Transforming probability intervals into other uncertainty models

Abstract

Probability intervals are imprecise probability assignments over elementary events. They constitute a very convenient tool to model uncertain information : two common cases are confidence intervals on parameters of multinomial distributions built from sample data and expert opinions provided in terms of such intervals. In this paper, we study how probability intervals can be transformed into other uncertainty models such as possibility distributions, Ferson's p-boxes, random sets and Neumaier's clouds.

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Transforming probability intervals into other uncertainty models
Sébastien Destercke
IRSN (bât 702)
CE Cadarache
13115 Saint Paul Lez Durance
sebastien.destercke@irsn.fr
desterck@irit.fr
Didier Dubois
IRIT
Université Paul Sabatier
118 route de Narbonne
31062 Toulouse Cedex 9, France
dubois@irit.fr
Eric Chojnacki
IRSN (bât 702)
CE Cadarache
13115 Saint Paul Lez Durance
eric.chojnacki@irsn.fr
Abstract
Probability intervals are imprecise proba-
bility assignments over elementary events.
They constitute a very convenient tool to
model uncertain information : two common
cases are confidence intervals on parameters
of multinomial distributions built from sam-
ple data and expert opinions provided in
terms of such intervals.
study how probability intervals can be trans-
formed into other uncertainty models such
as possibility distributions, Ferson’s p-boxes,
random sets and Neumaier’s clouds.
In this paper, we
Keywords: Probability intervals, random
sets, possibility, p-boxes, clouds.
1Introduction
When modeling uncertainty, the theory of imprecise
probabilities [17] and the so-called lower previsions for-
mally subsume most of the existing uncertainty theo-
ries. Provided one accepts its behavioral interpreta-
tion, this theory offers a very appealing unifying and
highly expressive framework. Unfortunately, compu-
tational intractability is often the price to pay for such
generality and expressiveness. It is thus important to
study simpler models that will often be sufficient in
practice to solve a given problem tainted with uncer-
tainty. Even when a lot of information is available (but
not enough to give precise probabilities), one may want
to work with approximated models (e.g. for mathe-
matical or computational tractability considerations).
Probability intervals [1] are among these simpler mod-
els. They are easy to understand and computationally
tractable. They can be, for example, confidence inter-
vals coming from sample data or opinions given by an
expert. Nevertheless, it may happen that one wishes to
map the information given by probability intervals into
another model, because mathematical tools proper to
the latter model must be used or simply to present in-
formation in a specific way. This mapping is the object
of this paper.
In section 2, we briefly recall the various formalism
concerned by this paper. Section 3 then reviews some
existing results relating probability intervals and ran-
dom sets. The next section deals with the relations be-
tween probability intervals and p-boxes. Finally, sec-
tion 5 and 6 are respectively devoted to the transfor-
mation of probability intervals into, respectively, pos-
sibility distributions and clouds (a recent model pro-
posed by Neumaier [15]).
2 Notations and preliminaries
In the paper, we will restrict ourselves to probabil-
ity families denoted P and defined on a finite arbi-
trary space X of n elements {x1,...,xn}. We will now
briefly introduce the models studied in the sequel.
2.1Lower/upper probabilities
Lower (P(A)) and upper probabilities (P(A)) on
events are respectively defined s.t.
infP∈PP(A) and P(A) = supP∈PP(A).
PP,P(A) = {P|∀A ⊆ X measurable, P(A) ≤ P(A) ≤
P(A)}. Although this model is already a restriction
from more general ones (lower/upper probabilities can
be seen as projections of a family P on the subspace
of events, and in general we have P ⊆ PP,P), it is still
fairly general and subsumes all other models studied
here.
P(A)
We have
=
2.2Probability intervals
Probability intervals and their properties are exten-
sively studied in [1]. Probability intervals are defined
as lower and upper bounds of probabilities restricted
to singletons xi. They can be seen as a family of in-
tervals L = {[li,ui],i = 1,...,n} defining the family
PL= {P|li≤ p(xi) ≤ ui∀xi∈ X}.

