# The intersection probability: betting with probability intervals

Fabio Cuzzolin
Visual Artificial Intelligence Laboratory
Oxford Brookes University, Oxford, UK
<EMAIL_ADDRESS>
###### Abstract

Probability intervals are an attractive tool for reasoning under uncertainty.
Unlike belief functions, though, they lack a natural probability
transformation to be used for decision making in a utility theory framework.
In this paper we propose the use of the intersection probability, a transform
derived originally for belief functions in the framework of the geometric
approach to uncertainty, as the most natural such transformation. We recall
its rationale and definition, compare it with other candidate representives of
systems of probability intervals, discuss its credal rationale as focus of a
pair of simplices in the probability simplex, and outline a possible decision
making framework for probability intervals, analogous to the Transferable
Belief Model for belief functions.

##  1 Introduction

An estimation or a decision problem $Q$ usually involvesknowing in which state
we are, where the possible states of the world are often assumed to belong to
a finite set $\Theta=\\{x_{1},...,x_{n}\\}$. Our uncertainty about the outcome
of $Q$ can be described in many ways: the classical option is to assume a
probability distribution on $\Theta$. This, however, can only model the
_aleatory_ uncertainty about the problem, in which the outcome is random, but
the probability distribution that governs the process is fully known. For
instance, if a person plays a fair roulette wheel they will not, by any means,
know the outcome in advance, but they will nevertheless be able to predict the
long-term frequency with which each outcome manifests itself (1/36).

In opposition, in practical situations we are required to incorporate
imprecise measurements and people’s opinions in our knowledge state, or need
to cope with missing or scarce information. A more cautious approach is
therefore to assume that we have no access to the ‘correct’ probability
distribution, but that the available evidence provides us with some constraint
on this unknown distribution. This more fundamental type of uncertainty is
referred to as _epistemic_ [83], and is caused by lack of knowledge about the
very process that generates the data. Suppose that the player is presented
with ten different doors, which lead to rooms each containing a roulette wheel
modelled by a different probability distribution. They will then be uncertain
about the very game they are supposed to play. How will this affect their
betting behaviour, for instance?

Numerous mathematical theories of epistemic uncertainty have been proposed,
starting from de Finetti’s pioneering work on subjective probability [61]. To
list just the most impactful efforts, we could mention possibility theory
[120, 72], credal sets [90, 88], monotone capacities [117], random sets [94]
and imprecise probability theory [114]. New original foundations of subjective
probability in behavioural terms [115] or by means of game theory [97] have
been put forward.

One of the simplest such approaches is given by _probability intervals_ [109,
60, 82]: the probability values $p(x)$ of the elements of the decision space
$\Theta$ are assumed to belong to an interval $l(x)\leq p(x)\leq u(x)$
delimited by a lower bound $l(x)$ and an upper bound $u(x)$. This encodes a
possible (convex) set of probabilities, usually called _credal set_ [90], from
which decision can then be taken. When considering an associated decision
problem $Q$, many extensions of the classical expected utility rule proposes
to extract not one, but multiple potentially optimal decisions [111].

There are many situations, however, in which one must converge to a unique
decision. Decision rules producing one optimal decision in the imprecise-
probabilistic literature either focus on specific bounds (e.g., maxi-min [113,
111] or maxi-max rules), not accounting for the whole representation, or
require solving complex optimisation problems (e.g., selecting the maximal
entropy distribution).

An alternative approach which has been often studied within the belief
function theory [11] consists in approximating a complex uncertainty measure
(such as a belief function) by a single probability distribution, from which a
unique optimal decision can be deduced. This is, for instance, the case of the
Transferable Belief Model [98], in which the so called _pignistic
transformation_ [103] is employed for this purpose.

Similarly to the case of belief functions, it could be useful to apply such a
transformation to reduce a set of probability intervals to a single
probability distribution prior to actually making a decision. However, this
problem has been quite neglected so far. One could of course pick a
representative from the corresponding credal set, but it makes sense to wonder
whether a transformation inherently designed for probability intervals as such
could be found.

In this paper we argue that the natural candidate for the role of a
probability transform of probability intervals is the _intersection
probability_ , originally identified in the context of the geometric approach
to uncertainty [20]. Even though originally introduced as a probability
transform for belief functions [20], the intersection probability is (as we
show here) inherently associated with probability intervals, in which context
its rationale clearly emerges.
It can be shown that the intersection probability is the only probability
distribution that behave homogeneously in each element $x$ of the frame
$\Theta$, i.e., it assigns the same fraction of the available probability
interval to each element of the decision space.

###  1.1 Contributions and paper outline

We first recall the basic notions of the theories of lower and upper
probabilities, interval probabilities and belief functions (Section 2), to
then quickly review the prior art on probability transform and its geometric
interpretation (Section 3).

We then formally define the intersection probability and its rationale
(Section 4), showing that it can be defined for any interval probability
system as the unique probability distribution obtained by assigning the same
fraction of the uncertainty interval to all the elements of the domain. We
compare it with other possible representatives of interval probability
systems, and recall its geometric interpretation in the space of belief
functions and the justification for its name that derives from it (Section 5).

In Section 6 we extensively illustrate the credal rationale for the
intersection probability as focus of the pair of lower and upper simplices
associated with the interval probability system.

As a belief function determines itself an interval probability system, the
intersection probability exists for belief functions too and can therefore be
compared with classical approximations of belief functions like the pignistic
function [98] and relative plausibility and belief [24, 27] of singletons, or
more recent approximations proposed by Sudano [105]. In Section 7 we thus
analyse the relations of intersection probability with other probability
transforms of belief functions, while in Section 8 we discuss its properties
with respect to affine combination and convex closure.

The potential use of the intersection probability to bet on interval
probability systems, in a framework analogous to the Transferable Belief
Model, is outlined in Section 9, which concludes the paper.

##  2 Uncertainty theories

###  2.1 Lower and upper probabilities

A _lower probability_ ${\underline{P}}$ is a function from $2^{\Theta}$, the
power set of $\Theta$, to the unit interval $[0,1]$. A lower probability is
associated with a dual _upper probability_ ${\overline{P}}$, defined for any
$A\subseteq\Theta$ as $\overline{P}(A)=1-\underline{P}(A^{c})$, where $A^{c}$
is the complement of $A$ (this means that we can simply focus on one of the
two measures, provided it is defined for all subsets). A lower probability
${\underline{P}}$ can also be associated with a (closed convex) set

${\mathcal{P}}({\underline{P}})=\Big{\\{}p:P(A)\geq{\underline{P}}(A),\forall
A\subseteq\Theta\Big{\\}}$ (1)

of probability distributions $p$ whose measure $P$ dominates
${\underline{P}}$. Such a polytope or convex set of probability distributions
is usually called a _credal set_. A lower probability ${\underline{P}}$ will
be called _consistent_ if ${\mathcal{P}}({\underline{P}})$ and _tight_ if
$\inf_{p\in{\mathcal{P}}({\underline{P}})}P(A)={\underline{P}}(A)$
(respectively it ‘avoids sure loss’ and is ‘coherent’ in Peter Walley’s [114]
terms). Consistency means that lower bound constraints ${\underline{P}}(A)$
can be satisfied, while tightness means that ${\underline{P}}$ is the lower
envelope on subsets of ${\mathcal{P}}({\underline{P}})$. Note that not all
convex sets of probabilities can be described by only focusing on events (see
Walley [114]), however they will be sufficient here. The notation
$\mathcal{P}$ will simply denote the set of all possible probabilities on
$\Theta$.

###  2.2 Probability intervals

Dealing with general lower probabilities defined on $2^{\Theta}$ can be
difficult when $\Theta$ is big, and it may be interesting in applications to
focus on simpler models. One popular and practical model used to model such
kind of uncertainty are probability intervals.

A _set of probability intervals_ or _interval probability system_ is a system
of constraints on the probability values of a probability distribution
$p:\Theta\rightarrow[0,1]$ on a finite domain $\Theta$ of the form

$\mathcal{P}(l,u)\doteq\Big{\\{}p:l(x)\leq p(x)\leq u(x),\forall
x\in\Theta\Big{\\}}.$ (2)

Probability intervals have been introduced as a tool for uncertain reasoning
in [60], where combination and marginalization of intervals were studied in
detail. In [60] the authors also studied the specific constraints for such
intervals to be consistent and tight.

As pointed out for instance in [84], a typical way in which probability
intervals arise is through measurement errors. As a matter of fact,
measurements can be inherently of interval nature (due to the finite
resolution of the instruments). In that case the _probability_ interval of
interest is the class of probability measures consistent with the _measured_
interval. A set of constraints of the form (2) determines a credal set, which
is just a sub-class of sets described by lower and upper probabilities.

The lower and upper probabilities induced by $\mathcal{P}(l,u)$ on any subset
$A\subseteq\Theta$ from bounds $(l,u)$ can be obtained using the simple
formulas:

${\underline{P}}(A)=\max\left\\{\sum_{x\in A}l(x),1-\sum_{x\not\in
A}u(x)\right\\},\;{\underline{P}}(A)=\min\left\\{\sum_{x\in
A}u(x),1-\sum_{x\not\in A}l(x)\right\\}.$ (3)

###  2.3 Belief functions

Let us denote by $\Omega$ and $\Theta$ the sets of outcomes of two different
but related problems $Q_{1}$ and $Q_{2}$, respectively. Given a probability
measure $P$ on $\Omega$, we want to derive a ‘degree of belief’ $Bel(A)$ that
$A\subset\Theta$ contains the correct response to $Q_{2}$. If we call
$\Gamma(\omega)$ the subset of outcomes of $Q_{2}$ compatible with
$\omega\in\Omega$, $\omega$ tells us that the answer to $Q_{2}$ is in $A$
whenever $\Gamma(\omega)\subset A$ (see Figure 1). The _degree of belief_
$Bel(A)$ of an event $A\subset\Theta$ is then the total probability (in
$\Omega$) of all the outcomes $\omega$ of $Q_{1}$ that satisfy the above
condition [Dempster67]:

$Bel(A)=P(\\{\omega|\Gamma(\omega)\subset
A\\})=\sum_{\omega\in\Omega:\Gamma(\omega)\subset A}P(\\{\omega\\}).$ (4)

The map $\Gamma:\Omega\rightarrow 2^{\Theta}=\\{A\subseteq\Theta\\}$ is called
a _multivalued mapping_ from $\Omega$ to $\Theta$. Such a mapping, together
with a probability measure $P$ on $\Omega$, induces a _belief function_ on
$2^{\Theta}$.

Figure 1:  A probability measure $P$ on $\Omega$ induces via a multivalued
mapping $\Gamma$ a belief function $Bel$ on $\Theta$ whose values are given in
(4).

In Dempster’s original formulation, then, belief functions are objects induced
by a source probability measure in a decision space for which we do not have a
probability, as long as there exists a 1-many mapping between the two.

####  2.3.1 Belief and plausibility measures

A special class of lower and upper probabilities is provided by belief and
plausibility measures. Namely, a _basic probability assignment_ (BPA) [96] is
a set function [65, 69] $m:2^{\Theta}\rightarrow[0,1]$ such that

$m(\emptyset)=0,\quad\sum_{A\subset\Theta}m(A)=1.$

Subsets of $\Theta$ whose mass values are non-zero are called _focal elements_
of $m$. The _belief function_ (BF) associated with a BPA
$m:2^{\Theta}\rightarrow[0,1]$ is the set function
$Bel:2^{\Theta}\rightarrow[0,1]$ defined as

$Bel(A)=\sum_{B\subseteq A}m(B).$ (5)

The corresponding _plausibility function_ is

$Pl(A)\doteq\sum_{B\cap A\neq\emptyset}m(B)\geq Bel(A).$

Note that belief functions can also be equivalently defined in axiomatic terms
[96].

Classical probability measures on $\Theta$ are a special case of belief
functions (those assigning mass to singletons only), termed _Bayesian belief
functions_. A BF is said to be _consonant_ if its focal elements
$A_{1},...,A_{m}$ are nested: $A_{1}\subset\cdots\subset A_{m}$, and
corresponds to a possibility measure [70, 73, 96, 43].

####  2.3.2 Combination

In belief theory conditioning is replaced by the notion of (associative)
_combination_ of any number of belief functions.

The _Dempster combination_ $Bel_{1}\oplus Bel_{2}$ of two belief functions on
$\Theta$ is the unique BF there with as focal elements all the non-empty
intersections of focal elements of $Bel_{1}$ and $Bel_{2}$, and basic
probability assignment

$m_{\oplus}(A)=\frac{m_{\cap}(A)}{1-m_{\cap}(\emptyset)},$ (6)

where

$m_{\cap}(A)=\sum_{B\cap C=A}m_{1}(B)m_{2}(C)$ (7)

and $m_{i}$ is the BPA of the input belief function $Bel_{i}$.

Nevertheless, Dempster’s combination naturally induces a conditioning
operator. Given a conditioning event $A\subset\Theta$, the ‘logical’ or
_categorical_ belief function $Bel_{A}$ such that $m(A)=1$ is combined via
Dempster’s rule with the a-priori belief function $Bel$. The resulting BF
$Bel\oplus Bel_{A}$ is the conditional belief function given $A$ _a la
Dempster_ , denoted by $Bel_{\oplus}(A|B)$.

Many alternative combination rules have since been defined [86, 119, 71, 67],
often associated with a distinct approach to conditioning [64, 74, 108]. An
exhaustive review of these proposals can be found in [43], Section 4.3.

Rather than normalising (as in Dempster’s rule) or reassigning the conflicting
mass $m_{\cap}(\emptyset)$ to other non-empty subsets, Philippe Smets’s
_conjunctive rule_ leaves the conflicting mass with the empty set,

$m_{\text{\text{$\scriptstyle{\cap}$⃝}}}(A)=\left\\{\begin{array}[]{ll}m_{\cap}(A)&\emptyset\neq
A\subseteq\Theta,\\\ m_{\cap}(\emptyset)&A=\emptyset,\end{array}\right.$ (8)

and thus is applicable to _unnormalised_ belief functions [100].

In Dempster’s original random-set idea, consensus between two sources is
expressed by the intersection of the supported events (7). When the _union_ of
the supported propositions is taken to represent such a consensus instead, we
obtain what Smets called the _disjunctive_ rule of combination,

$m_{\text{\text{$\scriptstyle{\cup}$⃝}}}(A)=\sum_{B\cup C=A}m_{1}(B)m_{2}(C).$
(9)

It is interesting to note that under disjunctive combination,

$Bel_{1}$\scriptstyle{\cup}$⃝Bel_{2}(A)=Bel_{1}(A)\ast Bel_{2}(A),$

i.e., the belief values of the input belief functions are simply multiplied.

####  2.3.3 Belief functions and other measures

Each belief function $Bel$ uniquely identifies a credal set [88]

$\mathcal{P}[Bel]=\\{P\in\mathcal{P}:P(A)\geq Bel(A)\\}$ (10)

(where $\mathcal{P}$ is the set of all probabilities one can define on
$\Theta$), of which it is its lower envelope: $Bel(A)=\underline{P}(A)$.
Belief functions are thus a special case of lower probabilities (Section 2.1).
The corresponding plausibility measure is the upper probability of an event
$A$: $Pl(A)=\overline{P}(A)$. The probability intervals resulting from
Dempster’s updating of the credal set associated with a BF, however, are
included in those resulting from Bayesian updating [88].

