{ is } k\text{-additive} \iff (\forall B, |B| > k \implies M_E(B) = [0, 0]).$$
<table><caption>**Table 10**
$E_3$'s support count and proportion functions.</caption><thead><tr><th>A</th><th>Ø</th><th>{r}</th><th>{b}</th><th>{r, b}</th></tr></thead><tbody><tr><td>#<sup>-</sup><sub>E<sub>1</sub></sub>(A)</td><td>0</td><td>1</td><td>0</td><td>3</td></tr><tr><td>#<sup>+</sup><sub>E<sub>3</sub></sub>(A)</td><td>0</td><td>3</td><td>2</td><td>3</td></tr><tr><td>P<sup>-</sup><sub>E<sub>1</sub></sub>(A)</td><td>0/3</td><td>1/3</td><td>0/3</td><td>3/3</td></tr><tr><td>P<sup>+</sup><sub>E<sub>1</sub></sub>(A)</td><td>0/3</td><td>3/3</td><td>2/3</td><td>3/3</td></tr></tbody></table>

As for BPAs, in general, evidence functions may have up to $2^{|\Omega|} - 1$ focal elements. Hence their representations may require exponential space, and their dedicated algorithms may require exponential time. However, by restricting algorithms to k-additive evidence functions, we ensure that the corresponding complexities will be polynomial, in $O(|\Omega|^k)$.

Since k-additive BPAs are a special case of k-additive evidence functions (which are recovered when the evidence is certain), algorithms dedicated to evidence functions will be more expressive than, yet as tractable as those devoted to BPAs. In particular, note that 1-additive evidence functions are more expressive than probabilities, as shown by Example 4: $M_{E_2}$ is 1-additive but does not correspond to a precise probability distribution.

### 4.3. Pools of evidence having equal proportions

Pools of evidence, evidence functions and count functions can be recovered from one another: they are different views of the same knowledge. In contrast, it is not possible to recover them from the proportion function. Several distinct pools of evidence may indeed have identical proportion functions. This is especially true for proportional pools: $P_E = P_{kE}$ for any E and $k > 0$. It seems intuitive since E and $kE$ convey similar knowledge.

**Lemma 6.** Let E be a pool of evidence over $\Omega$. Then for all $k \neq 0$, $P_E = P_{kE}$.

When restricting the modeling to certain evidence, only proportional pools of evidence result in equal proportions.

**Theorem 4.** Let E and E' be two pools of certain evidence. Then:

$$P_E = P_{E'} \iff \exists k, E = kE'$$

Classical evidential models typically make the (implicit) assumption that the evidence is certain (i.e., it is known how reliable it is and how much it should be taken in account). In such models, it is appropriate to consider the knowledge is captured by a belief function, or equivalently, by its BPA, as it encodes proportional pools of evidence which all convey similar knowledge and lead to equal conclusions. Studying the belief function or the BPA amounts to studying all such pools at once.

Conversely, when certain evidence is not assumed, relying on the belief function or BPA may be problematic. There are indeed pools of evidence that have equal proportions of supporting evidence—i.e., equal belief functions and BPA—but which are not proportional. These pools do not convey similar knowledge and generally do not lead to identical conclusions. To distinguish between them, one cannot represent them using belief functions or BPAs. Consider the pool of evidence $E_3 = \{\{r\}!, \{r, b\}!, \{r, b\}!\}$ (the careful reader will recognize its similarity to $m(\cdot |_{\Omega_3} C)$ in Table 5). Its proportion function is equal to that of $E_2 = \{\{r\}!, \{b\}?, \{b\}?\}$ even though $E_2$ and $E_3$ are not proportional! Representing $P_{E_2}$ and $P_{E_3}$ with unsimplified fractions (Tables 9 and 10) helps explain this phenomenon. For example, $P_{E_2}^+(\{r\}) = 1/1$ and $P_{E_3}^+(\{r\}) = 3/3$, both equating to 1, but for different reasons. It is indeed possible that there is only one reliable piece of evidence in $E_2$ (the one supporting $\{r\}$); whereas in $E_3$, there are necessarily three reliable pieces of evidence, all may support $\{r\}$.

Even though this example is very simple, it demonstrates that fundamentally different pieces of knowledge can have equal proportions of supporting evidence when uncertain evidence is considered. In such case, neither the belief function nor the BPA are sufficient to describe it. This observation reinforces the claim that mass functions are inadequate for handling uncertain evidence. It also applies to the credal interpretation of BPAs in terms of random sets. Moreover, it may be problematic for other credal interpretations of Belief Function Theory that rely only on the belief and plausibility functions, since there is no explicit assumption of certain evidence in these cases. (This issue also extends to the more general theory of lower and upper probabilities, although it falls outside the scope of this paper.)

### 5. Credal understanding of uncertain evidence

As for the conventional framework presented in Section 3, evidential and credal interpretations of uncertain evidence rely on the same mathematical definitions. They only differ in semantics.
## 5.1. Credal understanding of uncertain evidence

Urns filled with colored balls provide good examples of discrete probabilities, aligning with various views of probability: as proportions of individuals (statistics), as the likelihood of events occurring (propensities), and as long-term trends (frequencies). The contents of urns are essentially populations, and random draws lead to propensities and frequencies. Moreover, probability values are easily obtained by counting the considered balls and normalizing by the total number of balls in the urn. In this discussion, we'll reason in terms of contents—hence, probabilities equate to proportions—while random draws are left for the reader to consider.

The framework for uncertain evidence presented in the previous section is easily understood credally using the urn analogy. Let $\Omega$ be a set of colors. Pools of evidence and evidence functions over $\Omega$ can be interpreted as describing the placements of colored balls in an initially empty urn. If the knowledge is both certain and precise, each piece of evidence $\{\omega\}$! represents the placement of a ball with color $\omega$; the content is perfectly known. Generally, any piece of certain evidence $B$! represents the placement of a ball where we only know that its color is $\omega \in B$, while any piece of uncertain evidence $B$? indicates that we are not sure whether the ball has been placed (but if it was, its color would be $\omega \in B$). Following this analogy, it follows from the definitions that the count function $\#_E$ provides the minimum and maximum numbers of balls for each color or set of colors, while the proportion function $P_E$ provides the corresponding minimum and maximum proportions. These values are essentially lower and upper probabilities.

**Example 5 (The Up-To-Three-Balls Urn).** An urn has been filled with one ball taken from BAG I and up to two other balls taken from BAG III. BAG I contains only red balls, and BAG III contains only black balls.

While being credal, this example clearly allows for the same modeling as evidential Example 4: $E_2 = \{\{(r)\!1, \{b\}\!, \{b\}\?\}\}$ is the set of balls that are, or may be, in the urn: it contains one red ball and between zero and two black balls. $M_{E_2}$ provides the same information:

$$M_{E_2}(\{r\}) = [1, 1] \qquad M_{E_2}(\{b\}) = [0, 2] \qquad M_{E_2}(B) = [0, 0] \text{ otherwise.}$$

$M_{E_2}$ can be understood as an “unnormalized interval-valued random set”, induced by the mapping BAG I $\mapsto \{r\}$, BAG III $\mapsto \{b\}$ and the function $p(\text{BAG I}) = [1, 1]$, $p(\text{BAG III}) = [0, 2]$. From $M_{E_2}$ we deduce $\#_{E_2}$, which provides the ranges of the number of balls. For instance, there are between 1 and 3 balls in the urn: $\#_{E_2}(\Omega) = [1, 3]$. Finally, one can compute the minimum and maximum proportions. For instance, there is at least $1/3$ of the balls that are red: $P_{E_2}(\{r\}) = \left[\frac{1}{3}, \frac{1}{1}\right]$.

As for pieces of evidence, it is important to note that $M_{E_2}$'s intervals denote uncertainty not in terms of proportions, but in terms of numbers, e.g., $M_{E_2}(\{r\}) = [1, 1]$ represent the placement of exactly one red ball in the urn. The uncertainty about the proportion of red balls is a consequence of normalization, because the number of black balls is uncertain.

## 5.2. Credal sets associated with evidence functions

Possible content configurations can be easily identified from $E$ or $M_E$ by considering the possible placements indicated by the evidence. For each $B$, $M_E(B) = [x, y]$ represents between $x$ and $y$ placements of balls with color $\omega \in B$. Formally, a content configuration $c: \Omega \to \mathbb{N}$ is possible according to $E$ if and only if there exist allocations $\gamma^B: B \to \Omega$ such that for all $B \subseteq \Omega$, $\sum_{\omega \in B} \gamma^B(\omega) \in M_E(B)$, and $c(\omega) = \sum_{B \subseteq \Omega} \gamma^B(\omega)$. (It is closely related to the allocation of probabilities in random sets.) Possible configurations are, of course, constrained by the possible numbers of balls provided by $\#_P$: any possible configuration $c$ satisfies $\#_E^-(A) \le \sum_{\omega \in A} c(\omega) \le \#_E^+(A)$, for all $A \subseteq \Omega$. Non-empty configurations are unnormalized probability distributions. Letting $Cf: 2^\Omega \to \mathbb{N}$ denote additive measures<fnref n="10" /> (i.e. $Cf(A) = \sum_{\omega \in A} c(\omega)$), the above expression can be rewritten as $\#_E^-(A) \le Cf(A) \le \#_E^+(A)$.

**Definition 19 (Configuration Set).** Let $E$ be a pool of evidence over $\Omega$. Its configuration set is:

$$C_E = \{ \text{Cf} | \forall A, \#_E^-(A) \le \text{Cf}(A) \le \#_E^+(A) \}.$$

Definition 19 resembles Definition 4 of maximal credal sets. However, as noted in the previous section, $\#_E^-$ and $\#_E^+$ are not dual functions, unless the evidence is certain. In contrast to Definition 4, the two inequalities are required in this definition, one is not implied by the other.

From the content configurations allowed by $E$, one can compute its associated credal set by simply normalizing each configuration. Because urns contain only finite discrete numbers of balls, there are only finite number of possible configurations. Hence their corresponding credal sets are discrete and contain only $\mathbb{Q}$-valued probability measures.

**Definition 20 (Associated Discrete Credal Set).** Let $E$ be a pool of evidence over $\Omega$. Its discrete credal set is:

$$\mathcal{F}_E = \{ B \mapsto \text{Cf}(B)/\text{Cf}(\Omega) | \text{Cf} \in C_E \}.$$

<footnote n="10">¹⁰ In contrast to the approach developed in [33], all possible additive measures are equally “likely.” Cf($\Omega$) is nothing more than the total number of balls if Cf is the actual configuration; it does not provide insights into the likelihood of Cf.</footnote>
<table><caption>**Table 11**
Evidence, count and proportion functions for $E_2$ and $E_3$.</caption><thead><tr><th>A</th><th>θ</th><th>{r}</th><th>{b}</th><th>{r, b}</th></tr></thead><tbody><tr><td>M<sub>E<sub>2</sub></sub>(A)</td><td></td><td>[1, 1]</td><td>[0, 2]</td><td></td></tr><tr><td>#<sup>-</sup><sub>E<sub>2</sub></sub>(A)</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>#<sup>+</sup><sub>E<sub>2</sub></sub>(A)</td><td>0</td><td>1</td><td>2</td><td>3</td></tr><tr><td>P<sup>-</sup><sub>E<sub>2</sub></sub>(A)</td><td>0/3</td><td>1/3</td><td>0/1</td><td>1/1</td></tr><tr><td>P<sup>+</sup><sub>E<sub>2</sub></sub>(A)</td><td>0/1</td><td>1/1</td><td>2/3</td><td>3/3</td></tr></tbody></table>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 3.** $p_r$ as a function of $p_b$ for all possible probability distributions $(p_r, p_b)$ in $\mathcal{F}_{5E_2}$ (red, on the left) and in $\mathcal{F}_{5E_3}$ (blue, on the right).</figcaption></figure>

As already noted for the evidential understanding, the count functions $\#_E^-$ and $\#_E^+$ are more informative than the proportions $P_E^-$ and $P_E^+$ they induce. Consider the two examples closing the previous section. They convey different knowledge: credally, $E_2 = \{\{\{r\}\}, \{\{b\}\}, \{\{b\}\}\}$ describes the Up-To-Three-Balls Urn (Example 5) while $E_3 = \{\{\{r\}\}, \{\{r, b\}\}, \{\{r, b\}\}\}$ describes an urn with exactly 3 balls: one is red and the two others may be either red or black. Yet they have the same proportion functions. Their evidence, count and proportion functions are recalled in Table 11.

