#  Reasoning with fuzzy and uncertain evidence using epistemic random fuzzy
sets: general framework and practical models

Thierry Denœux<EMAIL_ADDRESS>Université de technologie de Compiègne,
CNRS
UMR 7253 Heudiasyc, Compiègne, France  Institut universitaire de France,
Paris, France

###### Abstract

We introduce a general theory of epistemic random fuzzy sets for reasoning
with fuzzy or crisp evidence. This framework generalizes both the Dempster-
Shafer theory of belief functions, and possibility theory. Independent
epistemic random fuzzy sets are combined by the generalized product-
intersection rule, which extends both Dempster’s rule for combining belief
functions, and the product conjunctive combination of possibility
distributions. We introduce Gaussian random fuzzy numbers and their multi-
dimensional extensions, Gaussian random fuzzy vectors, as practical models for
quantifying uncertainty about scalar or vector quantities. Closed-form
expressions for the combination, projection and vacuous extension of Gaussian
random fuzzy numbers and vectors are derived.

###### keywords:

Belief functions, evidence theory, possibility theory, random sets,
uncertainty.

††journal: Fuzzy Sets and Systems

##  1 Introduction

The Dempster-Shafer (DS) theory of belief functions [29] and possibility
theory [38] were introduced independently in the late 1970’s as non-
probabilistic frameworks for reasoning with uncertainty [11, 10]. The former
approach is based on the idea of representing elementary pieces of evidence as
completely monotone capacities, or _belief functions_ , and combining them
using an operator known as the product-intersection rule or Dempster’s rule.
As probability measures are special belief functions, and Dempster’s rule
extends Bayesian conditioning, DS can be seen as an extension of Bayesian
probability theory, particularly suitable to reasoning with severe
uncertainty. There is also a strong relation between DS theory and the theory
of random sets [23]: specifically, any random set induces a belief function
and, conversely, any belief function can be seen as being induced by some
random set [26]. In DS theory, a random set underlying a belief function does
not represent a random mechanism for generating sets of outcomes, but the
imprecise meanings of a piece of evidence under different interpretations with
known probabilities [30]. To avoid confusion, we use the term _epistemic
random set_ for random sets representing evidence in DS theory.

In contrast, possibility theory originates from the theory of fuzzy sets [36].
In this approach, a fuzzy statement about the variable of interest, seen as a
flexible constraint on its precise but unknown value in some domain $\Theta$,
induces a possibility measure and a dual necessity measure on $\Theta$.
Interestingly, a necessity measure is a belief function, and the dual
possibility measure is the corresponding plausibility function, but the
converse is not true (a belief function is not, in general, a necessity
measure). For this reason, possibility theory has sometimes been presented as
“a special branch of evidence theory” (another name for DS theory) [21, page
187]. However, combining two necessity measures by Dempster’s rule yields a
belief function that is no longer a necessity measure: this combination rule
is, thus, not compatible with possibilistic reasoning. In contrast,
possibility theory has its own conjunctive combination operators based on
triangular norms (or t-norms) [16]. Possibility and DS theory are, thus, two
distinct models of uncertain reasoning based on related knowledge
representation languages but different information processing mechanisms.

In a companion paper [9], we have revisited Zadeh’s notion of “evidence of the
second kind”, defined as a pair $(X,\Pi_{(Y\mid X)})$ in which $X$ is a
discrete random variable on a set $\Omega$ and $\Pi_{(Y\mid X)}$ a collection
of conditional possibility distributions of a variable $Y$ given $X=x$, for
all $x\in\Omega$. If random variable $X$ is constant, we get a unique
possibility distribution for variable $Y$; if the conditional possibility
distributions $\Pi_{(Y\mid X)}$ take values in $\\{0,1\\}$, then the pair
$(X,\Pi_{(Y\mid X)})$ defines a random set equivalent to a DS mass function.
The mappings associating, to each event, its expected necessity and its
expected possibility are, respectively, belief and plausibility functions. In
this framework, a possibility distribution thus represents certain but fuzzy
evidence, while a DS mass function is a model of uncertain and crisp evidence.
In general, a pair $(X,\Pi_{(Y\mid X)})$ defines an _epistemic random fuzzy
set_ , allowing us to describe evidence that is both uncertain and fuzzy. (The
term “epistemic” emphasizes the distinction between this interpretation and
that of random fuzzy sets as mechanisms for generating fuzzy data considered,
for instance in [28, 17]). In [9], we have proposed a family of combination
rules for epistemic random fuzzy sets in the finite setting, generalizing both
Dempster’s rule and the conjunctive combination rules of possibility theory.
One of these rules, based on the product t-norm, is associative and arguably
well suited for combining independent evidence. Equipped with this combination
rule (called here the _generalized product-intersection rule_), the theory of
epistemic random fuzzy sets can be seen as an extension of both DS theory and
possibility theory, making it possible to combine evidence of various types,
including expert assessments (possibly expressed in natural language), sensor
information, and statistical evidence about a model parameter.

In this paper, drawing from mathematical results presented by Couso and
Sánchez in [2], we give a more general exposition of the theory of epistemic
fuzzy sets, considering arbitrary probability and measurable spaces. We define
combination, marginalization and vacuous extension operations of random fuzzy
sets in this general setting, laying the foundations of a wide-ranging theory
of uncertainty encompassing DS and possibility theories as special cases.
Finally, for the important case where the frame of discernment is
$\mathbb{R}^{p}$, we propose Gaussian random fuzzy numbers and vectors as a
practical model, generalizing both Gaussian random variables and vectors on
the one hand, and Gaussian possibility distributions on the other hand.

The rest of this paper is organized as follows. Classical models (including
random sets, fuzzy sets and possibility theory) are first recalled in Section
2. Epistemic random fuzzy sets are then introduced in a general setting in
Section 3. Finally, Gaussian random fuzzy numbers and vectors are studied,
respectively, in Sections 4 and 5, and Section 6 concludes the paper.

##  2 Classical models

In this section, we recall the main definitions and results pertaining to the
two models of uncertainty generalized in this paper: random sets and belief
functions on the one hand (Section 2.1), fuzzy sets and possibility theory on
the other hand (Section 2.2).

###  2.1 Random sets and belief functions

Whereas belief functions in the finite setting can be introduced without any
reference to random sets [29], the mathematical framework of random sets is
useful to analyze belief functions in more general spaces, and to define the
practical models needed, e.g., in statistical applications. Important
references about the link between random sets and belief functions include
[26] and [2].

Let $(\Omega,\sigma_{\Omega},P)$ be a probability space,
$(\Theta,\sigma_{\Theta})$ a measurable space, and ${\overline{X}}$ a mapping
from $\Omega$ to $2^{\Theta}$. The _upper_ _and lower inverses_ of
${\overline{X}}$ are defined, respectively, as follows:

$\displaystyle{\overline{X}}^{*}(B)$
$\displaystyle=B^{*}=\\{\omega\in\Omega:{\overline{X}}(\omega)\cap
B\neq\emptyset\\}$ (1a) $\displaystyle{\overline{X}}_{*}(B)$
$\displaystyle=B_{*}=\\{\omega\in\Omega:\emptyset\neq{\overline{X}}(\omega)\subseteq
B\\}$ (1b)

for all $B\subseteq\Theta$. It is easy to check that

$B^{*}\cap(B^{c})_{*}=\emptyset$

and

$B^{*}\cup(B^{c})_{*}=\\{\omega\in\Omega:{\overline{X}}(\omega)\neq\emptyset\\}=\Theta^{*},$

where $B^{c}$ denotes the complement of $B$ in $\Theta$.

The mapping ${\overline{X}}$ is said to be $\sigma_{\Omega}-\sigma_{\Theta}$
_strongly measurable_ [26] if, for all $B\in\sigma_{\Theta}$,
$B^{*}\in\sigma_{\Omega}$ (or, equivalently, if for all $B\in\sigma_{\Theta}$,
$B_{*}\in\sigma_{\Omega}$). The tuple
$(\Omega,\sigma_{\Omega},P,\Theta,\sigma_{\Theta},{\overline{X}})$ is called a
_random set_. When there is no confusion about the domain and co-domain, we
will call the $\sigma_{\Omega}-\sigma_{\Theta}$ strongly measurable mapping
${\overline{X}}$ itself a _random set_.

In the special case where $|{\overline{X}}(\omega)|=1$ for all
$\omega\in\Omega$, we can define the mapping $X:\Omega\rightarrow\Theta$ such
that ${\overline{X}}(\omega)=\\{X(\omega)\\}$ for all $\omega\in\Omega$. We
then have $B^{*}=B_{*}=X^{-1}(B)$ for all $B\subseteq\Theta$, and $X$ is
$\sigma_{\Omega}-\sigma_{\Theta}$ measurable. The notion of random set thus
extends that of random variable.

#### Belief and plausibility functions

From now on, we will assume, for simplicity, that $P(\Theta^{*})=1$. (If not
verified, this property can be enforced by conditioning $P$ on $\Theta^{*}$).
Let $P^{*}$ and $P_{*}$ be the lower and upper probability measures associated
with random set ${\overline{X}}$, defined as the mappings from
$\sigma_{\Theta}$ to $[0,1]$ such that

$P_{*}(B)=P(B_{*})$ (2)

and

$P^{*}(B)=P(B^{*})=1-P_{*}(B^{c}),$ (3)

for all $B\in\sigma_{\Theta}$. Mapping $P_{*}$ is a completely monotone
capacity, i.e., a _belief function_ , and $P^{*}$ is the dual _plausibility
function_ [26, Proposition 1]. In the following, they will be denoted,
respectively, as $Bel_{\overline{X}}$ and $Pl_{\overline{X}}$. The
corresponding _contour function_ is defined as the mapping $pl_{\overline{X}}$
from $\Theta$ to $[0,1]$ such that

$pl_{\overline{X}}(\theta)=Pl_{\overline{X}}(\\{\theta\\})$

for all $\theta\in\Theta$. The subsets
${\overline{X}}(\omega)\subseteq\Theta$, for all $\omega\in\Omega$, are called
the _focal sets_ of $Bel_{\overline{X}}$.

#### Interpretation

In DS theory, $\Omega$ represents a set of interpretations of a piece of
evidence about a variable $\boldsymbol{\theta}$ taking values in set $\Theta$
(called the _frame of discernment_). If interpretation $\omega\in\Omega$
holds, we know that $\boldsymbol{\theta}\in{\overline{X}}(\omega)$, and
nothing more. For any $A\in\sigma_{\Omega}$, $P(A)$ is the (subjective)
probability that the true interpretation lies in $A$. For any
$B\in\sigma_{\Theta}$, the degree of belief $Bel_{\overline{X}}(B)$ is then a
measure of support of the proposition “$\boldsymbol{\theta}\in B$” given the
evidence, while the degree of plausibility $Pl_{\overline{X}}(B)$ is a measure
of lack of support for the proposition “$\boldsymbol{\theta}\not\in B$”. Under
this interpretation, the random set ${\overline{X}}$ represents a state of
knowledge: it can be said to be _epistemic_.

#### Vacuous random set

A constant random set
$(\Omega,\sigma_{\Omega},P,\Theta,\sigma_{\Theta},{\overline{X}})$ such that
${\overline{X}}(\omega)=\Theta$ for all $\omega\in\Omega$ is said to be
_vacuous_. For such a random set, we have $Bel_{\overline{X}}(A)=0$ for all
$A\in\sigma_{\Theta}\setminus\\{\Theta\\}$ and $Pl_{\overline{X}}(A)=1$ for
all $A\in\sigma_{\Theta}\setminus\\{\emptyset\\}$. A vacuous random set
represents complete ignorance about $\boldsymbol{\theta}$.

#### Finite case

Assume that $\Theta$ is finite, and $\sigma_{\Theta}=2^{\Theta}$. The _Möbius
inverse_ of $Bel_{\overline{X}}$ is the mapping $m_{\overline{X}}$ from
$2^{\Theta}$ to [0,1] such that

$m_{\overline{X}}(B)=\sum_{C\subseteq B}(-1)^{|B|-|C|}Bel_{\overline{X}}(C),$

for all $B\subseteq\Theta$. It verifies $m(B)\geq 0$ for all
$B\subseteq\Omega$, $\sum_{B\subseteq\Omega}m(B)=1$ and $m(\emptyset)=0$. The
belief and plausibility can be computed from $m_{\overline{X}}$, respectively,
as

$Bel_{\overline{X}}(B)=\sum_{C\subseteq
B}m_{\overline{X}}(C)\quad\text{and}\quad Pl_{\overline{X}}(B)=\sum_{C\cap
B\neq\emptyset}m_{\overline{X}}(C),$

for all $B\subseteq\Theta$.

#### Random closed intervals

Random closed intervals are particularly simple models allowing us to define
belief functions on the real line [4, 34, 6]. Let $(\Omega,\sigma_{\Omega},P)$
be a probability space and $X,Y$ two random variables
$\Omega\rightarrow\mathbb{R}$ such that $P(\\{\omega\in\Omega:X(\omega)\leq
Y(\omega)\\})=1$. Then, the mapping ${\overline{X}}:\Omega\rightarrow
2^{\mathbb{R}}$ defined by ${\overline{X}}(\omega)=[X(\omega),Y(\omega)]$ is
$\sigma_{\Omega}-\beta_{\mathbb{R}}$ strongly measurable, where
$\beta_{\mathbb{R}}$ is the Borel $\sigma$-algebra on $\mathbb{R}$ (see a
formal proof in [22]). This mapping defines a _random closed interval_. For a
random closed interval ${\overline{X}}=[X,Y]$, we have [4]

$Bel_{\overline{X}}([x,y])=P([X,Y]\subseteq[x,y])=P(X\geq x;Y\leq y)$ (4a) and
$Pl_{\overline{X}}([x,y])=P([X,Y]\cap[x,y]\neq\emptyset)=1-P(X>y)-P(Y<x),$
(4b)

for all $(x,y)\in\mathbb{R}^{2}$ such that $x\leq y$. In particular, by
letting $x$ tend to $-\infty$ in (4), we obtain the _lower and upper
cumulative distribution functions (cdf’s)_ of ${\overline{X}}$ as

$F_{*}(y)=Bel_{\overline{X}}((-\infty,y])=P(Y\leq y)=F_{Y}(y)$ (5a) and
$F^{*}(y)=Pl_{\overline{X}}((-\infty,y])=P(X\leq y)=F_{X}(y).$ (5b)

#### Lower and upper expectation

Let ${\overline{X}}$ be a random set from $(\Omega,\sigma_{\Omega},P)$ to
$(\mathbb{R},\beta_{\mathbb{R}})$. Following Dempster [3], we can define its
_lower and upper expectations_ , respectively, as

$\mathbb{E}_{*}({\overline{X}})=\int_{-\infty}^{+\infty}x\,dF^{*}(x)$

and

$\mathbb{E}^{*}({\overline{X}})=\int_{-\infty}^{+\infty}x\,dF_{*}(x),$

where $F_{*}(x)=Bel_{\overline{X}}((-\infty,x])$ and
$F^{*}(x)=Pl_{\overline{X}}((-\infty,x])$ are the lower and upper cdf’s of
${\overline{X}}$. When ${\overline{X}}$ is a random closed interval $[X,Y]$,
it follows from (5) that $\mathbb{E}_{*}({\overline{X}})=\mathbb{E}(X)$ and
$\mathbb{E}^{*}({\overline{X}})=\mathbb{E}(Y)$.

#### Dempster’s rule

Consider two pieces of evidence represented by random sets

$(\Omega_{1},\sigma_{1},P_{1},\Theta,\sigma_{\Theta},{\overline{X}}_{1})\quad\text{and}\quad(\Omega_{2},\sigma_{2},P_{2},\Theta,\sigma_{\Theta},{\overline{X}}_{2}),$

and the mapping ${\overline{X}}_{\cap}$ from $\Omega_{1}\times\Omega_{2}$ to
$2^{\Theta}$ defined by
${\overline{X}}_{\cap}(\omega_{1},\omega_{2})={\overline{X}}_{1}(\omega_{1})\cap{\overline{X}}_{2}(\omega_{2})$.
If interpretations $\omega_{1}\in\Omega_{1}$ and $\omega_{2}\in\Omega_{2}$
both hold, we know that
$\boldsymbol{\theta}\in{\overline{X}}_{\cap}(\omega_{1},\omega_{2})$, provided
that
${\overline{X}}_{1}(\omega_{1})\cap{\overline{X}}_{2}(\omega_{2})\neq\emptyset$.
Assume that ${\overline{X}}_{\cap}$ is
$(\sigma_{1}\otimes\sigma_{2})-\sigma_{\Theta}$ strongly measurable, where
$\sigma_{1}\otimes\sigma_{2}$ is the tensor product $\sigma$-algebra over the
Cartesian product $\Omega_{1}\times\Omega_{2}$. The two pieces of evidence are
said to be _independent_ if, for any $A\in\sigma_{1}\otimes\sigma_{2}$, the
probability that $A$ contains the true interpretations of the two pieces of
evidence is the conditional probability

$P_{12}(A)=(P_{1}\times P_{2})(A\mid\Theta^{*})=\frac{(P_{1}\times
P_{2})(A\cap\Theta^{*})}{(P_{1}\times P_{2})(\Theta^{*})},$ (6)

where $P_{1}\times P_{2}$ is the product measure satisfying $(P_{1}\times
P_{2})(A_{1}\times A_{2})=P_{1}(A_{1})P_{2}(A_{2})$ for all
$A_{1}\in\sigma_{1}$, $A_{2}\in\sigma_{2}$, and

$\Theta^{*}=\\{(\omega_{1},\omega_{2})\in\Omega_{1}\times\Omega_{2}:{\overline{X}}_{\cap}(\omega_{1},\omega_{2})\neq\emptyset\\}$

is the set of noncontradictory pairs of interpretations. The quantity

$\kappa=1-(P_{1}\times P_{2})(\Theta^{*})=(P_{1}\times
P_{2})(\\{(\omega_{1},\omega_{2})\in\Omega_{1}\times\Omega_{2}:{\overline{X}}_{\cap}(\omega_{1},\omega_{2})=\emptyset\\})$

is called the _degree of conflict_ between the two pieces of evidence. The
combined random set

$(\Omega_{1}\times\Omega_{2},\sigma_{1}\otimes\sigma_{2},P_{12},\Theta,\sigma_{\Theta},{\overline{X}}_{\cap})$

is called the _orthogonal sum_ of the two pieces of evidence, and is denoted
by ${\overline{X}}_{1}\oplus{\overline{X}}_{2}$. This combination rule, first
introduced by Dempster in [3], is called the _product-intersection rule_ , or
_Dempster’s rule of combination_.

We can remark that Dempster’s rule is usually viewed as an operation to
combine belief functions, whereas it is defined here as an operation to
combine _random sets_. This distinction is immaterial in the standard setting,
as the orthogonal sum of two belief functions does not depend on their
particular random set representations and can be defined without reference to
the random set framework [32]. However, it becomes crucial when considering
random fuzzy sets as a model for generating belief functions, as done in this
paper. We will come back to this important point in Section 3.2.

Any vacuous random set is obviously a neutral element for Dempster’s rule. The
following important proposition states that pieces of evidence can be combined
by Dempster’s rule in any order.

######  Proposition 1.

Dempster’s rule is commutative and associative.

###### Proof.

See A. ∎

######  Example 1.

Let $X_{1}\sim N(\mu_{1},\sigma_{1}^{2})$ and $X_{2}\sim
N(\mu_{2},\sigma_{2}^{2})$ be two independent normal random variables and
consider the random intervals ${\overline{X}}_{1}=[X_{1},+\infty)$ and
${\overline{X}}_{2}=(-\infty,X_{2}]$. The degree of conflict between
${\overline{X}}_{1}$ and ${\overline{X}}_{2}$ is

$\kappa=P(X_{1}>X_{2})=P(X_{2}-X_{1}<0)=\Phi\left(\frac{\mu_{1}-\mu_{2}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}}\right),$

where $\Phi$ is the standard normal cdf. The orthogonal sum of
${\overline{X}}_{1}$ and ${\overline{X}}_{2}$ is the random closed interval
$[X^{\prime}_{1},X^{\prime}_{2}]$, where $(X^{\prime}_{1},X^{\prime}_{2})$ is
the two-dimensional random vector with distribution equal the conditional
distribution of $(X_{1},X_{2})$ given $X_{1}\leq X_{2}$. Its density is

$f_{X^{\prime}_{1},X^{\prime}_{2}}(x_{1},x_{2})=\frac{\sigma_{1}^{-1}\sigma_{2}^{-1}\phi\left(\frac{x_{1}-\mu_{1}}{\sigma_{1}}\right)\phi\left(\frac{x_{2}-\mu_{2}}{\sigma_{2}}\right)I(x_{1}\leq
x_{2})}{\Phi\left(\frac{\mu_{2}-\mu_{1}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}}\right)},$

where $\phi$ is the standard normal probability density function (pdf) and
$I(\cdot)$ is the indicator function.