Belief functions are also infinitely _monotone capacities_ [8, 107], and a
special case of _coherent lower previsions_ [114, 115]. Finally, every belief
function specifies a unique _probability box_ [75], i.e., a class of
cumulative distribution functions delimited by two lower and upper bounds.

###  2.4 The geometry of uncertainty measures

_Geometry_ has been proposed by this author and others as a unifying language
for the field [42, 53, 54, 55, 58, 76, 77, 5, 85, 78, 116], possibly in
conjunction with an algebraic view [51, 12, 23, 57, 16, 18, 38].

Indeed, uncertainty measures can be seen as points of a suitably complex
geometric space, and there manipulated (e.g. combined, conditioned and so on)
[13, 21, 43]. Much work has been focusing on the geometry of belief functions,
which live in a convex space termed the _belief space_ , which can be
described both in terms of a simplex (a higher-dimensional triangle) and in
terms of a recursive bundle structure [56, 45, 41, 52]. The analysis can be
extended to Dempster’s rule of combination by introducing the notion of a
conditional subspace and outlining a geometric construction for Dempster’s sum
[49, 14]. The combinatorial properties of plausibility and commonality
functions, as equivalent representations of the evidence carried by a belief
function, have also been studied [22, 34]. The corresponding spaces are
simplices which are congruent to the belief space.
Subsequent work extended the geometric approach to other uncertainty measures,
focusing in particular on possibility measures (consonant belief functions)
[32] and consistent belief functions [36, 48, 25], in terms of simplicial
complexes [15]. Analyses of belief functions in terms credal sets have also
been conducted [26, 1, 6].

The geometry of the relationship between measures of different kinds has also
been extensively studied [50, 30, 17, 31], with particular attention to the
problem of transforming a belief function into a classical probability measure
[9, 112, 99] (see Section 3). One can distinguish between an _affine_ family
of probability transformations [20] (those which commute with affine
combination in the belief space), and an _epistemic_ family of transforms
[19], formed by the relative belief and relative plausibility of singletons
[28, 27, 37, 46, 33], which possess dual properties with respect to Dempster’s
sum [24]. The problem of finding the possibility measure which best
approximates a given belief function [2] can also be approached in geometric
terms [29, 47, 39, 40]. In particular, approximations induced by classical
Minkowski norms can be derived and compared with classical outer consonant
approximations [72]. Minkowski consistent approximations of belief functions
in both the mass and the belief space representations can also be derived
[36].

The geometric approach to uncertainty can also be applied to the conditioning
problem [89]. Conditional belief functions can be defined as those which
minimise an appropriate distance between the original belief function and the
‘conditioning simplex’ associated with the conditioning event [44, 35].

Recent papers on this topic include [93, 95, 91].

##  3 Probability transform

###  3.1 Probability transforms of belief functions

The relation between belief and probability in the theory of evidence has been
and continues to be an important subject of study[118, 87, 3, 4, 66, 68, 81].
A probability transform mapping belief functions to probability measures can
be instrumental in addressing a number of issues: mitigating the inherently
exponential complexity of belief calculus [3], making decisions via the
probability distributions obtained in a utility theory framework [99] and
obtaining pointwise estimates of quantities of interest from belief functions
(e.g., the pose of an articulated object in computer vision: see [52], Chapter
8, or [54]).

As both belief and probability measures can be assimilated into points of a
Cartesian space [43], the problem can (as mentioned) be posed in a geometric
setting. Without loss of generality, we can define a _probability transform_
as a mapping from the space of belief functions $\mathcal{B}$ on the domain of
interest to the probability simplex $\mathcal{P}$ there,

$\begin{array}[]{lllll}\mathcal{PT}&:&\mathcal{B}&\rightarrow&\mathcal{P},\\\
&&Bel\in\mathcal{B}&\mapsto&\mathcal{PT}[Bel]\in\mathcal{P},\end{array}$

such that an appropriate distance function or similarity measure $d$ from
$Bel$ is minimised [59]:

$\mathcal{PT}[Bel]=\arg\min_{P\in\mathcal{P}}d(Bel,P).$ (11)

A minimal, sensible requirement is for the probability which results from the
transform to be compatible with the upper and lower bounds that the original
belief function $Bel$ enforces on the singletons only, rather than on all the
focal sets. Thus, this does not require probability transforms to adhere to
the upper–lower probability semantics of belief functions. As a matter of
fact, some important transforms of this kind are not compatible with such
semantics.

Many such transformations have been proposed, according to different criteria
[92, 110, 4, 66, 68, 3]. In Smets’s transferable belief model [98, 104], in
particular, decisions are made by resorting to the _pignistic probability_ :

$BetP[Bel](x)=\sum_{A\supseteq\\{x\\}}\frac{m(A)}{|A|},$ (12)

which is the output of the _pignistic transform_.

An interesting approach to the problem seeks approximations which enjoy
commutativity properties with respect to a specific combination rule, in
particular Dempster’s sum [63, 62]. This is the case of the _relative
plausibility of singletons_ [112], the unique probability that, given a belief
function $Bel$ with plausibility $Pl(A)=1-Bel(A^{c})$, assigns to each
singleton its normalized plausibility:111With a harmless abuse of notation, we
will often denote the values of belief functions and plausibility functions on
a singleton $x$ by $m(x),Pl(x)$ rather than by $m(\\{x\\}),Pl(\\{x\\})$.

$\tilde{Pl}[Bel](x)=\frac{Pl(x)}{\sum_{y\in\Theta}Pl(y)}.$ (13)

Its properties have been later analyzed by Cobb and Shenoy [9, 10]. Voorbraak
proved that his (in our terminology) relative plausibility of singletons
$\tilde{Pl}[Bel]$ is a perfect representative of $Bel$ when combined with
other probabilities $P\in\mathcal{P}$ through Dempster’s rule $\oplus$:

$\tilde{Pl}[Bel]\oplus P=Bel\oplus P\quad\forall P\in\mathcal{P}.$ (14)

Dually, a _relative belief transform_
$\tilde{Bel}:\mathcal{B}\rightarrow\mathcal{P}$, $Bel\mapsto\tilde{Bel}[Bel]$
mapping each belief function $Bel$ to the corresponding _relative belief of
singletons_ [24, 28, 80, 59],

$\tilde{Bel}[Bel](x)=\frac{Bel(x)}{\sum_{y\in\Theta}Bel(y)},$ (15)

can be defined. The notion of a relative belief transform (under the name of
‘normalised belief of singletons’) was first proposed by Daniel in [59]. Some
analyses of the relative belief transform and its close relationship with the
(relative) plausibility transform were presented in [24, 28].

###  3.2 Geometric approaches

Only a few authors have in the past posed the study of the connections between
belief functions and probabilities in a geometric setting. In particular, Ha
and Haddawy [79] proposed an ‘affine operator’, which can be considered a
generalisation of both belief functions and interval probabilities, and can be
used as a tool for constructing convex sets of probability distributions. In
their work, uncertainty is modelled as sets of probabilities represented as
‘affine trees’, while actions (modifications of the uncertain state) are
defined as tree manipulators. In a later publication [78], the same authors
presented an interval generalisation of the probability cross-product
operator, called the ‘convex-closure’ (cc) operator, analysed the properties
of the cc operator relative to manipulations of sets of probabilities and
presented interval versions of Bayesian propagation algorithms based on it.
Probability intervals were represented there in a computationally efficient
fashion by means of a data structure called a ‘pcc-tree’, in which branches
are annotated with intervals, and nodes with convex sets of probabilities.

The intersection probability introduced in this paper is somewhat related to
Ha’s cc operator, as it commutes (at least under certain conditions) with
affine combination, and is therefore part of the _affine family_ of Bayesian
transforms of which Smets’s pignistic transform [102] is the foremost
representative.

##  4 The intersection probability

When our uncertainty is described by imprecise probabilities, it may be
desirable for some reasons to transform this knowledge into a classical unique
probability. Such reasons include the need to take a unique optimal decision,
the need to use classical probabilistic calculus (e.g., for efficiency), or
more simply the will to obtain a unique probability from partial probabilistic
information. Existing proposals are general in scope but rather complex, e.g.,
they imply solving a convex optimization problem.

###  4.1 Definition

There are clearly many ways of selecting a single measure to represent a
collection of probability intervals (2). Note, however, that each of the
intervals $[l(x),u(x)]$, $x\in\Theta$, carries the same weight within the
system of constraints (2), as there is no reason for the different elements
$x$ of the domain to be treated differently. It is then sensible to require
that the desired representative probability should behave homogeneously in
each element $x$ of the frame $\Theta$.

Mathematically, this translates into seeking a probability distribution
$p:\Theta\rightarrow[0,1]$ such that

$p(x)=l(x)+\alpha(u(x)-l(x))$

for all the elements $x$ of $\Theta$, and some constant value $\alpha\in[0,1]$
(see Fig. 2). This value needs to be between 0 and 1 in order for the sought
probability distribution $p$ to belong to the interval.

Figure 2: An illustration of the notion of the intersection probability for an
interval probability system $(l,u)$ on $\Theta=\\{x,y,z\\}$ (2).

It is easy to see that there is indeed a _unique_ solution to this problem. It
suffices to enforce the normalisation constraint

$\sum_{x}p(x)=\sum_{x}\Big{[}l(x)+\alpha(u(x)-l(x))\Big{]}=1$

to understand that the unique value of $\alpha$ is given by

$\alpha=\beta[(l,u)]\doteq\frac{1-\sum_{x\in\Theta}l(x)}{\sum_{x\in\Theta}\big{(}u(x)-l(x)\big{)}}.$
(16)

######  Definition 1.

The _intersection probability_ $p[(l,u)]:\Theta\rightarrow[0,1]$ associated
with the interval probability system (2) is the probability distribution

$p[(l,u)](x)=\beta[(l,u)]u(x)+(1-\beta[(l,u)])l(x),$ (17)

with $\beta[(l,u)]$ given by (16).

The ratio $\beta[(l,u)]$ (16) measures the fraction of each interval
$[l(x),u(x)]$ which we need to add to the lower bound $l(x)$ to obtain a valid
probability function (adding up to one).

It is easy to see that when $(l,u)$ are a pair of belief/plausibility measures
$(Bel,Pl)$, we can define the intersection probability for belief functions as
well. Although originally defined by geometric means [20], the intersection
probability is thus in fact ‘the’ rational probability transform for general
interval probability systems.

Note that $p[(l,u)]$ can also be written as

$p[(l,u)](x)=l(x)+\left(1-\sum_{x}l(x)\right)R[(l,u)](x),$ (18)

where

$R[(l,u)](x)\doteq\frac{u(x)-l(x)}{\sum_{y\in\Theta}(u(y)-l(y))}=\frac{\Delta(x)}{\sum_{y\in\Theta}\Delta(y)}.$
(19)

Here $\Delta(x)$ measures the width of the probability interval for $x$,
whereas $R[(l,u)]:\Theta\rightarrow[0,1]$ measures how much the uncertainty in
the probability value of each singleton ‘weighs’ on the total width of the
interval system (2). We thus term it the _relative uncertainty_ of singletons.
Therefore, we can say that $p[(l,u)]$ distributes the mass $(1-\sum_{x}l(x))$
to each singleton $x\in\Theta$ according to the relative uncertainty
$R[(l,u)](x)$ it carries for the given interval.

######  Example 1.

Consider as an example an interval probability system on a domain
$\Theta=\\{x,y,z\\}$ of size 3:

$\begin{array}[]{lll}0.2\leq p(x)\leq 0.8,&0.4\leq p(y)\leq 1,&0.3\leq
p(z)\leq 0.3.\end{array}$ (20)

Notice that there is no uncertainty at all on the value of $p(z)=0.3$. The
widths of the corresponding intervals are $\Delta(x)=0.6$, $\Delta(y)=0.6$,
$\Delta(z)=0$ respectively. The relative uncertainty on each singleton (19) is
therefore:

$\begin{array}[]{lll}R[(l,u)](x)&=&\frac{\Delta(x)}{\sum_{w\in\Theta}\Delta(w)}=\frac{0.6}{1.2}=\frac{1}{2},\\\
R[(l,u)](y)&=&\frac{1}{2},\\\
R[(l,u)](z)&=&\frac{\Delta(z)}{\sum_{w\in\Theta}\Delta(w)}=\frac{0}{1.2}=0.\end{array}$
(21)

Computing the intersection probability is then really easy. By Equation (16)
the fraction of the uncertainty $u(x)-l(x)$ on $p(x)$ we need to add to the
lower bound $l(x)$ to get an admissible, normalized probability is

$\beta=\frac{1-0.2-0.4-0.3}{0.6+0.6}=\frac{0.1}{1.2}=\frac{1}{12}.$

The intersection probability (18) has therefore values:

$\begin{array}[]{c}p[(l,u)](x)=0.2+\frac{1}{12}0.6=0.25,\hskip
14.22636ptp[(l,u)](y)=0.4+\frac{1}{12}0.6=0.45,\\\ \\\
p[(l,u)](z)=0.3+\frac{1}{12}0=0.3.\end{array}$

Notice that the fact of having a zero-width interval for one of the singletons
does not pose a problem for the intersection probability, which falls as
expected inside the probability interval for all the elements of the domain.

According to its interpretation of Equation (18), $p[(l,u)]$ is also the
result of distributing the necessary mass $(1-\sum_{x}l(x))=1-0.2-0.4-0.3=0.1$
to each singleton in proportion to the relative uncertainty $R[(l,u)]$
(Equation (21)) of their intervals:

$\begin{array}[]{c}p[(l,u)](x)=0.2+0.1\frac{1}{2}=0.25,\hskip
14.22636ptp[(l,u)](y)=0.4+0.1\frac{1}{2}=0.45,\\\ \\\ p[(l,u)](z)=0.3+0.1\cdot
0=0.3.\end{array}$

###  4.2 Comparison with other interval representatives

It can be useful to briefly compare the proposed intersection probability with
other possible representatives of an interval probability system (2).

####  4.2.1 Comparison with the center of mass of $P[(l,u)]$

The naive choice of picking the barycenter of each interval $[l(x),u(x)]$ to
represent an interval probability system $(l,u)$, for instance, does not yield
in general a valid probability function, for

$\sum_{x\in\Theta}\left[l(x)+\frac{1}{2}(u(x)-l(x))\right]\neq 1.$

This marks the difference with the case of belief functions, for which the
pignistic function has a strong interpretation as barycenter of the associated
credal set.

####  4.2.2 Comparison normalised lower and upper bounds

For the probability interval system (22) determined by a belief function,

$(Bel,Pl)\doteq\big{\\{}p\in\mathcal{P}:Bel(x)\leq p(x)\leq Pl(x),\forall
x\in\Theta\big{\\}},$ (22)

the probabilities we obtain by normalizing lower
$\tilde{l}(x)=l(x)/\sum_{y}l(y)$ or upper bound
$\tilde{u}(x)=u(x)/\sum_{y}u(y)$ are _not_ guaranteed to be consistent with
the interval itself.