Their respective discrete credal sets differ. Let tuples $(p_r, p_b)$ denote probability distributions over $\Omega = \{r, b\}$. It holds that $(\frac{1}{2}, \frac{1}{2}) \in \mathcal{F}_{E_2}$ but $(\frac{1}{2}, \frac{1}{2}) \notin \mathcal{F}_{E_3}$, and that $(\frac{1}{3}, \frac{2}{3}) \in \mathcal{F}_{E_3}$ but $(\frac{1}{3}, \frac{2}{3}) \notin \mathcal{F}_{E_2}$. In fact, these two credal sets are distributed differently. This becomes evident when considering proportional pools of evidence $kE_2$ and $kE_3$ (describing urns with $k$ times more balls), as shown in Fig. 3 using $k=5$.

Because credal sets associated with pools of evidence only contain $\Omega$-valued probabilities, they are never convex, even considering proportional pools of evidence $kE$ when $k \to \infty$. To consider $\mathbb{R}$-valued credal sets, one may extend them by continuity. More simply, one may directly allow content configurations to involve real numbers by letting evidence functions yield real-valued intervals (for instance, one may suppose they provide amounts of liquid instead of counts of balls, e.g., one may rephrase Example 5 with 1 liter of red water and up to 2 liters of black oil).<fnref n="11" />

**Definition 21** (Associated Convex Credal Set). Let $E$ be a pool of evidence over $\Omega$. Its convex credal set is:

$$ \mathcal{F}_E^* = \{ \Pr \in \mathrm{Prob}(\Omega) \mid \exists k, \forall A, \#_E^-(A) \le k \Pr(A) \le \#_E^+(A) \}. $$

Considering convex credal sets, it holds that $\mathcal{F}_{E_2}^*$ and $\mathcal{F}_{E_3}^*$ contain the same probability distributions, that is, $(p_r, p_b) \in \mathcal{F}_{E_2}^* \iff (p_r, p_b) \in \mathcal{F}_{E_3}^*$. They are both maximal in the sense of Definition 4. However, one might argue they should be treated distinctly because they differ in density, as illustrated above in the rational setting. They indeed do not convey similar knowledge: the evidence is certain in $E_3$ but uncertain in $E_2$, making lower values of $\Pr(\{r\})$ more common in $\mathcal{F}_{E_2}^*$ than in $\mathcal{F}_{E_3}^*$.

When $|\Omega| = 2$, convex credal sets are necessarily maximal. Conversely, when dealing with higher dimensions, credal sets induced by uncertain evidence are generally not maximal. Consider for example an urn $E_5$ containing four red balls, four black balls and up to four other balls that may be either black or white. The evidence function is:

$$ M_{E_5}(\{r\}) = [4, 4] \qquad M_{E_5}(\{b\}) = [4, 4] \qquad M_{E_5}(\{b, w\}) = [0, 4]. $$

The count and proportion functions of $E_5$ are given in Table 12.

<footnote n="11">¹¹ Considering a continuum of amounts of liquid is an approximation: liquids are made of discrete number of molecules. To the current state of knowledge, it is not sure real numbers exists in the real world, it is possible Nature is made of natural numbers only, e.g., discrete numbers of molecules or of energy quanta.</footnote>
<table><caption>**Table 12**
Count and proportion functions for $E_5$.</caption><thead><tr><th>A</th><th>∅</th><th>{r}</th><th>{b}</th><th>{w}</th><th>{r, b}</th><th>{r, w}</th><th>{b, w}</th><th>{r, b, w}</th></tr></thead><tbody><tr><td>#<sup>-</sup><sub>E<sub>5</sub></sub>(A)</td><td>0</td><td>4</td><td>4</td><td>0</td><td>8</td><td>4</td><td>4</td><td>8</td></tr><tr><td>#<sup>+</sup><sub>E<sub>5</sub></sub>(A)</td><td>0</td><td>4</td><td>8</td><td>4</td><td>12</td><td>8</td><td>8</td><td>12</td></tr><tr><td>P<sup>-</sup><sub>E<sub>5</sub></sub>(A)</td><td>0</td><td>1/3</td><td>1/3</td><td>0</td><td>2/3</td><td>1/3</td><td>1/2</td><td>1</td></tr><tr><td>P<sup>+</sup><sub>E<sub>5</sub></sub>(A)</td><td>0</td><td>1/2</td><td>2/3</td><td>1/3</td><td>1</td><td>2/3</td><td>2/3</td><td>1</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Geometric representation of the discrete credal set (red dots) and the convex credal set (red triangle) of $E_5$.</figcaption></figure>

$P_{E_5}^-$ and $P_{E_5}^+$ are dual belief and plausibility functions. Fig. 4 illustrates the discrete and convex credal sets of $E_5$, the latter being non-maximal. It is indeed the non-maximal credal set we referred to in Introduction.

# 6. Updating uncertain evidence

## 6.1. Three transfer-based evidence updates

We now turn to how to update uncertain evidence when acquiring new information. Back to the evidential interpretation, let $C \subseteq \Omega$ be the new information meaning “only the propositions $\omega \in C$ are possible.” Taking $C$ in account involves updating the prior pool of evidence $E$ to a new one $E \mid C$ in which the frame of discernment $\Omega$ is restricted to $C$. As discussed in the introduction, there are various ways to account for the new information, depending on its nature. In all cases, however, the posterior evidence will only cover propositions $B \subseteq C$, since the atoms $\omega \notin C$ are now impossible.

Specifically, the prior pieces of evidence will be retained, removed or modified depending on whether they are fully compatible, incompatible or partially compatible with the new information $C$. Consider a piece of evidence $Bx$. If $B \subseteq C$, it is fully compatible with $C$ and should be retained as is. If $B \subseteq C^c$, the piece of evidence is fully incompatible and should be removed. Otherwise, $B \cap C \neq \emptyset$ and $B \cap C^c \neq \emptyset$: the piece of evidence is only partially compatible with $C$, as we don't know if it asserts for $\omega \in C$ or $\omega \notin C$. Depending on the nature of $C$, one may want to correct $Bx$, to discard it, or to consider both these two options.

Consider Example 1, where one witness seen Rose (the only female suspect) and the other two witnesses seen either Bryan or Walter (the two male suspects). We previously discuss that if one learns Walter has an alibi, it is clear the only suspects are Rose and Bryan ($C = \{r, b\}$), but it is unclear what is the appropriate update: it depends on the nature of the alibi. If Walter was abroad at the time the murder was committed, one wants to correct the two imprecise testimonies (the witnesses surely saw Bryan, i.e. $\{b, w\}$! becomes $\{b\}$!). Conversely, if Walter was present but is innocent, one wants to ignore these two testimonies (the witnesses surely saw Walter but this is not relevant anymore, i.e. $\{b, w\}$! is removed). One may also hesitate between these two updates, as for one doesn't know the nature of the alibi. In this case, one should consider that these testimonies may either accuse Bryan or be irrelevant (i.e. $\{b, w\}$! becomes $\{b\}$?).

Considering uncertain evidence, we can express these three updates in terms of transfer.

**Definition 22 (Update by Transfer).** Let $E$ be a pool of evidence over $\Omega$, $f: \mathfrak{P}(\Omega) \times 2^\Omega \to \mathfrak{P}(\Omega)$ a function and $C \subseteq \Omega$ a new information. The $f$-update of $E$ given $C$ is defined as:

$$ \mathrm{upd}_f(E, C) = f(E \mid C) \setminus \{\emptyset!, \emptyset?\}, $$

where $f(E \mid C)$ denotes the image of $E$ under $f(\cdot \mid C)$, i.e. $\mathrm{upd}_f(E, C)$ is the multiset consisting of non-vacuous pieces of evidence $f(Bx \mid C)$ for all $Bx$ in $E$.

This transfer algorithm is parameterized by a function $f$, which we will refer to as the transfer function, and which transforms each piece of evidence in the prior into a piece of evidence in the posterior—evidence stating a vacuous claim is then removed. Since
<table><caption>**Table 13**
The $\mathcal{D}$-update of $E_1$ given $C = \{r, b\}$.</caption><thead><tr><th>A</th><th>∅</th><th>{r}</th><th>{b}</th><th>{r, b}</th></tr></thead><tbody><tr><td>M<sub>E<sub>i|D</sub>C</sub>(A)</td><td></td><td>[1, 1]</td><td>[2, 2]</td><td></td></tr><tr><td>#<sup>-</sup><sub>E<sub>i|D</sub>C</sub>(A) = #<sup>+</sup><sub>E<sub>i|D</sub>C</sub>(A)</td><td>0</td><td>1</td><td>2</td><td>3</td></tr><tr><td>P<sup>-</sup><sub>E<sub>i|D</sub>C</sub>(A) = P<sup>+</sup><sub>E<sub>i|D</sub>C</sub>(A)</td><td>0</td><td>1/3</td><td>2/3</td><td>3/3</td></tr></tbody></table>

this is merely a mapping of pieces of evidence, any update by transfer may be implemented by a linear-time algorithm (relative to the number of focal elements and assuming $O(f) = 1$). Update behaviors will be captured depending on the transfer function $f$. The first update corrects the partially incompatible pieces of evidence by restricting them to $C$.

**Definition 23 (Correction Transfer Function).** The correction transfer function given $C$ is denoted $\mathcal{D}(\cdot | C)$ and defined as follows:

$$ \mathcal{D}(Bx | C) = (B \cap C)x. $$

Let $E|_{\mathcal{D}} C = \text{upd}_{\mathcal{D}}(E, C)$ denote the $\mathcal{D}$-update of $E$ given $C$. Its evidence function is easily characterized.

**Lemma 7.** The evidence function of $E|_{\mathcal{D}} C$ is such as:

$$ \begin{aligned} M_{E|_{\mathcal{D}} C}(B) &= \left[ \sum_{B' \cap C=B} M_{E}^{-}(B'), \sum_{B' \cap C=B} M_{E}^{+}(B') \right] & \text{if } \emptyset \neq B \subseteq C \\ M_{E|_{\mathcal{D}} C}(B) &= [0, 0] & \text{otherwise.} \end{aligned} $$

The notation $E|_{\mathcal{D}} C$ is justified by the following theorem stating that the $\mathcal{D}$-update generalizes Dempster's conditioning, which is recovered when the prior evidence is certain.

**Theorem 5.** Let $E$ be a non-empty pool of certain evidence over $\Omega$ and $C \subseteq \Omega$ such that $P_E^+(C) > 0$. Then $E|_{\mathcal{D}} C$ is certain, non-empty and satisfies for all $A$:

$$ P_{E|_{\mathcal{D}} C}^{-}(A) = P_{E}^{-}(A|_{\mathcal{D}} C) \quad \text{and} \quad P_{E|_{\mathcal{D}} C}^{+}(A) = P_{E}^{+}(A|_{\mathcal{D}} C). $$

$$ M_{E|_{\mathcal{D}} C}^{-} = M_{E|_{\mathcal{D}} C}^{+} \text{ is then the unnormalized BPA of } P_{E|_{\mathcal{D}} C}^{-}. $$

That is, $\mathcal{D}$-updating certain evidence amounts to conditioning the corresponding belief and plausibility functions (or equivalently, their BPA) using Dempster's conditioning. This leads to certain posterior evidence. Consider the $\mathcal{D}$-update of $E_1 = \{(\{r\}!, \{b, w\}!, \{b, w\}!\}$ given $C = \{r, b\}$: its pieces of evidence are mapped to $\mathcal{D}(\{r\}! | C) = \{r\}!$ and $\mathcal{D}(\{b, w\}! | C) = \{b\}!$; hence $E_1|_{\mathcal{D}} C = \{(\{r\}!, \{b\}!, \{b\}!\}$ (Table 13, to be compared with Table 3).

The second presented update retains only the totally compatible pieces of evidence while the others are removed. To achieve this, incompatible pieces of evidence are transferred to the empty set.

**Definition 24 (Removal Transfer Function).** The removal transfer function given $C$ is denoted $\mathcal{G}(\cdot | C)$ and defined as follows:

$$ \mathcal{G}(Bx | C) = \begin{cases} Bx & \text{if } B \subseteq C \\ \emptyset x & \text{otherwise.} \end{cases} $$

Let $E|_{\mathcal{G}} C = \text{upd}_{\mathcal{G}}(E, C)$ denote the $\mathcal{G}$-update of $E$ given $C$. Its evidence function is easily characterized.