The following proposition states that the contour function of the orthogonal
sum of two independent random sets ${\overline{X}}_{1}$ and
${\overline{X}}_{2}$ is proportional to the product of the contour functions
of ${\overline{X}}_{1}$ and ${\overline{X}}_{2}$.

######  Proposition 2.

Let ${\overline{X}}_{1}$ and ${\overline{X}}_{2}$ be two independent random
sets on the same domain $\Theta$, with contour functions
$pl_{{\overline{X}}_{1}}$ and $pl_{{\overline{X}}_{2}}$. For any
$\theta\in\Theta$,

$pl_{{\overline{X}}_{1}\oplus{\overline{X}}_{2}}(\theta)=\frac{pl_{{\overline{X}}_{1}}(\theta)pl_{{\overline{X}}_{2}}(\theta)}{1-\kappa},$
(7)

where $\kappa$ is the degree of conflict between ${\overline{X}}_{1}$ and
${\overline{X}}_{2}$.

###### Proof.

We have

$\displaystyle pl_{{\overline{X}}_{1}\oplus{\overline{X}}_{2}}(\theta)$
$\displaystyle=\frac{(P_{1}\times
P_{2})(\\{(\omega_{1},\omega_{2})\in\Omega_{1}\times\Omega_{2}:\theta\in{\overline{X}}_{\cap}(\omega_{1},\omega_{2})\\})}{1-\kappa}$
$\displaystyle=\frac{(P_{1}\times
P_{2})(\\{\omega_{1}\in\Omega_{1}:\theta\in{\overline{X}}_{1}(\omega_{1})\\}\times\\{\omega_{2}\in\Omega_{2}:\theta\in{\overline{X}}_{2}(\omega_{2})\\})}{1-\kappa}$
$\displaystyle=\frac{P_{1}(\\{\omega_{1}\in\Omega_{1}:\theta\in{\overline{X}}_{1}(\omega_{1})\\})\cdot
P_{2}(\\{\omega_{2}\in\Omega_{2}:\theta\in{\overline{X}}_{2}(\omega_{2})\\})}{1-\kappa}$
$\displaystyle=\frac{pl_{{\overline{X}}_{1}}(\theta)pl_{{\overline{X}}_{2}}(\theta)}{1-\kappa}.$

∎

######  Example 2.

Let us consider again the two random intervals of Example 1. The contour
functions of ${\overline{X}}_{1}$ and ${\overline{X}}_{2}$ are, respectively,

$pl_{{\overline{X}}_{1}}(x)=P(X_{1}\leq
x)=\Phi\left(\frac{x-\mu_{1}}{\sigma_{1}}\right)$

and

$pl_{{\overline{X}}_{2}}(x)=P(X_{2}\geq
x)=1-\Phi\left(\frac{x-\mu_{2}}{\sigma_{2}}\right).$

Now, the contour function of ${\overline{X}}_{1}\oplus{\overline{X}}_{2}$ is

$\displaystyle pl_{{\overline{X}}_{1}\oplus{\overline{X}}_{2}}(x)$
$\displaystyle=P(X^{\prime}_{1}\leq x\leq X^{\prime}_{2})$
$\displaystyle=\int_{-\infty}^{x}\int_{x}^{+\infty}f_{X^{\prime}_{1},X^{\prime}_{2}}(x_{1},x_{2})dx_{2}dx_{1}$
$\displaystyle=\left[\Phi\left(\frac{\mu_{2}-\mu_{1}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}}\right)\right]^{-1}\int_{-\infty}^{x}\int_{x}^{+\infty}\sigma_{1}^{-1}\sigma_{2}^{-1}\phi\left(\frac{x_{1}-\mu_{1}}{\sigma_{1}}\right)\phi\left(\frac{x_{2}-\mu_{2}}{\sigma_{2}}\right)dx_{2}dx_{1}$
$\displaystyle=\frac{\Phi\left(\frac{x-\mu_{1}}{\sigma_{1}}\right)\left[1-\Phi\left(\frac{x-\mu_{2}}{\sigma_{2}}\right)\right]}{\Phi\left(\frac{\mu_{2}-\mu_{1}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}}\right)}=\frac{pl_{{\overline{X}}_{1}}(x)pl_{{\overline{X}}_{2}}(x)}{1-\kappa}.$

#### Marginalization and vacuous extension

Let us now consider the case where we have two variables
$\boldsymbol{\theta}_{1}$ and $\boldsymbol{\theta}_{2}$ with domains
$\Theta_{1}$ and $\Theta_{2}$. (The case of $n$ variables is not more
difficult conceptually but it requires heavier notations). Let
$\sigma_{\Theta_{1}}$ and $\sigma_{\Theta_{2}}$ be $\sigma$-algebras defined,
respectively, on $\Theta_{1}$ and $\Theta_{2}$. Let
$\Theta_{12}=\Theta_{1}\times\Theta_{2}$ and
$\sigma_{\Theta_{12}}=\sigma_{\Theta_{1}}\otimes\sigma_{\Theta_{2}}$. Let
${\overline{X}}_{12}$ be a random set from $(\Omega,\sigma_{\Omega},P)$ to
$(\Theta_{12},\sigma_{\Theta_{12}})$, and ${\overline{X}}_{1}$ the mapping
from $\Omega$ to $2^{\Theta_{1}}$ that maps each $\omega\in\Omega$ to the
_projection_ of ${\overline{X}}_{12}(\omega)$ onto $\Theta_{1}$:

${\overline{X}}_{1}(\omega)={\overline{X}}_{12}(\omega)\downarrow\Theta_{1}=\\{\theta_{1}\in\Theta_{1}:\exists\theta_{2}\in\Theta_{2},(\theta_{1},\theta_{2})\in{\overline{X}}_{12}(\omega)\\}.$

It is easy to see that ${\overline{X}}_{1}$ is
$\sigma_{\Omega}-\sigma_{\Theta_{1}}$ measurable: for any
$B\in\sigma_{\Theta_{1}}$,

$\displaystyle{\overline{X}}_{1}^{*}(B)$
$\displaystyle=\\{\omega\in\Omega:{\overline{X}}_{1}(\omega)\cap
B\neq\emptyset\\}$
$\displaystyle=\\{\omega\in\Omega:{\overline{X}}_{12}(\omega)\cap(B\times\Theta_{2})\neq\emptyset\\}$
$\displaystyle={\overline{X}}_{12}^{*}(B\times\Theta_{2}).$

As $B\times\Theta_{2}\in\sigma_{\Theta_{12}}$ and ${\overline{X}}_{12}$ is
$\sigma_{\Omega}-\sigma_{\Theta_{12}}$ strongly measurable, it thus follows
that ${\overline{X}}_{1}^{*}(B)\in\sigma_{\Omega}$. The random set
${\overline{X}}_{1}$ will be called the _marginal_ of ${\overline{X}}_{12}$ on
$\Theta_{1}$.

Conversely, let ${\overline{X}}_{1}$ be a random set from
$(\Omega,\sigma_{\Omega})$ to $(\Theta_{1},\sigma_{\Theta_{1}})$ and let
${\overline{X}}_{1\uparrow 2}$ be the mapping from $\Omega$ to $\Theta_{12}$
defined by

${\overline{X}}_{1\uparrow(1,2)}(\omega)={\overline{X}}_{1}(\omega)\times\Theta_{2}.$

For any $B\in\sigma_{\Theta_{12}}$,

$\displaystyle{\overline{X}}_{1\uparrow(1,2)}^{*}(B)$
$\displaystyle=\\{\omega\in\Omega:{\overline{X}}_{1\uparrow 2}(\omega)\cap
B\neq\emptyset\\}$
$\displaystyle=\\{\omega\in\Omega:{\overline{X}}_{1}(\omega)\cap(B\downarrow\Theta_{1})\neq\emptyset\\}$
$\displaystyle={\overline{X}}_{1}^{*}(B\downarrow\Theta_{1}).$

If for all $B\in\sigma_{\Theta_{12}}$,
${\overline{X}}_{1}^{*}(B\downarrow\Theta_{1})\in\sigma_{\Omega}$, then
${\overline{X}}_{1\uparrow(1,2)}$ is $\sigma_{\Omega}-\sigma_{\Theta_{12}}$
strongly measurable. It is said to be the _vacuous extension_ of
${\overline{X}}_{1}$ in $\Theta_{1}\times\Theta_{2}$.

We say that a random set ${\overline{X}}_{12}$ from
$(\Omega,\sigma_{\Omega},P)$ to $(\Theta_{12},\sigma_{\Theta_{12}})$ with
marginals ${\overline{X}}_{1}$ and ${\overline{X}}_{2}$ is _non-interactive_
if it is equal to the orthogonal sum of its marginals, i.e.,

${\overline{X}}_{12}={\overline{X}}_{1\uparrow(1,2)}\oplus{\overline{X}}_{2\uparrow(1,2)}\quad\text{denoted
by}\quad{\overline{X}}_{1}\oplus{\overline{X}}_{2}.$

######  Example 3.

Let $(X_{1},X_{2})$ be a two dimensional random vector from
$(\Omega,\sigma_{\Omega},P)$ to $(\mathbb{R}^{2},\beta_{\mathbb{R}^{2}})$ and
consider the mapping ${\overline{X}}_{12}:\Omega\rightarrow
2^{\mathbb{R}^{2}}$ defined as

${\overline{X}}_{12}(\omega)=(-\infty,X_{1}(\omega)]\times(-\infty,X_{2}(\omega)].$

This mapping defines a random set [23, page 3]. Its marginals are the random
closed intervals $(-\infty,X_{1}]$ and $(-\infty,X_{2}]$. If $X_{1}$ and
$X_{2}$ are independent, then
${\overline{X}}_{12}=(-\infty,X_{1}]\oplus(-\infty,X_{2}]$ and
${\overline{X}}_{12}$ is non-interactive.

###  2.2 Fuzzy sets and possibility theory

A _fuzzy subset_ of a set $\Theta$ is a pair
${\widetilde{F}}=(\Theta,\mu_{{\widetilde{F}}})$, where
$\mu_{{\widetilde{F}}}$ is a mapping from $\Theta$ to $[0,1]$, called the
_membership function_ of ${\widetilde{F}}$. Each number
$\mu_{{\widetilde{F}}}(\theta)$ is interpreted as a degree of membership of
element $\theta$ to the fuzzy set ${\widetilde{F}}$. In the following, to
simplify the notation, we will identify fuzzy sets to their membership
functions and write ${\widetilde{F}}(\theta)$ for
$\mu_{{\widetilde{F}}}(\theta)$. The height of fuzzy set ${\widetilde{F}}$ is
defined as

$0pt({\widetilde{F}})=\sup_{\theta\in\Theta}{\widetilde{F}}(\theta).$

If $0pt({\widetilde{F}})=1$, ${\widetilde{F}}$ is said to be _normal_. For any
$\alpha\in[0,1]$, the (weak) $\alpha$-cut of ${\widetilde{F}}$ is the set

${}^{\alpha}{\widetilde{F}}=\\{\theta\in\Theta:{\widetilde{F}}(\theta)\geq\alpha\\}.$

#### Possibility and necessity measures

Let $\boldsymbol{\theta}$ be a variable taking values in $\Theta$. Assume that
we receive a piece of evidence telling us that “$\boldsymbol{\theta}$ is
${\widetilde{F}}$”, where ${\widetilde{F}}$ is a normal fuzzy subset of
$\Theta$. This evidence induces a _possibility measure_ $\Pi_{\widetilde{F}}$
from $2^{\Theta}$ to $[0,1]$ defined by

$\Pi_{\widetilde{F}}(B)=\sup_{\theta\in B}{\widetilde{F}}(\theta),$ (8)

for all $B\subseteq\Theta$. The number $\Pi_{\widetilde{F}}(B)$ is interpreted
as the degree of possibility that $\boldsymbol{\theta}\in B$, given that
$\boldsymbol{\theta}$ is ${\widetilde{F}}$ [38]. The corresponding
_possibility distribution_ is the mapping from $\Theta$ to $[0,1]$ defined by

$\pi_{\widetilde{F}}(\theta)=\Pi_{\widetilde{F}}(\\{\theta\\})={\widetilde{F}}(\theta),$

i.e., it is identical to the membership function ${\widetilde{F}}$. The dual
_necessity measure_ is defined as

$N_{\widetilde{F}}(B)=1-\Pi_{\widetilde{F}}(B^{c})=\inf_{\theta\not\in
B}\left[1-{\widetilde{F}}(\theta)\right].$ (9)

It can easily be shown that mapping
$N_{\widetilde{F}}:2^{\Omega}\rightarrow[0,1]$ is completely monotone, i.e.,
it is a belief function, and $\Pi_{\widetilde{F}}$ is the dual plausibility
function [15]. These belief and plausibility functions are formally induced by
the random set
$([0,1],\beta_{[0,1]},\lambda,\Theta,2^{\Theta},{\overline{X}})$, where
$\beta_{[0,1]}$ is the Borel $\sigma$-algebra on $[0,1]$, $\lambda$ is the
uniform probability measure, and ${\overline{X}}$ is the mapping
$[0,1]\rightarrow 2^{\Theta}$ defined by
${\overline{X}}(\alpha)={}^{\alpha}{\widetilde{F}}$. However, as we will see
in Section 3.2, it is important, when combining evidence, to distinguish
between possibility distributions induced by fuzzy sets, and consonant belief
functions induced by random sets.

#### Conjunctive combination of possibility distributions

Assume that we receive two independent pieces of information telling us that
“$\boldsymbol{\theta}$ is ${\widetilde{F}}$” and “$\boldsymbol{\theta}$ is
${\widetilde{G}}$”, where ${\widetilde{F}}$ and ${\widetilde{G}}$ are two
fuzzy subsets of $\Theta$. The conjunctive combination of these two pieces of
evidence requires some notion of intersection between fuzzy sets. As reviewed
in [13], the intersection operation can be extended to fuzzy sets using
triangular norms (or t-norms for short). Given a t-norm $\top$, the
$\top$-intersection of two fuzzy subsets ${\widetilde{F}}$ and
${\widetilde{G}}$ of the same domain $\Theta$ can be defined as

$({\widetilde{F}}\cap_{\top}{\widetilde{G}})(\theta)={\widetilde{F}}(\theta)\top{\widetilde{G}}(\theta)$

for all $\theta\in\Theta$. The most common choices for $\top$ are the minimum
and product t-norms, as originally proposed by Zadeh [36]; the corresponding
operations are called, respectively, the _minimum_ and _product_
intersections. However, the intersection of two normal fuzzy sets is generally
not normal. To obtain a normal fuzzy set, as needed for the definitions of
possibility and necessity measures in (8)-(9), we define the normalized
$\top$-intersection as

$({\widetilde{F}}\cap^{*}_{\top}{\widetilde{G}})(\theta)=\begin{cases}\displaystyle\frac{{\widetilde{F}}(\theta)\top{\widetilde{G}}(\theta)}{0pt({\widetilde{F}}\cap_{\top}{\widetilde{G}})}&\text{if
}0pt({\widetilde{F}}\cap_{\top}{\widetilde{G}})>0\\\
0&\text{otherwise.}\end{cases}$

The fuzzy set ${\widetilde{F}}\cap^{*}_{\top}{\widetilde{G}}$ is normal
provided that $0pt({\widetilde{F}}\cap_{\top}{\widetilde{G}})>0$. In general,
the normalized intersection $\cap^{*}_{\top}$ associated with a t-norm $\top$
is not associative. A notable exception is the case where $\top$ is the
product t-norm: the normalized product intersection, denoted by $\varodot$, is
associative (see [16], and a simple proof in [9]). By abuse of notation, we
can use the same symbol to denote the conjunctive combination of possibility
measures and the normalized product intersection of fuzzy sets, and write

$\Pi_{\widetilde{F}}\varodot\Pi_{\widetilde{G}}=\Pi_{{\widetilde{F}}\varodot{\widetilde{G}}}.$

As noted by Dubois and Prade [16, page 352], product intersection has a
reinforcement effect that is appropriate when the information sources are
assumed to be independent. The choice of the normalized product intersection
for combining possibility distributions makes possibility theory fit in the
framework of valuation-based systems [33] and allows for possibilistic
reasoning with a large number of variables. The normalized product
intersection operator also has an interesting property with respect to
Gaussian fuzzy numbers, as recalled in the next paragraph.

#### Gaussian fuzzy numbers

A _fuzzy number_ (or fuzzy interval) can be defined as a normal and convex
fuzzy subset of the real line. In particular, a _Gaussian fuzzy number (GFN)_
is a normal fuzzy subset of $\mathbb{R}$ with membership function

$\varphi(x;m,h)=\exp\left(-\frac{h}{2}(x-m)^{2}\right),$

where $m\in\mathbb{R}$ is the mode and $h\in[0,+\infty]$ is the precision.
Such a fuzzy number will be denoted by $\textsf{GFN}(m,h)$. If $h=0$,
$\varphi(x;m,h)=1$ for all $x\in\mathbb{R}$: $\textsf{GFN}(m,0)$ is then
maximally imprecise and identical to the whole real line, whatever the value
of $m$. If $h=+\infty$, $\varphi(x;m,h)=I(x=m)$, where $I(\cdot)$ is the
indicator function; the fuzzy number $\textsf{GFN}(m,+\infty)$ is then
maximally precise and equivalent to the real number $m$.

It can easily be shown that the family of GFN’s is closed under the normalized
product intersection (see, e.g., [1]). More precisely, we have the following
proposition, proved in [1].

######  Proposition 3.

For any $x\in\mathbb{R}$,

$\varphi(x;m_{1},h_{1})\cdot\varphi(x;m_{2},h_{2})=\exp\left(-\frac{h_{1}h_{2}(m_{1}-m_{2})^{2}}{2(h_{1}+h_{2})}\right)\varphi(x;m_{12},h_{12}),$

with

$m_{12}=\frac{h_{1}m_{1}+h_{2}m_{2}}{h_{1}+h_{2}}\quad\text{and}\quad
h_{12}=h_{1}+h_{2}.$

Consequently,

$\textsf{GFN}(m_{1},h_{1})\varodot\textsf{GFN}(m_{2},h_{2})=\textsf{GFN}(m_{12},h_{12}),$

and

$0pt\left[\textsf{GFN}(m_{1},h_{1})\cdot\textsf{GFN}(m_{2},h_{2})\right]=\exp\left(-\frac{h_{1}h_{2}(m_{1}-m_{2})^{2}}{2(h_{1}+h_{2})}\right).$
(10)

#### Marginalization and cylindrical extension

Let us now assume that we have two variables $\boldsymbol{\theta}_{1}$ and
$\boldsymbol{\theta}_{2}$ jointly constrained by a possibility distribution
$\pi_{\widetilde{F}}$, where ${\widetilde{F}}$ is a fuzzy subset of
$\Theta_{12}=\Theta_{1}\times\Theta_{2}$. As a result of (8), variable
$\boldsymbol{\theta}_{1}$ alone is constrained by the possibility distribution

$\pi_{1}(\theta_{1})=\Pi(\\{\theta_{1}\\}\times\Theta_{2})=\sup_{\theta_{2}\in\Theta_{2}}\pi_{\widetilde{F}}(\theta_{1},\theta_{2})=\sup_{\theta_{2}\in\Theta_{2}}{\widetilde{F}}(\theta_{1},\theta_{2})=({\widetilde{F}}\downarrow\Theta_{1})(\theta_{1}),$

where ${\widetilde{F}}\downarrow\Theta_{1}$ is the _projection_ of
${\widetilde{F}}$ on $\Theta_{1}$. We say that $\pi_{1}$ is the _marginal_ of
$\pi_{\widetilde{F}}$ on $\Theta_{1}$. Conversely, given a possibility
distribution $\pi_{{\widetilde{F}}_{1}}$, where ${\widetilde{F}}_{1}$ is a
fuzzy subset of $\Theta_{1}$, its _cylindrical extension_ in
$\Theta_{1}\times\Theta_{2}$ is the possibility distribution
$\pi_{{\widetilde{F}}_{1}\times\Theta_{2}}$ defined as

$\pi_{{\widetilde{F}}_{1}\times\Theta_{2}}(\theta_{1},\theta_{2})=\pi_{{\widetilde{F}}_{1}}(\theta_{1})$

for all $(\theta_{1},\theta_{2})\in\Theta_{1}\times\Theta_{2}$. We say that
the joint possibility distribution $\pi_{{\widetilde{F}}}$ on $\Theta_{12}$ is
_non-interactive_ with respect to the product intersection if it is the
product of its marginals:

$\pi_{{\widetilde{F}}}(\theta_{1},\theta_{2})=\pi_{{\widetilde{F}}\downarrow\Theta_{1}}(\theta_{1})\cdot\pi_{{\widetilde{F}}\downarrow\Theta_{2}}(\theta_{2}).$

######  Example 4.

Let $\pi_{12}$ be the possibility distribution on $\mathbb{R}^{2}$ defined by

$\displaystyle\pi_{12}(x_{1},x_{2})$
$\displaystyle=\exp\left(-\frac{h_{1}}{2}(x_{1}-m_{1})^{2}-\frac{h_{2}}{2}(x_{2}-m_{2})^{2}\right)$
$\displaystyle=\exp\left(-\frac{h_{1}}{2}(x_{1}-m_{1})^{2}\right)\exp\left(-\frac{h_{2}}{2}(x_{2}-m_{2})^{2}\right).$

Its marginals are

$\pi_{1}(x_{1})=\max_{\theta_{2}}\pi_{12}(x_{1},x_{2})=\exp\left(-\frac{h_{1}}{2}(x_{1}-m_{1})^{2}\right)$

and

$\pi_{2}(x_{2})=\max_{\theta_{1}}\pi_{12}(x_{1},x_{2})=\exp\left(-\frac{h_{2}}{2}(x_{2}-m_{2})^{2}\right).$

Consequenty, $\pi_{12}$ is non-interactive with respect to the product
intersection.