For instance, if there exists an element $x\in\Theta$ such that $Bel(x)=Pl(x)$
(the interval has width zero for that element) we have that

$\tilde{Bel}(x)=\frac{m(x)}{\sum_{y}m(y)}>Pl(x),\quad\tilde{Pl}(x)=\frac{Pl(x)}{\sum_{y}Pl(y)}<Bel(x).$

Therefore, both relative belief and plausibility of singletons fall outside
the interval system (22). This holds for a general collection of probability
intervals (2), again marking the contrast with the behavior of the
intersection probability.

####  4.2.3 Comparison with Sudano’s proposal

In the belief functions framework, Sudano proposed in [106] the following four
probability transforms:

$\displaystyle\begin{array}[]{lll}PrPl[Bel](x)&\doteq&\displaystyle\sum_{A\supseteq\\{x\\}}m(A)\frac{Pl(x)}{\sum_{y\in
A}Pl(y)},\end{array}$ (24)
$PrBel[Bel](x)\doteq\sum_{A\supseteq\\{x\\}}m(A)\frac{Bel(x)}{\sum_{y\in
A}Bel(y)}=\sum_{A\supseteq\\{x\\}}m(A)\frac{m(x)}{\sum_{y\in A}m(y)},$ (25)
$PrNPl[Bel](x)\doteq\frac{1}{\Delta}\sum_{A\cap\\{x\\}\neq\emptyset}m(A)=\tilde{Pl}[Bel](x),$
(26) $PraPl[Bel](x)\doteq Bel(x)+\epsilon\cdot
Pl(x),\;\;\;\epsilon=\frac{1-\sum_{y\in\Theta}Bel(y)}{\sum_{y\in\Theta}Pl(y)}=\frac{1-k_{Bel}}{k_{Pl}},$
(27)

where

$k_{Bel}\doteq\sum_{x\in\Theta}Bel(x),\quad
k_{Pl}\doteq\sum_{x\in\Theta}Pl(x).$ (28)

The first two transformations are clearly inspired by the pignistic function
(12). While in the latter case the mass $m(A)$ of each focal element is
redistributed homogeneously to all its elements $x\in A$, $PrPl[Bel]$ (24)
redistributes $m(A)$ proportionally to the relative plausibility of a
singleton $x$ _inside $A$_. Similarly, $PrBel[Bel]$ (25) redistributes $m(A)$
proportionally to the relative _belief_ of a singleton $x$ within $A$.

The fourth transformation (27), $PraPl[Bel]$, is more related to the case of
probability intervals and to the intersection probability. By Equation (17),

$\begin{array}[]{lll}p[Bel](x)&=&\displaystyle(1-\beta[Bel])\tilde{Bel}(x)k_{Bel}+\beta[Bel]\tilde{Pl}(x)k_{Pl},\end{array}$
(29)

where

$(1-\beta[Bel])k_{Bel}+\beta[Bel]k_{Pl}=\frac{k_{Pl}-1}{k_{Pl}-k_{Bel}}k_{Bel}+\frac{1-k_{Bel}}{k_{Pl}-k_{Bel}}k_{Pl}=1,$

i.e., $p[Bel]$ lies on the line joining the relative plausibility $\tilde{Pl}$
and the relative belief $\tilde{Bel}$ of singletons. Here $\beta[Bel]$ is the
value of (16) for a system of probability intervals associated with a belief
function $Bel$, namely:

$\beta[Bel]\doteq\frac{1-\sum_{x\in\Theta}Bel(x)}{\sum_{x\in\Theta}\big{(}Pl(x)-Bel(x)\big{)}}.$
(30)

Just like the intersection probability (29) and the relative uncertainty of
singletons [20], $PraPl[b]$ can also be expressed as an affine combination of
relative belief and plausibility of singletons:

$PraPl[Bel](x)=m(x)+\frac{1-k_{Bel}}{k_{Pl}}Pl(x)=k_{Bel}\tilde{Bel}(x)+(1-k_{Bel})\tilde{Pl}(x).$
(31)

More to the point, as its definition only involves belief and plausibility
values _of singletons_ , it is more correct to think of $PraPl[b]$ as of a
probability transformation of a probability interval system (rather than an
approximation of a belief function)

$PraPl[(l,u)]\doteq l(x)+\frac{1-\sum_{y}l(y)}{\sum_{y}u(y)}u(x)$

just like the intersection probability.

However, it is easier to point out its weakness as a representative of
probability intervals when put in the above form. Just like in the case of
relative belief and plausibility of singletons, $PraPl[(l,u)]$ is not in
general consistent with the original probability interval system $(l,u)$.
If there exists an element $x\in\Theta$ such that $l(x)=u(x)$ (the interval
has width $\Delta(x)$ equal to zero for that element) we have that

$\begin{array}[]{lll}PraPl[(l,u)](x)&=&\displaystyle
l(x)+\frac{1-\sum_{y}l(y)}{\sum_{y}u(y)}u(x)=u(x)+\frac{1-\sum_{y}l(y)}{\sum_{y}u(y)}u(x)\\\
&=&\displaystyle
u(x)\cdot\frac{\sum_{y}u(y)+1-\sum_{y}l(y)}{\sum_{y}u(y)}>u(x)\end{array}$

as $\frac{\sum_{y}u(y)+1-\sum_{y}l(y)}{\sum_{y}u(y)}>1$, and $PraPl[(l,u)]$
falls outside the interval.

Another fundamental objection against $PraPl[(l,u)]$ arises when we compare it
to $p[(l,u)]$. While the latter adds to the lower bound $l(x)$ an equal
fraction of the uncertainty $u(x)-l(x)$ for all singletons (18),
$PraPl[(l,u)]$ adds to the lower bound $l(x)$ an equal fraction _of the upper
bound_ $u(x)$, effectively counting twice the evidence represented by the
lower bound $l(x)$ (31).

In the case of belief functions, this amounts to adding to the mass value
$m(x)$ of $x$ yet another fraction of $m(x)$ itself, instead of distributing
only the remaining mass $Pl(x)-m(x)$ allowed to be assigned to $x$.

##  5 Geometric interpretation

Despite having being defined as a representative for systems of probability
intervals, the intersection probability was first identified in the context of
the geometric analysis of belief measures [21, 13, 56, 52]. As we briefly
recall here, its very name derives from its geometry in the space of belief
functions, or _belief space_ [21, 49].

###  5.1 Geometry in the belief space

####  5.1.1 Belief space

Given a frame of discernment $\Theta$, a belief function
$Bel:2^{\Theta}\rightarrow[0,1]$ is completely specified by its $N-2$ belief
values $\\{Bel(A),\emptyset\subsetneq A\subsetneq\Theta\\}$, $N\doteq
2^{|\Theta|}$ (as $Bel(\emptyset)=0$, $Bel(\Theta)=1$ for all BFs), and can
then be seen as a point of $\mathbb{R}^{N-2}$.

The _belief space_ associated with $\Theta$ is the set of points
$\mathcal{B}\subset\mathbb{R}^{N-2}$ which correspond to admissible belief
functions [21]. This turns out to be the simplex determined by the convex
closure of all the categorical belief functions $Bel_{A}$, namely

$\mathcal{B}=Cl(Bel_{A},\;\emptyset\subsetneq A\subseteq\Theta),$

($Bel_{\Theta}$ included). The _faces_ of a simplex are all the simplices
generated by a subset of its vertices. The set of all the Bayesian belief
functions on $\Theta$, $\mathcal{P}=Cl(Bel_{x},x\in\Theta)$, is then a face of
$\mathcal{B}$.

Plausibility functions, also determined by their $N-2$ values
$\\{Pl(A),\emptyset\subsetneq A\subsetneq\Theta\\}$, can too be seen as points
of $\mathbb{R}^{N-2}$. We call _plausibility space_ [24, 22] the corresponding
region $\mathcal{PL}$ of $\mathbb{R}^{N-2}$, again, a simplex [45].

Figure 3: The geometry of the line $a(Bel,Pl)$ and the relative locations of
$p[Bel]$, $\varsigma[Bel]$ and of the _orthogonal projection_ $\pi[Bel]$ [20]
for a frame of discernment of arbitrary size. Each belief function $Bel$ and
the related plausibility function $Pl$ lie on opposite sides of the hyperplane
$\mathcal{P}^{\prime}$ of all Bayesian _pseudo belief functions_ [20], which
divides the space $\mathbb{R}^{N-2}$ of all such functions into two halves.
The line $a(Bel,Pl)$ connecting them always intersects $\mathcal{P}^{\prime}$,
but not necessarily $a(\mathcal{P})$ (vertical line). This intersection
$\varsigma[Bel]$ is naturally associated with a probability $p[Bel]$ (in
general distinct from the orthogonal projection $\pi[Bel]$ of $Bel$ onto
$\mathcal{P}$), having the same components in the base
$\\{Bel_{x},x\in\Theta\\}$ of $a(\mathcal{P})$. $\mathcal{P}$ is a simplex (a
segment in the figure) in $a(\mathcal{P})$: $\pi[Bel]$ and $p[Bel]$ are both
‘true’ probabilities.

####  5.1.2 Dual line

The intersection probability for a belief function $Bel$ can be shown to be
the unique probability distribution determined by the intersection of the
_dual line_ joining a belief function $Bel$ and the related plausibility
function $Pl$ with the region of Bayesian (pseudo) belief functions [20].

It can be shown that this dual line $a(Bel,Pl)$ is always orthogonal to
$\mathcal{P}$, but it does not intersect the probabilistic subspace in
general. It does always intersect, however, the region of Bayesian pseudo
belief functions (or ‘normalised sum functions’) in a point

$\varsigma[Bel]\doteq Bel+\beta[Bel](Pl-
Bel)=a(Bel,Pl)\cap\mathcal{P}^{\prime}$ (32)

(where $\mathcal{P}^{\prime}$ denotes the set of all Bayesian normalised sum
functions in $\mathbb{R}^{N-2}$). $\varsigma[Bel]$ is a Bayesian pseudo BF but
is not guaranteed to be a ‘proper’ Bayesian belief function.

But, of course, since $\sum_{x}m_{\varsigma[Bel]}(x)=1$, $\varsigma[Bel]$ is
naturally associated with a Bayesian belief function assigning an equal amount
of mass to each singleton and 0 to each $A:|A|>1$. Namely, we can define the
probability measure

$p[Bel]\doteq\sum_{x\in\Theta}m_{\varsigma[Bel]}(x)Bel_{x},$ (33)

where $m_{\varsigma[Bel]}(x)$ is given by

$m_{\varsigma[Bel]}(x)=m(x)+\beta[Bel]\sum_{A\supsetneq x}m(A).$ (34)

This Bayesian BF $p[Bel]$ is nothing but the intersection probability
associated with the probability interval system induced by $Bel$. The relative
geometry of $\varsigma[Bel]$ and $p[Bel]$ with respect to the regions of
Bayesian belief and normalised sum functions, respectively, is outlined in
Fig. 3.

###  5.2 Justification for the name

The pseudo probability $\varsigma[Bel]$ (32) provides the justification for
the name ‘intersection probability’. It turns out that $p[Bel]$ and
$\varsigma[Bel]$ are _equivalent_ when combined with a Bayesian belief
function.

We first need to recall the following result [14].

######  Proposition 1.

The orthogonal sum $Bel\oplus(\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2})$ of a
belief function $Bel$ and any affine combination
$\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}$, $\alpha_{1}+\alpha_{2}=1$ of other two
belief functions $Bel_{1}$, $Bel_{2}$ on the same frame reads as

$Bel\oplus(\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2})=\gamma_{1}(Bel\oplus
Bel_{1})+\gamma_{2}(Bel\oplus Bel_{2}),$ (35)

where

$\gamma_{i}=\frac{\alpha_{i}k(Bel,Bel_{i})}{\alpha_{1}k(Bel,Bel_{1})+\alpha_{2}k(Bel,Bel_{2})}$

and $k(Bel,Bel_{i})$ is the normalisation factor of the orthogonal sum
$Bel\oplus Bel_{i}$.

Similar results can be proven for both conjunctive and disjunctive rules.

######  Lemma 1.

Affine combination commutes with both conjunctive and disjunctive rules:

$\begin{array}[]{l}Bel$\scriptstyle{\cap}$⃝(\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2})=\alpha_{1}(Bel$\scriptstyle{\cap}$⃝Bel_{1})+\alpha_{2}(Bel$\scriptstyle{\cap}$⃝Bel_{2}),\\\
Bel$\scriptstyle{\cup}$⃝(\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2})=\alpha_{1}(Bel$\scriptstyle{\cup}$⃝Bel_{1})+\alpha_{2}(Bel$\scriptstyle{\cup}$⃝Bel_{2})\end{array}$

whenever $\alpha_{1}+\alpha_{2}=1$.

###### Proof.

By definition (9), we have that
$Bel$\scriptstyle{\cap}$⃝(\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2})$ has basic
probability assignment:

$\begin{array}[]{lll}&&m_{Bel$\scriptstyle{\cap}$⃝(\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2})}(A)\\\
\\\ &=&\displaystyle\sum_{B\cap
C=A}m(B)m_{\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}}(C)\\\
&=&\displaystyle\sum_{B\cap
C=A}m(B)\big{(}\alpha_{1}m_{1}(C)+\alpha_{2}m_{2}(C)\big{)}\\\
&=&\displaystyle\alpha_{1}\sum_{B\cap C=A}m(B)m_{1}(C)+\alpha_{2}\sum_{B\cap
C=A}m(B)m_{2}(C)\\\ \\\
&=&\alpha_{1}m_{Bel$\scriptstyle{\cap}$⃝Bel_{1}}(A)+\alpha_{2}m_{Bel$\scriptstyle{\cap}$⃝Bel_{2}}(A).\end{array}$

An analogous proof holds for (8). ∎

######  Theorem 1.

The combinations of $p[Bel]$ and $\varsigma[Bel]$ with any probability
function $p\in\mathcal{P}$ coincide under both the Dempster (6) and
conjunctive (9) rules:

$\begin{array}[]{ccc}p[Bel]\oplus p=\varsigma[Bel]\oplus p,&p[Bel]\cap
p=\varsigma[Bel]\cap p,&\forall p\in\mathcal{P}.\end{array}$ (36)

###### Proof.

Let us define by

$\mu(A)=\sum_{B\subseteq A}(-1)^{|A-B|}Pl(B)$ (37)

the Moebius inverse of a plausibility function $Pl$ (see [22]). It can be
proven that [34]:

$\sum_{A\supseteq\\{x\\}}\mu(A)=m(x).$ (38)

Now, applying Equation (35) to $\varsigma\oplus p$ yields

$\begin{array}[]{lll}\varsigma\oplus
p&=&\big{[}\beta[Bel]Pl+(1-\beta[Bel])Bel\big{]}\oplus p\\\ \\\
&=&\displaystyle\frac{\beta[Bel]k(p,Pl)Pl\oplus
p+(1-\beta[Bel])k(p,Bel)Bel\oplus
p}{\beta[Bel]k(p,Pl)+(1-\beta[Bel])k(p,Bel)}\end{array}$ (39)

where

$\begin{array}[]{llll}k(p,Pl)&=&\displaystyle\sum_{x\in\Theta}p(x)\left(\sum_{A\supseteq\\{x\\}}\mu(A)\right)&=\displaystyle\sum_{x\in\Theta}p(x)m(x),\\\
\\\
k(p,Bel)&=&\displaystyle\sum_{x\in\Theta}p(x)\left(\sum_{A\supseteq\\{x\\}}m(A)\right)&\displaystyle=\sum_{x\in\Theta}p(x)Pl(x).\end{array}$

by Equation (38), and by the definition of the plausibility of singletons
($\sum_{A\supseteq\\{x\\}}m(A)=Pl(x)$). On the other hand, recalling Equation
(29), we can write

$p[Bel]=\beta[Bel]\bar{Pl}+(1-\beta[Bel])\bar{Bel}$ (40)

where we call the quantities

$\begin{array}[]{ccc}\displaystyle\bar{Pl}\doteq\sum_{x\in\Theta}Pl(x)Bel_{x}&&\displaystyle\bar{Bel}=\sum_{x\in\Theta}m(x)Bel_{x}\end{array}$
(41)

_plausibility of singletons_ and _belief of singletons_ , respectively [27,
24, 33, 37], for sake of consistency of nomenclature. However, $\bar{Pl}$ is
traditionally referred to as the _contour function_.