**Lemma 8.** The evidence function of the $\mathcal{G}$-update of $E$ given $C$ is such as:

$$ M_{E|_{\mathcal{G}} C}(B) = M_E(B) \quad \text{if } B \subseteq C $$

$$ M_{E|_{\mathcal{G}} C}(B) = [0, 0] \quad \text{otherwise.} $$

The notation $E|_{\mathcal{G}} C$ is justified by the following theorem stating that the $\mathcal{G}$-update generalizes geometric conditioning, which is recovered when the prior evidence is certain.
<table><caption>**Table 14**
The $\mathfrak{B}$-update of $E_1$ given $C = \{r, b\}$.</caption><thead><tr><th>A</th><th>Ø</th><th>{r}</th><th>{b}</th><th>{r, b}</th></tr></thead><tbody><tr><td>M<sub>E<sub>1|Ø</sub> C</sub>(A)</td><td></td><td>[1, 1]</td><td></td><td></td></tr><tr><td>#<sup>-</sup><sub>E<sub>1|Ø</sub> C</sub>(A) = #<sup>+</sup><sub>E<sub>1|Ø</sub> C</sub>(A)</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>P<sup>-</sup><sub>E<sub>1|Ø</sub> C</sub>(A) = P<sup>+</sup><sub>E<sub>1|Ø</sub> C</sub>(A)</td><td>0</td><td>1/1</td><td>0/1</td><td>1/1</td></tr></tbody></table>

**Theorem 6.** Let $E$ be a non-empty pool of certain evidence over $\Omega$, and $C \subseteq \Omega$ such that $P_E^-(C) > 0$. Then $E|_{\mathfrak{B}} C$ is certain, non-empty and satisfies for all $A$:

$$P_{E|\mathfrak{B}}^{-} C(A) = P_{E}^{-}(A|\mathfrak{B}, C) \quad \text{and} \quad P_{E|\mathfrak{B}}^{+} C(A) = P_{E}^{+}(A|\mathfrak{B}, C).$$

$$M_{E|\mathfrak{B}}^{-} C = M_{E|\mathfrak{B}}^{+} C \text{ is then the unnormalized BPA of } P_{E|\mathfrak{B}}^{-} C.$$

That is, $\mathfrak{B}$-updating certain evidence amounts to conditioning the corresponding belief and plausibility functions (or equivalently, their BPA) using geometric conditioning. This leads to certain posterior evidence. Consider $\mathfrak{B}$-updating $E_1$ given $C = \{r, b\}$: the pieces of evidence are mapped to $\mathfrak{B}(\{r\} | C) = \{r\}$! and $\mathfrak{B}(\{b, w\} | C) = \emptyset$!; hence $E_1|_{\mathfrak{B}} C = \{\{r\}\}$ (Table 14, to be compared with Table 4).

Finally, to consider a hybrid approach between these two update behaviors, pieces of evidence that are partially incompatible with $C$ should be both restricted to $C$ and made uncertain.

**Definition 25 (Hybrid Transfer Function).** The hybrid transfer function is denoted $\mathfrak{B}(\cdot|C)$ and defined as follows:

$$\mathfrak{B}(Bx|C) = \begin{cases} Bx & \text{if } B \subseteq C \\ (B \cap C)? & \text{otherwise.} \end{cases}$$

Let $E|_{\mathfrak{B}} C = \text{upd}_{\mathfrak{B}}(E, C)$ denote the $\mathfrak{B}$-update of $E$ given $C$. Its evidence function is easily characterized.

**Lemma 9.** The evidence function of the $\mathfrak{B}$-update of $E$ given $C$ is such as:

$$M_{E|\mathfrak{B}} C(B) = \begin{cases} M_E^-(B), & \forall \emptyset \neq B \subseteq C \\ \sum_{B' \cap C=B} M_E^+(B') & \end{cases} \\ M_{E|\mathfrak{B}} C(B) = [0, 0] \qquad \text{otherwise.}$$

Note that it captures well the extreme cases of both the $\mathfrak{D}$-update and the $\mathfrak{B}$-update: $\forall B$, $M_{E|\mathfrak{B}}^-(B) = M_{E|\mathfrak{B}}^-(B)$ and $M_{E|\mathfrak{B}}^+(B) = M_{E|\mathfrak{B}}^+(B)$.

The notation $E|_{\mathfrak{B}} C$ is justified by the following theorem stating that the $\mathfrak{B}$-update generalizes Bayesian conditioning, which is recovered when the prior evidence is certain. Note that it generally does not allow recovering the conditioned BPA $m(\cdot|_{\mathfrak{B}} C)$ since the posterior pool of evidence $E|_{\mathfrak{B}} C$ is generally uncertain, even when the prior is certain.

**Theorem 7.** Let $E$ be a non-empty pool of certain evidence over $\Omega$ and $C \subseteq \Omega$ such that $P_E^-(C) > 0$. Then $E|_{\mathfrak{B}} C$ is non-empty and verifies $\forall A$:

$$P_{E|\mathfrak{B}}^{-} C(A) = P_{E}^{-}(A|\mathfrak{B}, C) \quad \text{and} \quad P_{E|\mathfrak{B}}^{+} C(A) = P_{E}^{+}(A|\mathfrak{B}, C).$$

That is, $\mathfrak{B}$-updating certain evidence amounts to conditioning the corresponding belief and plausibility functions using Bayesian conditioning, but this does not amount to conditioning their BPA since the posterior evidence is generally uncertain. Consider $\mathfrak{B}$-updating $E_1$ given $C = \{r, b\}$: the pieces of evidence are mapped to $\mathfrak{B}(\{r\} | C) = \{r\}$! and $\mathfrak{B}(\{b, w\} | C) = \{b\}$?; hence $E_1|_{\mathfrak{B}} C = \{\{r\}\!, \{b\}\?, \{b\}\?\} = E_2$, from which it is not clear how to recover $m_2(\cdot|_{\mathfrak{B}} \{r, b\})$ (Table 15, to be compared with Table 5).

The $\mathfrak{D}$-, $\mathfrak{B}$- and $\mathfrak{B}$-updates all are updates by transfer. Since they are merely mappings of pieces of evidence, they all can be implemented by linear-time algorithms (relative to the number of focal elements and assuming set operations in $O(1)$). In particular, the $\mathfrak{B}$-update is linear while Bayesian conditioning is exponential. Moreover, by applying one of these three updates, focal elements are either left unmodified, transferred to subsets of removed. Hence the number of focal elements cannot increase, as for the $k$-additivity degree. Again, note that the $\mathfrak{B}$-update preserves the number of focal elements and the degree of $k$-additivity while Bayesian conditioning does not.
<table><caption>**Table 15**
The $\mathfrak{B}$-update of $E_1$ given $C = \{r, b\}$.</caption><thead><tr><th>A</th><th>θ</th><th>{r}</th><th>{b}</th><th>{r, b}</th></tr></thead><tbody><tr><td>M<sub>E<sub>|<sub>θ</sub></sub>C</sub></sub>(A)</td><td></td><td>[1, 1]</td><td>[0, 2]</td><td></td></tr><tr><td>#<sup>-</sup><sub>E<sub>i</sub>|<sub>θ</sub></sub>C(A)</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>#<sup>+</sup><sub>E<sub>i</sub>|<sub>θ</sub></sub>C(A)</td><td>0</td><td>1</td><td>2</td><td>3</td></tr><tr><td>P<sup>-</sup><sub>E<sub>i</sub>|<sub>θ</sub></sub>C(A)</td><td>0</td><td>1/3</td><td>0/1</td><td>1/1</td></tr><tr><td>P<sup>+</sup><sub>E<sub>i</sub>|<sub>θ</sub></sub>C(A)</td><td>0</td><td>1/1</td><td>2/3</td><td>3/3</td></tr></tbody></table>

**Theorem 8.** Let $E$ be a pool of evidence over $\Omega$, $C \subseteq \Omega$ and $f \in \{\mathfrak{D}, \mathfrak{G}, \mathfrak{B}\}$ be a transfer function. Then:

$M_E$ has $n$ focal elements $\Rightarrow$ $M_{E|f}C$ has fewer than $n$ focal elements;

$M_E$ is $k$-additive $\Rightarrow$ $M_{E|f}C$ is $k$-additive.

Proportional pools of evidence convey similar knowledge (Definition 14). For this reason, their respective updates should also convey similar knowledge. It is ensured by the following theorem.

**Theorem 9.** Let $E$ be a pool of evidence over $\Omega$, $k > 0$ and $C \subseteq \Omega$. Then:

$$(kE)|_{\mathfrak{D}} C = k(E)|_{\mathfrak{D}} C$$

$$(kE)|_{\mathfrak{G}} C = k(E)|_{\mathfrak{G}} C$$

$$(kE)|_{\mathfrak{B}} C = k(E)|_{\mathfrak{B}} C$$

The updated pools remain proportional (with the same proportionality constant $k$), hence their respective proportions are equal.

Theorems 5, 6 and 7 demonstrate that the $\mathfrak{D}$-, $\mathfrak{G}$- and $\mathfrak{B}$-updates generalize Dempster's, geometric and Bayesian conditioning respectively: when the prior evidence $E$ is certain, the minimum and maximum proportion functions $P_{E|f}^-$ and $P_{E|f}^+$ of the posterior equate to the conditioned belief and plausibility functions $P_E^-(\cdot|f)C$ and $P_E^+(\cdot|f)C$. This follows from the evidence conveyed by BPAs is certain. Conversely, when the prior evidence is uncertain, the posterior proportion functions are generally different from the conditioned belief and plausibility functions. Even if $E$'s extrema are proper belief and plausibility functions, the knowledge $E$ conveys is not properly expressible in terms of mass or by upper and lower probabilities.

When the prior evidence is certain, the $\mathfrak{D}$-update yields certain posterior evidence. After sequential $\mathfrak{D}$-updates, obtained evidence hence still is certain. Theorem 5 ensures that sequential $\mathfrak{D}$-updates of certain evidence amounts to sequential Dempster's conditioning. Similarly, since the $\mathfrak{G}$-update of certain evidence yields certain evidence, Theorem 6 ensures that sequential $\mathfrak{G}$-updates of certain evidence amounts to sequential geometric conditioning. On the contrary, the $\mathfrak{B}$-update generally yields uncertain evidence, even if the prior evidence is certain. Hence sequential $\mathfrak{B}$-updates is not under the scope of Theorem 7 and generally does not amount to sequential Bayesian conditioning. This is a positive result since sequential Bayesian conditioning is not safe because Bayesian conditioning does not commute [18,9]. On the contrary, the $\mathfrak{B}$-update, as for the $\mathfrak{D}$- and $\mathfrak{G}$-updates, does commute, hence is suitable for sequential updating.

**Theorem 10.** Let $E$ be a pool of evidence over $\Omega$ and let $C_1 \subseteq \Omega$ and $C_2 \subseteq \Omega$ be two sets. Then:

$$E|_{\mathfrak{D}} C_1|_{\mathfrak{D}} C_2 = E|_{\mathfrak{D}} C_2|_{\mathfrak{D}} C_1 = E|_{\mathfrak{D}} C_1 \cap C_2$$

$$E|_{\mathfrak{G}} C_1|_{\mathfrak{G}} C_2 = E|_{\mathfrak{G}} C_2|_{\mathfrak{G}} C_1 = E|_{\mathfrak{G}} C_1 \cap C_2$$

$$E|_{\mathfrak{B}} C_1|_{\mathfrak{B}} C_2 = E|_{\mathfrak{B}} C_2|_{\mathfrak{B}} C_1 = E|_{\mathfrak{B}} C_1 \cap C_2$$

## 6.2. Credal understanding of the $\mathfrak{D}$-, $\mathfrak{G}$- and $\mathfrak{B}$-updates

We first presented the three updates in terms of evidence. Interestingly, the $\mathfrak{B}$-update amounts to considering both the $\mathfrak{D}$- and $\mathfrak{G}$-updates as possible updating rules, offering a clear evidential interpretation. As it generalizes Bayesian conditioning, the latter inherits this evidential interpretation. Bayesian conditioning is usually not considered in evidential interpretations, and rightly so, because the resulting BPA cannot be interpreted in terms of mass. Our framework enables its use by allowing uncertain evidence.