##  3 Epistemic random fuzzy sets

The proposed epistemic random fuzzy set model is introduced in this section.
The main definitions are first given in Section 3.1, and the generalized
product-intersection rule is introduced in Section 3.2. Marginalization and
vacuous extension are then addressed in Section 3.3, and an application to
statistical inference is briefly discussed in Section 3.4.

###  3.1 General definitions

As before, let $(\Omega,\sigma_{\Omega},P)$ be a probability space and let
$(\Theta,\sigma_{\Theta})$ be a measurable space. Let ${\widetilde{X}}$ by a
mapping from $\Omega$ to the set $[0,1]^{\Theta}$ of fuzzy subsets of
$\Theta$. For any $\alpha\in[0,1]$, let ${}^{\alpha}{\widetilde{X}}$ be the
mapping from $\Omega$ to $2^{\Theta}$ defined as

${}^{\alpha}{\widetilde{X}}(\omega)={}^{\alpha}[{\widetilde{X}}(\omega)],$

where ${}^{\alpha}[{\widetilde{X}}(\omega)]$ is the weak $\alpha$-cut of
${\widetilde{X}}(\omega)$. If for any $\alpha\in[0,1]$,
${}^{\alpha}{\widetilde{X}}$ is $\sigma_{\Omega}-\sigma_{\Theta}$ strongly
measurable, the tuple
$(\Omega,\sigma_{\Omega},P,\Theta,\sigma_{\Theta},{\widetilde{X}})$ is said to
be a _random fuzzy set_ (also called a _fuzzy random variable_) [2]. It is
clear that the class of random fuzzy sets includes that of random sets, just
as the class of fuzzy sets includes that of classical (crisp) sets.

######  Example 5.

Let $M$ be a Gaussian random variable from $(\Omega,\sigma_{\Omega},P)$ to
$(\mathbb{R},\beta_{\mathbb{R}})$, with mean $\mu$ and standard deviation
$\sigma$, and let ${\widetilde{X}}$ be the mapping from $\Omega$ to
$[0,1]^{\mathbb{R}}$ that maps each $\omega\in\Omega$ to the triangular fuzzy
number with mode $M(\omega)$ and support $[M(\omega)-a,M(\omega)+a]$:

${\widetilde{X}}(\omega)(x)=\begin{cases}\frac{a-|x-M(\omega)|}{a}&\text{if
}|x-M(\omega)|\leq a\\\ 0&\text{otherwise. }\end{cases}$

for some $a>0$. For any $\alpha\in[0,1]$, the $\alpha$-cut of
${\widetilde{X}}(\omega)$ is

${}^{\alpha}{\widetilde{X}}(\omega)=\left[M(\omega)-a(1-\alpha),M(\omega)+a(1-\alpha)\right].$

The random set
${}^{\alpha}{\widetilde{X}}:\omega\rightarrow{}^{\alpha}{\widetilde{X}}(\omega)$
is $\sigma_{\Omega}-\beta_{\mathbb{R}}$ strongly measurable (it is a random
closed interval). Consequently, ${\widetilde{X}}$ is a random fuzzy set. In
the following, such random fuzzy sets with domain $[0,1]^{\mathbb{R}}$ will be
called _random fuzzy numbers_.

#### Interpretation

Here, as in [9], we use random fuzzy sets as a model of unreliable and fuzzy
evidence. In this model, we see $\Omega$ as a _set of interpretations_ of a
piece of evidence about a variable $\boldsymbol{\theta}$ taking values in
$\Theta$. If interpretation $\omega\in\Omega$ holds, we know that
“$\boldsymbol{\theta}\text{ is }{\widetilde{X}}(\omega)$”, i.e.,
$\boldsymbol{\theta}$ is constrained by the possibility distribution
$\pi_{{\widetilde{X}}(\omega)}$. We qualify such random fuzzy sets as
_epistemic_ , because they encode a state of knowledge about some variable
$\boldsymbol{\theta}$. It should be noted that this semantics of random fuzzy
sets is different from those reviewed in [2]. The conditional possibility
interpretation developed in [2] is the closest to ours, since we also see the
fuzzy sets ${\widetilde{X}}(\omega)$ as defining conditional possibility
measures. However, in [2], the authors use the random fuzzy set formalism to
model a situation in which we have two random experiments, one of which is
completely determined; the family of possibility distributions
$\\{\pi_{{\widetilde{X}}(\omega)}:\omega\in\Omega\\}$ then models our
knowledge about the relationship between the outcomes $\omega$ of the first
experiment and the possible outcomes of the second one. This formalism allows
the authors of [2] to compute lower and upper bounds on the probability of any
event related to the second experiment. In contrast, our model does not rely
on the notion of random experiment. In particular, we do not postulate the
existence of an objective probability measure on $\Theta$, and the belief and
plausibility measures introduced below are not interpreted as lower and upper
bounds on “true” probabilities.

#### Belief and plausibility

We say that random fuzzy set ${\widetilde{X}}$ is _normalized_ if it verifies
the following conditions:

  1. 1.

For all $\omega\in\Omega$, ${\widetilde{X}}(\omega)$ is either the empty set,
or a normal fuzzy set, i.e., $0pt({\widetilde{X}}(\omega))\in\\{0,1\\}$.

  2. 2.

$P(\\{\omega\in\Omega:{\widetilde{X}}(\omega)=\emptyset\\})=0$.

These conditions will be assumed in the rest of this section. For any
$\omega\in\Omega$, let $\Pi_{\widetilde{X}}(\cdot\mid\omega)$ be the
possibility measure on $\Theta$ induced by ${\widetilde{X}}(\omega)$:

$\Pi_{\widetilde{X}}(B\mid\omega)=\sup_{\theta\in
B}{\widetilde{X}}(\omega)(\theta),$ (11)

and let $N_{\widetilde{X}}(\cdot\mid\omega)$ be the dual necessity measure:

$N_{\widetilde{X}}(B\mid\omega)=\begin{cases}1-\Pi_{\widetilde{X}}(B^{c}\mid\omega)&\text{if
}{\widetilde{X}}(\omega)\neq\emptyset\\\ 0&\text{otherwise. }\end{cases}$

Let $Bel_{\widetilde{X}}$ and $Pl_{\widetilde{X}}$ be the mappings from
$\sigma_{\Theta}$ to $[0,1]$ defined as

$Bel_{\widetilde{X}}(B)=\int_{\Omega}N(B\mid\omega)dP(\omega)$ (12)

and

$Pl_{\widetilde{X}}(B)=\int_{\Omega}\Pi(B\mid\omega)dP(\omega).$ (13)

Function $Bel_{\widetilde{X}}$ is a belief function, and $Pl_{\widetilde{X}}$
is the dual plausibility function. As shown in [2, Lemma 6.2], they are
induced by the random set
$(\Omega\times[0,1],\sigma_{\Omega}\otimes\beta_{[0,1]},P\otimes\lambda,\Theta,\sigma_{\Theta},{\overline{X}})$,
where ${\overline{X}}:\Omega\times[0,1]\rightarrow 2^{\Theta}$ is the multi-
valued mapping defined as

${\overline{X}}(\omega,\alpha)={}^{\alpha}{\widetilde{X}}(\omega).$ (14)

As a consequence, $Bel_{\widetilde{X}}(B)$ and $Pl_{\widetilde{X}}(B)$ can
also be written as follows:

$Bel_{\widetilde{X}}(B)=\int_{0}^{1}Bel_{{}^{\alpha}{\widetilde{X}}}(B)d\alpha$
(15a) and
$Pl_{\widetilde{X}}(B)=\int_{0}^{1}Pl_{{}^{\alpha}{\widetilde{X}}}(B)d\alpha.$
(15b)

#### Lower and upper expectations of a random fuzzy number

Let ${\widetilde{X}}$ be a random fuzzy number (i.e., a random fuzzy set with
domain $[0,1]^{\mathbb{R}}$), and let ${\overline{X}}$ be the corresponding
random set defined by (14). We define the lower and upper expectations of
${\widetilde{X}}$ as the lower and upper expectations of ${\overline{X}}$,
i.e., $\mathbb{E}_{*}({\widetilde{X}})=\mathbb{E}_{*}({\overline{X}})$ and
$\mathbb{E}^{*}({\widetilde{X}})=\mathbb{E}^{*}({\overline{X}})$. It follows
from (15) that

$\mathbb{E}_{*}({\widetilde{X}})=\int_{0}^{1}\mathbb{E}_{*}({}^{\alpha}{\widetilde{X}})d\alpha\quad\text{and}\quad\mathbb{E}^{*}({\widetilde{X}})=\int_{0}^{1}\mathbb{E}^{*}({}^{\alpha}{\widetilde{X}})d\alpha.$
(16)

######  Example 6.

Let us consider again the random fuzzy number of Example 5. Its lower and
upper cdf’s are, respectively, the mappings $x\rightarrow
Bel_{\widetilde{X}}((-\infty,x])$ and $x\rightarrow
Pl_{\widetilde{X}}((-\infty,x])$. Let us illustrate the calculation of the
upper cdf first, using two methods.

#### Method 1

From (11),

$\Pi\left((-\infty,x]\mid\omega\right)=\sup_{x^{\prime}\leq
x}{\widetilde{X}}(\omega)(x^{\prime})=\begin{cases}1&\text{if }M(\omega)\leq
x\\\ \frac{x-M(\omega)+a}{a}&\text{if }x<M(\omega)\leq x+a\\\
0&\text{otherwise.}\end{cases}$

Using (13), we get

$\displaystyle Pl_{\widetilde{X}}((-\infty,x])$ $\displaystyle=P(M\leq
x)\times 1+P(x<M\leq x+a)\mathbb{E}\left[\frac{x-M+a}{a}\mid x<M\leq
x+a\right]$
$\displaystyle=\Phi\left(\frac{x-\mu}{\sigma}\right)+\left[\Phi\left(\frac{x+a-\mu}{\sigma}\right)-\Phi\left(\frac{x-\mu}{\sigma}\right)\right]\times$
$\displaystyle\hskip 227.62204pt\left(\frac{x+a}{a}-\mathbb{E}\left[M\mid
x<M\leq x+a\right]\right).$

Now, using a well-known result about the truncated normal distribution,

$\mathbb{E}\left[M\mid x<M\leq
x+a\right]=\mu+\sigma\frac{\phi\left(\frac{x-\mu}{\sigma}\right)-\phi\left(\frac{x+a-\mu}{\sigma}\right)}{\Phi\left(\frac{x+a-\mu}{\sigma}\right)-\Phi\left(\frac{x-\mu}{\sigma}\right)}.$

After rearranging the terms, we finally obtain

$Pl_{\widetilde{X}}((-\infty,x])=\left(\frac{x+a-\mu}{a}\right)\Phi\left(\frac{x+a-\mu}{\sigma}\right)-\left(\frac{x-\mu}{a}\right)\Phi\left(\frac{x-\mu}{\sigma}\right)+\\\
\frac{\sigma}{a}\left[\phi\left(\frac{x+a-\mu}{\sigma}\right)-\phi\left(\frac{x-\mu}{\sigma}\right)\right].$
(17)

#### Method 2

Let us now use (15b). We have

$\displaystyle Pl_{\widetilde{X}}((-\infty,x])$
$\displaystyle=\int_{0}^{1}P(M-a(1-\alpha)\leq x)d\alpha$
$\displaystyle=\int_{0}^{1}\Phi\left(\frac{x+a(1-\alpha)-\mu}{\sigma}\right)d\alpha.$

Using the formula

$\int\Phi(u+vx)dx=\frac{1}{v}\left[(u+vx)\Phi(u+vx)+\phi(u+vx)\right]+C,$

we get the same result as (17). Using any of the two methods demonstrated
above, we obtain the following expression for the lower cdf:

$Bel_{\widetilde{X}}((-\infty,x])=\left(\frac{x-\mu}{a}\right)\Phi\left(\frac{x-\mu}{\sigma}\right)-\left(\frac{x-a-\mu}{a}\right)\Phi\left(\frac{x-a-\mu}{\sigma}\right)+\\\
\frac{\sigma}{a}\left[\phi\left(\frac{x-\mu}{\sigma}\right)-\phi\left(\frac{x-a-\mu}{\sigma}\right)\right].$
(18)

It can easily be checked that, when $a=0$,

$Bel_{\widetilde{X}}((-\infty,x])=Pl_{\widetilde{X}}((-\infty,x])=\Phi\left(\frac{x-\mu}{\sigma}\right).$

Examples of functions $Bel_{\widetilde{X}}((-\infty,x])$ and
$Pl_{\widetilde{X}}((-\infty,x])$ for different values of $a$ are shown in
Figure 1.

Figure 1: Lower and upper cdf’s for the random fuzzy numbers studied in
Examples 5 and 6, with $\mu=0$, $\sigma=1$, and $a=0.5$ (blue curves) or
$a=1.5$ (red curves). The Gaussian cdf corresponding to $a=0$ is shown as a
broken line.

Now, the lower and upper expectations of ${\widetilde{X}}$ can be computed
from (16) as

$\mathbb{E}_{*}({\widetilde{X}})=\int_{0}^{1}\mathbb{E}_{*}({}^{\alpha}{\widetilde{X}})d\alpha=\int_{0}^{1}[\mu-a(1-\alpha)]d\alpha=\mu-\frac{a}{2},$

and

$\mathbb{E}^{*}({\widetilde{X}})=\int_{0}^{1}\mathbb{E}^{*}({}^{\alpha}{\widetilde{X}})d\alpha=\int_{0}^{1}[\mu+a(1-\alpha)]d\alpha=\mu+\frac{a}{2}.$

###  3.2 Generalized product-intersection rule

Dempster’s rule and the possibilistic product intersection rule recalled,
respectively, in Sections 2.1 and 2.2 can be generalized to combine epistemic
random fuzzy sets. Consider two epistemic random fuzzy sets
$(\Omega_{1},\sigma_{1},P_{1},\Theta,\sigma_{\Theta},{\widetilde{X}}_{1})$ and
$(\Omega_{2},\sigma_{2},P_{2},\Theta,\sigma_{\Theta},{\widetilde{X}}_{2})$
encoding independent pieces of evidence. The independence assumption means
here that the relevant probability measure on the joint measurable space
$(\Omega_{1}\times\Omega_{2},\sigma_{1}\otimes\sigma_{2}$) is the product
measure $P_{1}\times P_{2}$.

If interpretations $\omega_{1}\in\Omega_{1}$ and $\omega_{2}\in\Omega_{2}$
both hold, we know that “$\boldsymbol{\theta}\text{ is
}{\widetilde{X}}_{1}(\omega_{1})$” and “$\boldsymbol{\theta}\text{ is
}{\widetilde{X}}_{2}(\omega_{2})$”. It is then natural to combine the fuzzy
sets ${\widetilde{X}}_{1}(\omega_{1})$ and ${\widetilde{X}}_{2}(\omega_{2})$
by an intersection operator. As discussed in Section 2.2, normalized product
intersection is a good candidate as it suitable for combining fuzzy
information from independent sources and it is associative. We will thus
consider the mapping
${\widetilde{X}}_{\varodot}(\omega_{1},\omega_{2})={\widetilde{X}}_{1}(\omega_{1})\varodot{\widetilde{X}}_{2}(\omega_{2})$,
which we will assume to be $\sigma_{1}\otimes\sigma_{2}$-$\sigma_{\Theta}$
strongly measurable.

As in the crisp case recalled in Section 2.1, if
$0pt({\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{2}))=0$, the
two interpretations $\omega_{1}$ and $\omega_{2}$ are inconsistent and they
must be discarded. If
$0pt({\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{2}))=1$, the
two interpretations are fully consistent. If
$0<0pt({\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{2}))<1$,
$\omega_{1}$ and $\omega_{2}$ are _partially consistent_. As proposed in [9],
instead of simply discarding only fully inconsistent pairs
$(\omega_{1},\omega_{2})$, it makes sense to give all pairs
$(\omega_{1},\omega_{2})$ a weight proportional to the _degree of consistency_
between ${\widetilde{X}}_{1}(\omega_{1})$ and
${\widetilde{X}}_{2}(\omega_{2})$. This can be achieved by conditioning
$P_{1}\times P_{2}$ on the _fuzzy set_ ${\widetilde{\Theta}}^{*}$ of
consistent pairs of interpretations, with membership function

${\widetilde{\Theta}}^{*}(\omega_{1},\omega_{2})=0pt\left({\widetilde{X}}_{1}(\omega_{1})\cdot{\widetilde{X}}_{2}(\omega_{2})\right).$

Using Zadeh’s definition of a fuzzy event [37], we get the following
expression for the conditional probability measure
${\widetilde{P}}_{12}=(P_{1}\times P_{2})(\cdot\mid{\widetilde{\Theta}}^{*})$,
for any $B\in\sigma_{1}\otimes\sigma_{2}$:

${\widetilde{P}}_{12}(B)=\frac{(P_{1}\times
P_{2})(B\cap{\widetilde{\Theta}}^{*})}{(P_{1}\times
P_{2})({\widetilde{\Theta}}^{*})}=\frac{\int_{\Omega_{1}}\int_{\Omega_{2}}B(\omega_{1},\omega_{2})0pt\left({\widetilde{X}}_{1}(\omega_{1})\cdot{\widetilde{X}}_{2}(\omega_{2})\right)dP_{2}(\omega_{2})dP_{1}(\omega_{1})}{\int_{\Omega_{1}}\int_{\Omega_{2}}0pt\left({\widetilde{X}}_{1}(\omega_{1})\cdot{\widetilde{X}}_{2}(\omega_{2})\right)dP_{2}(\omega_{2})dP_{1}(\omega_{1})},$

where $B(\cdot,\cdot)$ denotes the indicator function of $B$. This
conditioning operation, called _soft normalization_ was first proposed in [35]
in the finite case and with a different justification.

The combined random fuzzy set

$(\Omega_{1}\times\Omega_{2},\sigma_{1}\otimes\sigma_{2},{\widetilde{P}}_{12},\Theta,\sigma_{\Theta},{\widetilde{X}}_{\varodot})$

is called the _orthogonal sum_ of the two pieces of evidence. This operation
generalizes both Dempster’s rule and the normalized product of possibility
distribution. We will refer to it as the _generalized product-intersection
rule_ , and it will be denoted by the same symbol $\oplus$ as Dempster’s rule.
It is clear that ${\widetilde{X}}\oplus{\overline{X}}_{0}={\widetilde{X}}$ for
any random fuzzy set ${\widetilde{X}}$ and any vacuous random set
${\overline{X}}_{0}$ on the same domain $\Theta$. The degree of conflict
between two random fuzzy sets ${\widetilde{X}}_{1}$ and ${\widetilde{X}}_{2}$
is naturally defined as

$\kappa=1-(P_{1}\times
P_{2})({\widetilde{\Theta}}^{*})=1-\int_{\Omega_{1}}\int_{\Omega_{2}}0pt\left({\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{2})\right)dP_{2}(\omega_{2})dP_{1}(\omega_{1}).$
(19)

The associativity of $\oplus$ was proved in [9] in the finite case; we give a
similar proof in the general case.

######  Proposition 4.

The generalized product-intersection rule $\oplus$ for random fuzzy sets is
commutative and associative.