Therefore when we apply (35) to $p[Bel]\oplus p$, instead, we get (by Equation
(40)):

$\begin{array}[]{lll}p[Bel]\oplus
p&=&\big{[}\beta[Bel]\bar{Pl}+(1-\beta[Bel])\bar{Bel}\big{]}\oplus p\\\ \\\
&=&\displaystyle\frac{\beta[Bel]k(p,\bar{Pl})\bar{Pl}\oplus
p+(1-\beta[Bel])k(p,\bar{Bel})\bar{Bel}\oplus
p}{\beta[Bel]k(p,\bar{Pl})+(1-\beta[Bel])k(p,\bar{Bel})}.\end{array}$ (42)

By definition of Dempster’s combination (6):

$\begin{array}[]{c}\displaystyle\bar{Pl}\oplus
p=\frac{\displaystyle\sum_{x\in\Theta}Bel_{x}p(x)(Pl(x)+1-k_{Pl})}{k(p,\bar{Pl})},\\\
\\\ \displaystyle\bar{Bel}\oplus
p=\frac{\displaystyle\sum_{x\in\Theta}Bel_{x}p(x)(m(x)+1-k_{Bel})}{k(p,\bar{Bel})}.\end{array}$

Hence

$\begin{array}[]{ll}\displaystyle k(p,\bar{Pl})\bar{Pl}\oplus
p&\displaystyle=\sum_{x\in\Theta}Bel_{x}p(x)Pl(x)+(1-k_{Pl})\sum_{x\in\Theta}Bel_{x}p(x)\\\
&=\displaystyle k(Bel,p)Bel\oplus p+(1-k_{Pl})p,\\\ \\\ \displaystyle
k(p,\bar{Bel})\bar{Bel}\oplus
p&\displaystyle=\sum_{x\in\Theta}Bel_{x}p(x)m(x)+(1-k_{Bel})\sum_{x\in\Theta}Bel_{x}p(x)\\\
&=\displaystyle k(Pl,p)Pl\oplus p+(1-k_{Bel})p,\end{array}$

as:

  1. 1.

$\sum_{x\in\Theta}Bel_{x}p(x)=p$;

  2. 2.

in the calculation of $Bel\oplus p$, each singleton $x$ is assigned mass

$p(x)\sum_{A\supseteq\\{x\\}}m(A)=p(x)Pl(x);$

  3. 3.

in the calculation of $Pl\oplus p$, each singleton $x$ is assigned mass

$p(x)\sum_{A\supseteq\\{x\\}}\mu(A)=p(x)m(x)$

(again by Equation (38)).

After replacing these expressions in the numerator of (42) we can notice that,
as

$\begin{array}[]{ccc}\displaystyle\beta[Bel]=\frac{1-k_{Bel}}{k_{Pl}-k_{Bel}},&&\displaystyle
1-\beta[Bel]=\frac{k_{Pl}-1}{k_{Pl}-k_{Bel}},\end{array}$

the contributions of $p$ vanish, leaving expression (39) for $\varsigma\oplus
p$.

As conjunctive rule and affine combination commute, and $k(Bel_{1},Bel_{2})=1$
for each pair of pseudo belief functions $Bel_{1},Bel_{2}$ under conjunctive
combination, the proof holds for $$\scriptstyle{\cap}$⃝$ too. ∎

Even though $p[Bel]$ is _not_ the actual intersection $\varsigma[Bel]$ of the
line $a(Bel,Pl)$ with the region of pseudo probabilities in the belief space,
it behaves exactly like it when aggregated to a probability distribution.

Notice that Theorem 1 is _not_ a simple consequence of Voorbraak’s
representation theorem [24]:

$Bel\oplus p=\tilde{Pl}\oplus p.$

It is indeed easy to prove that the relative plausibility of singletons of
$\varsigma[Bel]$ _is not_ $p[Bel]$. As
$\varsigma[Bel](A)=Bel(A)+\beta[Bel][Pl-Bel](A)$ we have:

$\begin{array}[]{lll}&&Pl_{\varsigma[Bel]}(x)\\\ &=&\displaystyle
1-\varsigma[Bel](\\{x\\}^{c})=1-Bel(\\{x\\}^{c})-\beta[Bel]\left[Pl(\\{x\\}^{c})-Bel(\\{x\\}^{c})\right]\\\
&=&\displaystyle Pl(x)-\beta[Bel]\left[1-Bel(x)-1+Pl(x)\right]\\\
&=&Pl(x)-\beta[Bel]\left[Pl(x)-Bel(x)\right]\\\
&=&\beta[Bel]Bel(x)+(1-\beta[Bel])Pl(x)=\beta[Bel]m(x)+(1-\beta[Bel])Pl(x).\end{array}$

Its normalization factor is

$\begin{array}[]{lll}\sum_{x}Pl_{\varsigma[Bel]}(x)&=&\displaystyle\sum_{x}Pl(x)-\beta[Bel]\left[\sum_{x}Pl(x)-\sum_{x}m(x)\right]\\\
&=&\displaystyle\sum_{x}Pl(x)-1+\sum_{x}m(x).\end{array}$

Clearly then

$\tilde{Pl}_{\varsigma[Bel]}(x)=\frac{Pl_{\varsigma[Bel]}(x)}{\sum_{y}Pl_{\varsigma[Bel]}(y)}=\frac{\beta[Bel]m(x)+(1-\beta[Bel])Pl(x)}{\sum_{y}Pl(y)-1+\sum_{y}m(y)}$

is different from $p[Bel](x)=\beta[Bel]Pl(x)+(1-\beta[Bel])m(x)$.

##  6 Credal rationale

Probability interval systems admit a credal representation, which for
intervals associated with belief functions is also strictly related to the
credal set $\mathcal{P}[Bel]$ of all consistent probabilities [31, 43].

By the definition (10) of $\mathcal{P}[Bel]$, it follows that the polytope of
consistent probabilities can be decomposed into a number of component
polytopes, namely

$\mathcal{P}[Bel]=\bigcap_{i=1}^{n-1}\mathcal{P}^{i}[Bel],$ (43)

where $\mathcal{P}^{i}[Bel]$ is the set of probabilities that satisfy the
lower probability constraint for size-$i$ events,

$\mathcal{P}^{i}[Bel]\doteq\Big{\\{}P\in\mathcal{P}:P(A)\geq Bel(A),\forall
A:|A|=i\Big{\\}}.$ (44)

Note that for $i=n$ the constraint is trivially satisfied by all probability
measures $P$: $\mathcal{P}^{n}[Bel]=\mathcal{P}$.

###  6.1 Lower and upper simplices

A simple and elegant geometric description of interval probability systems can
be provided if, instead of considering the polytopes (44), we focus on the
credal sets

$T^{i}[Bel]\doteq\Big{\\{}P^{\prime}\in\mathcal{P}^{\prime}:P^{\prime}(A)\geq
Bel(A),\forall A:|A|=i\Big{\\}}.$

Here $\mathcal{P}^{\prime}$ denotes the set of all _pseudo-probability
measures_ $P^{\prime}$ on $\Theta$, whose distribution
$p^{\prime}:\Theta\rightarrow\mathbb{R}$ satisfy the normalisation constraint
$\sum_{x\in\Theta}p^{\prime}(x)=1$ but not necessarily the non-negativity one
– there may exist an element $x$ such that $p^{\prime}(x)<0$. In particular,
we focus here on the set of pseudo-probability measures which satisfy the
lower constraint on singletons,

$T^{1}[Bel]\doteq\Big{\\{}p^{\prime}\in\mathcal{P}^{\prime}:p^{\prime}(x)\geq
Bel(x)\;\;\forall x\in\Theta\Big{\\}},$ (45)

and the set $T^{n-1}[Bel]$ of pseudo-probability measures which satisfy the
analogous constraint on events of size $n-1$,

$\begin{array}[]{lll}T^{n-1}[Bel]&\doteq&\Big{\\{}P^{\prime}\in\mathcal{P}^{\prime}:P^{\prime}(A)\geq
Bel(A)\;\;\forall A:|A|=n-1\Big{\\}}\\\ \\\
&=&\Big{\\{}P^{\prime}\in\mathcal{P}^{\prime}:P^{\prime}(\\{x\\}^{c})\geq
Bel(\\{x\\}^{c})\;\;\forall x\in\Theta\Big{\\}}\\\ \\\
&=&\Big{\\{}p^{\prime}\in\mathcal{P}^{\prime}:p^{\prime}(x)\leq
Pl(x)\;\;\forall x\in\Theta\Big{\\}},\end{array}$ (46)

i.e., the set of pseudo-probabilities which satisfy the upper constraint on
singletons.

###  6.2 Simplicial form

The extension to pseudo-probabilities allows us to prove that the credal sets
(45) and (46) have the form of simplices (see [31] and [43], Chapter 16).

######  Theorem 2.

The credal set $T^{1}[Bel]$, or _lower simplex_ , can be written as

$T^{1}[Bel]=Cl(t^{1}_{x}[Bel],x\in\Theta),$ (47)

namely as the convex closure of the vertices

$t^{1}_{x}[Bel]=\sum_{y\neq x}m(y)Bel_{y}+\left(1-\sum_{y\neq
x}m(y)\right)Bel_{x}.$ (48)

Dually, the _upper simplex_ $T^{n-1}[Bel]$ reads as the convex closure

$T^{n-1}[Bel]=Cl(t^{n-1}_{x}[Bel],x\in\Theta)$ (49)

of the vertices

$t^{n-1}_{x}[Bel]=\sum_{y\neq x}Pl(y)Bel_{y}+\left(1-\sum_{y\neq
x}Pl(y)\right)Bel_{x}.$ (50)

By (48), each vertex $t^{1}_{x}[Bel]$ of the lower simplex is a pseudo-
probability that adds the total mass $1-k_{Bel}$ of non-singletons to that of
the element $x$, leaving all the others unchanged:

$m_{t^{1}_{x}[Bel]}(x)=m(x)+1-k_{Bel},\quad
m_{t^{1}_{x}[Bel]}(y)=m(y)\;\forall y\neq x.$

In fact, as $m_{t^{1}_{x}[Bel]}(z)\geq 0$ for all $z\in\Theta$ and for all
$x\in\Theta$ (all $t^{1}_{x}[Bel]$ are actual probabilities), we have that

$T^{1}[Bel]=\mathcal{P}^{1}[Bel],$ (51)

and $T^{1}[Bel]$ is completely included in the probability simplex.

On the other hand, the vertices (50) of the upper simplex are not guaranteed
to be valid probabilities.
Each vertex $t^{n-1}_{x}[Bel]$ assigns to each element of $\Theta$ different
from $x$ its plausibility $Pl(y)=pl(y)$, while it subtracts from $Pl(x)$ the
‘excess’ plausibility $k_{Pl}-1$:

$\begin{array}[]{llll}m_{t^{n-1}_{x}[Bel]}(x)&=&Pl(x)+(1-k_{Pl}),&\\\
m_{t^{n-1}_{x}[Bel]}(y)&=&Pl(y)&\forall y\neq x.\end{array}$

Now, as $1-k_{Pl}$ can be a negative quantity, $m_{t^{n-1}_{x}[Bel]}(x)$ can
also be negative and $t^{n-1}_{x}[Bel]$ is not guaranteed to be a ‘true’
probability.

We will have confirmation of this fact in the example in Section 6.4.

###  6.3 Lower and upper simplices and probability intervals

By comparing (2), (45) and (46), it is clear that the credal set
$\mathcal{P}[(l,u)]$ associated with a set of probability intervals $(l,u)$ is
nothing but the intersection

$\mathcal{P}[(l,u)]=T[l]\cap T[u]$

of the lower and upper simplices (52) associated with its lower- and upper-
bound constraints, respectively:

$T[l]\doteq\Big{\\{}p:p(x)\geq l(x)\;\forall x\in\Theta\Big{\\}},\quad
T[u]\doteq\Big{\\{}p:p(x)\leq u(x)\;\;\forall x\in\Theta\Big{\\}}.$ (52)

In particular, when these lower and upper bounds are those enforced by a pair
of belief and plausibility measures on the singleton elements of a frame of
discernment, $l(x)=Bel(x)$ and $u(x)=Pl(x)$, we get

$\mathcal{P}[(Bel,Pl)]=T^{1}[Bel]\cap T^{n-1}[Bel].$

###  6.4 Ternary case

Let us consider the case of a frame of cardinality 3, $\Theta=\\{x,y,z\\}$,
and a belief function $Bel$ with mass assignment

$\begin{array}[]{ccc}m(x)=0.2,&m(y)=0.1,&m(z)=0.3,\\\ \\\
m(\\{x,y\\})=0.1,&m(\\{y,z\\})=0.2,&m(\Theta)=0.1.\end{array}$ (53)

Figure 4 illustrates the geometry of the related credal set $\mathcal{P}[Bel]$
in the simplex, denoted by $Cl(P_{x},P_{y},P_{z})$, of all the probability
measures on $\Theta$.

It is well known [7, 26] that the credal set associated with a belief function
is a polytope whose vertices are associated with all possible permutations of
singletons.