Credally, the urn analogy developed in Section 5 makes $E$ and $M_E$ describe colored balls in an urn, possibly understood as an “unnormalized interval-valued random set.” They define the possible content configurations and proportions, leading to a credal set. When acquiring new information $C \subseteq \Omega$ (meaning “only colors $\omega \in C$ are possible”), knowledge is updated depending on $C$'s nature. In the examples above, we learn $C = \{r, b\}$, that is, only red and black balls are in the urn.
<table><caption>**Table 16**
Support count functions of $E_1$ and of its $\mathfrak{B}$-update given $C = \{r, b\}$.</caption><thead><tr><th>A</th><th>Ø</th><th>{r}</th><th>{b}</th><th>{w}</th><th>{r, b}</th><th>{r, w}</th><th>{b, w}</th><th>{r, b, w}</th></tr></thead><tbody><tr><td>#<sub>E<sub>1</sub></sub>(A)</td><td>0</td><td>1</td><td>0</td><td>0</td><td>1</td><td>1</td><td>2</td><td>3</td></tr><tr><td>#<sub>E<sub>1</sub></sub>(A)</td><td>0</td><td>1</td><td>2</td><td>2</td><td>3</td><td>3</td><td>2</td><td>3</td></tr><tr><td>#<sup>2</sup><sub>E<sub>1|<sub>3</sub></sub></sub>C(A)</td><td>0</td><td>1</td><td>0</td><td>1</td><td></td><td></td><td></td><td></td></tr><tr><td>#<sup>2</sup><sub>E<sub>1|<sub>3</sub></sub></sub>C(A)</td><td>0</td><td>1</td><td>2</td><td>3</td><td></td><td></td><td></td><td></td></tr></tbody></table>

If $C$ holds because there never was any white balls in the urn, the content of the urn itself had not changed but our knowledge has to be refined. If prior placements $\{b, w\}$! are trusted, $C$ ensures they represent black balls, so the $\mathfrak{D}$-update applies. Conversely, if these prior placements are deemed incorrect, noise, or a mistake, they should be discarded, so $\mathfrak{B}$-update applies.

On the contrary, $C$ may result from a change in the urn's content: white balls, if any, were removed. Prior placements $\{b, w\}$! become ambiguous: they may denote black balls or removed white ones. Therefore, $\{b, w\}! \mapsto \{b\}$. This corresponds to the $\mathfrak{B}$-update.

As explained, the $\mathfrak{D}$- and $\mathfrak{B}$-updates revise the knowledge based on the information $C$ that pertains to the knowledge itself: the situation has not changed, and $C$ allows $M_E$ to be made more accurate. This is knowledge revision. Conversely, the semantics proposed for the $\mathfrak{B}$-update is about a change in the situation: the prior evidence was valid at one point, and $C$ does not serve to make it more accurate. Rather, $C$ represents a modification, and the $\mathfrak{B}$-update reflects this modification by focusing on the sub-situation.

This observation may be useful for the discussion of the distinction between focusing and knowledge revision [13,14,11]: revision amounts to acquiring information about the evidence, while focusing involves acquiring information about changes in the situation— whether hypothetical or not. Note that both the philosophical interpretation of this point and its applicability outside Belief Function Theory remain to be addressed.

### 6.3. The $\mathfrak{B}$-update as a “no-assumption” update: dilation is only apparent

The $\mathfrak{B}$-update operates without introducing assumptions beyond those encoded in the prior. It merely restricts the frame of discernment to the observed evidence. What might appear to be “dilation” does not reflect an increase in uncertainty but instead arises from the changes in normalization.

**Theorem 11.** Let $E$ be a pool of evidence over $\Omega$ and $C$ a subset of $\Omega$. Then:

$$ \#_{E|_{\mathfrak{B}} C}(A) = \#_{E}(A \cap C). $$

This ensures $\#_{E|_{\mathfrak{B}} C}(A) = \#_{E}(A)$ for all $A \subseteq C$ and $\#_{E|_{\mathfrak{B}} C}(A) = [0, 0]$ for all $A \subseteq C^c$: the $\mathfrak{B}$-update thus restricts the prior count function from $\Omega$ to $C$ without introducing new assumptions. Consider the Three-Ball-Urn example (Example 2): the urn contains one red ball and two balls that could be either black or white. If white balls are removed ($C = \{r, b\}$), it does not impact the possible number of red and black balls (Table 16).

Restricting the frame of discernment, however, affects normalized proportions. For instance, while $P_{E_1}(\{r\}) = [\frac{1}{3}, \frac{1}{3}]$ is precise in the prior, it becomes $P_{E_1|_{\mathfrak{B}} C}(\{r\}) = [\frac{1}{3}, 1]$ after the update. This shift does not indicate increased uncertainty but rather reflects increased specificity: the total number of balls has changed from $\#_{E_1}(\Omega)$ to $\#_{E_1}(C)$.

This example does not illustrate dilation in the formal sense (Definition 11), although it highlights how increase in knowledge can transform precise prior beliefs into imprecise posterior ones. By contrast, the Fight and Coin Flip example (Example 3) involves imprecise prior beliefs on the considered events. In terms of urns, the frame of discernment $\Omega = \{wh, wt, bh, bt\}$ represents four colors, each denoting a winner ($w$ or $b$) and a coin toss result ($h$ or $t$). The prior model assumes equal numbers of balls for heads and tails:

$$ M_E(\{wh, bh\}) = [k, k] \qquad M_E(\{wt, bt\}) = [k, k] \qquad M_E(B) = [0, 0] \text{ otherwise.} $$

The prior model allows for correlations: for instance, $\#_{E}(\{wh\}) = \#_{E}(\{wt\}) = [0, k]$ means that the wrestler may win only on heads, only on tails, on both, or on neither. It follows, e.g., that $P_E(\{wh\}) = [0, \frac{1}{2}]$.

If the wrestler wins ($C = \{wh, wt\}$), the $\mathfrak{B}$-update retains only the balls consistent with $C$: $M_{E|_{\mathfrak{B}} C}(\{wh\}) = M_{E|_{\mathfrak{B}} C}(\{wt\}) = [0, k]$. Consequently, $\#_{E|_{\mathfrak{B}} C}(\{wh\}) = \#_{E|_{\mathfrak{B}} C}(\{wt\}) = [0, k]$, just as in the prior. The count function is merely restricted to $C$.

Thus, $P_{E|_{\mathfrak{B}} C}(\{wh\}) = [0, 1]$. Since correlations are permitted by the prior model and the $\mathfrak{B}$-update makes no additional assumptions, the posterior reflects these possible correlations. The wrestler may never win on heads (making $wh$ impossible given $C$) or never win on tails (making $wh$ necessary given $C$), resulting in the obtained range of possible proportions. Note that it is also possible for the wrestler to never win at all, as the posterior pool of evidence may be empty.

### 6.4. The $\mathfrak{B}$-update as a proper credal update

Bayesian conditioning aims to capture the credal update by providing the bounds of conditioned credal sets [18], as explained in Section 3. However, conditioned credal sets are generally non-maximal (Definition 4), making conditioned belief and plausibility
<table><caption>**Table 17**
Count and proportion functions for $E_5$.</caption><thead><tr><th>A</th><th>0</th><th>{r}</th><th>{b}</th><th>{w}</th><th>{y}</th><th>{r, b}</th><th>{r, w}</th><th>{r, y}</th><th>{b, w}</th><th>{b, y}</th><th>{w, y}</th><th>{r, b, w}</th><th>{r, b, y}</th><th>{r, w, y}</th><th>{b, w, y}</th><th>Ω</th></tr></thead><tbody><tr><th>#<sub>E<sub>5</sub></sub><sup>-</sup>(A)</th><td>0</td><td>2</td><td>2</td><td>0</td><td>0</td><td>4</td><td>2</td><td>2</td><td>2</td><td>2</td><td>0</td><td>4</td><td>4</td><td>2</td><td>4</td><td>6</td></tr><tr><th>#<sub>E<sub>5</sub></sub><sup>+</sup>(A)</th><td>0</td><td>2</td><td>4</td><td>2</td><td>2</td><td>6</td><td>4</td><td>4</td><td>4</td><td>4</td><td>2</td><td>6</td><td>6</td><td>4</td><td>4</td><td>6</td></tr><tr><th>P<sub>E<sub>5</sub></sub><sup>-</sup>(A)</th><td>0</td><td>1/3</td><td>1/3</td><td>0</td><td>0</td><td>2/3</td><td>1/3</td><td>1/3</td><td>1/3</td><td>1/3</td><td>0</td><td>2/3</td><td>2/3</td><td>1/3</td><td>2/3</td><td>1</td></tr><tr><th>P<sub>E<sub>5</sub></sub><sup>+</sup>(A)</th><td>0</td><td>1/3</td><td>2/3</td><td>1/3</td><td>1/3</td><td>1</td><td>2/3</td><td>2/3</td><td>2/3</td><td>2/3</td><td>1/3</td><td>1</td><td>1</td><td>2/3</td><td>2/3</td><td>1</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Discrete credal set (red dots) and convex one (red triangle) associated with $E_5$.</figcaption></figure>

functions unable to delimit them accurately. The $\mathfrak{B}$-update, which generalizes Bayesian conditioning, provides support count functions delimiting these non-maximal conditioned credal sets.

**Theorem 12.** Let E be a pool of evidence over $\Omega$ and $C \subseteq \Omega$ a set such that $P_E^+(C) > 0$. Then:

$$F_E^* | C = F_{E|\Omega_3 C}^*$$

Thus, the convex credal set $F_{E|\Omega_3 C}^*$ associated with $E|\Omega_3 C$ contains exactly all the conditioned probability measures of $F_E^*$. This result enables sequential credal conditioning. Let us illustrate this desired behavior.

Let $\Omega = \{r, b, w, y\}$ and $E_5 = \{(\{r\}!, \{r\}!, \{b\}!, \{b, w, y\}!, \{b, w, y\}!\}$. Credally, it is interpreted as an urn containing 6 balls: two red, two black and two which may be either black, white or yellow. Its evidence function $M_{E_5}$ is analogous to the BPA $m_5$:

$$M_{E_5}(\{r\}) = [2, 2] \qquad m_5(\{r\}) = 1/3$$

$$M_{E_5}(\{b\}) = [2, 2] \qquad m_5(\{b\}) = 1/3$$

$$M_{E_5}(\{b, w, y\}) = [2, 2] \qquad m_5(\{b, w, y\}) = 1/3$$

Count and proportion functions are listed in Table 17. Obviously, equal proportions are obtained from the BPA $m_5$, i.e., $P_{E_5}^- = \text{Bel}_5$ and $P_{E_5}^+ = \text{Pl}_5$.

Considering discrete number of balls, one can enumerate the six possible content configurations: *rrbbbb*, *rrbbbw*, *rrbbbty*, *rrbbwww*, *rrbbwy* and *rrbbyy*, each leading to a probability distribution. The discrete credal set (Definition 20) associated with $E_5$ is $F_{E_5} =$

$$\{(\frac{2}{6}, \frac{4}{6}, \frac{0}{6}, \frac{0}{6}), (\frac{2}{6}, \frac{3}{6}, \frac{1}{6}, \frac{0}{6}), (\frac{2}{6}, \frac{3}{6}, \frac{0}{6}, \frac{1}{6}), (\frac{2}{6}, \frac{2}{6}, \frac{0}{6}, \frac{2}{6}), (\frac{2}{6}, \frac{2}{6}, \frac{1}{6}, \frac{1}{6}), (\frac{2}{6}, \frac{2}{6}, \frac{0}{6}, \frac{2}{6})\}$$

The corresponding convex credal set (Definition 21) is:

$$F_{E_5}^* = \left\{ \left( \frac{2}{6}, \frac{2+x_b}{6}, \frac{x_w}{6}, \frac{x_y}{6} \right) \middle| x_b + x_w + x_y = 2 \right\}$$

In the expression above, $x_b$, $x_w$ and $x_y$ are non-negative real numbers encoding the several ways to allocate the amount $M_{E_5}(\{b, w, y\}) = [2, 2]$. Since $|\Omega| = 4$, these discrete and convex credal sets are subsets of the standard 4-simplex, which is a tetrahedron (Fig. 5). $F_{E_5}^*$ is a triangle parallel to facet *bwy* since $\text{Pr}(r) = 1/3$ is fixed. One of its edges belongs to facet *rby*, one other to facet *rbw*. Note the parallelism with edges and facets of the standard 4-simplex: $F_{E_5}^*$ is maximal in the sense of Definition 4.