###### Proof.

See B ∎

The following proposition states that a counterpart of Proposition 2 is still
valid when combining independent random fuzzy sets, i.e., the combined contour
function is still proportional to the product of the contour functions.

######  Proposition 5.

Let ${\widetilde{X}}_{1}$ and ${\widetilde{X}}_{2}$ be two random fuzzy sets
on the same domain $\Theta$, with contour functions $pl_{{\widetilde{X}}_{1}}$
and $pl_{{\widetilde{X}}_{2}}$ and with degree of conflict $\kappa$ defined by
(19). The contour function $pl_{{\widetilde{X}}_{1}\oplus{\widetilde{X}}_{2}}$
of ${\widetilde{X}}_{1}\oplus{\widetilde{X}}_{2}$ verifies

$(pl_{{\widetilde{X}}_{1}\oplus{\widetilde{X}}_{2}})(\theta)=\frac{pl_{{\widetilde{X}}_{1}}(\theta)pl_{{\widetilde{X}}_{2}}(\theta)}{1-\kappa},$
(20)

for all $\theta\in\Theta$.

###### Proof.

We have

$\displaystyle(pl_{{\widetilde{X}}_{1}\oplus{\widetilde{X}}_{2}})(\theta)$
$\displaystyle=\frac{\int_{\Omega_{1}}\int_{\Omega_{2}}0pt\left({\widetilde{X}}_{1}(\omega_{1})\cdot{\widetilde{X}}_{2}(\omega_{2})\right){\widetilde{X}}_{\varodot}(\omega_{1},\omega_{2})(\theta)dP_{2}(\omega_{2})dP_{1}(\omega_{1})}{1-\kappa}$
$\displaystyle=\frac{\int_{\Omega_{1}}\int_{\Omega_{2}}0pt\left({\widetilde{X}}_{1}(\omega_{1})\cdot{\widetilde{X}}_{2}(\omega_{2})\right)\frac{{\widetilde{X}}_{1}(\omega_{1})(\theta){\widetilde{X}}_{2}(\omega_{2})(\theta)}{0pt\left({\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{2})\right)}dP_{2}(\omega_{2})dP_{1}(\omega_{1})}{1-\kappa}$
$\displaystyle=\frac{\left(\int_{\Omega_{1}}{\widetilde{X}}_{1}(\omega_{1})(\theta)dP_{1}(\omega_{1})\right)\left(\int_{\Omega_{2}}{\widetilde{X}}_{2}(\omega_{2})(\theta)dP_{2}(\omega_{2})\right)}{1-\kappa}$
$\displaystyle=\frac{pl_{{\widetilde{X}}_{1}}(\theta)pl_{{\widetilde{X}}_{2}}(\theta)}{1-\kappa}.$

∎

As remarked in Section 2.2, a belief function induced by a random fuzzy set is
also induced by a random (crisp) set. However, combining random fuzzy sets or
random crisp sets does not result in the same belief function in general. In
particular, it is well-known that Dempster’s rule does not preserve
consonance. To combine two belief functions, we must, therefore, examine the
evidence on which they are based, not only to determine whether the bodies of
evidence are independent or not, but also to determine whether the evidence is
fuzzy or crisp. This point is illustrated by the following example.

######  Example 7.

Consider the following two mappings from $\mathbb{R}$ to $[0,1]$ represented
in Figure 2a:

$\pi_{1}(x)=\textsf{GFN}(0,0.3),\quad\pi_{2}(x)=\textsf{GFN}\left(1,0.5\right).$

If these two mappings are possibility distributions encoding fully reliable
but fuzzy evidence, they correspond to “constant random fuzzy sets”, i.e.,
mappings ${\widetilde{X}}_{1}(\omega)=\pi_{1}$ and
${\widetilde{X}}_{2}(\omega)=\pi_{2}$ with $P(\\{\omega\\})=1$. The combined
random fuzzy set ${\widetilde{X}}_{1}\oplus{\widetilde{X}}_{2}$ is then
defined by
$({\widetilde{X}}_{1}\oplus{\widetilde{X}}_{2})(\omega)=\pi_{1}\varodot\pi_{2}$.
From Proposition 3, the normalized product of two GFN’s is a GFN. Here, we get
the combined possibility distribution plotted as a red broken curve in Figure
2a:

$(\pi_{1}\varodot\pi_{2})(x)=\textsf{GFN}(0.625,0.8).$

(a) (b)

Figure 2: (a): Two Gaussian possibility distributions (black solid curves)
with their normalized product intersection (red broken curve) and the contour
function of the combined random set (blue solid curve). (b): Lower and upper
cdf’s of the combined possibility distribution (red broken curves) and of the
combined random set (blue solid curves).

The corresponding lower and upper cumulative distribution functions (cdf’s)
are, respectively

$Bel_{{\widetilde{X}}_{1}\oplus{\widetilde{X}}_{2}}((-\infty,x])=\begin{cases}0&\text{if
}x\leq 0.625\\\ 1-\exp\left(-0.4(x-0.625)^{2}\right)&\text{if
}x>0.625\end{cases}$

and

$Pl_{{\widetilde{X}}_{1}\oplus{\widetilde{X}}_{2}}((-\infty,x])=\begin{cases}\exp\left(-0.4(x-0.625)^{2}\right)&\text{if
}x\leq 0.625\\\ 1&\text{if }x>0.625.\end{cases}$

These two functions are plotted as red broken curves in Figure 2b.
Alternatively, as explained in Section 2.1, we may see $\pi_{1}$ and $\pi_{2}$
as encoding crisp but partially reliable evidence, in which case they define
two independent consonant random intervals
${\overline{X}}_{1}(\alpha_{1})={}^{\alpha_{1}}\pi_{1}$ and
${\overline{X}}(\alpha_{2})={}^{\alpha_{2}}\pi_{2}$ , where
$(\alpha_{1},\alpha_{2})$ has a uniform distribution on $[0,1]^{2}$. These two
random intervals can be combined numerically using Monte-Carlo simulation, as
explained in [20]. The contour function and the lower and upper cdf’s are
plotted as solid blue lines in Figures 2a and 2b, respectively. We notice that
the contour functions are proportional, as a consequence of Proposition 2.

###  3.3 Marginalization and vacuous extension

Let us now consider again the case where we have two variables
$\boldsymbol{\theta}_{1}$ and $\boldsymbol{\theta}_{2}$ with respective
domains $\Theta_{1}$ and $\Theta_{2}$. Let ${\widetilde{X}}_{12}$ be a random
fuzzy set from a probability space $(\Omega,\sigma_{\Omega},P)$ to the
measurable space $(\Theta_{12},\sigma_{\Theta_{12}})$ with
$\Theta_{12}=\Theta_{1}\times\Theta_{2}$ and
$\sigma_{\Theta_{12}}=\sigma_{\Theta_{1}}\otimes\sigma_{\Theta_{2}}$, where
$\sigma_{\Theta_{1}}$ and $\sigma_{\Theta_{2}}$ are $\sigma$-algebras on
$\Theta_{1}$ and $\Theta_{2}$, respectively. Let ${\widetilde{X}}_{1}$ be the
mapping from $\Omega$ to $[0,1]^{\Theta_{1}}$ defined by

${\widetilde{X}}_{1}(\omega)={\widetilde{X}}_{12}(\omega)\downarrow\Theta_{1},$

where, as before, $\downarrow$ denotes fuzzy set projection. If, for all
$\alpha\in[0,1]$, the mapping ${}^{\alpha}{\widetilde{X}}_{1}$ is
$\sigma_{\Omega}-\sigma_{\Theta_{1}}$ strongly measurable, then the random
fuzzy set ${\widetilde{X}}_{1}$ is called the _marginal_ of
${\widetilde{X}}_{12}$ on $\Theta_{1}$.

Conversely, given a random fuzzy set ${\widetilde{X}}_{1}$ from
$(\Omega,\sigma_{\Omega},P)$ to $(\Theta_{1},\sigma_{\Theta_{1}})$, let
${\widetilde{X}}_{1\uparrow(1,2)}$ be the mapping from $\Omega$ to
$[0,1]^{\Theta_{12}}$ that maps each $\omega\in\Omega$ to the cylindrical
extension of ${\widetilde{X}}_{1}(\omega)$ in $\Theta_{12}$

${\widetilde{X}}_{1\uparrow(1,2)}(\omega)={\widetilde{X}}_{1}(\omega)\times\Theta_{2},$

i.e., for all $(\theta_{1},\theta_{2})\in\Theta_{12}$,

${\widetilde{X}}_{1\uparrow(1,2)}(\omega)(\theta_{1},\theta_{2})={\widetilde{X}}_{1}(\omega)(\theta_{1}).$

If the mapping ${\widetilde{X}}_{1\uparrow(1,2)}$ is
$\sigma_{\Omega}-\sigma_{\Theta_{12}}$ strongly measurable, then the random
fuzzy set ${\widetilde{X}}_{1\uparrow(1,2)}$ is called the _vacuous extension_
of ${\widetilde{X}}_{1}$ in $\Theta_{12}$.

We say that a joint random fuzzy set is _non-interactive_ if it is equal to
the orthogonal sum of the vacuous extensions of its projections:

${\widetilde{X}}_{12}={\widetilde{X}}_{1\uparrow(1,2)}\oplus{\widetilde{X}}_{2\uparrow(1,2)}\quad\text{denoted
as}\quad{\widetilde{X}}_{1}\oplus{\widetilde{X}}_{2}.$

A particular kind of non-interactive random fuzzy sets will be studied in
Section 5.3.

###  3.4 Application to statistical inference

Epistemic random fuzzy sets naturally arise in the context of statistical
inference. As proposed by Shafer [29] and formally justified in [7][8], the
information conveyed by the likelihood function in statistical inference
problems can be represented by a consonant belief function, whose contour
function is equal to the relative likelihood function. For a statistical model
$f({\boldsymbol{x}},\theta)$, where ${\boldsymbol{x}}\in{\cal X}$ is the
observation and $\theta\in\Theta$ is the unknown parameter, the likelihood-
based belief function $Bel(\cdot,{\boldsymbol{x}})$ on $\Theta$ after
observing ${\boldsymbol{x}}$ is, thus, consonant and defined by the contour
function

$pl(\theta;{\boldsymbol{x}})=\frac{L(\theta;{\boldsymbol{x}})}{\sup_{\theta^{\prime}\in\Theta}L(\theta^{\prime};{\boldsymbol{x}})},$
(21)

where $L(\cdot,{\boldsymbol{x}}):\theta\rightarrow f({\boldsymbol{x}};\theta)$
is the likelihood function, and it is assumed that the denominator in (21) is
finite. The corresponding plausibility function is, thus, defined by

$Pl(A;{\boldsymbol{x}})=\sup_{\theta\in A}pl(\theta;{\boldsymbol{x}})$

for any $A\subset\Theta$, i.e., it is a possibility measure. However, as
noticed by Shafer in [29] and [31], and also discussed in [7], this
construction is not compatible with Dempster’s rule: if we consider two
independent observations ${\boldsymbol{x}}$ and ${\boldsymbol{x}}^{\prime}$,
the belief function $Bel(\cdot;{\boldsymbol{x}},{\boldsymbol{x}}^{\prime})$ is
not equal to the orthogonal sum $Bel(\cdot;{\boldsymbol{x}})\oplus
Bel(\cdot;{\boldsymbol{x}}^{\prime})$, which is not consonant. As argued in
[9], this problem disappears if we do not consider the likelihood-based belief
function to be induced by a consonant random crisp set, but by a constant
random fuzzy set ${\widetilde{\theta}}_{\boldsymbol{x}}$ with membership
function
${\widetilde{\theta}}_{\boldsymbol{x}}(\theta)=pl(\theta;{\boldsymbol{x}})$.
We can interpret ${\widetilde{\theta}}_{\boldsymbol{x}}$ as the fuzzy set of
likely values of $\theta$ after observing ${\boldsymbol{x}}$. Combining the
contour functions (21) by the normalized product intersection rule then yields
the correct result, i.e., the constant random fuzzy set
${\widetilde{\theta}}_{{\boldsymbol{x}},{\boldsymbol{x}}^{\prime}}$ with
membership function
${\widetilde{\theta}}_{{\boldsymbol{x}},{\boldsymbol{x}}^{\prime}}(\theta)={\widetilde{\theta}}_{\boldsymbol{x}}(\theta)\varodot{\widetilde{\theta}}_{{\boldsymbol{x}}^{\prime}}(\theta)$.

Now, consider a prediction problem, where we want to predict the value of a
random variable $Y$ whose distribution also depends on $\theta$. We can always
write $Y$ in the form $Y=\varphi(\theta,U)$, where $U$ is a pivotal random
variable with known distribution [19, 20]. After observing the data
${\boldsymbol{x}}$, our knowledge of $\theta$ is represented by the fuzzy set
${\widetilde{\theta}}_{\boldsymbol{x}}$. Conditionally on $U=u$, our knowledge
of $Y$ is, thus, represented by the fuzzy set
${\widetilde{Y}}(u)=\varphi({\widetilde{\theta}}_{\boldsymbol{x}},u)$, with
membership function

${\widetilde{Y}}(u)(y)=\sup_{\theta:\varphi(\theta,u)=y}{\widetilde{\theta}}_{\boldsymbol{x}}(\theta).$

The mapping ${\widetilde{Y}}:u\rightarrow{\widetilde{Y}}(u)$ is, then, a
random fuzzy set representing statistical evidence about $Y$.

######  Example 8.

Let ${\boldsymbol{X}}=(X_{1},\ldots,X_{n})$ be an independent and identically
distributed (iid) Gaussian sample with parent distribution $N(\theta,1)$, and
let $Y\sim N(\theta,1)$. After observing a realization ${\boldsymbol{x}}$ of
${\boldsymbol{X}}$, the likelihood function is

$L(\theta;{\boldsymbol{x}})=(2\pi)^{-n/2}\exp\left(-\frac{1}{2}\sum_{i=1}^{n}(x_{i}-\theta)^{2}\right).$

Denoting by $\widehat{\theta}$ the sample mean, the fuzzy set
${\widetilde{\theta}}_{\boldsymbol{x}}$ of likely values of $\theta$ after
observing ${\boldsymbol{x}}$ is the relative likelihood

${\widetilde{\theta}}_{\boldsymbol{x}}(\theta)=\frac{L(\theta;{\boldsymbol{x}})}{L(\widehat{\theta};{\boldsymbol{x}})}=\exp\left(-\frac{n}{2}(\theta-\widehat{\theta})^{2}\right).$

It is the Gaussian fuzzy number $\textsf{GFN}(\widehat{\theta},n)$ with mode
$\widehat{\theta}$ and precision $n$. Now, $Y$ can be written as $Y=\theta+U$,
with $U\sim N(0,1)$. Consequently, the conditional possibility distribution on
$Y$ given $U=u$ is the Gaussian fuzzy number
${\widetilde{\theta}}_{\boldsymbol{x}}+u=\textsf{GFN}(\widehat{\theta}+u,n)$,
and our knowledge of $Y$ is described by the random fuzzy set
$U\rightarrow\textsf{GFN}(\widehat{\theta}+U,n)$, with $U\sim N(0,1)$. This is
a Gaussian fuzzy number with fixed precision $h=n$ and normal random mode
$M=\widehat{\theta}+U\sim N(\widehat{\theta},1)$. This important class of
random fuzzy sets will be studied in the next section.

##  4 Gaussian random fuzzy numbers

In this section, we introduce Gaussian random fuzzy numbers (GRFN’s) as a
practical model for representing uncertainty on a real variable. As we will
see, this model encompasses Gaussian random variables and Gaussian fuzzy
numbers as special cases. A GRFN can be seen, equivalently, as a Gaussian
random variable with fuzzy mean, or as a Gaussian fuzzy number with random
mode. The definition and main properties will first be presented in Section
4.1. The expression of the orthogonal sum of two GRFN’s will then be derived
in Section 4.2. Finally, arithmetic operations on GRFN’s will be addressed in
Section 4.3.

###  4.1 Definition and main properties

######  Definition 1.

Let $(\Omega,\sigma_{\Omega},P)$ be a probability space and let
$M:\Omega\rightarrow\mathbb{R}$ be a Gaussian random variable with mean $\mu$
and variance $\sigma^{2}$. The random fuzzy set
${\widetilde{X}}:\Omega\rightarrow[0,1]^{\mathbb{R}}$ defined as

${\widetilde{X}}(\omega)=\textsf{GFN}(M(\omega),h)$

is called a Gaussian random fuzzy number (GRFN) with mean $\mu$, variance
$\sigma^{2}$ and precision $h$, which we write
${\widetilde{X}}\sim{\widetilde{N}}(\mu,\sigma^{2},h)$.

In the definition of a GRFN, $\mu$ is a location parameter, while parameters
$h$ and $\sigma^{2}$ correspond, respectively, to possibilistic and
probabilistic uncertainty. If $h=0$, imprecision is maximal whatever the
values of $\mu$ and $\sigma^{2}$: the GRFN ${\widetilde{X}}$ then induces the
vacuous belief function on $\mathbb{R}$, in which case
$Bel_{\widetilde{X}}(A)=0$ for all $A\subset\mathbb{R}$, and
$Pl_{\widetilde{X}}(A)=1$ for all $A\subseteq\mathbb{R}$ such that
$A\neq\emptyset$; such a GRFN will be said to be _vacuous_ and will be denoted
by ${\widetilde{X}}\sim{\widetilde{N}}(0,1,0)$. If $h=+\infty$, each fuzzy
number $\textsf{GFN}(M(\omega),h)$ is reduced to a point: the GRFN
${\widetilde{X}}$ is then equivalent to a Gaussian random variable with mean
$\mu$ and variance $\sigma^{2}$, which we can write:
${\widetilde{N}}(\mu,\sigma^{2},+\infty)=N(\mu,\sigma^{2})$. Another special
case of interest is that where $\sigma^{2}=0$, in which case $M$ is a constant
random variable taking value $\mu$, and ${\widetilde{X}}$ is a possibilistic
variable with possibility distribution $\textsf{GFN}(\mu,h)$.

The following proposition gives the expression of the contour functions
$pl_{\widetilde{X}}(x)$ associated to ${\widetilde{X}}$.

######  Proposition 6.

The contour function of GRFN
${\widetilde{X}}\sim{\widetilde{N}}(\mu,\sigma^{2},h)$ is

$pl_{\widetilde{X}}(x)=\frac{1}{\sqrt{1+h\sigma^{2}}}\exp\left(-\frac{h(x-\mu)^{2}}{2(1+h\sigma^{2})}\right).$
(22)

###### Proof.

See C. ∎

A shown by Proposition 6, the contour function $pl_{\widetilde{X}}$ is
constant in two cases: if $h=0$, ${\widetilde{X}}$ is vacuous, and
$pl_{\widetilde{X}}(x)=1$ for all $x\in\mathbb{R}$; if $h=+\infty$,
${\widetilde{X}}$ is a random variable, and $pl_{\widetilde{X}}(x)=0$ for all
$x\in\mathbb{R}$. We also note that, if $\sigma^{2}=0$, $pl_{\widetilde{X}}$
is equal to the possibility distribution $\textsf{GFN}(\mu,h)$. When
$\sigma^{2}\rightarrow+\infty$ and $h>0$, $pl_{\widetilde{X}}(x)\rightarrow 0$
for all $x$. The next proposition gives the expressions of the belief and
plausibility of any real interval.

######  Proposition 7.

For any real interval $[x,y]$, the degrees of belief and plausibility of
$[x,y]$ induced by the GRFN
${\widetilde{X}}\sim{\widetilde{N}}(\mu,\sigma^{2},h)$ are, respectively,

$Bel_{\widetilde{X}}([x,y])=\Phi\left(\frac{y-\mu}{\sigma}\right)-\Phi\left(\frac{x-\mu}{\sigma}\right)-\\\
pl_{\widetilde{X}}(x)\left[\Phi\left(\frac{(x+y)/2-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)-\Phi\left(\frac{x-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)\right]-\\\
pl_{\widetilde{X}}(y)\left[\Phi\left(\frac{y-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)-\Phi\left(\frac{(x+y)/2-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)\right],$
(23)

and

$Pl_{\widetilde{X}}([x,y])=\Phi\left(\frac{y-\mu}{\sigma}\right)-\Phi\left(\frac{x-\mu}{\sigma}\right)+pl_{\widetilde{X}}(x)\Phi\left(\frac{x-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)+\\\
pl_{\widetilde{X}}(y)\left[1-\Phi\left(\frac{y-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)\right].$
(24)

###### Proof.

See D. ∎

######  Corollary 1.

The lower and upper cdf’s of the GRFN
${\widetilde{X}}\sim{\widetilde{N}}(\mu,\sigma^{2},h)$ are, respectively

$Bel_{\widetilde{X}}((-\infty,y])=\Phi\left(\frac{y-\mu}{\sigma}\right)-pl_{\widetilde{X}}(y)\Phi\left(\frac{y-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)$
(25)

and

$Pl_{\widetilde{X}}((-\infty,y])=\Phi\left(\frac{y-\mu}{\sigma}\right)+pl_{\widetilde{X}}(y)\left[1-\Phi\left(\frac{y-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)\right].$
(26)

###### Proof.

Immediate from Proposition 7 by letting $x$ tend to $-\infty$ in (23) and (24)
∎

We can easily check from (23) and (24) that $Bel_{\widetilde{X}}([x,y])$ and
$Pl_{\widetilde{X}}([x,y])$ both tend to
$\Phi\left(\frac{y-\mu}{\sigma}\right)-\Phi\left(\frac{x-\mu}{\sigma}\right)$
when $h\rightarrow\infty$, which is consistent with the fact that a GRFN with
infinite precision is equivalent to a Gaussian random variable. Finally, the
following proposition gives the expressions of the lower and upper
expectations of a GRFN.