######  Proposition 2.

Given a belief function $Bel:2^{\Theta}\rightarrow[0,1]$, the simplex
$\mathcal{P}[Bel]$ of the probability measures consistent with $Bel$ is the
polytope

$\mathcal{P}[Bel]=Cl(P^{\rho}[Bel]\;\forall\rho),$

where $\rho$ is any permutation $\\{x_{\rho(1)},\ldots,x_{\rho(n)}\\}$ of the
singletons of $\Theta$, and the vertex $P^{\rho}[Bel]$ is the Bayesian belief
function such that

$P^{\rho}[Bel](x_{\rho(i)})=\sum_{A\ni x_{\rho}(i),A\not\ni
x_{\rho}(j)\;\forall j<i}m(A).$ (54)

By Proposition 2, for the example belief function (53), $\mathcal{P}[Bel]$ has
as vertices the probabilities $P^{\rho^{1}},P^{\rho^{2}},P^{\rho^{3}}$,
$P^{\rho^{4}}$, $P^{\rho^{5}}[Bel]$ identified by purple squares in Fig. 4,
namely

$\begin{array}[]{llll}\rho^{1}=(x,y,z):&\quad
P^{\rho^{1}}[Bel](x)=.4,&P^{\rho^{1}}[Bel](y)=.3,&P^{\rho^{1}}[Bel](z)=.3,\\\
\rho^{2}=(x,z,y):&\quad
P^{\rho^{2}}[Bel](x)=.4,&P^{\rho^{2}}[Bel](y)=.1,&P^{\rho^{2}}[Bel](z)=.5,\\\
\rho^{3}=(y,x,z):&\quad
P^{\rho^{3}}[Bel](x)=.2,&P^{\rho^{3}}[Bel](y)=.5,&P^{\rho^{3}}[Bel](z)=.3,\\\
\rho^{4}=(z,x,y):&\quad
P^{\rho^{4}}[Bel](x)=.3,&P^{\rho^{4}}[Bel](y)=.1,&P^{\rho^{4}}[Bel](z)=.6,\\\
\rho^{5}=(z,y,x):&\quad
P^{\rho^{5}}[Bel](x)=.2,&P^{\rho^{5}}[Bel](y)=.2,&P^{\rho^{5}}[Bel](z)=.6\end{array}$
(55)

(as the permutations $(y,x,z)$ and $(y,z,x)$ yield the same probability
distribution).

Figure 4: The polytope $\mathcal{P}[Bel]$ of probabilities consistent with the
belief function $Bel$ (53) defined on $\\{x,y,z\\}$ is shown in a darker shade
of purple, in the probability simplex $\mathcal{P}=Cl(P_{x},P_{y},P_{z})$
(shown in light blue). Its vertices (purple squares) are given in (55). The
intersection probability, relative belief and relative plausibility of
singletons (green squares) are the foci of the pairs of simplices
$\\{T^{1}[Bel],T^{2}[Bel]\\}$, $\\{T^{1}[Bel],\mathcal{P}\\}$ and
$\\{\mathcal{P},T^{2}[Bel]\\}$, respectively. In the ternary case,
$T^{1}[Bel]$ and $T^{2}[Bel]$ (light purple) are regular triangles.
Geometrically, their focus is the intersection of the lines joining their
corresponding vertices (dashed lines for
$\\{T^{1}[Bel],\mathcal{P}\\}$,$\\{\mathcal{P},T^{2}[Bel]\\}$; solid lines for
$\\{T^{1}[Bel],T^{2}[Bel]\\}$).

We can notice a number of relevant facts:

  1. 1.

As pointed out above, $\mathcal{P}[Bel]$ (the polygon delimited by the purple
squares) is the intersection of the two triangles (two-dimensional simplices)
$T^{1}[Bel]$ and $T^{2}[Bel]$.

  2. 2.

The relative belief of singletons,

$\tilde{Bel}(x)=\frac{.2}{.6}=\frac{1}{3},\quad\tilde{Bel}(y)=\frac{.1}{.6}=\frac{1}{6},\quad\tilde{Bel}(z)=\frac{.3}{.6}=\frac{1}{2},$

is the intersection of the lines joining the corresponding vertices of the
probability simplex $\mathcal{P}$ and the lower simplex $T^{1}[Bel]$.

  3. 3.

The relative plausibility of singletons,

$\begin{array}[]{lll}\tilde{Pl}(x)&=&\displaystyle\frac{m(x)+m(\\{x,y\\})+m(\Theta)}{k_{Pl}-k_{Bel}}=\frac{.4}{.4+.5+.6}=\frac{4}{15},\\\
\\\
\tilde{Pl}(y)&=&\displaystyle\frac{.5}{.4+.5+.6}=\frac{1}{3},\quad\tilde{Pl}(z)=\frac{2}{5},\end{array}$

is the intersection of the lines joining the corresponding vertices of the
probability simplex $\mathcal{P}$ and the upper simplex $T^{2}[Bel]$.

  4. 4.

Finally, the intersection probability,

$\begin{array}[]{lll}p[Bel](x)&=&\displaystyle
m(x)+\beta[Bel]\big{(}m(\\{x,y\\})+m(\Theta)\big{)}=.2+\frac{.4\ast
0.2}{1.5-0.4}=.27,\\\ p[Bel](y)&=&\displaystyle.1+\frac{.4}{1.1}0.4=.245,\quad
p[Bel](z)=.485,\end{array}$

is the unique intersection of the lines joining the corresponding vertices of
the upper and lower simplices $T^{2}[Bel]$ and $T^{1}[Bel]$.

Although Fig. 4 suggests that $\tilde{Bel}$, $\tilde{Pl}$ and $p[Bel]$ might
be consistent with $Bel$, this is a mere artefact of this ternary example, for
it can be proved ([43], Chapter 12) that neither the relative belief of
singletons nor the relative plausibility of singletons necessarily belongs to
the credal set $\mathcal{P}[Bel]$. Indeed, the point here is that the
epistemic transforms $\tilde{Bel}$, $\tilde{Pl}$, $p[Bel]$ are instead
consistent with the interval probability system $\mathcal{P}[(Bel,Pl)]$
associated with the original belief function $Bel$:

$\tilde{Bel},\tilde{Pl},p[Bel]\in\mathcal{P}[(Bel,Pl)]=T^{1}[Bel]\cap
T^{n-1}[Bel].$

Their geometric behaviour, as described by facts 2, 3 and 4, holds in the
general case as well.

###  6.5 Focus of a pair of simplices

######  Definition 2.

Consider two simplices in $\mathbb{R}^{n-1}$, denoted by
$S=Cl(s_{1},\ldots,s_{n})$ and $T=Cl(t_{1},\ldots,t_{n})$, with the same
number of vertices. If there exists a permutation $\rho$ of $\\{1,\ldots,n\\}$
such that the intersection

$\bigcap_{i=1}^{n}a(s_{i},t_{\rho(i)})$ (56)

of the lines joining corresponding vertices of the two simplices exists and is
unique, then $p=f(S,T)\doteq\bigcap_{i=1}^{n}a(s_{i},t_{\rho(i)})$ is termed
the _focus_ of the two simplices $S$ and $T$.

Not all pairs of simplices admit a focus. For instance, the pair of simplices
(triangles)
$S=Cl(s_{1}=[2,2]^{\prime},s_{2}=[5,2]^{\prime},s_{3}=[3,5]^{\prime})$ and
$T=Cl(t_{1}=[3,1]^{\prime},t_{2}=[5,6]^{\prime},t_{3}=[2,6]^{\prime})$ in
$\mathbb{R}^{2}$ does not admit a focus, as no matter what permutation of the
order of the vertices we consider, the lines joining corresponding vertices do
not intersect. Geometrically, all pairs of simplices admitting a focus can be
constructed by considering all possible stars of lines, and all possible pairs
of points on each line of each star as pairs of corresponding vertices of the
two simplices.

######  Definition 3.

We call a focus _special_ if the affine coordinates of $f(S,T)$ on the lines
$a(s_{i},t_{\rho(i)})$ all coincide, namely $\exists\alpha\in\mathbb{R}$ such
that

$f(S,T)=\alpha s_{i}+(1-\alpha)t_{\rho(i)}\quad\forall i=1,\ldots,n.$ (57)

Not all foci are special. As an example, the simplices
$S=Cl(s_{1}=[-2,-2]^{\prime},s_{2}=[0,3]^{\prime},s_{3}=[1,0]^{\prime})$ and
$T=Cl(t_{1}=[-1,0]^{\prime},t_{2}=[0,-1]^{\prime},t_{3}=[2,2]^{\prime})$ in
$\mathbb{R}^{2}$ admit a focus, in particular for the permutation
$\rho(1)=3,\rho(2)=2,\rho(3)=1$ (i.e., the lines $a(s_{1},t_{3})$,
$a(s_{2},t_{2})$ and $a(s_{3},t_{1})$ intersect in
$\mathbf{0}=[0,0]^{\prime}$). However, the focus $f(S,T)=[0,0]^{\prime}$ is
not special, as its simplicial coordinates in the three lines are
$\alpha=\frac{1}{2}$, $\alpha=\frac{1}{4}$ and $\alpha=\frac{1}{2}$,
respectively.

On the other hand, given any pair of simplices in $\mathbb{R}^{n-1}$
$S=Cl(s_{1},\ldots,s_{n})$ and $T=Cl(t_{1},\ldots,t_{n})$, for any permutation
$\rho$ of the indices there always exists a linear variety of points which
have the same affine coordinates in both simplices,

$\left\\{p\in\mathbb{R}^{n-1}\Bigg{|}p=\sum_{i=1}^{n}\alpha_{i}s_{i}=\sum_{j=1}^{n}\alpha_{j}t_{\rho(j)},\;\sum_{i=1}^{n}\alpha_{i}=1.\right\\}$
(58)

Indeed, the conditions on the right-hand side of (58) amount to a linear
system of $n$ equations in $n$ unknowns ($\alpha_{i}$, $i=1,\ldots,n$). Thus,
there exists a linear variety of solutions to such a system, whose dimension
depends on the rank of the matrix of constraints in (58).

It is rather easy to prove the following theorem.

######  Theorem 3.

Any special focus $f(S,T)$ of a pair of simplices $S,T$ has the same affine
coordinates in both simplices, i.e.,

$f(S,T)=\sum_{i=1}^{n}\alpha_{i}s_{i}=\sum_{j=1}^{n}\alpha_{j}t_{\rho(j)},\quad\sum_{i=1}^{n}\alpha_{i}=1,$

where $\rho$ is the permutation of indices for which the intersection of the
lines $a(s_{i},t_{\rho(i)})$, $i=1,\ldots,n$ exists.

Note that the affine coordinates $\\{\alpha_{i},i\\}$ associated with a focus
can be negative, i.e., the focus may be located outside one or both simplices.

Notice also that the barycentre itself of a simplex is a special case of a
focus. In fact, the centre of mass $b$ of a $d$-dimensional simplex $S$ is the
intersection of the medians of $S$, i.e., the lines joining each vertex with
the barycentre of the opposite (($d-1$)-dimensional) face (see Fig. 5). But
those barycentres, for all ($d-1$)-dimensional faces, themselves constitute
the vertices of a simplex $T$.

Figure 5: The barycentre of a simplex is itself a special case of a focus, as
shown here for a two-dimensional example.

###  6.6 Intersection probability as a focus

######  Theorem 4.

For each belief function $Bel$, the intersection probability $p[Bel]$ has the
same affine coordinates in the lower and upper simplices $T^{1}[Bel]$ and
$T^{n-1}[Bel]$, respectively.

######  Theorem 5.

The intersection probability is the special focus of the pair of lower and
upper simplices $\\{T^{1}[Bel],T^{n-1}[Bel]\\}$, with its affine coordinate on
the corresponding intersecting lines equal to $\beta[Bel]$ (30).

The fraction $\alpha=\beta[Bel]$ of the width of the probability interval that
generates the intersection probability can be read in the probability simplex
as its coordinate on any of the lines determining the special focus of
$\\{T^{1}[Bel],T^{n-1}[Bel]\\}$.

Similar results hold for the relative belief and relative plausibility of
singletons, which are the (special) foci associated with the lower and upper
simplices $T^{1}[Bel]$ and $T^{n-1}[Bel]$, the geometric incarnations of the
lower and upper constraints on singletons. Those two simplices can also be
interpreted as the sets of probabilities consistent with the plausibility and
belief of singletons, respectively (see [43], Chapter 12).

Just as the pignistic function adheres to sensible rationality principles and,
as a consequence, it has a clear geometrical interpretation as the centre of
mass of the credal set associated with a belief function $Bel$, the
intersection probability has an elegant geometric behaviour with respect to
the credal set associated with an interval probability system, being the
(special) focus of the related upper and lower simplices.

Now, selecting the special focus of two simplices representing two different
constraints (i.e., the point with the same convex coordinates in the two
simplices) means adopting the single probability distribution which satisfies
both constraints in exactly the same way. If we assume homogeneous behaviour
in the two sets of constraints $\\{P(x)\geq Bel(x)\;\forall x\\}$,
$\\{P(x)\leq Pl(x)\;\forall x\\}$ as a rationality principle for the
probability transformation of an interval probability system, then the
intersection probability necessarily follows as the unique solution to the
problem.

Another interesting results stems from the fact that the pignistic function
and affine combination commute:

$BetP[\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}]=\alpha_{1}BetP[Bel_{1}]+\alpha_{2}BetP[Bel_{2}]$

whenever $\alpha_{1}+\alpha_{2}=1$.

######  Proposition 3.

[43] The intersection probability is the convex combination of the barycentres
$t^{1}[Bel],t^{n-1}[Bel]$ of the lower and upper simplices
$T^{1}[Bel],T^{n-1}[Bel]$ associated with a belief function $Bel$, with
coefficient (30)

$p[Bel]=\beta[Bel]t^{n-1}[Bel]+(1-\beta[Bel])t^{1}[Bel].$

##  7 Relations with other probability transforms

In Section 4.2 we discussed how the intersection probability relates to
alternative probability transforms for probability intervals, including
Sudano’s proposals.

Since the intersection probability can also be interpreted (as we have seen)
as a probability transform which applies to belief functions, it is quite
natural to wonder how it relates to the two main classes of such transforms:
the epistemic family and the affine family (see [43], Chapters 11 and 12).

###  7.1 Epistemic transforms

From (18) it follows that, for a belief function $Bel$,

$p[Bel](x)=m(x)+(1-k_{Bel})\frac{Pl(x)-m(x)}{k_{Pl}-k_{Bel}},$ (59)

which can be rewritten as:

$p[Bel]=k_{Bel}\;\tilde{Bel}+(1-k_{Bel})R[Bel].$ (60)

Since $k_{Bel}=\sum_{x\in\Theta}m(x)\leq 1$, (60) implies that the
intersection probability $p[Bel]$ belongs to the segment linking the relative
uncertainty of singletons $R[Bel]$ to the relative belief of singletons
$\tilde{Bel}$. Its convex coordinate on this segment is the total mass of
singletons $k_{Bel}$.