Suppose one learns that yellow balls, if any, have been removed, so $C = \{r, b, w\}$ are the only possible colors. The $\mathfrak{B}$-update of $E_5$ is easily achieved: $\mathfrak{B}(\{r\}! | C) = \{r\}!$, $\mathfrak{B}(\{b\}! | C) = \{b\}!$ and $\mathfrak{B}(\{b, w, y\}! | C) = \{b, w\}?$. Conversely, the values of the BPA $m_5(\cdot | \Omega_3 C)$ are harder to compute.
<figure><img src="image_10.png" /><figcaption>Fig. 6. The discrete and convex credal sets associated with $E_5|_{\mathfrak{B}} C$ (on the left) and those associated with the fictitious urn $U$ that $m_5(\cdot|_{\mathfrak{B}} C)$ describes (on the right).</figcaption></figure>

$$M_{E_5|_{\mathfrak{B}} C}(\{r\}) = [2, 2] \qquad m_5(\{r\} |_{\mathfrak{B}} C) = 1/3$$

$$M_{E_5|_{\mathfrak{B}} C}(\{b\}) = [2, 2] \qquad m_5(\{b\} |_{\mathfrak{B}} C) = 1/3$$

$$M_{E_5|_{\mathfrak{B}} C}(\{b, w\}) = [0, 2] \qquad m_5(\{b, w\} |_{\mathfrak{B}} C) = 1/6$$

$$m_5(\{r, b, w\} |_{\mathfrak{B}} C) = 1/6$$

$E_5|_{\mathfrak{B}} C$ clearly describes the possible updated content configurations: $rrbbb, rrbbw, rrbb, rrbwww, rrbbw$ and $rrbb$; each corresponding to a prior configuration where $y$ is removed. Conversely, the meaning of $m_5(\cdot|_{\mathfrak{B}} C)$ is not clear: it describes the fictitious urn $U = \{\{r\}, \{r\}, \{b\}, \{b\}, \{b, w\}, \{r, b, w\}\}$ which is not related to $E_5$ because it enables, e.g., the configuration $rrbbw$ which is not possible according to the prior.

The discrete and convex credal sets of $E_5|_{\mathfrak{B}} C$ are:

$$F_{E_5|_{\mathfrak{B}} C} = \left\{ \left( \frac{2}{6}, \frac{4}{6}, \frac{0}{6} \right), \left( \frac{2}{6}, \frac{3}{6}, \frac{1}{6} \right), \left( \frac{2}{5}, \frac{3}{5}, \frac{0}{5} \right), \left( \frac{2}{6}, \frac{2}{6}, \frac{2}{6} \right), \left( \frac{2}{5}, \frac{2}{5}, \frac{1}{5} \right), \left( \frac{2}{4}, \frac{2}{4}, \frac{0}{4} \right) \right\}$$

$$F_{E_5|_{\mathfrak{B}} C}^* = \left\{ \left( \frac{2}{4 + x_b + x_w}, \frac{2 + x_b}{4 + x_b + x_w}, \frac{x_w}{x_b + x_w} \right) \mid x_b + x_w \le 2 \right\}.$$

Theorem 12 ensures $F_{E_5|_{\mathfrak{B}} C}$ and $F_{E_5|_{\mathfrak{B}} C}^*$ are the conditioned discrete and convex credal sets, respectively. They are illustrated on the left side of Fig. 6; those associated with the fictitious urn $U$ are illustrated on the right side. As discussed in Sections 3 and 5, $F_U^*$ is maximal while $F_{E_5|_{\mathfrak{B}} C}^*$ is not.

Suppose one learns additional information: white balls, if any, have also been removed, so $C' = \{r, b, y\}$. Again, the $\mathfrak{B}$-update of $E_5|_{\mathfrak{B}} C$ given $C'$ is straightforward, while the values of $m_5(\cdot|_{\mathfrak{B}} C|_{\mathfrak{B}} C')$ are difficult to compute.

$$M_{kE_5|_{\mathfrak{B}} C|_{\mathfrak{B}} C'}(\{r\}) = [2, 2] \qquad m_5(\{r\} |_{\mathfrak{B}} C|_{\mathfrak{B}} C') = 1/3$$

$$M_{kE_5|_{\mathfrak{B}} C|_{\mathfrak{B}} C'}(\{b\}) = [2, 4] \qquad m_5(\{b\} |_{\mathfrak{B}} C|_{\mathfrak{B}} C') = 2/5$$

$$m_5(\{r, b\} |_{\mathfrak{B}} C|_{\mathfrak{B}} C') = 4/15$$

It is clear that $M_{E_5|_{\mathfrak{B}} C|_{\mathfrak{B}} C'}$ describes the possible content of the urn if yellow and white balls are removed. One can enumerate the possible content configurations: $rrbb, rrbbb and rrbbbb$; which amounts to removing the occurrences of $w$ in $E_5|_{\mathfrak{B}} C$'s possible configurations, that is, to removing occurrences of both $w$ and $y$ from $E_5$'s ones. The discrete and convex credal sets correspond to the conditioning of those of $E_5|_{\mathfrak{B}} C$:

$$F_{E_5|_{\mathfrak{B}} C|_{\mathfrak{B}} C'} = \left\{ \left( \frac{2}{4}, \frac{2}{4} \right), \left( \frac{2}{5}, \frac{3}{5} \right), \left( \frac{2}{6}, \frac{4}{6} \right) \right\}$$

$$F_{E_5|_{\mathfrak{B}} C|_{\mathfrak{B}} C'}^* = \left\{ \left( \frac{2}{4 + x_b}, \frac{2 + x_b}{4 + x_b} \right) \mid x_b \le 2 \right\}.$$

Conversely, the meaning of $m_5(\cdot|_{\mathfrak{B}} C|_{\mathfrak{B}} C')$ is unclear since the common denominator is 15. Do the balls split? Do they replicate? Moreover, $Bel_5(\cdot|_{\mathfrak{B}} C|_{\mathfrak{B}} C')$ does not equate to $P_{E_5|_{\mathfrak{B}} C|_{\mathfrak{B}} C'}^{-}(\text{Table 18})$. Bayesian conditioning of $Bel_5(\cdot|_{\mathfrak{B}} C)$ indeed amounts to conditioning the maximal credal set it delimits, which actually corresponds to the fictitious $U$, not to $E_5|_{\mathfrak{B}} C$.

It is easy to check that $E_5|_{\mathfrak{B}} C|_{\mathfrak{B}} C' = E_5|_{\mathfrak{B}} C' |_{\mathfrak{B}} C = E_5|_{\mathfrak{B}} C \cap C'$: the $\mathfrak{B}$-update commutes (Theorem 10). On the contrary, Bayesian conditioning does not, and is not suited for sequential conditioning. Moreover, this long example illustrates that the $\mathfrak{B}$-update does capture well the conditioning of credal sets whose extrema are belief and plausibility functions (Theorem 12). Enabling uncertain evidence, count functions may indeed delimit non-maximal credal sets, hence capture the whole generality of the credal interpretation of Belief Function Theory.

## 7. Conclusion

The proposed framework extends the evidential interpretation of Belief Function Theory by incorporating uncertain evidence. It also refines the credal interpretation and resolves a major limitation: the inability to distinguish distinct credal sets with identical upper
<caption>**Table 18**</caption>

Minimum and maximum number and proportions of balls for $E_5|_{\mathfrak{B}_3}$
$C|_{\mathfrak{B}_3} C'$. $\text{Bel}_5(\cdot|_{\mathfrak{B}_3} C|_{\mathfrak{B}_3} C')$ and $\text{Pl}_5(\cdot|_{\mathfrak{B}_3} C|_{\mathfrak{B}_3} C')$ do not yield the
expected proportions.

<table><thead><tr><th>A</th><th>Ø</th><th>{r}</th><th>{b}</th><th>{r, b}</th></tr></thead><tbody><tr><td>#<sub>E<sub>5</sub>|<sub>\B</sub> C|<sub>\B</sub> C'</sub>(A)</td><td>0</td><td>2</td><td>2</td><td>4</td></tr><tr><td>#<sup>+</sup><sub>E<sub>5</sub>|<sub>\B</sub> C|<sub>\B</sub> C'</sub>(A)</td><td>0</td><td>2</td><td>4</td><td>6</td></tr><tr><td>P<sub>E<sub>5</sub>|<sub>\B</sub> C|<sub>\B</sub> C'</sub>(A)</td><td>0</td><td>1/3</td><td>1/2</td><td>1</td></tr><tr><td>P<sup>+</sup><sub>E<sub>5</sub>|<sub>\B</sub> C|<sub>\B</sub> C'</sub>(A)</td><td>0</td><td>1/2</td><td>2/3</td><td>1</td></tr><tr><td>Bel<sub>5</sub>(A|<sub>\B</sub> C|<sub>\B</sub> C')</td><td>0</td><td>1/3</td><td>2/5</td><td>1</td></tr><tr><td>Pl<sub>5</sub>(A|<sub>\B</sub> C|<sub>\B</sub> C')</td><td>0</td><td>3/5</td><td>2/3</td><td>1</td></tr></tbody></table>

and lower probabilities. It enables capturing credal conditioning with a commutative, low-complexity update-by-transfer algorithm. Additionally, it explains the apparent dilation effect, showing how increased knowledge may reduce precision.

This framework opens several research directions. One first direction is its application to Smets' generalized belief functions [40], allowing reasoning under the open world assumption. While straightforward in principle, its well-definedness remains an open question.

From a credal perspective, identifying the extreme points of non-maximal credal sets induced by uncertain evidence could provide an alternative, potentially tractable representation of knowledge. Moreover, defining updates relative to these points may yield additional insights.

Considering refining the representation of reliability is another direction. Moving beyond binary states of “certain” and “uncertain” may improve expressiveness; but feasibility, semantics and alignment with the scope of Belief Function Theory need to be explored.

From a practical point of view, k-additive evidence functions are more expressive than, but as complex as k-additive BPAs, so tractable algorithms could rely on them. In particular, the expressiveness and practical utility of 1-additive evidence functions need to be studied.

Moreover, the framework seems suited to learn from, or identify with data: pools are datasets which are easy to extend with incoming data; and $\mathfrak{B}$-update may serve for prediction. Experiments are needed to check whether such an approach leads to relevant predictions in case of uncertain data.

The framework also suggests connections with alternative conditioning methods, such as likelihood-based approaches [32,2] or less common ones [10,5,4]. Investigating their compatibility with uncertain evidence and their interpretation in evidential and credal terms could provide new insights.

Importantly, a decision theory suited to uncertain evidence remains to be developed. While lower-measure-based axiomatics are straightforwardly applicable to proportions functions, some other approaches rely on the focal elements [25,19] or on average probability measures [42], the definitions of which are impacted by uncertain evidence. Their extension will improve their applicability and could provide new insights.