######  Proposition 8.

Let ${\widetilde{X}}\sim{\widetilde{N}}(\mu,\sigma^{2},h)$ be a GRFN with
$h>0$. Its lower and upper expectations are, respectively,

$\mathbb{E}_{*}({\widetilde{X}})=\mu-\sqrt{\frac{\pi}{2h}}\quad\text{and}\quad\mathbb{E}^{*}({\widetilde{X}})=\mu+\sqrt{\frac{\pi}{2h}}.$
(27)

###### Proof.

See E. ∎

As expected, we can see from (27) that the lower and upper expectations boil
down to the usual expectation $\mu$ when $h=+\infty$.

###  4.2 Orthogonal sum of Gaussian fuzzy random numbers

In this section, we derive the expression of the orthogonal sum
${\widetilde{X}}_{1}\oplus{\widetilde{X}}_{2}$ of two GRFN’s
${\widetilde{X}}_{1}$ and ${\widetilde{X}}_{2}$. We start with the following
lemma.

######  Lemma 1.

Let $M_{1}\sim N(\mu_{1},\sigma_{1}^{2})$ and $M_{2}\sim
N(\mu_{2},\sigma_{2}^{2})$ be two independent Gaussian random variables, and
let ${\widetilde{F}}$ be the fuzzy subset of $\mathbb{R}^{2}$ with membership
function

${\widetilde{F}}(m_{1},m_{2})=0pt\left(\textsf{GFN}(m_{1},h_{1})\cdot\textsf{GFN}(m_{2},h_{2})\right)=\exp\left(-\frac{h_{1}h_{2}(m_{1}-m_{2})^{2}}{2(h_{1}+h_{2})}\right).$

The conditional probability distribution of $(M_{1},M_{2})$ given
${\widetilde{F}}$ is two-dimensional Gaussian with mean vector
${\widetilde{\boldsymbol{\mu}}}=({\widetilde{\mu}}_{1},{\widetilde{\mu}}_{2})^{T}$
and covariance matrix

${\widetilde{\boldsymbol{\Sigma}}}=\begin{pmatrix}{\widetilde{\sigma}}_{1}^{2}&\rho{\widetilde{\sigma}}_{1}{\widetilde{\sigma}}_{2}\\\
\rho{\widetilde{\sigma}}_{1}{\widetilde{\sigma}}_{2}&{\widetilde{\sigma}}_{2}^{2}\end{pmatrix},$

with

$\displaystyle{\widetilde{\mu}}_{1}$
$\displaystyle=\frac{\mu_{1}(1+{\overline{h}}\sigma_{2}^{2})+\mu_{2}{\overline{h}}\sigma_{1}^{2}}{1+{\overline{h}}(\sigma_{1}^{2}+\sigma_{2}^{2})}$
(28a) $\displaystyle{\widetilde{\mu}}_{2}$
$\displaystyle=\frac{\mu_{2}(1+{\overline{h}}\sigma_{1}^{2})+\mu_{1}{\overline{h}}\sigma_{2}^{2}}{1+{\overline{h}}(\sigma_{1}^{2}+\sigma_{2}^{2})}$
(28b) $\displaystyle{\widetilde{\sigma}}_{1}^{2}$
$\displaystyle=\frac{\sigma_{1}^{2}(1+{\overline{h}}\sigma_{2}^{2})}{1+{\overline{h}}(\sigma_{1}^{2}+\sigma_{2}^{2})}$
(28c) $\displaystyle{\widetilde{\sigma}}_{2}^{2}$
$\displaystyle=\frac{\sigma_{2}^{2}(1+{\overline{h}}\sigma_{1}^{2})}{1+{\overline{h}}(\sigma_{1}^{2}+\sigma_{2}^{2})}$
(28d) $\displaystyle\rho$
$\displaystyle=\frac{{\overline{h}}\sigma_{1}\sigma_{2}}{\sqrt{(1+{\overline{h}}\sigma_{1}^{2})(1+{\overline{h}}\sigma_{2}^{2})}},$
(28e) where ${\overline{h}}=\frac{h_{1}h_{2}}{h_{1}+h_{2}}.$ (28f)

Furthermore, the degree of conflict between two independent GRFN’s
${\widetilde{X}}_{1}\sim{\widetilde{N}}(\mu_{1},\sigma_{1}^{2},h_{1})$ and
${\widetilde{X}}_{2}\sim{\widetilde{N}}(\mu_{2},\sigma_{2}^{2},h_{2})$ is

$\kappa=1-\iint f(m_{1},m_{2}){\widetilde{F}}(m_{1},m_{2})dm_{1}dm_{2}=\\\
1-\frac{{\widetilde{\sigma}}_{1}{\widetilde{\sigma}}_{2}}{\sigma_{1}\sigma_{2}}\sqrt{1-\rho^{2}}\exp\left\\{-\frac{1}{2}\left[\frac{\mu_{1}^{2}}{\sigma_{1}^{2}}+\frac{\mu_{2}^{2}}{\sigma_{2}^{2}}\right]+\frac{1}{2(1-\rho^{2})}\left[\frac{{\widetilde{\mu}}_{1}^{2}}{{\widetilde{\sigma}}_{1}^{2}}+\frac{{\widetilde{\mu}}_{2}^{2}}{{\widetilde{\sigma}}_{2}^{2}}-2\rho\frac{{\widetilde{\mu}}_{1}{\widetilde{\mu}}_{2}}{{\widetilde{\sigma}}_{1}{\widetilde{\sigma}}_{2}}\right]\right\\},$

where $f(m_{1},m_{2})$ is the pdf of random vector $(M_{1},M_{2})$.

###### Proof.

See F. ∎

######  Proposition 9.

Let ${\widetilde{X}}_{1}\sim{\widetilde{N}}(\mu_{1},\sigma_{1}^{2},h_{1})$ and
${\widetilde{X}}_{2}\sim{\widetilde{N}}(\mu_{2},\sigma_{2}^{2},h_{2})$ be two
independent GRFN’s, and assume that $h_{1}>0$ or $h_{2}>0$. We have

${\widetilde{X}}_{1}\oplus{\widetilde{X}}_{2}\sim{\widetilde{N}}({\widetilde{\mu}}_{12},{\widetilde{\sigma}}_{12}^{2},h_{12}),$

with

$h_{12}=h_{1}+h_{2},$ (29)
${\widetilde{\mu}}_{12}=\frac{h_{1}{\widetilde{\mu}}_{1}+h_{2}{\widetilde{\mu}}_{2}}{h_{1}+h_{2}},$
(30)

and

${\widetilde{\sigma}}_{12}^{2}=\frac{h^{2}_{1}{\widetilde{\sigma}}^{2}_{1}+h^{2}_{2}{\widetilde{\sigma}}^{2}_{2}+2\rho
h_{1}h_{2}{\widetilde{\sigma}}_{1}{\widetilde{\sigma}}_{2}}{(h_{1}+h_{2})^{2}},$
(31)

where ${\widetilde{\mu}}_{1}$, ${\widetilde{\mu}}_{2}$,
${\widetilde{\sigma}}_{1}$, ${\widetilde{\sigma}}_{2}$ and $\rho$ are given by
(28) in Lemma 1.

###### Proof.

Let $M_{1}$ and $M_{2}$ be the Gaussian random variables from
$(\Omega_{1},\sigma_{1},P_{1})$ and $(\Omega_{2},\sigma_{2},P_{2})$ to
$(\mathbb{R},\beta_{\mathbb{R}})$ corresponding, respectively, to GRFN’s
${\widetilde{X}}_{1}\sim{\widetilde{N}}(\mu_{1},\sigma_{1}^{2},h_{1})$ and
${\widetilde{X}}_{2}\sim{\widetilde{N}}(\mu_{2},\sigma_{2}^{2},h_{2})$. The
orthogonal sum of ${\widetilde{X}}_{1}$ and ${\widetilde{X}}_{2}$ is the
random fuzzy set
$(\Omega_{1}\times\Omega_{2},\sigma_{1}\otimes\sigma_{2},{\widetilde{P}}_{12},\mathbb{R},\beta_{\mathbb{R}},{\widetilde{X}}_{\varodot})$,
where ${\widetilde{X}}_{\varodot}$ is the mapping

${\widetilde{X}}_{\varodot}:(\omega_{1},\omega_{2})\rightarrow\textsf{GFN}(M_{12}(\omega_{1},\omega_{2}),h_{1}+h_{2}),$

with

$M_{12}(\omega_{1},\omega_{2})=\frac{h_{1}M_{1}(\omega_{1})+h_{2}M_{2}(\omega_{2})}{h_{1}+h_{2}},$

and ${\widetilde{P}}_{12}$ is the probability measure on
$\Omega_{1}\times\Omega_{2}$ obtained by conditioning $P_{1}\times P_{2}$ on
the fuzzy set
${\widetilde{\Theta}}^{*}(\omega_{1},\omega_{2})=0pt\left(\textsf{GFN}(M_{1}(\omega_{1}),h_{1}),\textsf{GFN}(M_{2}(\omega_{2}),h_{2})\right)$.
From Lemma 1, the pushforward measure of ${\widetilde{P}}_{12}$ by the random
vector $(M_{1},M_{2})$ is the two-dimensional Gaussian distribution with
parameters
$({\widetilde{\mu}}_{1},{\widetilde{\mu}}_{2},{\widetilde{\sigma}}_{1},{\widetilde{\sigma}}_{2},\rho)$.
Consequently, $M_{12}$ is a Gaussian random variable with mean

$\mathbb{E}(M_{12})=\frac{h_{1}\mathbb{E}(M_{1})+h_{2}\mathbb{E}(M_{2})}{h_{1}+h_{2}}=\frac{h_{1}{\widetilde{\mu}}_{1}+h_{2}{\widetilde{\mu}}_{2}}{h_{1}+h_{2}},$

and variance

$\displaystyle\text{Var}(M_{12})$
$\displaystyle=\frac{h_{1}^{2}\text{Var}(M_{1})+h_{2}^{2}\text{Var}(M_{2})+2h_{1}h_{2}\text{Cov}(M_{1},M_{2})}{(h_{1}+h_{2})^{2}}$
$\displaystyle=\frac{h^{2}_{1}{\widetilde{\sigma}}^{2}_{1}+h^{2}_{2}{\widetilde{\sigma}}^{2}_{2}+2\rho
h_{1}h_{2}{\widetilde{\sigma}}_{1}{\widetilde{\sigma}}_{2}}{(h_{1}+h_{2})^{2}}.$

∎

Let us now consider some special cases in which one of two GRFN’s is a
Gaussian random variable. The next proposition states that the orthogonal sum
of a Gaussian random variable and an arbitrary GRFN with finite precision is a
Gaussian random variable.

######  Proposition 10.

Let $X_{1}\sim N(\mu_{1},\sigma_{1}^{2})$ be a Gaussian random variable and
${\widetilde{X}}_{2}\sim{\widetilde{N}}(\mu_{2},\sigma_{2}^{2},h_{2})$ a GRFN
with finite precision $h_{2}<+\infty$. Their orthogonal sum is a Gaussian
random variable $X_{1}\oplus{\widetilde{X}}_{2}\sim
N({\widetilde{\mu}}_{12},{\widetilde{\sigma}}_{12}^{2})$ with

${\widetilde{\mu}}_{12}=\frac{\mu_{1}(1+h_{2}\sigma_{2}^{2})+\mu_{2}h_{2}\sigma_{1}^{2}}{1+h_{2}(\sigma_{1}^{2}+\sigma_{2}^{2})},$
(32)
${\widetilde{\sigma}}_{12}^{2}=\frac{\sigma_{1}^{2}(1+h_{2}\sigma_{2}^{2})}{1+h_{2}(\sigma_{1}^{2}+\sigma_{2}^{2})},$
(33)

and the probability density of $X_{1}\oplus{\widetilde{X}}_{2}$ is
proportional to the product of the pdf of $X_{1}$ and the contour function of
${\widetilde{X}}_{2}$.

###### Proof.

See G. ∎

The following corollary addresses the special case where ${\widetilde{X}}_{2}$
is a possibilistic GRFN.

######  Corollary 2.

Let $X_{1}\sim N(\mu_{1},\sigma_{1}^{2})$ be a Gaussian random variable and
${\widetilde{X}}_{2}\sim{\widetilde{N}}(\mu_{2},0,h_{2})$ a possibilistic
GRFN. Their orthogonal sum $X_{1}\oplus{\widetilde{X}}_{2}$ is a Gaussian
random variable and its distribution is the conditional distribution of
$X_{1}$ given the fuzzy event $\textsf{GFN}(\mu_{2},h_{2})$.

###### Proof.

From Proposition 10,
$X_{1}\oplus{\widetilde{X}}_{2}\sim{\widetilde{N}}({\widetilde{\mu}}_{12},{\widetilde{\sigma}}_{12}^{2})$
with

${\widetilde{\mu}}_{12}=\frac{\mu_{1}+\mu_{2}h_{2}\sigma_{1}^{2}}{1+h_{2}\sigma_{1}^{2}}\quad\text{and}\quad{\widetilde{\sigma}}_{12}^{2}=\frac{\sigma_{1}^{2}}{1+h_{2}\sigma_{1}^{2}}.$

Now, we know from Proposition 10 that the density of
$X_{1}\oplus{\widetilde{X}}_{2}$ is proportional to the product of the density
of $X_{1}$ and the contour function of ${\widetilde{X}}_{2}$, which is
$\varphi(x;\mu_{2},h_{2})$. Consequently, we have

$f_{X_{1}\oplus{\widetilde{X}}_{2}}(x)=\frac{\frac{1}{\sigma_{1}^{2}}\exp\left(-\frac{1}{2}\frac{(x-\mu_{1})^{2}}{\sigma_{1}^{2}}\right)\exp\left(-\frac{h_{2}(x-\mu_{2})^{2}}{2(1+h_{2}\sigma_{2}^{2})}\right)}{\int\frac{1}{\sigma_{1}^{2}}\exp\left(-\frac{1}{2}\frac{(x-\mu_{1})^{2}}{\sigma_{1}^{2}}\right)\exp\left(-\frac{h_{2}(x-\mu_{2})^{2}}{2(1+h_{2}\sigma_{2}^{2})}\right)dx},$

which is the conditional density $f_{X_{1}}(x|\textsf{GFN}(\mu_{2},h_{2}))$. ∎

Finally, another special case of interest is when both GRFN’s are Gaussian
random variables. This case is addressed by the following corollary.

######  Corollary 3.

Let $X_{1}\sim N(\mu_{1},\sigma_{1}^{2})$ and $X_{2}\sim
N(\mu_{2},\sigma_{2}^{2})$ be two Gaussian random variables. We have
$X_{1}\oplus X_{2}\sim N({\widetilde{\mu}}_{12},\sigma_{12}^{2})$ with

${\widetilde{\mu}}_{12}=\frac{\mu_{1}\sigma_{2}^{2}+\mu_{2}\sigma_{1}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}\quad\text{and}\quad{\widetilde{\sigma}}_{12}^{2}=\frac{\sigma_{1}^{2}\sigma_{2}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}.$

###### Proof.

Immediate from Proposition 10 by letting $h_{2}$ tend to $+\infty$ in (32) and
(33). ∎

###  4.3 Arithmetic operations on GRFN’s

Arithmetic operations can be extended to fuzzy numbers using Zadeh’s extension
principles [14, 12]. More precisely, let ${\widetilde{A}}$ and
${\widetilde{B}}$ be two fuzzy numbers, and let $*$ be a binary operation on
reals. Then the fuzzy number ${\widetilde{C}}={\widetilde{A}}*{\widetilde{B}}$
is defined as

${\widetilde{C}}(c)=\sup_{c=a*b}\min({\widetilde{A}}(a),{\widetilde{B}}(b)).$

The membership function ${\widetilde{C}}$ is equal to the possibility
distribution on $c=a*b$, if $a$ and $b$ are constrained, respectively, by
possibility distributions ${\widetilde{A}}$ and ${\widetilde{B}}$. Unary or
$n$-ary operations can be extended from real to fuzzy numbers in the same way.
For a certain class of fuzzy number called LR-fuzzy numbers [14, page 54],
closed-form expressions exist for the addition, subtraction and scalar
multiplication of fuzzy numbers. In particular, Gaussian fuzzy numbers with
positive precision are LR fuzzy numbers and they verify the following
equalities [25]:

$\displaystyle\textsf{GFN}(m_{1},h_{1})+\textsf{GFN}(m_{2},h_{2})$
$\displaystyle=\textsf{GFN}(m_{1}+m_{2},(h_{1}^{-1/2}+h_{2}^{-1/2})^{-2})$
$\displaystyle\textsf{GFN}(m_{1},h_{1})-\textsf{GFN}(m_{2},h_{2})$
$\displaystyle=\textsf{GFN}(m_{1}-m_{2},(h_{1}^{-1/2}+h_{2}^{-1/2})^{-2})$
$\displaystyle\lambda\cdot\textsf{GFN}(m,h)$
$\displaystyle=\textsf{GFN}(\lambda
m,h/\lambda^{2}),\quad\forall\lambda\in\mathbb{R}.$

As addition of fuzzy numbers is associative, we can express the linear
combination of $n$ GFN’s as

$\sum_{i=1}^{n}\lambda_{i}\cdot\textsf{GFN}(m_{i},h_{i})=\textsf{GFN}\left(\sum_{i=1}^{n}\lambda_{i}m_{i},\left(\sum_{i=1}^{n}|\lambda_{i}|h_{i}^{-1/2}\right)^{-2}\right).$
(34)

Now, let us consider $n$ independent GRFN’s ${\widetilde{X}}_{i}$ from
probability spaces $(\Omega_{i},\sigma_{i},P_{i})$ to $[0,1]^{\mathbb{R}}$
defined by

${\widetilde{X}}_{i}(\omega)=\textsf{GFN}(M_{i}(\omega),h_{i})$

for all $\omega\in\Omega_{i}$, where $M_{i}$ is a Gaussian random variable
with mean $\mu_{i}$ and standard deviation $\sigma_{i}$, and $h_{i}>0$. Let

${\widetilde{X}}=\sum_{i=1}^{n}\lambda_{i}{\widetilde{X}}_{i}$

be the random fuzzy set from
$(\Omega_{1}\times\ldots\times\Omega_{n},\sigma_{1}\otimes\ldots\otimes\sigma_{n},P_{1}\times\ldots\times
P_{n})$ to $[0,1]^{\mathbb{R}}$ defined by

${\widetilde{X}}(\omega_{1},\ldots,\omega_{n})=\sum_{i=1}^{n}\lambda_{i}\cdot\textsf{GFN}(M_{i}(\omega_{i}),h_{i}).$

If each GRFN ${\widetilde{X}}_{i}$ represents our knowledge about the value of
some quantity $X_{i}$, ${\widetilde{X}}$ represents our knowledge about
$X=\sum_{i=1}^{n}\lambda_{i}X_{i}$. From (34),
${\widetilde{X}}\sim{\widetilde{N}}(\mu,\sigma,h)$ with

$\mu=\sum_{i=1}^{n}\lambda_{i}\mu_{i},\quad\sigma^{2}=\sum_{i=1}^{n}\lambda_{i}^{2}\sigma_{i}^{2},\quad\text{and}\quad
h=\left(\sum_{i=1}^{n}|\lambda_{i}|h_{i}^{-1/2}\right)^{-2}.$

##  5 Gaussian random fuzzy vectors

In this section, we introduce Gaussian random fuzzy vectors (GRFV’s), an
extension of the model presented in Section 4 allowing us to describe
knowledge about multidimensional quantities. The main definitions and
properties are first introduced in Section 5.1. The expression of the
orthogonal sum of two GRFV’s is then given in Section 5.2, after which the
marginalization and vacuous extension of GRFV’s are described in Section 5.3.
Finally, our model is compared to Dempster’s normal belief function model in
Section 5.4.