The relative plausibility function $\tilde{Pl}$ can also be written in terms
of $\tilde{Bel}$ and $R[Bel]$ as, by definition (13),

$\begin{array}[]{lll}R[Bel](x)&=&\displaystyle\frac{Pl(x)-m(x)}{k_{Pl}-k_{Bel}}=\frac{Pl(x)}{k_{Pl}-k_{Bel}}-\frac{m(x)}{k_{Pl}-k_{Bel}}\\\
\\\
&=&\displaystyle\tilde{Pl}(x)\frac{k_{Pl}}{k_{Pl}-k_{Bel}}-\tilde{Bel}(x)\frac{k_{Bel}}{k_{Pl}-k_{Bel}},\end{array}$

since $\tilde{Pl}(x)=Pl(x)/k_{Pl}$ and $\tilde{Bel}(x)=m(x)/k_{Bel}$.
Therefore,

$\tilde{Pl}=\left(\frac{k_{Bel}}{k_{Pl}}\right)\tilde{Bel}+\left(1-\frac{k_{Bel}}{k_{Pl}}\right)R[Bel].$
(61)

In summary, both the relative plausibility of singletons $\tilde{Pl}$ and the
intersection probability $p[Bel]$ belong to the segment
$Cl(R[Bel],\tilde{Bel})$ joining the relative belief $\tilde{Bel}$ and the
relative uncertainty of singletons $R[Bel]$ (see Fig. 6). The convex
coordinate of $\tilde{Pl}$ in $Cl(R[Bel],\tilde{Bel})$ (61) measures the ratio
between the total mass and the total plausibility of the singletons, while
that of $\tilde{Bel}$ measures the total mass of the singletons $k_{Bel}$.
Since $k_{Pl}=\sum_{A\subset\Theta}m(A)|A|\geq 1$, we have that
$k_{Bel}/k_{Pl}\leq k_{Bel}$: hence the relative plausibility function of
singletons $\tilde{Pl}$ is closer to $R[Bel]$ than $p[Bel]$ is (Figure 6
again).

Figure 6:  Location in the probability simplex $\mathcal{P}$ of the
intersection probability $p[Bel]$ and the relative plausibility of singletons
$\tilde{Pl}$ with respect to the relative uncertainty of singletons $R[Bel]$.
They both lie on the segment joining $R[Bel]$ and the relative belief of
singletons $\tilde{Bel}$, but $\tilde{Pl}$ is closer to $R[Bel]$ than $p[Bel]$
is.

Obviously, when $k_{Bel}=0$ (the relative belief of singletons $\tilde{Bel}$
does not exist, for $Bel$ assigns no mass to singletons), the remaining
probability approximations coincide: $p[Bel]=\tilde{Pl}=R[Bel]$ by (59).

###  7.2 Pignistic function

To shed even more light on $p[Bel]$, and to get an alternative interpretation
of the intersection probability, it is useful to compare $p[Bel]$ as expressed
in (59) with the pignistic function,

$BetP[Bel](x)\doteq\sum_{A\supseteq x}\frac{m(A)}{|A|}=m(x)+\sum_{A\supset
x,A\neq x}\frac{m(A)}{|A|}.$

In $BetP[Bel]$ the mass of each event $A$, $|A|>1$, is considered _separately_
, and its mass $m(A)$ is shared _equally_ among the elements of $A$. In
$p[Bel]$, instead, it is the _total_ mass $\sum_{|A|>1}m(A)=1-k_{Bel}$ of non-
singleton focal sets which is considered, and this total mass is distributed
_proportionally_ to their non-Bayesian contribution to each element of
$\Theta$.

Now, if $|A|>1$,

$\beta[Bel_{A}]=\frac{\sum_{|B|>1}m(B)}{\sum_{|B|>1}m(B)|B|}=\frac{1}{|A|},$

so that both $p[Bel](x)$ and $BetP[Bel](x)$ assume the form

$m(x)+\sum_{A\supset x,A\neq x}m(A)\beta_{A},$

where $\beta_{A}=\text{const}=\beta[Bel]$ for $p[Bel]$, while
$\beta_{A}=\beta[Bel_{A}]$ in the case of the pignistic function.

Under what conditions do the intersection probability and pignistic function
coincide? A sufficient condition can be easily given for a special class of
belief functions.

######  Theorem 6.

The intersection probability and pignistic function coincide for a given
belief function $Bel$ whenever the focal elements of $Bel$ have size 1 or $k$
only.

###### Proof.

The desired equality $p[Bel]=BetP[Bel]$ is equivalent to

$m(x)+\sum_{A\supsetneq x}m(A)\beta[Bel]=m(x)+\sum_{A\supsetneq
x}\frac{m(A)}{|A|},$

which in turn reduces to

$\sum_{A\supsetneq x}m(A)\beta[Bel]=\sum_{A\supsetneq x}\frac{m(A)}{|A|}.$

If $\exists$ $k:m(A)=0$ for $|A|\neq k$, $|A|>1$, then $\beta[Bel]=1/k$ and
the equality is satisfied. ∎

In particular, this is true when $Bel$ is 2-additive (its focal elements have
cardinality $\leq 2$).

######  Example 1.

Let us briefly discuss these two interpretations of $p[Bel]$ in a simple
example. Consider a ternary frame $\Theta=\\{x,y,z\\}$, and a belief function
$Bel$ with BPA

$\begin{array}[]{c}\begin{array}[]{ccc}m(x)=0.1,&m(y)=0,&m(z)=0.2,\end{array}\\\
\begin{array}[]{cccc}m(\\{x,y\\})=0.3,&m(\\{x,z\\})=0.1,&m(\\{y,z\\})=0,&m(\Theta)=0.3.\end{array}\end{array}$
(62)

The related basic plausibility assignment is, according to (37),

$\begin{array}[]{c}\begin{array}[]{lll}\mu(x)&=&\displaystyle(-1)^{|x|+1}\sum_{B\supseteq\\{x\\}}m(B)\\\
&=&\displaystyle m(x)+m(\\{x,y\\})+m(\\{x,z\\})+m(\Theta)=0.8,\end{array}\\\
\\\ \begin{array}[]{lllll}\mu(y)=0.6,&&\mu(z)=0.6,&&\mu(\\{x,y\\})=-0.6,\\\
\\\
\mu(\\{x,z\\})=-0.4,&&\mu(\\{y,z\\})=-0.3,&&\mu(\Theta)=0.3.\end{array}\end{array}$

Figure 7 depicts the subsets of $\Theta$ with non-zero BPA (left) and BPlA
(middle) induced by the belief function (62): dashed ellipses indicate a
negative mass. The total mass that (62) accords to singletons is
$k_{Bel}=0.1+0+0.2=0.3$. Thus, the line coordinate $\beta[Bel]$ of the
intersection $\varsigma[Bel]$ of the line $a(Bel,Pl)$ with
$\mathcal{P}^{\prime}$ is

$\beta[Bel]=\frac{1-k_{Bel}}{m(\\{x,y\\})|\\{x,y\\}|+m(\\{x,z\\})|\\{x,z\\}|+m(\Theta)|\Theta|}=\frac{0.7}{1.7}.$

By (34), the mass assignment of $\varsigma[Bel]$ is therefore

$\begin{array}[]{l}\displaystyle
m_{\varsigma[Bel]}(x)=m(x)+\beta[Bel](\mu(x)-m(x))=0.1+0.7\cdot\frac{0.7}{1.7}=0.388,\\\
\\\ \displaystyle
m_{\varsigma[Bel]}(y)=0+0.6\cdot\frac{0.7}{1.7}=0.247,\;\;\;\;\;\;\;\;\;m_{\varsigma[Bel]}(z)=0.2+0.4\cdot\frac{0.7}{1.7}=0.365,\\\
\\\ \displaystyle
m_{\varsigma[Bel]}(\\{x,y\\})=0.3-0.9\cdot\frac{0.7}{1.7}=-0.071,\\\ \\\
\displaystyle
m_{\varsigma[Bel]}(\\{x,z\\})=0.1-0.5\cdot\frac{0.7}{1.7}=-0.106,\\\ \\\
\displaystyle
m_{\varsigma[Bel]}(\\{y,z\\})=0-0.3\cdot\frac{0.7}{1.7}=-0.123,\;\;\;\;m_{\varsigma[Bel]}(\Theta)=0.3+0\cdot\frac{0.7}{1.7}=0.3.\end{array}$

We can verify that all singleton masses are indeed non-negative and add up to
one, while the masses of the non-singleton events add up to zero,

$-0.071-0.106-0.123+0.3=0,$

confirming that $\varsigma[Bel]$ is a Bayesian normalised sum function (pseudo
belief function). Its mass assignment has signs which are still described by
Fig. 7 (middle) although, as $m_{\varsigma[Bel]}$ is a weighted average of $m$
and $\mu$, its mass values are closer to zero.

Figure 7: Signs of non-zero masses assigned to events by the functions
discussed in Example 1. Left: BPA of the belief function (62), with five focal
elements. Middle: the associated BPlA assigns positive masses (solid ellipses)
to all events of size 1 and 3, and negative ones (dashed ellipses) to all
events of size 2. This is the case for the mass assignment associated with
$\varsigma$ (34) too. Right: the intersection probability $p[Bel]$ (33)
retains, among the latter, only the masses assigned to singletons.

In order to compare $\varsigma[Bel]$ with the intersection probability, we
need to recall (59): the non-Bayesian contributions of $x,y,z$ are,
respectively,

$\begin{array}[]{lll}Pl(x)-m(x)&=&m(\Theta)+m(\\{x,y\\})+m(\\{x,z\\})=0.7,\\\
Pl(y)-m(y)&=&m(\\{x,y\\})+m(\Theta)=0.6,\\\
Pl(z)-m(z)&=&m(\\{x,z\\})+m(\Theta)=0.4,\end{array}$

so that the relative uncertainty of singletons is

$R(x)=0.7/1.7,\quad R(y)=0.6/1.7,\quad R(z)=0.4/1.7.$

For each singleton $\theta$, the value of the intersection probability results
from adding to the original BPA $m(\theta)$ a share of the mass of the non-
singleton events $1-k_{Bel}=0.7$ proportional to the value of $R(\theta)$ (see
Fig. 7, right):

$\begin{array}[]{lllll}p[Bel](x)&=&m(x)+(1-k_{Bel})R(x)=0.1+0.7*0.7/1.7&=&0.388,\\\
p[Bel](y)&=&m(y)+(1-k_{Bel})R(y)=0+0.7*0.6/1.7&=&0.247,\\\
p[Bel](z)&=&m(z)+(1-k_{Bel})R(z)=0.2+0.7*0.4/1.7&=&0.365.\end{array}$

We can see that $p[Bel]$ coincides with the restriction of $\varsigma[Bel]$ to
singletons.

Equivalently, $\beta[Bel]$ measures the share of $Pl(x)-m(x)$ assigned to each
element of the frame of discernment:

$\begin{array}[]{lllll}p[Bel](x)&=&m(x)+\beta[Bel](Pl(x)-m(x))&=&0.1+0.7/1.7*0.7,\\\
p[Bel](y)&=&m(y)+\beta[Bel](Pl(y)-m(y))&=&0+0.7/1.7*0.6,\\\
p[Bel](z)&=&m(z)+\beta[Bel](Pl(z)-m(z))&=&0.2+0.7/1.7*0.4.\end{array}$

##  8 Operators

To complete our analysis, it can be useful to understand the way the
intersection probability relates to two major (geometric) operators in the
space of belief functions: affine combination and convex closure.

###  8.1 Affine combination

We have seen in Section 7 that $p[Bel]$ and $BetP[Bel]$ are closely related
probability transforms, linked by the role of the quantity $\beta[Bel]$. It is
natural to wonder whether $p[Bel]$ exhibits a similar behaviour with respect
to the convex closure operator (see [43], Chapter 4). Indeed, although the
situation is a little more complex in this second case, $p[Bel]$ turns also
out to be related to $Cl(.)$ in a rather elegant way.

Let us introduce the notation $\beta[Bel_{i}]=N_{i}/D_{i}$.

######  Theorem 7.

Given two arbitrary belief functions $Bel_{1},Bel_{2}$ defined on the same
frame of discernment, the intersection probability of their affine combination
$\alpha_{1}\;Bel_{1}$ $+\;\alpha_{2}\;Bel_{2}$ is, for any
$\alpha_{1}\in[0,1]$, $\alpha_{2}=1-\alpha_{1}$,

$\begin{array}[]{lll}p[\alpha_{1}\;Bel_{1}+\alpha_{2}\;Bel_{2}]&=&\displaystyle\widehat{\alpha_{1}D_{1}}\big{(}\alpha_{1}p[Bel_{1}]+\alpha_{2}T[Bel_{1},Bel_{2}]\big{)}\\\
\\\
&&\displaystyle+\widehat{\alpha_{2}D_{2}}\big{(}\alpha_{1}T[Bel_{1},Bel_{2}])+\alpha_{2}p[Bel_{2}]\big{)},\end{array}$
(63)

where
$\widehat{\alpha_{i}D_{i}}=\frac{\alpha_{i}D_{i}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}$,
$T[Bel_{1},Bel_{2}]$ is the probability with values

$T[Bel_{1},Bel_{2}](x)\doteq\hat{D}_{1}p[Bel_{2},Bel_{1}]+\hat{D}_{2}p[Bel_{1},Bel_{2}],$
(64)

with $\hat{D}_{i}\doteq\frac{D_{i}}{D_{1}+D_{2}}$, and

$\begin{array}[]{c}p[Bel_{2},Bel_{1}](x)\doteq
m_{2}(x)+\beta[Bel_{1}]\big{(}Pl_{2}(x)-m_{2}(x)\big{)},\\\ \\\
p[Bel_{1},Bel_{2}](x)\doteq
m_{1}(x)+\beta[Bel_{2}]\big{(}Pl_{1}(x)-m_{1}(x)\big{)}.\end{array}$ (65)

Figure 8: Behaviour of the intersection probability $p[Bel]$ under affine
combination. $\alpha_{2}T+\alpha_{1}p[Bel_{1}]$ and
$\alpha_{1}T+\alpha_{2}p[Bel_{2}]$ lie in inverted locations on the segments
joining $T[Bel_{1},Bel_{2}]$ and $p[Bel_{1}]$, $p[Bel_{2}]$, respectively:
$\alpha_{i}p[Bel_{i}]+\alpha_{j}T$ is the intersection of the line
$Cl(T,p[Bel_{i}])$ with the line parallel to $Cl(T,p[Bel_{j}])$ passing
through $\alpha_{1}p[Bel_{1}]+\alpha_{2}p[Bel_{2}]$. The quantity
$p[\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}]$ is, finally, the point of the segment
joining them with convex coordinate $\widehat{\alpha_{i}D_{i}}$.

Geometrically, $p[\alpha_{1}\;Bel_{1}+\alpha_{2}\;Bel_{2}]$ can be constructed
as in Fig. 8 as a point of the simplex
$Cl(T[Bel_{1},Bel_{2}],p[Bel_{1}],p[Bel_{2}])$. The point

$\alpha_{1}T[Bel_{1},Bel_{2}]+\alpha_{2}p[Bel_{2}]$

is the intersection of the segment $Cl(T,p[Bel_{2}])$ with the line $l_{2}$
passing through $\alpha_{1}p[Bel_{1}]+\alpha_{2}p[Bel_{2}]$ and parallel to
$Cl(T,p[Bel_{1}])$. Dually, the point

$\alpha_{2}T[Bel_{1},Bel_{2}]+\alpha_{1}p[Bel_{1}]$

is the intersection of the segment $Cl(T,p[Bel_{1}])$ with the line $l_{1}$
passing through $\alpha_{1}p[Bel_{1}]+\alpha_{2}p[Bel_{2}]$ and parallel to
$Cl(T,p[Bel_{2}])$. Finally, $p[\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}]$ is the
point of the segment

$Cl(\alpha_{1}T+\alpha_{2}p[Bel_{2}],\alpha_{2}T+\alpha_{1}p[Bel_{1}])$

with convex coordinate $\widehat{\alpha_{1}D_{1}}$ (or, equivalently,
$\widehat{\alpha_{2}D_{2}}$).