Finally, this framework provides valuable insights to characterize the boundaries between Belief Function Theory and its sub- and super-frameworks. Probabilities emerge when the evidence is precise and certain, while imprecise probabilities can be reached when introducing dependencies among evidence. Whether such dependencies fully encompass imprecise probabilities and whether the obtained negative evidence functions could be provided clear interpretation remain open questions.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Appendix A. Proofs for Section 4 (Uncertain Evidence)

**Lemma 1.** Let $E$ be a non-empty pool of certain evidence. Then $M_E^- = M_E^+$ is an unnormalized BPA.

**Proof.** Since $E$ is certain, it holds that for all $B$, $M_E^+(B) = M_E^-(B) + I_E(B') = M_E^-(B) + 0$. Since $\emptyset \notin E$, $M_E^-(\emptyset) = 0$. Finally, for all $B$, $M_E^-(B) \in \mathbb{N}$ implies $M_E^-(B) \ge 0$. $\square$

**Lemma 2.** A pool of evidence over $\Omega$ is certain iff $\#_E^-(\Omega) = \#_E^+(\Omega)$.

**Proof.** If $E$ is a pool of certain evidence, then it follows from Lemma 1 that $M_E^-(\Omega) = M_E^+(\Omega)$. Conversely, suppose there exists $B' \in E$. Hence, $M_E^-(B) < M_E^+(B)$. Since $\forall B'$, $M_E^-(B') \le M_E^+(B')$, it follows that $\#_E^-(\Omega) < \#_E^+(\Omega)$. $\square$

**Lemma 3.** Let $E$ be a non-empty pool of certain evidence over $\Omega$. Then $\#_E^-$ and $\#_E^+$ are unnormalized dual belief and plausibility and their unnormalized BPA is $M_E^- = M_E^+$. The normalization factor is $\#_E^-(\Omega) = \#_E^+(\Omega)$.
**Proof.** Since $E$ is certain, $M_E^- = M_E^+$ is an unnormalized BPA (Lemma 1), and $K = \bar{\#}_E^-(\Omega) = \bar{\#}_E^+(\Omega)$ is its normalization factor. For any $B$, it is true that $\bar{\#}_E^-(A)/K = \sum_{B \subseteq A} M_E^-(B)/K$ and $\bar{\#}_E^+(A)/K = \sum_{B \cap A \neq \emptyset} M_E^-(B)/K$. That is, $\bar{\#}_E^+/K$ and $\bar{\#}_E^+/K$ are the dual belief and plausibility functions with BPA $M_E^-/K$. $\square$

**Theorem 2.** Let $E$ be a pool of evidence. The proportion of evidence supporting $A$ in $E$ is bounded by:

$$P_E^-(A) = \frac{\#_E^-(A)}{\#_E^-(A) + \#_E^+(A^c)} \quad \text{and} \quad P_E^+(A) = \frac{\#_E^+(A)}{\#_E^+(A) + \#_E^-(A^c)}.$$

**Proof.** All possible proportions $P_E(A)$ (and only these) are obtained by considering the several pools of precise and certain evidence compatible with $E$, where one can exactly compute the number of pieces of evidence supporting $A$. Any pool of precise and certain evidence $E^*$ compatible with $E$ is obtained using the following procedure: each piece of uncertain evidence $B \in E$ is either made certain as $B!$ or removed; then, each piece of evidence $B!$ is made precise as $\{\omega\}$! with $\omega \in B$. The fraction then becomes $P_{E^*}(A) = \frac{\sum_{\omega \in A} I_{E^*}(\{\omega\})!}{\sum_{\omega \in \Omega} I_{E^*}(\{\omega\})!}$. To minimize $P_{E^*}(A)$, one should minimize the numerator and maximize the denominator. This can be achieved by: (i) removing any $B?$ if $B \subseteq A$, or turning it into $B!$ otherwise; and (ii) making any $B!$ precise as $\{\omega\}$! with $\omega \notin A$, whenever possible. Conversely, to maximize $P_{E^*}(A)$, one should remove any $B?$ such that $B \subseteq A^c$ and make any $B!$ precise as $\{\omega\}$! with $\omega \in A$, whenever possible. It follows that $P_E(A) = \left[ \frac{\#_E^-(A)}{\#_E^-(A) + \#_E^+(A^c)}, \frac{\#_E^+(A)}{\#_E^+(A) + \#_E^-(A^c)} \right]$. $\square$

**Lemma 4.** Let $E$ be a pool of evidence. Then for all $A$, $P_E^-(A) + P_E^+(A^c) = 1$.

**Proof.** For all $A \subseteq \Omega$, it holds that $P_E^-(A) + P_E^+(A^c) = \frac{\#_E^-(A) + \#_E^+(A^c)}{\#_E^-(A) + \#_E^+(A^c)} = 1$. $\square$

**Lemma 5.** Let $E$ be a non-empty pool of certain evidence over $\Omega$. Then $P_E^-$ and $P_E^+$ are dual belief and plausibility functions; their BPA $m$ is defined by $m(B) = M_E^-(B)/\#_E^-(\Omega)$.

**Proof.** Directly from Lemma 3, since for all $A \subseteq \Omega$, it holds that $P_E(A) = \left[ \frac{\#_E^-(A)}{\#_E^-(\Omega)}, \frac{\#_E^+(A)}{\#_E^+(\Omega)} \right]$. $\square$

**Theorem 3.** Let $E$ be a non-empty pool of evidence. Then $P_E^-$ and $P_E^+$ are dual belief and plausibility functions.

**Proof.** If $E$ is certain, this directly follows from Lemma 5. Suppose now that $E$ is uncertain. Let $E_0$ be the pool of evidence on $\Omega \cup \{\omega_0\}$, where each piece of evidence is the image of one of the pieces in $E$ under the mapping $f$, such as $f(B!) = B!$ and $f(B?) = (B \cup \{\omega_0\})!$. Since $E_0$ is a pool of certain evidence, it follows from Lemma 5 that $P_{E_0}^-$ and $P_{E_0}^+$ are dual belief and plausibility functions.

Furthermore, it holds that $E = E_0|_{\Omega_3} \Omega$. Since $E_0$ is certain, it follows from Theorem 7 that for all $A \subseteq \Omega$, $P_E^-(A) = P_{E_0|_{\Omega_3}}^-(A) = P_{E_0}^-(A|_{\Omega_3} \Omega)$. From [18, Theorem 2.3], $P_E^-$ is a belief function since $P_{E_0}^-$ is a belief function. Finally, from Lemma 4, $P_E^+$ is its dual plausibility function. $\square$

**Lemma 6.** Let $E$ be a pool of evidence over $\Omega$. Then for all $k \neq 0$, $P_E = P_{kE}$.

**Proof.** For all $A$, $P_{kE}^-(A) = \frac{k \#_E^-(A)}{k \#_E^-(A) + k \#_E^+(A^c)} = \frac{\#_E^-(A)}{\#_E^-(A) + \#_E^+(A^c)} = P_E^-(A)$. Similarly, $P_{kE}^+(A) = P_E^+(A)$. $\square$

**Theorem 4.** Let $E$ and $E'$ be two pools of certain evidence. Then:

$$P_E = P_{E'} \iff \exists k, E = kE'$$

**Proof.** Since $E$ is a pool of certain evidence, it follows that $P_E^-$ and $P_E^+$ are dual belief and plausibility functions with BPA $m_E(B) = M_E^-(B)/\bar{\#}_E^-(\Omega)$ (Lemma 5).

Now suppose $E' = kE$. If $k = 0$ then both $E$ and $E'$ are empty and thus equal. Otherwise, $E'$ is certain, so $P_{E'}^-$ and $P_{E'}^+$ are dual belief and plausibility functions with BPA $m_{E'}(B) = M_{E'}^-(B)/\bar{\#}_{E'}^-(\Omega) = kM_E^-(B)/k\bar{\#}_E^-(\Omega) = m_E(B)$. Since the BPAs are equal, the corresponding belief and plausibility functions must also be equal, i.e., $P_E = P_{E'}$.

Conversely, let $E'$ be a pool of certain evidence such that $P_{E'} = P_E$. The BPA of $P_{E'}$ is defined by $m_{E'}(B) = M_{E'}^-(B)/\bar{\#}_{E'}^-(\Omega)$, so for all $B$, it holds that $M_{E'}^-(B)/\bar{\#}_{E'}^-(\Omega) = M_E^-(B)/\bar{\#}_E^-(\Omega)$. It directly follows that $E$ and $E'$ are proportional. $\square$
## Appendix B. Proofs for Section 6 (Updating Uncertain Evidence)

**Lemma 7.** The evidence function of $E|_D C$ is such as:

$$M_{E|_D C}(B) = \left[ \sum_{B' \cap C = B} M_E^-(B'), \sum_{B' \cap C = B} M_E^+(B') \right] \quad \text{if } \emptyset \neq B \subseteq C$$

$$M_{E|_D C}(B) = [0, 0] \qquad \text{otherwise.}$$

**Proof.** From Definition 23 it follows that any piece of evidence $Bx \in E|_D C$ satisfies $\emptyset \neq B \subseteq C$. Moreover, all pieces of evidence $B'x \in E$ such that $B' \cap C = B$ are mapped to $Bx$, which gives the expression for $M_{E|_D C}(B)$. $\square$

**Theorem 5.** Let $E$ be a non-empty pool of certain evidence over $\Omega$ and $C \subseteq \Omega$ such that $P_E^+(C) > 0$. Then $E|_D C$ is certain, non-empty and satisfies for all $A$:

$$P_{E|_D C}^-(A) = P_E^-(A|_D C) \quad \text{and} \quad P_{E|_D C}^+(A) = P_E^+(A|_D C).$$

$M_{E|_D C}^- = M_{E|_D C}^+$ is then the unnormalized BPA of $P_{E|_D C}^-$.

**Proof.** Since $E$ is certain and $P_E^+(C) > 0$, there exists a set $B \subseteq \Omega$ such that $B \cap C \neq \emptyset$ and $I_E(B') > 0$, implying that $I_{E|_D C}((B \cap C)') > 0$. Therefore, $E|_D C$ is non-empty. Additionally, it holds that $M_E^-(B) = M_E^+(B), \forall B$. From Lemma 7, it follows that $\forall B \subseteq \Omega$, $M_{E|_D C}^-(B) = \sum_{B' \cap C=B} M_E^-(B') = \sum_{B' \cap C=B} M_E^+(B') = M_{E|_D C}^+(B)$. Thus, $E|_D C$ is certain and $M_{E|_D C}^- = M_{E|_D C}^+$ equates to $m(\cdot|_D C)$, up to normalization (Definition 1). Finally, from Lemma 5 it follows that $P_{E|_D C}^-$ equates to $P_E^-(\cdot|_D C)$ and $P_{E|_D C}^+$ equates to $P_E^+(\cdot|_D C)$. $\square$

**Lemma 8.** The evidence function of the $\mathfrak{B}$-update of $E$ given $C$ is such as:

$$M_{E|_D C}(B) = M_E(B) \quad \text{if } B \subseteq C$$

$$M_{E|_D C}(B) = [0, 0] \qquad \text{otherwise.}$$

**Proof.** From Definition 24, it follows that $E|_D C = \{(Bx|Bx \in E \land B \subseteq C)\}$, which gives the expression for $M_{E|_D C}(B)$. $\square$

**Theorem 6.** Let $E$ be a non-empty pool of certain evidence over $\Omega$, and $C \subseteq \Omega$ such that $P_E^-(C) > 0$. Then $E|_D C$ is certain, non-empty and satisfies for all $A$:

$$P_{E|_D C}^-(A) = P_E^-(A|_D C) \quad \text{and} \quad P_{E|_D C}^+(A) = P_E^+(A|_D C).$$

$M_{E|_D C}^- = M_{E|_D C}^+$ is then the unnormalized BPA of $P_{E|_D C}^-$.