###  5.1 Definition and main properties

We consider a $p$-dimensional variable $\boldsymbol{\theta}$ taking values in
$\mathbb{R}^{p}$. Knowledge about $\boldsymbol{\theta}$ may be encoded as a
$p$-dimensional Gaussian fuzzy vector, defined as follows.

######  Definition 2.

We define the $p$-dimensional Gaussian fuzzy vector (GFV) with center
${\boldsymbol{m}}\in\mathbb{R}^{p}$ and $p\times p$ symmetric and positive
semidefinite precision matrix ${\boldsymbol{H}}$ as the normalized fuzzy
subset of $\mathbb{R}^{p}$ with membership function

$\varphi({\boldsymbol{x}};{\boldsymbol{m}},{\boldsymbol{H}})=\exp\left(-\frac{1}{2}({\boldsymbol{x}}-{\boldsymbol{m}})^{T}{\boldsymbol{H}}({\boldsymbol{x}}-{\boldsymbol{m}})\right),$

denoted as $\textsf{GFV}({\boldsymbol{m}},{\boldsymbol{H}})$.

As shown in [27], the normalized product of two GFV’s is still a GFV. The
following proposition generalizes Proposition 3.

######  Proposition 11.

Let $\textsf{GFV}({\boldsymbol{m}}_{1},{\boldsymbol{H}}_{1})$ and
$\textsf{GFV}({\boldsymbol{m}}_{2},{\boldsymbol{H}}_{2})$ be two
$p$-dimensional GFV’s with positive definite precision matrices
${\boldsymbol{H}}_{1}$ and ${\boldsymbol{H}}_{2}$. We have

$\varphi({\boldsymbol{x}};{\boldsymbol{m}}_{1},{\boldsymbol{H}}_{1})\cdot\varphi({\boldsymbol{x}};{\boldsymbol{m}}_{2},{\boldsymbol{H}}_{2})=\varphi({\boldsymbol{x}};{\boldsymbol{m}}_{12},{\boldsymbol{H}}_{12})\times\\\
\exp\left(-\frac{1}{2}({\boldsymbol{m}}_{1}-{\boldsymbol{m}}_{2})^{T}({\boldsymbol{H}}_{1}^{-1}+{\boldsymbol{H}}_{2}^{-1})^{-1}({\boldsymbol{m}}_{1}-{\boldsymbol{m}}_{2})\right),$

with

${\boldsymbol{m}}_{12}=({\boldsymbol{H}}_{1}+{\boldsymbol{H}}_{2})^{-1}({\boldsymbol{H}}_{1}{\boldsymbol{m}}_{1}+{\boldsymbol{H}}_{2}{\boldsymbol{m}}_{2})\quad\text{and}\quad{\boldsymbol{H}}_{12}={\boldsymbol{H}}_{1}+{\boldsymbol{H}}_{2}.$

Consequently, the following equation holds:

$\\\
GFV({\boldsymbol{m}}_{1},{\boldsymbol{H}}_{1})\varodot\textsf{GFV}({\boldsymbol{m}}_{2},{\boldsymbol{H}}_{2})=\textsf{GFV}({\boldsymbol{m}}_{12},{\boldsymbol{H}}_{12}),$

and the height of the product intersection between
$\textsf{GFV}({\boldsymbol{m}}_{1},{\boldsymbol{H}}_{1})$ and
$\textsf{GFV}({\boldsymbol{m}}_{1},{\boldsymbol{H}}_{2})$ is

$\displaystyle
0pt\left(\textsf{GFV}({\boldsymbol{m}}_{1},{\boldsymbol{H}}_{1}),\textsf{GFV}({\boldsymbol{m}}_{1},{\boldsymbol{H}}_{2})\right)$
$\displaystyle=\max_{\boldsymbol{x}}\varphi({\boldsymbol{x}};{\boldsymbol{m}}_{1},{\boldsymbol{H}}_{1})\varphi({\boldsymbol{x}};{\boldsymbol{m}}_{2},{\boldsymbol{H}}_{2})$
(35a)
$\displaystyle=\exp\left(-\frac{1}{2}({\boldsymbol{m}}_{1}-{\boldsymbol{m}}_{2})^{T}({\boldsymbol{H}}_{1}^{-1}+{\boldsymbol{H}}_{2}^{-1})^{-1}({\boldsymbol{m}}_{1}-{\boldsymbol{m}}_{2})\right).$
(35b)

Equipped with the notion of GFV, we can now introduce a model of random fuzzy
set that can be seen as a GFV whose mode is a multidimensional Gaussian random
variable. This model is defined formally as follows.

######  Definition 3.

Let $(\Omega,\sigma_{\Omega},P)$ be a probability space,
${\boldsymbol{M}}:\Omega\rightarrow\mathbb{R}^{p}$ a $p$-dimensional Gaussian
random vector with mean $\boldsymbol{\mu}$ and variance matrix
$\boldsymbol{\Sigma}$, and ${\boldsymbol{H}}$ a $p\times p$ symmetric and
positive semidefinite real matrix. The random fuzzy set
${\widetilde{X}}:\Omega\rightarrow[0,1]^{\mathbb{R}^{p}}$ defined as

${\widetilde{X}}(\omega)=\textsf{GFV}({\boldsymbol{M}}(\omega),{\boldsymbol{H}})$

is called a Gaussian random fuzzy vector (GRFV), which we denote as
${\widetilde{X}}\sim{\widetilde{N}}(\boldsymbol{\mu},\boldsymbol{\Sigma},{\boldsymbol{H}})$.

The following proposition generalizes Proposition 6.

######  Proposition 12.

The contour function of GRFV
${\widetilde{X}}\sim{\widetilde{N}}(\boldsymbol{\mu},\boldsymbol{\Sigma},{\boldsymbol{H}})$
with positive definite precision matrix ${\boldsymbol{H}}$ is

$pl_{\widetilde{X}}({\boldsymbol{x}})=\frac{1}{|{\boldsymbol{I}}_{p}+\boldsymbol{\Sigma}{\boldsymbol{H}}|^{1/2}}\exp\left(-\frac{1}{2}({\boldsymbol{x}}-\boldsymbol{\mu})^{T}({\boldsymbol{H}}^{-1}+\boldsymbol{\Sigma})^{-1}({\boldsymbol{x}}-\boldsymbol{\mu})\right),$

where ${\boldsymbol{I}}_{p}$ is the $p$-dimensional identity matrix.

###### Proof.

See H. ∎

###  5.2 Orthogonal sum of Gaussian random fuzzy vectors

The practical interest of GRFV’s arises from the fact that they can be easily
combined by the generalized product-intersection rule. The following lemma and
proposition, which generalize, respectively, Lemma 1 and Proposition 9, give
us the expression of the orthogonal sum of two GRFV’s.

######  Lemma 2.

Let ${\boldsymbol{M}}_{1}\sim{\cal
N}(\boldsymbol{\mu}_{1},\boldsymbol{\Sigma}_{1})$ and
${\boldsymbol{M}}_{2}\sim{\cal
N}(\boldsymbol{\mu}_{2},\boldsymbol{\Sigma}_{2})$ be two independent Gaussian
$p$-dimensional random vectors and let ${\boldsymbol{H}}_{1}$ and
${\boldsymbol{H}}_{2}$ be two symmetric and positive definite $p\times p$
matrices. Let ${\widetilde{F}}$ be the fuzzy subset of $\mathbb{R}^{2p}$ with
membership function

${\widetilde{F}}({\boldsymbol{m}}_{1},{\boldsymbol{m}}_{2})=0pt\left(\textsf{GFV}({\boldsymbol{m}}_{1},{\boldsymbol{H}}_{1})\cdot\textsf{GFV}({\boldsymbol{m}}_{2},{\boldsymbol{H}}_{2})\right),$

and let ${\boldsymbol{M}}$ be the $2p$-dimensional vector
$({\boldsymbol{M}}_{1},{\boldsymbol{M}}_{2})$. The conditional probability
distribution of ${\boldsymbol{M}}$ given ${\widetilde{F}}$ is $2p$-dimensional
Gaussian with mean vector ${\widetilde{\boldsymbol{\mu}}}$ and covariance
matrix ${\widetilde{\boldsymbol{\Sigma}}}$ defined as follows:

${\widetilde{\boldsymbol{\Sigma}}}=\begin{pmatrix}\boldsymbol{\Sigma}_{1}^{-1}+{\overline{{\boldsymbol{H}}}}&-{\overline{{\boldsymbol{H}}}}\\\
-{\overline{{\boldsymbol{H}}}}&\boldsymbol{\Sigma}_{1}^{-1}+{\overline{{\boldsymbol{H}}}}\end{pmatrix}^{-1},$
${\widetilde{\boldsymbol{\mu}}}=\begin{pmatrix}{\overline{{\boldsymbol{H}}}}^{-1}\boldsymbol{\Sigma}_{1}^{-1}+{\boldsymbol{I}}_{p}&-{\boldsymbol{I}}_{p}\\\
-{\boldsymbol{I}}_{p}&{\overline{{\boldsymbol{H}}}}^{-1}\boldsymbol{\Sigma}_{2}^{-1}+{\boldsymbol{I}}_{p}\end{pmatrix}^{-1}\begin{pmatrix}{\overline{{\boldsymbol{H}}}}^{-1}\boldsymbol{\Sigma}_{1}^{-1}&\boldsymbol{0}\\\
\boldsymbol{0}&{\overline{{\boldsymbol{H}}}}^{-1}\boldsymbol{\Sigma}_{2}^{-1}\end{pmatrix}\begin{pmatrix}\boldsymbol{\mu}_{1}\\\
\boldsymbol{\mu}_{2}\end{pmatrix},$ (36)

with

${\overline{{\boldsymbol{H}}}}=({\boldsymbol{H}}_{1}^{-1}+{\boldsymbol{H}}_{2}^{-1})^{-1}.$

Furthermore, the degree of conflict between two GRFV’s
${\widetilde{X}}_{1}\sim{\widetilde{N}}(\boldsymbol{\mu}_{1},\boldsymbol{\Sigma}_{1},{\boldsymbol{H}}_{1})$
and
${\widetilde{X}}_{2}\sim{\widetilde{N}}(\boldsymbol{\mu}_{2},\boldsymbol{\Sigma}_{2},{\boldsymbol{H}}_{2})$
is

$\kappa=1-\int_{\mathbb{R}^{2p}}f({\boldsymbol{m}}_{1},{\boldsymbol{m}}_{2}){\widetilde{F}}({\boldsymbol{m}}_{1},{\boldsymbol{m}}_{2})d{\boldsymbol{m}}_{1}d{\boldsymbol{m}}_{2}=\\\
1-\sqrt{\frac{|{\widetilde{\boldsymbol{\Sigma}}}|}{|\boldsymbol{\Sigma}_{1}||\boldsymbol{\Sigma}_{2}|}}\exp\left\\{-\frac{1}{2}\left[\boldsymbol{\mu}_{1}^{T}\boldsymbol{\Sigma}_{1}^{-1}\boldsymbol{\mu}_{1}+\boldsymbol{\mu}_{2}^{T}\boldsymbol{\Sigma}_{2}^{-1}\boldsymbol{\mu}_{2}-{\widetilde{\boldsymbol{\mu}}}^{T}{\widetilde{\boldsymbol{\Sigma}}}^{-1}{\widetilde{\boldsymbol{\mu}}}\right]\right\\}.$

###### Proof.

See I ∎

######  Proposition 13.

Let
${\widetilde{X}}_{1}\sim{\widetilde{N}}(\boldsymbol{\mu}_{1},\boldsymbol{\Sigma}_{1},{\boldsymbol{H}}_{1})$
and
${\widetilde{X}}_{2}\sim{\widetilde{N}}(\boldsymbol{\mu}_{2},\boldsymbol{\Sigma}_{2},{\boldsymbol{H}}_{2})$
be two independent GRFV’s. We have

${\widetilde{X}}_{1}\oplus{\widetilde{X}}_{2}\sim{\widetilde{N}}({\widetilde{\boldsymbol{\mu}}}_{12},{\widetilde{\boldsymbol{\Sigma}}}_{12},{\boldsymbol{H}}_{12})$

with

${\boldsymbol{H}}_{12}={\boldsymbol{H}}_{1}+{\boldsymbol{H}}_{2},$
${\widetilde{\boldsymbol{\mu}}}_{12}={\boldsymbol{A}}{\widetilde{\boldsymbol{\mu}}},$

and

${\widetilde{\boldsymbol{\Sigma}}}_{12}={\boldsymbol{A}}{\widetilde{\boldsymbol{\Sigma}}}{\boldsymbol{A}}^{T},$

where ${\boldsymbol{A}}$ is the constant $p\times 2p$ matrix defined as

${\boldsymbol{A}}={\boldsymbol{H}}_{12}^{-1}\begin{pmatrix}{\boldsymbol{H}}_{1}&{\boldsymbol{H}}_{2}\end{pmatrix}.$

###### Proof.

Let ${\boldsymbol{M}}_{1}$ and ${\boldsymbol{M}}_{2}$ be the Gaussian random
vector from $(\Omega_{1},\sigma_{1},P_{1})$ and
$(\Omega_{2},\sigma_{2},P_{2})$ to $(\mathbb{R}^{p},\beta_{\mathbb{R}^{p}})$
corresponding, respectively, to GRFV’s
${\widetilde{X}}_{1}\sim{\widetilde{N}}(\boldsymbol{\mu}_{1},\boldsymbol{\Sigma}_{1},{\boldsymbol{H}}_{1})$
and
${\widetilde{X}}_{2}\sim{\widetilde{N}}(\boldsymbol{\mu}_{2},\boldsymbol{\Sigma}_{2},{\boldsymbol{H}}_{2})$.
The orthogonal sum of ${\widetilde{X}}_{1}$ and ${\widetilde{X}}_{2}$ is
defined by the mapping

${\widetilde{X}}_{\varodot}:(\omega_{1},\omega_{2})\rightarrow\textsf{GFV}({\boldsymbol{M}}_{12}(\omega_{1},\omega_{2}),{\boldsymbol{H}}_{1}+{\boldsymbol{H}}_{2})$

with

${\boldsymbol{M}}_{12}=({\boldsymbol{H}}_{1}+{\boldsymbol{H}}_{2})^{-1}({\boldsymbol{H}}_{1}{\boldsymbol{M}}_{1}+{\boldsymbol{H}}_{2}{\boldsymbol{M}}_{2})={\boldsymbol{A}}\begin{pmatrix}{\boldsymbol{M}}_{1}\\\
{\boldsymbol{M}}_{2}\end{pmatrix},$

where ${\boldsymbol{A}}$ is the $p\times 2p$ matrix

${\boldsymbol{A}}=({\boldsymbol{H}}_{1}+{\boldsymbol{H}}_{2})^{-1}\begin{pmatrix}{\boldsymbol{H}}_{1}&{\boldsymbol{H}}_{2}\end{pmatrix},$

and the probability measure ${\widetilde{P}}_{12}$ on
$\Omega_{1}\times\Omega_{2}$ obtained by conditioning $P_{1}\times P_{2}$ on
the fuzzy set
${\widetilde{\Theta}}^{*}(\omega_{1},\omega_{2})=0pt\left(\textsf{GFV}({\boldsymbol{M}}_{1}(\omega_{1}),{\boldsymbol{H}}_{1}),\textsf{GFV}({\boldsymbol{M}}_{2}(\omega_{2}),{\boldsymbol{H}}_{2})\right)$.
From Lemma 2, the pushforward measure of ${\widetilde{P}}_{12}$ by the random
vector $({\boldsymbol{M}}_{1},{\boldsymbol{M}}_{2})$ is the $p$-dimensional
Gaussian distribution with parameters
$({\widetilde{\boldsymbol{\mu}}},{\widetilde{\boldsymbol{\Sigma}}})$.
Consequently, ${\boldsymbol{M}}_{12}$ is a Gaussian random vector with mean

$\mathbb{E}({\boldsymbol{M}}_{12})={\boldsymbol{A}}{\widetilde{\boldsymbol{\mu}}}$

and variance

$\text{Var}({\boldsymbol{M}}_{12})={\boldsymbol{A}}{\widetilde{\boldsymbol{\Sigma}}}{\boldsymbol{A}}^{T}.$

∎

###  5.3 Marginalization and vacuous extension

In this section, we consider the marginalization and vacuous extension
(defined in Section 3.3) of a GRFV. We assume that variable
$\boldsymbol{\theta}$ taking values in $\mathbb{R}^{p}$ is decomposed as
$\boldsymbol{\theta}=(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})$ with
$\boldsymbol{\theta}_{1}\in\Theta_{1}=\mathbb{R}^{p-k}$ and
$\boldsymbol{\theta}_{2}\in\Theta_{2}=\mathbb{R}^{k}$ for $0<k<p$.

#### Marginalization

We start with the following lemma.

######  Lemma 3.

Let ${\widetilde{F}}=\textsf{GFV}({\boldsymbol{m}},{\boldsymbol{H}})$ be a
$p$-dimensional Gaussian fuzzy vector with mode
${\boldsymbol{m}}=({\boldsymbol{m}}_{1},{\boldsymbol{m}}_{2})$, where
${\boldsymbol{m}}_{1}\in\Theta_{1}=\mathbb{R}^{p-k}$ and
${\boldsymbol{m}}_{2}\in\Theta_{2}=\mathbb{R}^{k}$ for $0<k<p$, and precision
matrix ${\boldsymbol{H}}$ with block decomposition

${\boldsymbol{H}}=\begin{pmatrix}{\boldsymbol{H}}_{11}&{\boldsymbol{H}}_{12}\\\
{\boldsymbol{H}}_{21}&{\boldsymbol{H}}_{22}\end{pmatrix}.$

Assume that ${\boldsymbol{H}}_{22}$ is nonsingular. The projection of
${\widetilde{F}}$ on $\Theta_{1}$, denoted as
${\widetilde{F}}\downarrow\Theta_{1}$ is the Gaussian fuzzy vector
$\textsf{GFV}({\boldsymbol{m}}_{1},{\boldsymbol{H}}^{\prime}_{11})$ with

${\boldsymbol{H}}^{\prime}_{11}={\boldsymbol{H}}_{1}-{\boldsymbol{H}}_{12}{\boldsymbol{H}}_{22}^{-1}{\boldsymbol{H}}_{21}.$

###### Proof.

See J ∎

Let us now consider a $p$-dimensional GRFV
${\widetilde{X}}\sim{\widetilde{N}}(\boldsymbol{\mu},\boldsymbol{\Sigma},{\boldsymbol{H}})$
representing partial knowledge about
$\boldsymbol{\theta}=(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})$. The
marginal RFS for $\boldsymbol{\theta}_{1}$ is given by the following
proposition, which follows directly from Lemma 3.