####  8.1.1 Location of $T[Bel_{1},Bel_{2}]$ in the binary case

As an example, let us consider the location of $T[Bel_{1},Bel_{2}]$ in the
binary belief space $\mathcal{B}_{2}$ (Fig. 9), where

$\beta[Bel_{1}]=\beta[Bel_{2}]=\frac{m_{i}(\Theta)}{2m_{i}(\Theta)}=\frac{1}{2}\quad\forall
Bel_{1},Bel_{2}\in\mathcal{B}_{2},$

and $p[Bel]$ always commutes with the convex closure operator. Accordingly,

$\begin{array}[]{lll}\displaystyle
T[Bel_{1},Bel_{2}](x)&=&\displaystyle\frac{m_{1}(\Theta)}{m_{1}(\Theta)+m_{2}(\Theta)}\left[m_{2}(x)+\frac{m_{2}(\Theta)}{2}\right]\\\
\\\
&&\displaystyle+\frac{m_{2}(\Theta)}{m_{1}(\Theta)+m_{2}(\Theta)}\left[m_{1}(x)+\frac{m_{1}(\Theta)}{2}\right]\\\
\\\
&=&\displaystyle\frac{m_{1}(\Theta)}{m_{1}(\Theta)+m_{2}(\Theta)}p[Bel_{2}]+\frac{m_{2}(\Theta)}{m_{1}(\Theta)+m_{2}(\Theta)}p[Bel_{1}].\end{array}$

Figure 9: Location of the probability function $T[Bel_{1},Bel_{2}]$ in the
binary belief space $\mathcal{B}_{2}$.

Looking at Figure 9, simple trigonometric considerations show that the segment
defined by $Cl(p[Bel_{i}],T[Bel_{1},Bel_{2}])$ has length
$m_{i}(\Theta)/(\sqrt{2}\tan{\phi})$, where $\phi$ is the angle between the
segments $Cl(Bel_{i},T)$ and $Cl(p[Bel_{i}],T)$. $T[Bel_{1},Bel_{2}]$ is then
the unique point of $\mathcal{P}$ such that the angles
$\widehat{Bel_{1}Tp[Bel_{1}]}$ and $\widehat{Bel_{2}Tp[Bel_{2}]}$ coincide,
i.e., $T$ is the intersection of $\mathcal{P}$ with the line passing through
$Bel_{i}$ and the reflection of $Bel_{j}$ through $\mathcal{P}$. As this
reflection (in $\mathcal{B}_{2}$) is nothing but $Pl_{j}$,

$T[Bel_{1},Bel_{2}]=Cl(Bel_{1},Pl_{2})\cap\mathcal{P}=Cl(Bel_{2},Pl_{1})\cap\mathcal{P}.$

###  8.2 Convex closure

Although the intersection probability does not commute with affine combination
(Theorem 7), $p[Bel]$ can still be assimilated into the orthogonal projection
and the pignistic function. Theorem 8 states the conditions under which
$p[Bel]$ and convex closure ($Cl$) commute.

######  Theorem 8.

The intersection probability and convex closure commute iff

$T[Bel_{1},Bel_{2}]=\hat{D}_{1}p[Bel_{2}]+\hat{D}_{2}p[Bel_{1}]$

or, equivalently, either $\beta[Bel_{1}]=\beta[Bel_{2}]$ or
$R[Bel_{1}]=R[Bel_{2}]$.

Geometrically, only when the two lines $l_{1},l_{2}$ in Fig. 8 are parallel to
the affine space $a(p[Bel_{1}],p[Bel_{2}])$ (i.e., $T[Bel_{1},Bel_{2}]\in
Cl(p[Bel_{1}],p[Bel_{2}])$; compare the above) does the desired quantity
$p[\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}]$ belong to the line segment
$Cl(p[Bel_{1}],p[Bel_{2}])$ (i.e., it is also a convex combination of
$p[Bel_{1}]$ and $p[Bel_{2}]$).

Theorem 8 reflects the two complementary interpretations of $p[Bel]$ we gave
in terms of $\beta[Bel]$ and $R[Bel]$:

$\begin{array}[]{l}p[Bel]=m(x)+(1-k_{Bel})R[Bel](x),\\\ \\\
p[Bel]=m(x)+\beta[Bel](Pl(x)-m(x)).\end{array}$

If $\beta[Bel_{1}]=\beta[Bel_{2}]$, both belief functions assign to each
singleton the same share of their non-Bayesian contribution. If
$R[Bel_{1}]=R[Bel_{2}]$, the non-Bayesian mass $1-k_{Bel}$ is distributed in
the same way to the elements of $\Theta$.

A sufficient condition for the commutativity of $p[.]$ and $Cl(.)$ can be
obtained via the following decomposition of $\beta[Bel]$:

$\begin{array}[]{l}\displaystyle\beta[Bel]=\frac{\sum_{|B|>1}m(B)}{\sum_{|B|>1}m(B)|B|}=\frac{\sum_{k=2}^{n}\sum_{|B|=k}m(B)}{\sum_{k=2}^{n}k\cdot\sum_{|B|=k}m(B)}=\frac{\sigma_{2}+\cdots+\sigma_{n}}{2\sigma_{2}+\cdots+n\sigma_{n}},\end{array}$
(66)

where $\sigma_{k}\doteq\sum_{|B|=k}m(B)$.

######  Theorem 9.

If the ratio between the total masses of focal elements of different
cardinality is the same for all the belief functions involved, namely

$\begin{array}[]{cccc}\displaystyle\frac{\sigma_{1}^{l}}{\sigma_{1}^{m}}=\frac{\sigma_{2}^{l}}{\sigma_{2}^{m}}&\quad\forall
l,m\geq 2&\text{s.t.}&\sigma_{1}^{m},\sigma_{2}^{m}\neq 0,\end{array}$ (67)

then the intersection probability (considered as an operator mapping belief
functions to probabilities) and convex combination commute.

##  9 Conclusions

The intersection probability possesses a simple credal interpretation in the
probability simplex (Section 6), as it can be linked to a pair of credal sets,
thus in turn extending the classical interpretation of the pignistic
transformation as the barycentre of the polygon of consistent probabilities.

Just like $\mathcal{P}[Bel]$ is the credal set associated with a belief
function $Bel$, the upper and lower simplices geometrically embody the
probability interval associated with $Bel$:

$\mathcal{P}[(Bel,Pl)]=\Big{\\{}p\in\mathcal{P}:Bel(x)\leq p(x)\leq
Pl(x),\forall x\in\Theta\Big{\\}}.$

The intersection probability turns out to be the focus to the pair
$\\{T^{1}[Bel],T^{n-1}[Bel]\\}$ of lower and upper simplices:

$f(T^{1}[Bel],T^{n-1}[Bel])=p[Bel].$ (68)

Its coordinates as foci encode major features of the underlying belief
function, in particular the fraction $\beta$ of the related probability
interval which yields the intersection probability.

This credal interpretation hints at the possible formulation of a decision
making framework for probability intervals analogous to Smets’s _transferable
belief model_ (TBM) [101]. We can think of the TBM as a pair
$\\{\mathcal{P}[Bel],BetP[Bel]\\}$ formed by a credal set linked to each
belief function $Bel$ (in this case the polytope of consistent probabilities)
and a probability transformation (the pignistic function). As the barycentre
of a simplex is a special case of a focus, the pignistic transformation is
just another probability transformation induced by the focus of two simplices.

The results in this paper therefore suggest a similar framework

$\Big{\\{}\big{\\{}T^{1}[Bel],T^{n-1}[Bel]\big{\\}},p[Bel]\Big{\\}},$

in which interval constraints on probability distributions on $\mathcal{P}$
are represented by a similar pair, formed by the above pair of simplices and
by the probability transformation identified by their focus. Decisions are
then made based on the appropriate focus probability, i.e., the intersection
probability.

In the TBM [101], disjunctive/conjunctive combination rules are applied to
belief functions to update or revise our state of belief according to new
evidence. The formulation of a similar alternative frameworks for interval
probability systems would then require us to design specific evidence
elicitation/revision operators for them. This elicits a number of questions:
how can we design such operator(s)? How are they related to combination rules
for belief functions in the case of probability intervals induced by belief
functions?

We will further explore the betting interpretation of the intersection
probability in the near future.

## Appendix

### Proof of Theorem 2

######  Lemma 2.

The points $\\{t_{x}^{1}[Bel],x\in\Theta\\}$ are affinely independent.

###### Proof.

Let us suppose, contrary to the thesis, that there exists an affine
decomposition of one of the points, say $t_{x}[Bel]$, in terms of the others:

$t_{x}^{1}[Bel]=\sum_{z\neq x}\alpha_{z}t_{z}^{1}[Bel],\quad\alpha_{z}\geq
0\;\forall z\neq x,\;\sum_{z\neq x}\alpha_{z}=1.$

But then we would have, by the definition of $t_{z}^{1}[Bel]$,

$\begin{array}[]{lll}t_{x}^{1}[Bel]&=&\displaystyle\sum_{z\neq
x}\alpha_{z}t_{z}^{1}[Bel]\\\ &=&\displaystyle\sum_{z\neq
x}\alpha_{z}\left(\sum_{y\neq z}m(y)Bel_{y}\right)+\sum_{z\neq
x}\alpha_{z}\big{(}m(z)+1-k_{Bel}\big{)}Bel_{z}\\\ \\\ &=&\displaystyle
m(x)Bel_{x}\sum_{z\neq x}\alpha_{z}+\sum_{z\neq
x}Bel_{z}m(z)(1-\alpha_{z})\end{array}$

$\begin{array}[]{lll}&&+\displaystyle\sum_{z\neq
x}\alpha_{z}m(z)Bel_{z}+(1-k_{Bel})\sum_{z\neq x}\alpha_{z}Bel_{z}\\\ \\\
&=&\displaystyle\sum_{z\neq x}m(z)Bel_{z}+m(x)Bel_{x}+(1-k_{Bel})\sum_{z\neq
x}\alpha_{z}Bel_{z}.\end{array}$

The latter is equal to (48)

$t_{x}^{1}[Bel]=\sum_{z\neq x}m(z)Bel_{z}+(m(x)+1-k_{Bel})Bel_{x}$

if and only if $\sum_{z\neq x}\alpha_{z}Bel_{z}=Bel_{x}.$ But this is
impossible, as the categorical probabilities $Bel_{x}$ are trivially affinely
independent. ∎

### Proof of Theorem 3

Suppose that $p$ is a special focus of $S$ and $T$, namely
$\exists\alpha\in\mathbb{R}$ such that

$p=\alpha s_{i}+(1-\alpha)t_{\rho(i)}\quad\forall i=1,\ldots,n,$

for some permutation $\rho$ of $\\{1,\ldots,n\\}$. Then, necessarily,

$t_{\rho(i)}=\frac{1}{1-\alpha}[p-\alpha s_{i}]\quad\forall i=1,\ldots,n.$

If $p$ has coordinates $\\{\alpha_{i},i=1,\ldots,n\\}$ in $T$,
$p=\sum_{i=1}^{n}\alpha_{i}t_{\rho(i)}$, it follows that

$\begin{array}[]{lll}p&=&\displaystyle\sum_{i=1}^{n}\alpha_{i}t_{\rho(i)}=\frac{1}{1-\alpha}\sum_{i}\alpha_{i}\big{[}p-\alpha
s_{i}\big{]}\\\ \\\
&=&\displaystyle\frac{1}{1-\alpha}\left[p\sum_{i}\alpha_{i}-\alpha\sum_{i}\alpha_{i}s_{i}\right]=\frac{1}{1-\alpha}\left[p-\alpha\sum_{i}\alpha_{i}s_{i}\right].\end{array}$

The latter implies that $p=\sum_{i}\alpha_{i}s_{i}$, i.e., $p$ has the same
simplicial coordinates in $S$ and $T$.

### Proof of Theorem 4

The common simplicial coordinates of $p[Bel]$ in $T^{1}[Bel]$ and
$T^{n-1}[Bel]$ turn out to be the values of the relative uncertainty function
(19) for $Bel$,

$R[Bel](x)=\frac{Pl(x)-m(x)}{k_{Pl}-k_{Bel}}.$ (69)

Recalling the expression (48) for the vertices of $T^{1}[Bel]$, the point of
the simplex $T^{1}[Bel]$ with coordinates (69) is

$\begin{array}[]{lll}&&\displaystyle\sum_{x}R[Bel](x)t_{x}^{1}[Bel]\\\
&=&\displaystyle\sum_{x}R[Bel](x)\left[\sum_{y\neq
x}m(y)Bel_{y}+\bigg{(}1-\sum_{y\neq x}m(y)\bigg{)}Bel_{x}\right]\\\ \\\
&=&\displaystyle\sum_{x}R[Bel](x)\left[\sum_{y\in\Theta}m(y)Bel_{y}+(1-k_{Bel})Bel_{x}\right]\\\
\\\
&=&\displaystyle\sum_{x}Bel_{x}\left[(1-k_{Bel})R[Bel](x)+m(x)\sum_{y}R[Bel](y)\right]\\\
\\\
&=&\displaystyle\sum_{x}Bel_{x}\Big{[}(1-k_{Bel})R[Bel](x)+m(x)\Big{]},\end{array}$

as $R[Bel]$ is a probability ($\sum_{y}R[Bel](y)=1$). By Equation (18), the
above quantity coincides with $p[Bel]$.

The point of $T^{n-1}[Bel]$ with the same coordinates,
$\\{R[Bel](x),x\in\Theta\\}$, is

$\begin{array}[]{lll}&&\displaystyle\sum_{x}R[Bel](x)t_{x}^{n-1}[Bel]\\\
&=&\displaystyle\sum_{x}R[Bel](x)\left[\sum_{y\neq
x}Pl(y)Bel_{y}+\bigg{(}1-\sum_{y\neq x}Pl(y)\bigg{)}Bel_{x}\right]\\\ \\\
&=&\displaystyle\sum_{x}R[Bel](x)\left[\sum_{y\in\Theta}Pl(y)Bel_{y}+(1-k_{Pl})Bel_{x}\right]\\\
\\\
&=&\displaystyle\sum_{x}Bel_{x}\left[(1-k_{Pl})R[Bel](x)+Pl(x)\sum_{y}R[Bel](y)\right]\\\
\\\ &=&\displaystyle\sum_{x}Bel_{x}\Big{[}(1-k_{Pl})R[Bel](x)+Pl(x)\Big{]}\\\
\\\
&=&\displaystyle\sum_{x}Bel_{x}\bigg{[}Pl(x)\frac{1-k_{Bel}}{k_{Pl}-k_{Bel}}-m(x)\frac{1-k_{Bel}}{k_{Pl}-k_{Bel}}\bigg{]},\end{array}$

which is equal to $p[Bel]$ by (69).