**Proof.** Since $E$ is certain and $P_E^-(C) > 0$, there exists a non-empty set $B \subseteq C$ such that $I_E(B') > 0$. Hence, $I_{E|_D C}(B') > 0$, which implies that $E|_D C$ is non-empty. It also follows that $M_E^-(B) = M_E^+(B), \forall B$. Then, from Lemma 8 it follows that for all $B \subseteq C$, $M_{E|_D C}^-(B) = M_E^-(B) = M_E^+(B) = M_{E|_D C}^+(B)$, and that $M_E^-(B) = M_E^+(B) = 0$ otherwise. Hence, $E|_D C$ is certain, and $M_{E|_D C}^- = M_{E|_D C}^+$ equates to $m(\cdot|_D C)$, up to normalization (Definition 2). Finally, from Lemma 5, it follows that $P_{E|_D C}^-$ equates to $P_E^-(\cdot|_D C)$ and $P_{E|_D C}^+$ equates to $P_E^+(\cdot|_D C)$. $\square$

**Lemma 9.** The evidence function of the $\mathfrak{B}$-update of $E$ given $C$ is such as:

$$M_{E|_D C}(B) = \left[ M_E^-(B), \sum_{B' \cap C = B} M_E^+(B') \right] \quad \forall \emptyset \neq B \subseteq C$$

$$M_{E|_D C}(B) = [0, 0] \qquad \text{otherwise.}$$

**Proof.** From Definition 25, it follows that any piece of evidence $Bx \in E|_D C$ satisfies $\emptyset \neq B \subseteq C$. Moreover, pieces of evidence $B'$ such that $B \subseteq C$ are retained as is, thus giving the expression for $M_{E|_D C}^-$; all other pieces of evidence $B'x \in E$ such that $B' \cap C = B$ are mapped to $B'$, thus giving the expression for $M_{E|_D C}^+(B)$. $\square$

**Theorem 7.** Let $E$ be a non-empty pool of certain evidence over $\Omega$ and $C \subseteq \Omega$ such that $P_E^-(C) > 0$. Then $E|_D C$ is non-empty and verifies $\forall A$:

$$P_{E|_D C}^-(A) = P_E^-(A|_D C) \quad \text{and} \quad P_{E|_D C}^+(A) = P_E^+(A|_D C).$$
**Proof.** Since $E$ is certain and $P_E^+(C) > 0$, there exists $B$ such that $B \cap C \neq \emptyset$ and $I_E(B') > 0$. Hence, either $I_{E|_B C}(B') > 0$ or $I_{E|_B C}(B') > 0$, meaning that $E|_B C$ is non-empty. It also holds that $I_E(B') = 0$ for all $B$, hence $\#_E^+(A) = \sum_{B \cap A \neq \emptyset} I_E(B!)$ for all $A$. Let $K = \#_E^-(\Omega) = \#_E^+(\Omega)$. From Definition 8 and Lemma 3, it follows that:

$$
\begin{align*}
\bar{P}_E^-(A|_B C) &= \frac{\bar{P}_E^-(A \cap C)}{\bar{P}_E^-(A \cap C) + \bar{P}_E^+(A^c \cap C)} \\
&= \frac{\#_E^-(A \cap C)/K}{\#_E^-(A \cap C)/K + \#_E^+(A^c \cap C)/K} \\
&= \frac{\sum_{B \subseteq A \cap C} I_E(B!)}{\sum_{B \subseteq A \cap C} I_E(B!) + \sum_{B \cap (A^c \cap C) \neq \emptyset} I_E(B!)}.
\end{align*}
$$

Let $x$ and $y$ denote these sums, so that $\bar{P}_E^-(A|_B C) = \frac{x}{x+y}$. On the other hand, it holds that:

$$
\begin{align*}
\bar{P}_{E|_B C}^-(A) &= \frac{\#_{E|_B C}^-(A)}{\#_{E|_B C}^-(A) + \#_{E|_B C}^+(A^c)} \\
&= \frac{\sum_{B \subseteq A} I_{E|_B C}(B!)}{\sum_{B \subseteq A} I_{E|_B C}(B!) + \sum_{B \cap A^c \neq \emptyset} I_{E|_B C}(B!) + \sum_{B \cap A^c \neq \emptyset} I_{E'}(B')}
\end{align*}
$$

Let $x', y'_1, y'_2$ denote these sums, so that $\bar{P}_{E|_B C}^-(A) = \frac{x'}{x'+y'_1+y'_2}$.

From Definition 25, it holds that $B' \in E|_B C \Leftrightarrow (B' \in E) \wedge (B \subseteq C)$. It follows that $x' = \sum_{B \subseteq A} I_{E|_B C}(B!) = \sum_{B \subseteq A \cap C} I_E(B!) = x$.

It also follows that $y'_1 = \sum_{B \cap A^c \neq \emptyset} I_{E|_B C}(B!) = \sum_{B \cap A^c \neq \emptyset, B \subseteq C} I_E(B!)$. Since $B \subseteq C$ implies $B \cap C \neq \emptyset$ or $B = \emptyset$ (the latter case being irrelevant as $I_E(\emptyset) = I_E(\emptyset') = 0$), it follows that $y'_1 = \sum_{B \cap A^c \neq \emptyset, B \cap C \neq \emptyset, B \subseteq C} I_E(B)$.

Moreover, since $E$ is certain, it holds that $B' \in E|_B C \Leftrightarrow \exists B' \neq B, (B' \not\subseteq C) \wedge (B'! \in E) \wedge (B' \cap C = B)$. It follows that $y'_2 = \sum_{B \cap A^c \neq \emptyset} I_{E|_B C}(B?) = \sum_{B \cap A^c \neq \emptyset, B \cap C \neq \emptyset, B \not\subseteq C} I_E(B)$.

Finally, note that $y = \sum_{B \cap (A^c \cap C) \neq \emptyset} I_E(B!) = \sum_{B \cap A^c \neq \emptyset, B \subseteq C} I_E(B!) + \sum_{B \cap A^c \neq \emptyset, B \cap C \neq \emptyset} I_E(B!) = y'_1 + y'_2$. Since $x = x'$ and $y = y'_1 + y'_2$, it follows that $\bar{P}_E^-(A|_B C) = \bar{P}_{E|_B C}^-(A)$. The equality also holds for $P^+$, either using the same reasoning or directly by duality (Lemma 4). $\square$

**Theorem 8.** Let $E$ be a pool of evidence over $\Omega$, $C \subseteq \Omega$ and $f \in \{\mathfrak{D}, \mathfrak{G}, \mathfrak{B}\}$ be a transfer function. Then:

$M_E$ has $n$ focal elements $\Rightarrow M_{E|_f C}$ has fewer than $n$ focal elements;

$M_E$ is $k$-additive $\Rightarrow M_{E|_f C}$ is $k$-additive.

**Proof.** These results directly follow from the definitions of the transfer functions $\mathfrak{D}$, $\mathfrak{G}$ and $\mathfrak{B}$. Obviously, the number of focal elements cannot increase since the transfer functions are many-to-one. Moreover, let $E$ be a pool of evidence over $\Omega$ such that $M_E$ is $k$-additive: $|B| > k \Rightarrow M_E(B) = [0,0]$ for all $B \subseteq \Omega$. Let $C \subseteq \Omega$. From $\mathfrak{G}$'s definition it holds that $M_{E|_\mathfrak{G} C}(B) \neq [0,0] \Rightarrow M_E(B) \neq [0,0]$, so $|B| \le k$. Hence $M_{E|_\mathfrak{G} C}$ is $k$-additive. From $\mathfrak{D}$'s definition it holds that $M_{E|_\mathfrak{D} C}(B) \neq [0,0] \Rightarrow \exists B', B \subseteq B' \wedge M_E(B') \neq [0,0]$, so $|B| \le |B'| \le k$. Hence $M_{E|_\mathfrak{D} C}$ is $k$-additive. The same proof holds for the $\mathfrak{B}$-update. $\square$

**Theorem 9.** Let $E$ be a pool of evidence over $\Omega$, $k > 0$ and $C \subseteq \Omega$. Then:

$$(kE)|_\mathfrak{D} C = k(E)|_\mathfrak{D} C$$

$$(kE)|_{\mathfrak{G}} C = k(E)|_{\mathfrak{G}} C$$

$$(kE)|_{\mathfrak{B}} C = k(E)|_{\mathfrak{B}} C$$

**Proof.** It directly follows from Lemmas 7, 8, and 9 that for all $B$ such that $\emptyset \neq B \subseteq C$:

$$
\begin{align*}
M_{kE|_B C}(B) &= \left[ \sum_{B' \cap C=B} \bar{M}_{kE}(B'), \sum_{B' \cap C=B} \bar{M}_{kE}^+(B') \right] \\
&= \left[ k \sum_{B' \cap C=B} \bar{M}_E(B'), k \sum_{B' \cap C=B} \bar{M}_E^+(B') \right] = k M_{E|_B C}(B) \\
M_{kE|_\mathfrak{G} C}(B) &= [\bar{M}_{kE}(B), \bar{M}_{kE}^+(B)] = [k \bar{M}_E(B), k \bar{M}_E^+(B)] = k M_{E|_\mathfrak{G} C}(B) \\
M_{kE|_\mathfrak{B} C}(B) &= \left[ \bar{M}_{kE}(B), \sum_{B' \cap C=B} \bar{M}_{kE}^+(B') \right]
\end{align*}
$$
$$ = \left[ k M_{E}^{-}(B), k \sum_{B' \cap C = B} M_{E}^{+}(B') \right] = k M_{E|_{\mathcal{D}} C}(B) $$

Otherwise, either $B = \emptyset$ or $B \not\subseteq C$ hence $M_{kE|C}(B) = [0,0] = kM_{E|C}(B)$ for the three updates. □

**Theorem 10.** Let E be a pool of evidence over $\Omega$ and let $C_1 \subseteq \Omega$ and $C_2 \subseteq \Omega$ be two sets. Then:

$$ E|_{\mathcal{D}} C_1|_{\mathcal{D}} C_2 = E|_{\mathcal{D}} C_2|_{\mathcal{D}} C_1 = E|_{\mathcal{D}} C_1 \cap C_2 $$

$$ E|_{\mathfrak{G}} C_1|_{\mathfrak{G}} C_2 = E|_{\mathfrak{G}} C_2|_{\mathfrak{G}} C_1 = E|_{\mathfrak{G}} C_1 \cap C_2 $$

$$ E|_{\mathfrak{B}} C_1|_{\mathfrak{B}} C_2 = E|_{\mathfrak{B}} C_2|_{\mathfrak{B}} C_1 = E|_{\mathfrak{B}} C_1 \cap C_2 $$

**Proof.** To conclude, we need to verify that for each update $|_f$ ($f \in \{\mathcal{D}, \mathfrak{G}, \mathfrak{B}\}$), any piece of evidence $Bx \in E$ is either transferred of removed in the same manner in both the sequential update $E|_f C_1|_f C_2$ and the joint update $E|_f C_1 \cap C_2$. We consider the following cases:

* Case 1: $B \subseteq C_1$ and $B \subseteq C_2$. In this case, $Bx \in E$ remains $Bx \in E|_f C_1$, and subsequently remains $Bx \in E|_f C_1|_f C_2$. Since $B \subseteq (C_1 \cap C_2)$, it also holds that $Bx \in E|_f C_1 \cap C_2$.

* Case 2: $B \subseteq C_1^c$ or $B \subseteq C_2^c$. In this case, $Bx$ will be removed during either the update given $C_1$ or the update given $C_2$. Since $B \subseteq (C_1 \cap C_2)^c$, it will also be removed during the joint update given $C_1 \cap C_2$.

* Case 3: $B \cap (C_1 \cap C_2) \neq \emptyset$ and $B \cap (C_1 \cap C_2)^c \neq \emptyset$. There are three sub-cases, depending on whether $B \subseteq C_1$, $B \subseteq C_2$ or $B \not\subseteq C_1$ and $B \not\subseteq C_2$.

- For the sequential $\mathcal{D}$-update, $Bx$ will be transferred to $(B \cap C_2)x$, $(B \cap C_1)x$ or $(B \cap C_1 \cap C_2)x$ respectively. In all cases, it is effectively transferred to $(B \cap C_1 \cap C_2)x$, as in the joint update;

- For the sequential $\mathfrak{G}$-update, $Bx$ will be removed either during the first or the second update, just as in the joint update;

- For the sequential $\mathfrak{B}$-update, $Bx$ will be transferred to $(B \cap C_2)^?$, $(B \cap C_1)^?$ or $(B \cap C_1 \cap C_2)^?$ respectively. In all cases, it is effectively transferred to $(B \cap C_1 \cap C_2)^?$, as in the joint update.

Since the treatment of each piece of evidence is equal in the sequential and joint updates, we conclude that the updated pools of evidence are the same.