######  Proposition 14.

Let
${\widetilde{X}}\sim{\widetilde{N}}(\boldsymbol{\mu},\boldsymbol{\Sigma},{\boldsymbol{H}})$
by a $p$-dimensional GRFV taking values in $2^{\Theta}$, with
$\Theta=\Theta_{1}\times\Theta_{2}$, where $\Theta_{1}=\mathbb{R}^{p-k}$ and
$\Theta_{2}=\mathbb{R}^{k}$ for $0<k<p$. Let
$\boldsymbol{\mu}=(\boldsymbol{\mu}_{1},\boldsymbol{\mu}_{2})$ with
$\boldsymbol{\mu}_{1}\in\Theta_{1}$ and $\boldsymbol{\mu}_{2}\in\Theta_{2}$,
and consider the block decompositions

$\boldsymbol{\Sigma}=\begin{pmatrix}\boldsymbol{\Sigma}_{11}&\boldsymbol{\Sigma}_{12}\\\
\boldsymbol{\Sigma}_{21}&\boldsymbol{\Sigma}_{22}\end{pmatrix}\quad\text{and}\quad{\boldsymbol{H}}=\begin{pmatrix}{\boldsymbol{H}}_{11}&{\boldsymbol{H}}_{12}\\\
{\boldsymbol{H}}_{21}&{\boldsymbol{H}}_{22}\end{pmatrix}.$

Assume that ${\boldsymbol{H}}_{22}$ is nonsingular. The marginal of
${\widetilde{X}}$ on $\Theta_{1}$ is the GRFV
${\widetilde{X}}_{1}\sim{\widetilde{N}}(\boldsymbol{\mu}_{1},\boldsymbol{\Sigma}_{11},{\boldsymbol{H}}^{\prime}_{11})$
with

${\boldsymbol{H}}^{\prime}_{11}={\boldsymbol{H}}_{11}-{\boldsymbol{H}}_{12}{\boldsymbol{H}}_{22}^{-1}{\boldsymbol{H}}_{21}.$

#### Vacuous extension

We now consider a Gaussian fuzzy vector
$\textsf{GFV}({\boldsymbol{m}}_{1},{\boldsymbol{H}}_{11})$ in
$\Theta_{1}=\mathbb{R}^{p-k}$ for $0<k<p$. Its cylindrical extension in
$\Theta=\Theta_{1}\times\Theta_{2}$, with $\Theta_{2}=\mathbb{R}^{k}$ has the
following membership function

$\varphi({\boldsymbol{x}})=\exp\left(-\frac{1}{2}({\boldsymbol{x}}_{1}-{\boldsymbol{m}}_{1})^{T}{\boldsymbol{H}}_{11}({\boldsymbol{x}}_{1}-{\boldsymbol{m}}_{1})\right),$

which can be written as

$\varphi({\boldsymbol{x}})=\exp\left(-\frac{1}{2}({\boldsymbol{x}}-{\boldsymbol{m}})^{T}{\boldsymbol{H}}({\boldsymbol{x}}-{\boldsymbol{m}})\right),$

where ${\boldsymbol{m}}$ is the $p$-dimensional vector

${\boldsymbol{m}}=\begin{pmatrix}{\boldsymbol{m}}_{1}\\\
\boldsymbol{0}\end{pmatrix}$

and ${\boldsymbol{H}}$ is the $p\times p$ matrix

${\boldsymbol{H}}=\begin{pmatrix}{\boldsymbol{H}}_{11}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{pmatrix}.$ (37)

Given a GRFV
${\widetilde{X}}_{1}\sim{\widetilde{N}}(\boldsymbol{\mu}_{1},\boldsymbol{\Sigma}_{11},{\boldsymbol{H}}_{11})$
taking values in $2^{\Theta_{1}}$, it follows immediately that its vacuous
extension in $\Theta=\Theta_{1}\times\Theta_{2}$ is the GRFV

${\widetilde{X}}_{1\uparrow(1,2)}\sim{\widetilde{N}}(\boldsymbol{\mu},\boldsymbol{\Sigma},{\boldsymbol{H}})$

with

$\boldsymbol{\mu}=\begin{pmatrix}\boldsymbol{\mu}_{1}\\\
\boldsymbol{0}\end{pmatrix},\quad\boldsymbol{\Sigma}=\begin{pmatrix}{\boldsymbol{H}}_{11}&\boldsymbol{0}\\\
\boldsymbol{0}&{\boldsymbol{I}}_{k}\end{pmatrix},$

where ${\boldsymbol{I}}_{k}$ is the $k\times k$ identity matrix, and
${\boldsymbol{H}}$ given by (37).

#### Noninteractivity

In Section 3.3, we defined the notion of noninteractive random fuzzy vector.
The following proposition gives a necessary and sufficient condition for a
GRFV to be noninteractive.

######  Proposition 15.

A $p$-dimensional GRFV
${\widetilde{X}}\sim{\widetilde{N}}(\boldsymbol{\mu},\boldsymbol{\Sigma},{\boldsymbol{H}})$
is non-interactive iff matrices $\boldsymbol{\Sigma}$ and ${\boldsymbol{H}}$
are diagonal.

###### Proof.

Let ${\widetilde{X}}_{1},\ldots,{\widetilde{X}}_{p}$ be the marginals of
${\widetilde{X}}$ on each of the $p$ coordinates. Let
$\sigma^{2}_{1},\ldots,\sigma^{2}_{p}$ and $h_{1},\ldots,h_{p}$ be the
diagonal elements of, respectively, $\boldsymbol{\Sigma}$ and
${\boldsymbol{H}}$. Let $\Omega$ be the set of departure of ${\widetilde{X}}$.
Let ${\widetilde{X}}_{i\uparrow(1:p)}$ denote the vacuous extension of
${\widetilde{X}}_{i}$ in $\mathbb{R}^{p}$, defined by

${\widetilde{X}}_{i\uparrow(1:p)}(\omega)({\boldsymbol{x}})=\exp\left(-\frac{h}{2}(x_{i}-M_{i}(\omega))^{2}\right)$

with $M_{i}\sim N(\mu_{i},\sigma^{2}_{i})$. The orthogonal sum

${\widetilde{X}}^{\prime}={\widetilde{X}}_{1\uparrow(1:p)}\oplus\ldots\oplus{\widetilde{X}}_{p\uparrow(1:p)}$

is given by

${\widetilde{X}}^{\prime}(\omega)({\boldsymbol{x}})=\prod_{i=1}^{p}\exp\left(-\frac{h}{2}(x_{i}-M_{i}(\omega))^{2}\right)=\exp\left(-\frac{1}{2}({\boldsymbol{x}}-{\boldsymbol{M}}^{\prime}(\omega))^{T}{\boldsymbol{H}}^{\prime}({\boldsymbol{x}}-{\boldsymbol{M}}^{\prime}(\omega))\right),$

where ${\boldsymbol{H}}^{\prime}$ is the diagonal matrix with diagonal
elements $h_{1},\ldots,h_{p}$, and ${\boldsymbol{M}}^{\prime}$ is a random
vector with mean $\boldsymbol{\mu}$ and diagonal covariance matrix
$\boldsymbol{\Sigma}^{\prime}$ with diagonal elements
$\sigma^{2}_{1},\ldots,\sigma^{2}_{p}$. We have
${\widetilde{X}}={\widetilde{X}}^{\prime}$ iff
${\boldsymbol{H}}={\boldsymbol{H}}^{\prime}$ and
$\boldsymbol{\Sigma}=\boldsymbol{\Sigma}^{\prime}$, i.e., if both
${\boldsymbol{H}}$ and $\boldsymbol{\Sigma}$ are diagonal. ∎

###  5.4 Comparison with Dempster’s normal belief functions

In [5], Dempster introduced another class of continuous belief functions in
$\mathbb{R}^{p}$, called _normal belief functions_ 111Ref. [5] was actually
available as a working paper from the Statistical Department of Harvard
University since 1990, but it only appeared as a book chapter in 2001.. It is
interesting to compare Dempster’s model with ours, as both models generalize
the multivariate Gaussian distribution. A normal belief function $Bel$ on
$\mathbb{R}^{p}$ as defined in [5] is specified by the following components:

  * 1.

An $n$-dimensional subspace ${\cal S}$ of $\mathbb{R}^{p}$;

  * 2.

A $q$-dimensional partition $\Pi$ of ${\cal S}$ into parallel $n-q$
dimensional subspaces; (If $q=0$, $\Pi=\\{{\cal S}\\}$);

  * 3.

A full-rank $q$-dimensional Gaussian distribution $N(\mu,\Sigma)$ on $\Pi$ if
$q>0$, or the discrete probability measure with mass function $m({\cal S})=1$
if $q=0$.

Belief function $Bel$ is then induced by a random set from $\Pi$, equipped
with the normal distribution $N(\mu,\Sigma)$ if $q>0$ or probability mass
function $m$ if $q=0$, to the corresponding family of parallel $n-q$
dimensional subspaces of ${\cal S}$. The following special cases are of
interest:

  1. 1.

If $p=n=q$, $Bel$ is a Gaussian probability distribution on $\mathbb{R}^{p}$;

  2. 2.

If $p>n=q$, $Bel$ is a Gaussian probability distribution limited to an
$n$-dimensional subspace of $\mathbb{R}^{p}$;

  3. 3.

If $p=n$ and $q=0$, $Bel$ is vacuous;

  4. 4.

If $q=0$ while $p>n>0$, $Bel$ is logical with ${\cal S}$ as its only focal
set; it is then equivalent to specifying $p-n$ linear equations;

  5. 5.

If $n=q=0$, the true point in $\mathbb{R}^{p}$ is known with certainty.

Like GRFV’s, Dempster’s normal belief functions thus include the vacuous
belief function, Gaussian probability distributions, as well as vacuous
extensions of marginal Gaussian distributions. However, the two models are
clearly distinct. Dempster’s model is based on the combination of Gaussian
probability distributions and linear equations, and is specially useful in
relation with linear statistical models such as the Kalman filter [5] or
linear regression [24]. In contrast, in the GRFV model, focal sets are fuzzy
subsets of $\mathbb{R}^{n}$ ($n\leq p$) with Gaussian membership functions, or
cylindrical extensions of such fuzzy subsets. This model allows us to
represent not only probabilistic and logical evidence, but also fuzzy
information. In particular, it includes Gaussian probability distribution and
Gaussian possibility distributions as special cases. We could attempt to
design an even more general model that would contain both Dempster’s normal
belief functions and belief functions induced by GRFV’s as special cases. Such
a model would allow us to reason with Gaussian probability and possibility
distributions as well as with linear equations. The rigorous development of
such a model is left for further research.

##  6 Conclusions

In this paper, continuing a study started in [9] with the finite case, we have
introduced a theory of epistemic random fuzzy sets in a general setting. An
epistemic random fuzzy set represents a piece of evidence, which may be crisp
or fuzzy. This framework generalizes both epistemic random sets as considered
in the Dempster-Shafer theory of belief functions, and possibility
distributions considered in possibility theory. Independent epistemic random
fuzzy sets are combined by the generalized product-intersection rule, which
extends both Dempster’s rule for combining belief functions and the product
intersection rule for combining possibility distributions.

In addition, we have also introduced Gaussian random fuzzy numbers (GRFN’s)
and their multidimensional extensions, Gaussian random fuzzy vectors (GRFV’s)
as practical models of random fuzzy subsets of, respectively, $\mathbb{R}$ and
$\mathbb{R}^{p}$ with $p\geq 2$. A GRFN is described by three parameters: its
mode $m$, its variance $\sigma^{2}$ and its precision $h$. In this setting, a
Gaussian random variable can be seen as an infinitely precise GRFN
($h=+\infty$), while a Gaussian possibility distribution is a constant GRFN
($\sigma^{2}=0$). A maximally imprecise GRFN such that $h=0$ is said to be
vacuous: it represents complete ignorance. In GRFV’s, the mode becomes a
$p$-dimensional vector, while the variance and precision become positive semi-
definite $p\times p$ square matrices. The practical convenience of GRFN’s and
GRFV’s arises from the fact that they can easily be combined by the
generalized product-intersection rule. Also, formulas for the projection and
marginal extension fo GRFV’s have been derived.

This work opens up several perspectives. Using random fuzzy sets and, in
particular, GRFN’s to represent expert knowledge about numerical quantities
will require the development of adequate elicitation procedures. We also
consider using this framework in machine learning, to quantify prediction
uncertainty in regression problems. Finally, the extension of the model
introduced in this paper to take into account linear equations, as well as the
development of computational procedures for reasoning with GRFV’s over many
variables are promising avenues for further research.

## References

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##  Appendix A Proof of Proposition 1

Commutativity is obvious. To prove associativity, let us consider three random
sets
$(\Omega_{i},\sigma_{i},P_{i},\Theta,\sigma_{\Theta},{\overline{X}}_{i})$,
$i=1,2,3$. Consider the combined random set

$(\Omega_{1}\times\Omega_{2}\times\Omega_{3},\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{3},P_{123},\Theta,\sigma_{\Theta},{\overline{X}}_{1\cap
2\cap 3}),$ (38)

where

${\overline{X}}_{1\cap 2\cap
3}(\omega_{1},\omega_{2},\omega_{3})={\overline{X}}_{1}(\omega_{1})\cap{\overline{X}}_{2}(\omega_{2})\cap{\overline{X}}_{3}(\omega_{3}),$
$P_{123}=(P_{1}\times P_{2}\times P_{3})(\cdot\mid\Theta^{*}_{123}),$

and

$\Theta^{*}_{123}=\\{(\omega_{1},\omega_{2},\omega_{3})\in\Omega_{1}\times\Omega_{2}\times\Omega_{2}:{\overline{X}}_{1\cap
2\cap 3}(\omega_{1},\omega_{2},\omega_{3})\neq\emptyset\\}.$

We will show that we get the same result by combining ${\overline{X}}_{1}$
with ${\overline{X}}_{2}$ first, and then combining the result with
${\overline{X}}_{3}$. Combining the first two random sets, we get

$(\Omega_{1}\times\Omega_{2},\sigma_{1}\otimes\sigma_{2},P_{12},\Theta,\sigma_{\Theta},{\overline{X}}_{1\cap
2}),$

with ${\overline{X}}_{1\cap
2}(\omega_{1},\omega_{2})={\overline{X}}_{1}(\omega_{1})\cap{\overline{X}}_{2}(\omega_{2})$,
$P_{12}=(P_{1}\times P_{2})(\cdot\mid\Theta^{*}_{12})$ and

$\Theta^{*}_{12}=\\{(\omega_{1},\omega_{2})\in\Omega_{1}\times\Omega_{2}:{\overline{X}}_{1\cap
2}(\omega_{1},\omega_{2})\neq\emptyset\\}.$

Combining it with ${\overline{X}}_{3}$ we get

$(\Omega_{1}\times\Omega_{2}\times\Omega_{3},\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{3},P_{(12)3},\Theta,\sigma_{\Theta},{\overline{X}}_{1\cap
2\cap 3}),$ (39)

with $P_{(12)3}=(P_{12}\times P_{3})(\cdot\mid\Theta^{*}_{123})$. Comparing
(38) and (39), we see that we only need to show that $P_{(12)3}=P_{123}$. For
any event $C\subseteq\Theta^{*}_{123}$ and any $\omega_{3}\in\Omega_{3}$, let
$C_{\omega_{3}}=\\{(\omega_{1},\omega_{2})\in\Omega_{1}\times\Omega_{2}:(\omega_{1},\omega_{2},\omega_{3})\in
C\\}$. By definition of the product measure $P_{12}\times P_{3}$ (see [18,
page 144]), we have

$P_{(12)3}(C)=\frac{(P_{12}\times P_{3})(C)}{(P_{12}\times
P_{3})(\Theta^{*}_{123})}=\frac{1}{(P_{12}\times P_{3})(\Theta^{*}_{123})}\int
P_{12}(C_{\omega_{3}})dP_{3}(\omega_{3})$ (40)

Now, as $C\subseteq\Theta^{*}_{123}$, for any $(\omega_{1},\omega_{2})\in
C_{\omega_{3}}$,
${\overline{X}}_{1}(\omega_{1})\cap{\overline{X}}_{2}(\omega_{2})\neq\emptyset$.
Consequently, $C_{\omega_{3}}\subseteq\Theta^{*}_{12}$, so

$P_{12}(C_{\omega_{3}})=\frac{(P_{1}\times
P_{2})(C_{\omega_{3}})}{(P_{1}\times P_{2})(\Theta^{*}_{12})}.$ (41)

From (40) and (41), we get

$\displaystyle P_{(12)3}(C)$ $\displaystyle=\frac{1}{(P_{12}\times
P_{3})(\Theta^{*}_{123})(P_{1}\times P_{2})(\Theta^{*}_{12})}\int(P_{1}\times
P_{2})(C_{\omega_{3}})dP_{3}(\omega_{3})$ (42a)
$\displaystyle=\frac{(P_{1}\times P_{2}\times P_{3})(C)}{(P_{12}\times
P_{3})(\Theta^{*}_{123})(P_{1}\times P_{2})(\Theta^{*}_{12})}.$ (42b)

Now,

$P_{123}(C)=\frac{(P_{1}\times P_{2}\times P_{3})(C)}{(P_{1}\times P_{2}\times
P_{3})(\Theta^{*}_{123})}.$ (43)

As $P_{(12)3}(\Theta^{*}_{123})=P_{123}(\Theta^{*}_{123})=1$, the denominators
in (42b) and (43) are equal, and $P_{(12)3}=P_{123}$.

##  Appendix B Proof of Proposition 4

Commutativity is obvious. To prove associativity, consider three random fuzzy
sets

$(\Omega_{i},\sigma_{i},P_{i},\Theta,\sigma_{\Theta},{\widetilde{X}}_{i}),\quad
i=1,2,3.$

Let ${\widetilde{\Theta}}^{*}_{12}$ be the fuzzy subset of
$\Omega_{1}\times\Omega_{2}$ with membership function

${\widetilde{\Theta}}_{12}^{*}(\omega_{1},\omega_{2})=0pt\left({\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{2})\right),$

and let ${\widetilde{\Theta}}^{*}_{(12)3}$ and
${\widetilde{\Theta}}^{*}_{123}$ be the fuzzy subsets of
$\Omega_{1}\times\Omega_{2}\times\Omega_{3}$ defined, respectively, as

${\widetilde{\Theta}}_{(12)3}^{*}(\omega_{1},\omega_{2},\omega_{3})=0pt\left(\left[{\widetilde{X}}_{1}(\omega_{1})\varodot{\widetilde{X}}_{2}(\omega_{1})\right]{\widetilde{X}}_{3}(\omega_{3})\right)$

and

${\widetilde{\Theta}}_{123}^{*}(\omega_{1},\omega_{2},\omega_{3})=0pt\left({\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{2}){\widetilde{X}}_{3}(\omega_{3})\right).$

Let ${\widetilde{P}}_{12}=(P_{1}\times
P_{2})(\cdot\mid{\widetilde{\Theta}}_{12}^{*})$,
${\widetilde{P}}_{(12)3}=({\widetilde{P}}_{12}\times
P_{3})(\cdot\mid{\widetilde{\Theta}}_{(12)3}^{*})$, and
${\widetilde{P}}_{123}=(P_{1}\times P_{2}\times
P_{3})(\cdot\mid{\widetilde{\Theta}}_{123}^{*})$. We only need to show that
${\widetilde{P}}_{(12)3}={\widetilde{P}}_{123}$. For any
$B\in\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{3}$, we have

$\displaystyle{\widetilde{P}}_{(12)3}(B)$
$\displaystyle\propto\int_{\Omega_{1}\times\Omega_{2}}\int_{\Omega_{3}}B(\omega_{1},\omega_{2},\omega_{3})0pt\left(\left[{\widetilde{X}}_{1}(\omega_{1})\varodot{\widetilde{X}}_{2}(\omega_{1})\right]{\widetilde{X}}_{3}(\omega_{3})\right)dP_{3}(\omega_{3})d{\widetilde{P}}_{12}(\omega_{1},\omega_{2})$
$\displaystyle\propto\int_{\Omega_{1}}\int_{\Omega_{2}}\int_{\Omega_{3}}B(\omega_{1},\omega_{2},\omega_{3})0pt\left(\left[{\widetilde{X}}_{1}(\omega_{1})\varodot{\widetilde{X}}_{2}(\omega_{1})\right]{\widetilde{X}}_{3}(\omega_{3})\right)\times$
$\displaystyle\hskip
142.26378pt0pt\left({\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{2})\right)dP_{3}(\omega_{3})dP_{2}(\omega_{2})dP_{1}(\omega_{1}).$

Now,

$\displaystyle
0pt\left(\left[{\widetilde{X}}_{1}(\omega_{1})\varodot{\widetilde{X}}_{2}(\omega_{1})\right]{\widetilde{X}}_{3}(\omega_{3})\right)$
$\displaystyle=0pt\left(\frac{{\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{1})}{0pt({\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{2}))}{\widetilde{X}}_{3}(\omega_{3})\right)$
$\displaystyle=\frac{0pt({\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{2}){\widetilde{X}}_{3}(\omega_{3}))}{0pt({\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{1}))}.$

Hence,

${\widetilde{P}}_{(12)3}(B)\propto\int_{\Omega_{1}}\int_{\Omega_{2}}\int_{\Omega_{3}}B(\omega_{1},\omega_{2},\omega_{3})0pt\left({\widetilde{X}}_{1}(\omega_{1}){\widetilde{X}}_{2}(\omega_{2}){\widetilde{X}}_{3}(\omega_{3})\right)dP_{3}(\omega_{3})dP_{2}(\omega_{2})dP_{1}(\omega_{1}),$

which proves that ${\widetilde{P}}_{(12)3}={\widetilde{P}}_{123}$, and the
associativity of $\oplus$.