### Proof of Theorem 5

Again, we need to impose the condition (57) on the pair
$\\{T^{1}[Bel],T^{n-1}[Bel]\\}$, namely

$p[Bel]=t_{x}^{1}[Bel]+\alpha(t_{x}^{n-1}[Bel]-t_{x}^{1}[Bel])=(1-\alpha)t_{x}^{1}[Bel]+\alpha
t_{x}^{n-1}[Bel]$

for all the elements $x\in\Theta$ of the frame, $\alpha$ being some constant
real number. This is equivalent to (after substituting the expressions (48),
(50) for $t_{x}^{1}[Bel]$ and $t_{x}^{n-1}[Bel]$)

$\begin{array}[]{lll}&&\displaystyle\sum_{x\in\Theta}Bel_{x}\Big{[}m(x)+\beta[Bel](Pl(x)-m(x))\Big{]}\\\
&=&\displaystyle(1-\alpha)\left[\sum_{y\in\Theta}m(y)Bel_{y}+(1-k_{Bel})Bel_{x}\right]\\\
\\\
&&\displaystyle+\alpha\left[\sum_{y\in\Theta}Pl(y)Bel_{y}+(1-k_{Pl})Bel_{x}\right]\\\
\\\ &=&\displaystyle Bel_{x}\Big{[}(1-\alpha)(1-k_{Bel})+(1-\alpha)m(x)+\alpha
Pl(x)+\alpha(1-k_{Pl})\Big{]}\\\ \\\ &&\displaystyle+\sum_{y\neq
x}Bel_{y}\Big{[}(1-\alpha)m(y)+\alpha Pl(y)\Big{]}\\\ \\\ &=&\displaystyle
Bel_{x}\Big{\\{}(1-k_{Bel})+m(x)+\alpha\big{[}Pl(x)+(1-k_{Pl})-m(x)-(1-k_{Bel})\big{]}\Big{\\}}\\\
\\\ &&\displaystyle+\sum_{y\neq
x}Bel_{y}\Big{[}m(y)+\alpha\big{(}Pl(y)-m(y)\big{)}\Big{]}.\end{array}$

If we set $\alpha=\beta[Bel]=(1-k_{Bel})/(k_{Pl}-k_{Bel})$, we get the
following for the coefficient of $Bel_{x}$ in the above expression (i.e., the
probability value of $x$):

$\begin{array}[]{lll}&&\displaystyle\frac{1-k_{Bel}}{k_{Pl}-k_{Bel}}\big{[}Pl(x)+(1-k_{Pl})-m(x)-(1-k_{Bel})\big{]}+(1-k_{Bel})+m(x)\\\
\\\
&=&\beta[Bel][Pl(x)-m(x)]+(1-k_{Bel})+m(x)-(1-k_{Bel})=p[Bel](x).\end{array}$

On the other hand,

$m(y)+\alpha(Pl(y)-m(y))=m(y)+\beta[Bel](Pl(y)-m(y))=p[Bel](y)$

for all $y\neq x$, no matter what the choice of $x$.

### Proof of Theorem 7

By definition, the quantity $p[\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}](x)$ can be
written as

$m_{\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}}(x)+\beta[\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}]\sum_{A\supsetneq\\{x\\}}m_{\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}}(A),$
(70)

where

$\begin{array}[]{lll}&&\displaystyle\beta[\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}]\\\
\\\
&=&\displaystyle\frac{\displaystyle\sum_{|A|>1}m_{\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}}(A)}{\displaystyle\sum_{|A|>1}m_{\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}}(A)|A|}=\frac{\displaystyle\alpha_{1}\sum_{|A|>1}m_{1}(A)+\alpha_{2}\sum_{|A|>1}m_{2}(A)}{\displaystyle\alpha_{1}\sum_{|A|>1}m_{1}(A)|A|+\alpha_{2}\sum_{|A|>1}m_{2}(A)|A|}\\\
\\\
&=&\displaystyle\frac{\alpha_{1}N_{1}+\alpha_{2}N_{2}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}=\frac{\alpha_{1}D_{1}\beta[Bel_{1}]+\alpha_{2}D_{2}\beta[Bel_{2}]}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}\\\
\\\
&=&\widehat{\alpha_{1}D_{1}}\beta[Bel_{1}]+\widehat{\alpha_{2}D_{2}}\beta[Bel_{2}],\end{array}$

once we introduce the notation $\beta[Bel_{i}]=N_{i}/D_{i}$.

Substituting this decomposition for
$\beta[\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}]$ into (70) yields

$\begin{array}[]{lll}&&\displaystyle\alpha_{1}m_{1}(x)+\alpha_{2}m_{2}(x)\\\
&&\displaystyle+\Big{(}\widehat{\alpha_{1}D_{1}}\beta[Bel_{1}]+\widehat{\alpha_{2}D_{2}}\beta[Bel_{2}]\Big{)}\left(\alpha_{1}\sum_{A\supsetneq\\{x\\}}m_{1}(A)+\alpha_{2}\sum_{A\supsetneq\\{x\\}}m_{2}(A)\right)\\\
\\\
&=&\displaystyle\frac{(\alpha_{1}D_{1}+\alpha_{2}D_{2})(\alpha_{1}m_{1}(x)+\alpha_{2}m_{2}(x))}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}\\\
\\\
&&\displaystyle+\frac{\alpha_{1}D_{1}\beta[Bel_{1}]+\alpha_{2}D_{2}\beta[Bel_{2}]}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}\left(\alpha_{1}\sum_{A\supsetneq\\{x\\}}m_{1}(A)+\alpha_{2}\sum_{A\supsetneq\\{x\\}}m_{2}(A)\right),\end{array}$
(71)

which can be reduced further to

$\begin{array}[]{lll}&&\displaystyle\frac{\alpha_{1}D_{1}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}\Big{(}\alpha_{1}m_{1}(x)+\alpha_{2}m_{2}(x)\Big{)}\\\
&&+\displaystyle\frac{\alpha_{1}D_{1}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}\beta[Bel_{1}]\left(\alpha_{1}\sum_{A\supsetneq\\{x\\}}m_{1}(A)+\alpha_{2}\sum_{A\supsetneq\\{x\\}}m_{2}(A)\right)\\\
&&\displaystyle+\frac{\alpha_{2}D_{2}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}\Big{(}\alpha_{1}m_{1}(x)+\alpha_{2}m_{2}(x)\Big{)}\\\
&&\displaystyle+\frac{\alpha_{2}D_{2}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}\beta[Bel_{2}]\left(\alpha_{1}\sum_{A\supsetneq\\{x\\}}m_{1}(A)+\alpha_{2}\sum_{A\supsetneq\\{x\\}}m_{2}(A)\right)\\\
\\\
&=&\displaystyle\frac{\alpha_{1}D_{1}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}\alpha_{1}\left(m_{1}(x)+\beta[Bel_{1}]\sum_{A\supsetneq\\{x\\}}m_{1}(A)\right)\end{array}$

$\begin{array}[]{lll}&&\displaystyle+\frac{\alpha_{1}D_{1}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}\alpha_{2}\left(m_{2}(x)+\beta[Bel_{1}]\sum_{A\supsetneq\\{x\\}}m_{2}(A)\right)\\\
\\\
&&+\displaystyle\frac{\alpha_{2}D_{2}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}\alpha_{1}\left(m_{1}(x)+\beta[Bel_{2}]\sum_{A\supsetneq\\{x\\}}m_{1}(A)\right)\\\
\\\
&&+\displaystyle\frac{\alpha_{2}D_{2}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}\alpha_{2}\left(m_{2}(x)+\beta[Bel_{2}]\sum_{A\supsetneq\\{x\\}}m_{2}(A)\right)\\\
\\\
&=&\displaystyle\frac{\alpha^{2}_{1}D_{1}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}p[Bel_{1}]+\frac{\alpha^{2}_{2}D_{2}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}p[Bel_{2}]\\\
\\\
&&\displaystyle+\frac{\alpha_{1}\alpha_{2}}{\alpha_{1}D_{1}+\alpha_{2}D_{2}}\Big{(}D_{1}p[Bel_{2},Bel_{1}](x)+D_{2}p[Bel_{1},Bel_{2}](x)\Big{)},\end{array}$
(72)

after recalling (65).

We can notice further that the function

$F(x)\doteq D_{1}p[Bel_{2},Bel_{1}](x)+D_{2}p[Bel_{1},Bel_{2}](x)$

is such that

$\begin{array}[]{ll}&\displaystyle\sum_{x\in\Theta}F(x)\\\
=&\displaystyle\sum_{x\in\Theta}\big{[}D_{1}m_{2}(x)+N_{1}(Pl_{2}-m_{2}(x))+D_{2}m_{1}(x)+N_{2}(Pl_{1}-m_{1}(x))\big{]}\\\
=&D_{1}(1-N_{2})+N_{1}D_{2}+D_{2}(1-N_{1})+N_{2}D_{1}=D_{1}+D_{2}\end{array}$

(making use of (30)). Thus, $T[Bel_{1},Bel_{2}](x)=F(x)/(D_{1}+D_{2})$ is a
probability (as $T[Bel_{1},Bel_{2}](x)$ is always non-negative), expressed by
(64). By (71), the quantity $p[\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}](x)$ can be
expressed as

$\frac{\alpha_{1}^{2}D_{1}p[Bel_{1}](x)+\alpha_{2}^{2}D_{2}p[Bel_{2}](x)+\alpha_{1}\alpha_{2}(D_{1}+D_{2})T[Bel_{1},Bel_{2}](x)}{\alpha_{1}D_{1}+\alpha_{2}D_{2}},$

i.e., (63).

### Proof of Theorem 8

By (63), we have that

$\begin{array}[]{lll}&&p[\alpha_{1}Bel_{1}+\alpha_{2}Bel_{2}]-\alpha_{1}p[Bel_{1}]-\alpha_{2}p[Bel_{2}]\\\
\\\
&=&\widehat{\alpha_{1}D_{1}}\alpha_{1}p[Bel_{1}]+\widehat{\alpha_{1}D_{1}}\alpha_{2}T+\widehat{\alpha_{2}D_{2}}\alpha_{1}T+\widehat{\alpha_{2}D_{2}}\alpha_{2}p[Bel_{2}]\\\
\\\ &&-\alpha_{1}p[Bel_{1}]-\alpha_{2}p[Bel_{2}]\\\ \\\
&=&\alpha_{1}p[Bel_{1}](\widehat{\alpha_{1}D_{1}}-1)+\widehat{\alpha_{1}D_{1}}\alpha_{2}T+\widehat{\alpha_{2}D_{2}}\alpha_{1}T+\alpha_{2}p[Bel_{2}](\widehat{\alpha_{2}D_{2}}-1)\\\
\\\
&=&-\alpha_{1}p[Bel_{1}]\widehat{\alpha_{2}D_{2}}+\widehat{\alpha_{1}D_{1}}\alpha_{2}T+\widehat{\alpha_{2}D_{2}}\alpha_{1}T-\alpha_{2}p[Bel_{2}]\widehat{\alpha_{1}D_{1}}\\\
\\\
&=&\widehat{\alpha_{1}D_{1}}\Big{(}\alpha_{2}T-\alpha_{2}p[Bel_{2}]\Big{)}+\widehat{\alpha_{2}D_{2}}\Big{(}\alpha_{1}T-\alpha_{1}p[Bel_{1}]\Big{)}\\\
\\\
&=&\displaystyle\frac{\displaystyle\alpha_{1}\alpha_{2}}{\displaystyle\alpha_{1}D_{1}+\alpha_{2}D_{2}}\Big{[}D_{1}(T-p[Bel_{2}])+D_{2}(T-p[Bel_{1}])\Big{]}.\end{array}$

This is zero iff

$T[Bel_{1},Bel_{2}](D_{1}+D_{2})=p[Bel_{1}]D_{2}+p[Bel_{2}]D_{1},$

which is equivalent to

$T[Bel_{1},Bel_{2}]=\hat{D}_{1}p[Bel_{2}]+\hat{D}_{2}p[Bel_{1}],$

as $(\alpha_{1}\alpha_{2})/(\alpha_{1}D_{1}+\alpha_{2}D_{2})$ is always non-
zero in non-trivial cases. This is equivalent to (after replacing the
expressions for $p[Bel]$ (LABEL:eq:interpretation2) and $T[Bel_{1},Bel_{2}]$
(64))

$D_{1}(Pl_{2}-m_{2}(x))(\beta[Bel_{2}]-\beta[Bel_{1}])+D_{2}(Pl_{1}-m_{1}(x))(\beta[Bel_{1}]-\beta[Bel_{2}])=0,$

which is in turn equivalent to

$\begin{array}[]{c}(\beta[Bel_{2}]-\beta[Bel_{1}])\Big{[}D_{1}(Pl_{2}(x)-m_{2}(x))-D_{2}(Pl_{1}(x)-m_{1}(x))\Big{]}=0.\end{array}$

Obviously, this is true iff $\beta[Bel_{1}]=\beta[Bel_{2}]$ _or_ the second
factor is zero, i.e.,

$\begin{array}[]{lll}&&\displaystyle
D_{1}D_{2}\frac{Pl_{2}(x)-m_{2}(x)}{D_{2}}-D_{1}D_{2}\frac{Pl_{1}(x)-m_{1}(x)}{D_{1}}\\\
\\\ &=&D_{1}D_{2}(R[Bel_{2}](x)-R[Bel_{1}](x))=0\end{array}$

for all $x\in\Theta$, i.e., $R[Bel_{1}]=R[Bel_{2}]$.

### Proof of Theorem 9

By (66), the equality $\beta[Bel_{1}]=\beta[Bel_{2}]$ is equivalent to

$(2\sigma_{2}^{2}+\cdots+n\sigma_{2}^{n})(\sigma_{1}^{2}+\cdots+\sigma_{1}^{n})=(2\sigma_{1}^{2}+\cdots+n\sigma_{1}^{n})(\sigma_{2}^{2}+\cdots+\sigma_{2}^{n}).$

Let us assume that there exists a cardinality $k$ such that
$\sigma_{1}^{k}\neq 0\neq\sigma_{2}^{k}$. We can then divide the two sides by
$\sigma_{1}^{k}$ and $\sigma_{2}^{k}$, obtaining

$\begin{array}[]{lll}&&\displaystyle\left(2\frac{\sigma_{2}^{2}}{\sigma_{2}^{k}}+\cdots+k+\cdots+n\frac{\sigma_{2}^{n}}{\sigma_{2}^{k}}\right)\left(\frac{\sigma_{1}^{2}}{\sigma_{1}^{k}}+\cdots+1+\cdots+\frac{\sigma_{1}^{n}}{\sigma_{1}^{k}}\right)\\\
\\\
&=&\displaystyle\left(2\frac{\sigma_{1}^{2}}{\sigma_{1}^{k}}+\cdots+k+\cdots+n\frac{\sigma_{1}^{n}}{\sigma_{1}^{k_{1}}}\right)\left(\frac{\sigma_{2}^{2}}{\sigma_{2}^{k}}+\cdots+1+\cdots+\frac{\sigma_{2}^{n}}{\sigma_{2}^{k}}\right).\end{array}$

Therefore, if $\sigma_{1}^{j}/\sigma_{1}^{k}=\sigma_{2}^{j}/\sigma_{2}^{k}$
$\forall j\neq k$, the condition $\beta[Bel_{1}]=\beta[Bel_{2}]$ is satisfied.
But this is equivalent to (67).

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