Finally, since $\cap$ is commutative, it follows that $E|_f C_1 \cap C_2 = E|_f C_2 \cap C_1 = E|_f C_2|_f C_1$. □

**Theorem 11.** Let E be a pool of evidence over $\Omega$ and C a subset of $\Omega$. Then:

$$ \#_{E|_{\mathfrak{B}} C}(A) = \#_{E}(A \cap C). $$

**Proof.** By definition, any piece of evidence $Bx \in E|_{\mathfrak{B}} C$ satisfies $B \subseteq C$, hence $\#_{E|_{\mathfrak{B}} C}(A) = \#_{E|_{\mathfrak{B}} C}(A \cap C)$. It is sufficient to show that $\#_{E|_{\mathfrak{B}} C}(A \cap C) = \#_{E}(A \cap C)$ for all $A \subseteq \Omega$, that is, $\#_{E|_{\mathfrak{B}} C}(A') = \#_{E}(A')$ for all $A' \subseteq C$. It follows from Lemma 9 that $M_{E|_{\mathfrak{B}} C}^{-}(B) = M_{E}^{-}(B)$ for all $B \subseteq C$, hence $\#_{E|_{\mathfrak{B}} C}(A') = \sum_{B \subseteq A'} M_{E|_{\mathfrak{B}} C}^{-}(B) = \sum_{B \subseteq A'} M_{E}^{-}(B) = \#_{E}^{-}(A')$. Moreover, it also follows from Lemma 9 that $M_{E|_{\mathfrak{B}} C}^{+}(B) = \sum_{B' \cap C=B} M_{E}^{+}(B')$ for all $B \subseteq C$. Hence, $\#_{E|_{\mathfrak{B}} C}^{+}(A') = \sum_{B \cap A' \neq \emptyset} M_{E|_{\mathfrak{B}} C}^{+}(B) = \sum_{B \cap A' \neq \emptyset} \sum_{B' \cap C=B} M_{E}^{+}(B') = \sum_{B' \cap A' \neq \emptyset} M_{E}^{+}(B') = \#_{E}^{+}(A')$. □

**Theorem 12.** Let E be a pool of evidence over $\Omega$ and C $\subseteq \Omega$ a set such that $P_{E}^{+}(C) > 0$. Then:

$$ F_{E}^{*} | C = F_{E|_{\mathfrak{B}} C}^{*}. $$

**Proof.** Let E be a pool of evidence over $\Omega$ and C $\subseteq \Omega$ such that $P_{E}^{+}(C) > 0$. To show that $F_{E}^{*} | C = F_{E|_{\mathfrak{B}} C}^{*}$ we show that (i) for each prior $Pr \in F_{E}^{*}$, its unique posterior $Pr(\cdot | C)$ belongs to $F_{E|_{\mathfrak{B}} C}^{*}$ and that (ii) for each posterior $Pr' \in F_{E|_{\mathfrak{B}} C}^{*}$, one of the several compatible priors Pr (such that $Pr' = Pr(\cdot | C)$) belongs to $F_{E}^{*}$. Hence $F_{E|_{\mathfrak{B}} C}^{*}$ contains all the conditioned probability measures of $F_{E}^{*}$, and only them. That is, $F_{E|_{\mathfrak{B}} C}^{*} = F_{E}^{*} | C$.

(i) Let Pr and k be such that $\forall A, \#_{E}^{-}(A) \le k Pr(A) \le \#_{E}^{+}(A)$. By definition, $Pr \in F_{E}^{*}$. For all $A$, $Pr(C)Pr(A | C) = Pr(A \cap C)$, hence $\#_{E}^{-}(A \cap C) \le k Pr(C)Pr(A | C) \le \#_{E}^{+}(A \cap C)$. From Theorem 11, it is true that $\#_{E|_{\mathfrak{B}} C}(A) = \#_{E}(A \cap C)$. Let $k' = k Pr(C)$. It follows that $\#_{E|_{\mathfrak{B}} C}(A) \le k' Pr(A | C) \le \#_{E|_{\mathfrak{B}} C}^{+}(A)$ for all $A \subseteq C$. That is, $Pr \in F_{E}^{*} \Rightarrow Pr(\cdot | C) \in F_{E|_{\mathfrak{B}} C}^{*}$.

(ii) Let $Pr'$ and $k'$ be such that $\#_{E|_{\mathfrak{B}} C}^{-}(A) \le k' Pr'(A) \le \#_{E|_{\mathfrak{B}} C}^{+}(A)$. By definition, $Pr' \in F_{E|_{\mathfrak{B}} C}^{*}$. Now, let $Cf : 2^{\Omega} \to \mathbb{N}$ be a configuration such that $Cf(A) = k' Pr'(A)$ for all $A \subseteq C$ and such that $\#_{E}^{-}(A) \le Cf(A) \le \#_{E}^{+}(A)$ for all $A \subseteq \Omega$. Cf exists since $\#_{E|_{\mathfrak{B}} C}(A) = \#_{E}(A)$ for all $A \subseteq C$ (Theorem 11): it is a possible configuration according to E that equates to $k' Pr'(A)$ whenever $A \subseteq C$. It is generally not unique. Note that $Cf(C) = k'$. Let $Pr_0$ be its normalization, such as $Pr_0(A) = Cf(A)/Cf(\Omega)$. $Pr_0$ is a probability measure and satisfies $\#_{E}(A) \le Cf(\Omega)Pr_0(A) \le \#_{E}^{+}(A)$ for all $A$, hence $Pr_0 \in F_{E}^{*}$. Moreover, $Pr_0(A | C) = \frac{Pr_0(\Lambda \cap C)}{Pr_0(C)} = \frac{Cf(\Lambda \cap C)}{Cf(C)} = \frac{k' Pr'(\Lambda \cap C)}{k'} = Pr'(A \cap C) = Pr'(A)$. That is, $Pr' = Pr_0(\cdot | C)$. □
# Data availability

No data was used for the research described in the article.

# References

[1] Andrey G. Bronevich, Alexander E. Lepskiy, Measures of conflict, basic axioms and their application to the clusterization of a body of evidence, Fuzzy Sets Syst. 446 (2022) 277-300.

[2] Andrey G. Bronevich, Igor N. Rozenberg, Updating information based on generalized credal sets. Part 1: finite case, Fuzzy Sets Syst. 481 (2024) 108891.

[3] Fabio Cuzzolin, The Geometry of Uncertainty: The Geometry of Imprecise Probabilities, Springer Nature, 2020.

[4] Fabio Cuzzolin, A geometric approach to conditioning belief functions, Computing Research Repository, arXiv, arXiv:2104.10651 [abs], 2021.

[5] Daniel Milan, Belief conditioning rules for classic belief functions, in: Workshops on Uncertainty Processing, WUPES'09, vol. 9, 2009, pp. 46-56.

[6] Luis M. De Campos, Juan F. Huete, Serafin Moral, Probability intervals: a tool for uncertain reasoning, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 2 (02) (1994) 167-196.

[7] Arthur P. Dempster, Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Stat. 38 (2) (1967) 325-339.

[8] Thierry Denœux, Modeling vague beliefs using fuzzy-valued belief structures, Fuzzy Sets Syst. 116 (2) (2000) 167-199.

[9] Sébastien Destercke, Didier Dubois, The role of generalised p-boxes in imprecise probability models, in: International Symposium on Imprecise Probabilities: Theories and Applications, ISIPTA'09, 2009, pp. 179-188.

[10] Jean Dezert, Florentin Smarandache, Belief conditioning rules, in: Advances and Applications of DSmT for Information Fusion (Collected Works), vol. 2, 2006, p. 237.

[11] Didier Dubois, Thierry Denœux, Conditioning in Dempster-Shafer theory: prediction vs. revision, in: International Conference on Belief Functions, BELIEF'12, vol. 164, 2012, pp. 385-392.

[12] Didier Dubois, Henri Prade, Possibility Theory, Springer US, Boston, MA, 1988.

[13] Didier Dubois, Henri Prade, Evidence, knowledge, and belief functions, Int. J. Approx. Reason. 6 (3) (1992) 295-319.

[14] Didier Dubois, Henri Prade, Focusing vs. belief revision: a fundamental distinction when dealing with generic knowledge, in: Qualitative and Quantitative Practical Reasoning, Berlin, Heidelberg, vol. 1244, 1997, pp. 96-107.

[15] Didier Dubois, Henri Prade, Possibility theory: qualitative and quantitative aspects, in: Quantified Representation of Uncertainty and Imprecision, Springer, 1998, pp. 169-226.

[16] Didier Dubois, Henri Prade, Philippe Smets, New semantics for quantitative possibility theory, in: Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU'01, Springer, 2001, pp. 410-421.

[17] Daniel Ellsberg, Risk, ambiguity, and the Savage axioms, Q. J. Econ. (1961) 643-669.

[18] Ronald Fagin, Joseph Y. Halpern, Uncertainty, belief, and probability 1, Comput. Intell. 7 (3) (1991) 160-173.

[19] Hélène Fargier, Pierre Pomeret-Coquot, Generalized expected utility as a universal decision rule: a step forward, in: Uncertainty in Artificial Intelligence, UAI'24, vol. 244, 2024, pp. 1323-1338.

[20] Terrence L. Fine, Lower probability models for uncertainty and nondeterministic processes, J. Stat. Plan. Inference 20 (3) (1988) 389-411.

[21] Andrew Gelman, The boxer, the wrestler, and the coin flip: a paradox of robust Bayesian inference and belief functions, Am. Stat. 60 (2) (2006) 146-150.

[22] Ruobin Gong, Xiao Li Meng, Judicious judgment meets unsettling updating: dilation, sure loss and Simpson's paradox, Stat. Sci. 36 (2) (2021) 169-190.

[23] Michel Grabisch, K-order additive discrete fuzzy measures and their representation, Fuzzy Sets Syst. 92 (2) (1997) 167-189.

[24] Joseph Y. Halpern, Ronald Fagin, Two views of belief: belief as generalized probability and belief as evidence, Artif. Intell. 54 (3) (1992) 275-317.

[25] Jean-Yves Jaffray, Linear utility theory for belief functions, Oper. Res. Lett. 8 (2) (1989) 107-112.

[26] Jean-Yves Jaffray, Bayesian updating and belief functions, IEEE Trans. Syst. Man Cybern. 22 (5) (1992) 1144-1152.

[27] David G. Kendall, Foundation of a Theory of Random Sets, Stochastic Geometry, 1974.

[28] Henry E. Kyburg Jr., Representing knowledge and evidence for decision, in: Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU'86, Springer, 1986, pp. 30-40.

[29] Isaac Levi, The Enterprise of Knowledge: An Essay on Knowledge, Credal Probability, and Chance, MIT Press, 1980.

[30] Georges Matheron, Random Sets and Integral Geometry, Wiley Series in Probability and Mathematical Statistics, 1975.

[31] Ilya Molchanov, Theory of Random Sets, vol. 19, Springer, 2005.

[32] Serafin Moral, Calculating uncertainty intervals from conditional convex sets of probabilities, in: Uncertainty in Artificial Intelligence, UAI'92, 1992, pp. 199-206.

[33] Serafin Moral, Luis M. de Campos, Updating uncertain information, in: Uncertainty in Knowledge Bases, Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU '90, Paris, France, July 2-6, 1990, Proceedings, 1990, pp. 58-67.

[34] A.F. Móbius, Über eine besondere art von Umkehrung der Reihen, J. Reine Angew. Math. 9 (1832) 105-123.

[35] Lalinha G. Polpitiya, Kamal Premaratne, Manohar N. Murthi, Stephen J. Murrell, Dilip Sarkar, Efficient computation of conditionals in the Dempster-Shafer belief theoretic framework, IEEE Trans. Cybern. 52 (5) (2020) 2931-2941.

[36] Teddy Seidenfeld, Larry Wasserman, Dilation for sets of probabilities, Ann. Stat. 21 (3) (1993) 1139-1154.

[37] Glenn Shafer, A Mathematical Theory of Evidence, Princeton University Press, 1976.

[38] Glenn Shafer, Amos Tversky, Languages and designs for probability judgment, Cogn. Sci. 9 (3) (1985) 309-339.

[39] Philippe Smets, Probability of provability and belief functions, Log. Anal. (1991) 177-195.

[40] Philippe Smets, The nature of the unnormalized beliefs encountered in the transferable belief model, in: Uncertainty in Artificial Intelligence, UAI'92, 1992, pp. 292-297.

[41] Philippe Smets, Jeffrey's rule of conditioning generalized to belief functions, in: Uncertainty in Artificial Intelligence, UAI'93, 1993, pp. 500-505.

[42] Philippe Smets, Robert Kennes, The transferable belief model, Artif. Intell. 66 (2) (1994) 191-234.

[43] Patrick Suppes, Mario Zanotti, On using random relations to generate upper and lower probabilities, Synthese 36 (4) (1977) 427-440.

[44] Peter Walley, Statistical Reasoning with Imprecise Probabilities, Chapman & Hall, 1991.

[45] Ying-Ming Wang, Jian-Bo Yang, Dong-Ling Xu, Kwai-Sang Chin, On the combination and normalization of interval-valued belief structures, Inf. Sci. 177 (5) (2007) 1230-1247.

[46] Xiaobin Xu, Zhen Zhang, Dongling Xu, Yuwang Chen, Interval-valued evidence updating with reliability and sensitivity analysis for fault diagnosis, Int. J. Comput. Intell. Syst. 9 (3) (2016) 396-415.

[47] Lotfi Asker Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst. 1 (1) (1978) 3-28.