##  Appendix C Proof of Proposition 6

We have

$\displaystyle pl_{\widetilde{X}}(x)=$
$\displaystyle\mathbb{E}_{M}[\varphi(x;M,h)]$ (44) $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}\varphi(x;m,h)\phi(m;\mu,\sigma)dm$ (45)
$\displaystyle=$
$\displaystyle\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{+\infty}\exp\left(-\frac{h}{2}(x-m)^{2}\right)\exp\left(-\frac{(m-\mu)^{2}}{2\sigma^{2}}\right)dm.$
(46)

From Proposition 3, the integrand can be written as

$\exp\left(-\frac{(m-\mu_{0})^{2}}{2\sigma_{0}^{2}}\right)\exp\left(-\frac{h(x-\mu)^{2}}{2(1+h\sigma^{2})}\right),$

with

$\mu_{0}=\frac{xh+\mu/\sigma^{2}}{h+1/\sigma^{2}}=\frac{xh\sigma^{2}+\mu}{h\sigma^{2}+1}$

and

$\sigma_{0}=\sqrt{\frac{1}{h+1/\sigma^{2}}}=\frac{\sigma}{\sqrt{1+h\sigma^{2}}}.$

Consequently,

$\displaystyle pl_{\widetilde{X}}(x)=$
$\displaystyle\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{h(x-\mu)^{2}}{2(1+h\sigma^{2})}\right)\underbrace{\int_{-\infty}^{+\infty}\exp\left(-\frac{(m-\mu_{0})^{2}}{2\sigma_{0}^{2}}\right)dm}_{\sigma_{0}\sqrt{2\pi}}$
(47)
$\displaystyle=\frac{1}{\sqrt{1+h\sigma^{2}}}\exp\left(-\frac{h(x-\mu)^{2}}{2(1+h\sigma^{2})}\right).$
(48)

##  Appendix D Proof of Proposition 7

If $h=0$, we have, trivially, $Bel_{\widetilde{X}}([x,y])=0$ and
$Pl_{\widetilde{X}}([x,y])=1$ for all $x\leq y$. Let us assume that $h>0$. We
have

$Pl_{\widetilde{X}}([x,y])=\mathbb{P}(M\leq x)\mathbb{E}[\varphi(x;M,h)\mid
M\leq x]+\\\ \mathbb{P}(x<M\leq y)\times
1+\mathbb{P}(M>y)\mathbb{E}[\varphi(y;M,h)\mid M>y],$ (49)

which can be written as

$Pl_{\widetilde{X}}([x,y])=\Phi\left(\frac{x-\mu}{\sigma}\right)\mathbb{E}[\varphi(x;M,h)\mid
M\leq x]+\\\
\Phi\left(\frac{y-\mu}{\sigma}\right)-\Phi\left(\frac{x-\mu}{\sigma}\right)+\\\
\left[1-\Phi\left(\frac{y-\mu}{\sigma}\right)\right]\mathbb{E}[\varphi(y;M,h)\mid
M>y].$ (50)

Conditionally on $M\leq x$, $M$ has a truncated normal distribution on
$(-\infty,x]$ with pdf

$f(m)=\frac{1}{\sigma\sqrt{2\pi}}\frac{\exp\left(\frac{-(m-\mu)^{2}}{2\sigma^{2}}\right)}{\Phi\left(\frac{x-\mu}{\sigma}\right)}\mathbf{1}_{(-\infty,x]}(m).$

Consequently,

$\mathbb{E}[\varphi(x;M,h)\mid M\leq
x]=\frac{1}{\sigma\sqrt{2\pi}}\frac{1}{\Phi\left(\frac{x-\mu}{\sigma}\right)}\underbrace{\int_{-\infty}^{x}\exp\left(-\frac{h}{2}(x-m)^{2}\right)\exp\left(-\frac{(m-\mu)^{2}}{2\sigma^{2}}\right)dm}_{\sigma_{0}\sqrt{2\pi}\Phi\left(\frac{x-\mu_{0}}{\sigma_{0}}\right)\exp\left(-\frac{(x-\mu)^{2}}{2(h^{-1}+\sigma^{2})}\right)},$
(51)

so

$\mathbb{E}[\varphi(x;M,h)\mid M\leq
x]=\frac{1}{\Phi\left(\frac{x-\mu}{\sigma}\right)}pl_{\widetilde{X}}(x)\Phi\left(\frac{x-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right).$

Using similar calculations, we find

$\mathbb{E}[\varphi(y;M,h)\mid
M>y]=\frac{1}{1-\Phi\left(\frac{y-\mu}{\sigma}\right)}pl_{\widetilde{X}}(y)\left[1-\Phi\left(\frac{y-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)\right],$

which concludes the proof of (24).

Now, let us consider (23). We have

$Bel_{\widetilde{X}}([x,y])=1-Pl_{\widetilde{X}}((-\infty,x]\cup[y,+\infty)),$

and

$Pl_{\widetilde{X}}((-\infty,x]\cup[y,+\infty))=\mathbb{P}(M\leq x)\times
1+\\\ \mathbb{P}(x<M\leq(x+y)/2)\mathbb{E}[\varphi(x;M,h)\mid
x<M\leq(x+y)/2]+\\\ \mathbb{P}((x+y)/2<M\leq
y)\mathbb{E}[\varphi(x;M,h)\mid(x+y)/2<M\leq y]+\mathbb{P}(M>y)\times 1,$ (52)

which can be written as

$Pl_{\widetilde{X}}((-\infty,x]\cup[y,+\infty))=\Phi\left(\frac{x-\mu}{\sigma}\right)+\\\
\left[\Phi\left(\frac{(x+y)/2-\mu}{\sigma}\right)-\Phi\left(\frac{x-\mu}{\sigma}\right)\right]\mathbb{E}[\varphi(x;M,h)\mid
x<M\leq(x+y)/2]+\\\
\left[\Phi\left(\frac{y-\mu}{\sigma}\right)-\Phi\left(\frac{(x+y)/2-\mu}{\sigma}\right)\right]\mathbb{E}[\varphi(x;M,h)\mid(x+y)/2<M\leq
y]+\\\ 1-\Phi\left(\frac{y-\mu}{\sigma}\right).$ (53)

Conditionally on $x<M\leq(x+y)/2$, $M$ has a truncated normal distribution on
$(x,(x+y)/2]$ with pdf

$f(m)=\frac{1}{\sigma\sqrt{2\pi}}\frac{\exp\left(\frac{-(m-\mu)^{2}}{2\sigma^{2}}\right)}{\Phi\left(\frac{(x+y)/2-\mu}{\sigma}\right)-\Phi\left(\frac{x-\mu}{\sigma}\right)}\mathbf{1}_{(x,(x+y)/2]}(m).$

Consequently,

$\mathbb{E}[\varphi(x;M,h)\mid
x<M\leq(x+y)/2]=\frac{1}{\sigma\sqrt{2\pi}}\frac{1}{\Phi\left(\frac{(x+y)/2-\mu}{\sigma}\right)-\Phi\left(\frac{x-\mu}{\sigma}\right)}\times\\\
\underbrace{\int_{x}^{(x+y)/2}\exp\left(-\frac{h}{2}(x-m)^{2}\right)\exp\left(-\frac{(m-\mu)^{2}}{2\sigma^{2}}\right)dm}_{\sigma_{0}\sqrt{2\pi}\left[\Phi\left(\frac{(x+y)/2-\mu_{0}}{\sigma_{0}}\right)-\Phi\left(\frac{x-\mu_{0}}{\sigma_{0}}\right)\right]\exp\left(-\frac{(x-\mu)^{2}}{2(h^{-1}+\sigma^{2})}\right)},$
(54)

or

$\mathbb{E}[\varphi(x;M,h)\mid x<M\leq(x+y)/2]=\\\
\frac{1}{\Phi\left(\frac{(x+y)/2-\mu}{\sigma}\right)-\Phi\left(\frac{x-\mu}{\sigma}\right)}pl_{\widetilde{X}}(x)\left[\Phi\left(\frac{(x+y)/2-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)-\Phi\left(\frac{x-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)\right].$
(55)

Similarly, we find

$\mathbb{E}[\varphi(y;M,h)\mid(x+y)/2<M\leq y]=\\\
\frac{1}{\Phi\left(\frac{y-\mu}{\sigma}\right)-\Phi\left(\frac{(x+y)/2-\mu}{\sigma}\right)}pl_{\widetilde{X}}(y)\left[\Phi\left(\frac{y-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)-\Phi\left(\frac{(x+y)/2-\mu}{\sigma\sqrt{h\sigma^{2}+1}}\right)\right].$
(56)

The expressions of $Pl_{\widetilde{X}}((-\infty,x]\cup[y,+\infty))$ and
$Bel_{\widetilde{X}}([x,y])$ follow.

##  Appendix E Proof of Proposition 8

Let ${\widetilde{X}}(\omega)=\textsf{GFN}(M(\omega),h)$ be the image of
$\omega\in\Omega$ by ${\widetilde{X}}$, with $M\sim N(\mu,\sigma^{2})$. For
any $\alpha\in(0,1]$, its alpha-cut is the random interval

${}^{\alpha}{\widetilde{X}}(\omega)=\left[M(\omega)-\sqrt{\frac{-2\ln\alpha}{h}},M(\omega)+\sqrt{\frac{-2\ln\alpha}{h}}\right].$

Consequently, from (16), the lower and upper expectation of ${\widetilde{X}}$
are

$\mathbb{E}_{*}({\widetilde{X}})=\mu-\int_{0}^{1}\sqrt{\frac{-2\ln\alpha}{h}}d\alpha,$

and

$\mathbb{E}^{*}({\widetilde{X}})=\mu+\int_{0}^{1}\sqrt{\frac{-2\ln\alpha}{h}}d\alpha.$

By the change of variable $\beta=\sqrt{-2(\ln\alpha)/h}$, we get

$\int_{0}^{1}\sqrt{\frac{-2\ln\alpha}{h}}d\alpha=h\int_{0}^{+\infty}\beta^{2}\exp\left(-\frac{h\beta^{2}}{2}\right)d\beta.$

Now, the second-order moment of the normal distribution $N(0,1/h)$ is

$\sqrt{\frac{h}{2\pi}}\int_{-\infty}^{+\infty}\beta^{2}\exp\left(-\frac{h\beta^{2}}{2}\right)d\beta=\frac{1}{h},$

from which we get

$h\int_{0}^{+\infty}\beta^{2}\exp\left(-\frac{h\beta^{2}}{2}\right)d\beta=h\cdot\frac{1}{h}\sqrt{\frac{\pi}{2h}}=\sqrt{\frac{\pi}{2h}}.$

##  Appendix F Proof of Lemma 1

The conditional density of $(M_{1},M_{2})$ is

$f(m_{1},m_{2}\mid{\widetilde{F}})=\frac{f(m_{1},m_{2}){\widetilde{F}}(m_{1},m_{2})}{\iint
f(m_{1},m_{2}){\widetilde{F}}(m_{1},m_{2})dm_{1}dm_{2}}.$ (57)

The numerator on the right-hand side of (57) is

$\frac{1}{2\pi\sigma_{1}\sigma_{2}}\exp\left\\{-\frac{1}{2}\left[\left(\frac{m_{1}-\mu_{1}}{\sigma_{1}}\right)^{2}+\left(\frac{m_{2}-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\\}\exp\left\\{-\frac{{\overline{h}}(m_{1}-m_{2})^{2}}{2}\right\\}\\\
=\frac{1}{2\pi\sigma_{1}\sigma_{2}}\exp\left\\{-\frac{1}{2}\left[m_{1}^{2}\left(\frac{1}{\sigma_{1}^{2}}+{\overline{h}}\right)-\frac{2m_{1}\mu_{1}}{\sigma_{1}^{2}}+\frac{\mu_{1}^{2}}{\sigma_{1}^{2}}+\right.\right.\\\
\left.\left.m_{2}^{2}\left(\frac{1}{\sigma_{2}^{2}}+{\overline{h}}\right)-\frac{2m_{2}\mu_{2}}{\sigma_{2}^{2}}+\frac{\mu_{2}^{2}}{\sigma_{2}^{2}}-2{\overline{h}}m_{1}m_{2}\right]\right\\}.$
(58)

Now, the two-dimensional Gaussian density with parameters
$({\widetilde{\mu}}_{1},{\widetilde{\mu}}_{2},{\widetilde{\sigma}}_{1},{\widetilde{\sigma}}_{2},\rho)$
equals

$\frac{1}{2\pi{\widetilde{\sigma}}_{1}{\widetilde{\sigma}}_{2}\sqrt{1-\rho^{2}}}\exp\left\\{-\frac{1}{2(1-\rho)^{2}}\left[\left(\frac{m_{1}-{\widetilde{\mu}}_{1}}{{\widetilde{\sigma}}_{1}}\right)^{2}-\right.\right.\\\
\left.\left.2\rho\left(\frac{m_{1}-{\widetilde{\mu}}_{1}}{{\widetilde{\sigma}}_{1}}\right)\left(\frac{m_{2}-{\widetilde{\mu}}_{2}}{{\widetilde{\sigma}}_{2}}\right)+\left(\frac{m_{2}-{\widetilde{\mu}}_{2}}{{\widetilde{\sigma}}_{2}}\right)^{2}\right]\right\\}.$
(59)

Equating the second and first-order terms inside the exponentials in (58) and
(59) gives us

$\displaystyle{\widetilde{\sigma}}_{1}$
$\displaystyle=\frac{1}{1-\rho^{2}}\left(\frac{1}{\sigma_{1}^{2}}+{\overline{h}}\right)^{-1}$
(60a) $\displaystyle{\widetilde{\sigma}}_{2}$
$\displaystyle=\frac{1}{1-\rho^{2}}\left(\frac{1}{\sigma_{2}^{2}}+{\overline{h}}\right)^{-1}$
(60b) $\displaystyle\rho$
$\displaystyle=\frac{{\overline{h}}\sigma_{1}\sigma_{2}}{\sqrt{(1+{\overline{h}}\sigma_{1}^{2})(1+{\overline{h}}\sigma_{2}^{2})}}$
(60c) $\displaystyle{\widetilde{\mu}}_{1}$
$\displaystyle=\frac{\mu_{1}{\widetilde{\sigma}}_{1}^{2}}{\sigma_{1}^{2}}+\rho\mu_{2}\frac{{\widetilde{\sigma}}_{1}{\widetilde{\sigma}}_{2}}{\sigma_{2}^{2}}$
(60d) $\displaystyle{\widetilde{\mu}}_{2}$
$\displaystyle=\frac{\mu_{2}{\widetilde{\sigma}}_{2}^{2}}{\sigma_{2}^{2}}+\rho\mu_{1}\frac{{\widetilde{\sigma}}_{1}{\widetilde{\sigma}}_{2}}{\sigma_{1}^{2}}.$
(60e)

Replacing $\rho$ by its expression (60c) in (60a) and (60b) yields (28c) and
(28d). Replacing $\rho$, ${\widetilde{\sigma}}_{1}$ and
${\widetilde{\sigma}}_{2}$ by their expressions in (60d) and (60e) gives (28a)
and (28b).

Finally, the degree of conflict between GRFN’s
${\widetilde{X}}_{1}\sim{\widetilde{N}}(\mu_{1},\sigma_{1}^{2},h_{1})$ and
${\widetilde{X}}_{2}\sim{\widetilde{N}}(\mu_{2},\sigma_{2}^{2},h_{2})$ is

$\kappa=1-(P_{1}\times P_{2})({\widetilde{\Theta}}^{*}),$

with

$(P_{1}\times P_{2})({\widetilde{\Theta}}^{*})=\iint
f(m_{1},m_{2}){\widetilde{F}}(m_{1},m_{2})dm_{1}dm_{2}.$

Taking the ratio of (58) to (59), we get

$\iint f(m_{1},m_{2}){\widetilde{F}}(m_{1},m_{2})dm_{1}dm_{2}=\\\
\frac{{\widetilde{\sigma}}_{1}{\widetilde{\sigma}}_{2}}{\sigma_{1}\sigma_{2}}\sqrt{1-\rho^{2}}\exp\left\\{-\frac{1}{2}\left[\frac{\mu_{1}^{2}}{\sigma_{1}^{2}}+\frac{\mu_{2}^{2}}{\sigma_{2}^{2}}\right]+\frac{1}{2(1-\rho^{2})}\left[\frac{{\widetilde{\mu}}_{1}^{2}}{{\widetilde{\sigma}}_{1}^{2}}+\frac{{\widetilde{\mu}}_{2}^{2}}{{\widetilde{\sigma}}_{2}^{2}}-2\rho\frac{{\widetilde{\mu}}_{1}{\widetilde{\mu}}_{2}}{{\widetilde{\sigma}}_{1}{\widetilde{\sigma}}_{2}}\right]\right\\}.$

##  Appendix G Proof of Proposition 10

From (29), $h_{12}=+\infty$ and the combined GRFN
${\widetilde{N}}({\widetilde{\mu}}_{12},{\widetilde{\sigma}}_{12}^{2},h_{12})$
is probabilistic. From (30) and (31),

${\widetilde{\mu}}_{12}=\lim_{h_{1}\rightarrow+\infty}\frac{{\widetilde{\mu}}_{1}+\frac{h_{2}}{h_{1}}{\widetilde{\mu}}_{2}}{1+\frac{h_{2}}{h_{1}}}={\widetilde{\mu}}_{1},$

and

${\widetilde{\sigma}}_{12}^{2}=\lim_{h_{1}\rightarrow+\infty}\frac{{\widetilde{\sigma}}^{2}_{1}+\frac{h^{2}_{2}}{h^{2}_{1}}{\widetilde{\sigma}}^{2}_{2}+2\rho\frac{h_{2}}{h_{1}}{\widetilde{\sigma}}_{1}{\widetilde{\sigma}}_{2}}{(1+\frac{h_{2}}{h_{1}})^{2}}={\widetilde{\sigma}}^{2}_{1}.$

From (28f),

${\overline{h}}=\lim_{h_{1}\rightarrow+\infty}\frac{h_{2}}{1+\frac{h_{2}}{h_{1}}}=h_{2}.$

From (28a) and (28c),

${\widetilde{\mu}}_{1}=\frac{\mu_{1}(1+h_{2}\sigma_{2}^{2})+\mu_{2}h_{2}\sigma_{1}^{2}}{1+h_{2}(\sigma_{1}^{2}+\sigma_{2}^{2})},$

and

${\widetilde{\sigma}}_{1}^{2}=\frac{\sigma_{1}^{2}(1+h_{2}\sigma_{2}^{2})}{1+h_{2}(\sigma_{1}^{2}+\sigma_{2}^{2})}.$

Now, using Proposition 3, the product of the probability density of $X_{1}$
and the contour function of ${\widetilde{X}}_{2}$ can be written as

$\displaystyle f_{X_{1}}(x)pl_{{\widetilde{X}}_{2}}(x)$
$\displaystyle\propto\exp\left(-\frac{1}{2}\frac{(x-\mu_{1})^{2}}{\sigma_{1}^{2}}\right)\exp\left(-\frac{h_{2}(x-\mu_{2})^{2}}{2(1+h_{2}\sigma_{2}^{2})}\right)$
$\displaystyle\propto\exp\left(-\frac{1}{2\sigma_{12}^{2}}(x-\mu_{12})^{2}\right),$

with

$\frac{1}{\sigma_{12}^{2}}=\frac{1}{\sigma_{1}^{2}}+\frac{h_{2}}{1+h_{2}\sigma_{2}^{2}}=\frac{1+h_{2}(\sigma_{1}^{2}+\sigma_{2}^{2})}{\sigma_{1}^{2}(1+h_{2}\sigma_{2}^{2})}$

and

$\mu_{12}=\frac{\frac{1}{\sigma_{1}^{2}}\mu_{1}+\frac{h_{2}}{1+h_{2}\sigma_{2}^{2}}\mu_{2}}{\frac{1}{\sigma_{1}^{2}}+\frac{h_{2}}{1+h_{2}\sigma_{2}^{2}}}=\frac{\mu_{1}(1+h_{2}\sigma_{2}^{2})+\mu_{2}h_{2}\sigma_{1}^{2}}{1+h_{2}(\sigma_{1}^{2}+\sigma_{2}^{2})}.$

We can check that $\mu_{12}={\widetilde{\mu}}_{1}$ and
$\sigma_{12}^{2}={\widetilde{\sigma}}_{1}^{2}$.

##  Appendix H Proof of Proposition 12

We have

$\displaystyle pl_{\widetilde{X}}({\boldsymbol{x}})$
$\displaystyle=\mathbb{E}_{\boldsymbol{M}}[\varphi({\boldsymbol{x}};{\boldsymbol{M}},{\boldsymbol{H}})]$
(61)
$\displaystyle=\int_{\mathbb{R}^{p}}\varphi({\boldsymbol{x}};{\boldsymbol{m}},{\boldsymbol{H}})\phi({\boldsymbol{m}};\boldsymbol{\mu},\Sigma)d{\boldsymbol{m}}$
(62)
$\displaystyle=\frac{1}{(2\pi)^{p/2}|\boldsymbol{\Sigma}|^{1/2}}\int_{\mathbb{R}^{p}}\exp\left(-\frac{1}{2}({\boldsymbol{x}}-{\boldsymbol{m}})^{T}{\boldsymbol{H}}({\boldsymbol{x}}-{\boldsymbol{m}})\right)\times$