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In this paper, we will focus on reachable sets L which
define non-empty families PL, since other ones have
little interest. A set L will be called reachable if, for
each xi, we can find a probability distribution P ∈ PL
s.t. P(xi) = liand another one for which P(xi) = ui
(in other words, each bound can be reached by at least
one distribution in the family). Non-emptiness and
reachability respectively correspond to the two condi-
tions
?n
i=1li≤ 1 ≤?n
i=1ui
?
j?=ilj+ ui≤ 1 and
?
j?=iuj+ li≥ 1
∀ i
Given intervals Li, general lower and upper probabil-
ities can be computed through the simple formulas
P(A) = max(?
P(A) = min(?
xi∈Ali,1 −?
xi∈Aui,1 −?
xi/ ∈Aui)
xi/ ∈Ali)
and this lower (upper) probability is an order 2 mono-
tone (alternate) Choquet capacity [2].
2.3Random sets
Formally, a random set is a set-valued mapping from
a (here finite) probability space to a set X. It induces
lower and upper probabilities on X [5]. Here, we use
mass functions [16] to represent random sets. A mass
function m is defined by a mapping from the power
set 2Xto the unit interval, s.t.?
m(∅) = 0. A set E with positive mass is called a focal
set. Two measures, a plausibility and a belief measure
can be defined from this mass function:
E⊆Xm(E) = 1 and
Belief measure:
Bel(A) =?
E,E⊆Am(E)
Plausibility measure:
Pl(A) = 1 − Bel(Ac)
The set PBel = {P|∀A ⊆ X measurable, Bel(A) ≤
P(A) ≤ Pl(A)} is the probability family induced by
the belief function.
2.4 (Generalized) P-boxes
A p-box is usually defined on the real line by a pair of
cumulative distributions [F,F], defining the probabil-
ity family P[F,F]= {P|F(x) ≤ F(x) ≤ F(x)
?}. The notion of cumulative distribution on the
real line is based on a natural ordering of numbers.
In order to generalize this notion to arbitrary finite
∀x ∈
sets, we need to define a weak order relation ≤R on
this space. Given ≤R, an R-downset is of the form
{xi: xi≤Rx}, and denoted (x]R. A generalized R-
cumulative distribution [7] is defined as the function
FR: X → [0,1] s.t. FR(x) = Pr((x]R), where Pr is a
probability measure on X. We can now define a gen-
eralized p-box as a pair [FR(x),FR(x)] of generalized
cumulative distributions defining a probability family
P[FR(x),FR(x)]= {P|∀x, FR(x) ≤ FR(x) ≤ FR(x)}.
Generalized P-boxes can also be represented by a set
of constraints
αi≤ P(Ai) ≤ βi
i = 1,...,n
(1)
where 0 ≤ α1≤ α2≤ ... ≤ αn≤ 1, 0 ≤ β1≤ β2≤
... ≤ βn≤ 1 and Ai= (xi]R,∀xi∈ X with xi≤Rxj
iff i < j (sets Aiform a sequence of nested confidence
sets ∅ ⊂ A1⊂ A2⊂ ... ⊂ An⊂ X). When X = ?
and Ai= (−∞,xi], we find back the usual definition
of p-boxes.
2.5Possibility distributions
A possibility distribution π is a mapping from X to the
unit interval (hence a fuzzy set) such that π(x) = 1 for
some x ∈ X. Several set-functions can be defined from
them [8]:
Possibility measures:
Π(A) = supx∈Aπ(x)
Necessity measures:
N(A) = 1 − Π(Ac)
Guaranteed poss. measures:
∆(A) = infx∈Aπ(x)
Possibility degrees express the extent to which an
event is plausible, i.e., consistent with a possible state
of the world, necessity degrees express the certainty of
events and ∆-measures the extent to which all states
of the world where A occurs are plausible. They ap-
ply to so-called guaranteed possibility distributions [8]
generally denoted by δ.
A possibility degree can be viewed as an upper bound
of a probability degree [9].
X measurable, P(A) ≤ Π(A)} be the set of probabil-
ity measures encoded by π. A necessity measure is a
special case of belief function when the focal sets are
nested.
Let Pπ = {P,∀A ⊆
2.6 Clouds
Formally, a cloud is described by an Interval-Valued
Fuzzy Set (IVF) s.t. (0,1) ⊆ ∪x∈XF(x) ⊆ [0,1], where
F(x) is an interval [δ(x),π(x)]. A cloud is called thin
when the two membership functions coincide (δ = π).
It is called fuzzy when the lower membership function
δ is 0 everywhere.Let αi be a sequence of α-cuts
s.t. 1 = α0 > α1 > α2 > ... > αn > αn+1 = 0

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with Aithe strong α-cut of π and Bj the α-cut of δ
(Ai = {xi,π(xi) > αi} and Bi = {xi,δ(xi) ≥ αi}).
Then, a random variable x is in a cloud if it satisfies
the constraints
P(Bi) ≤ 1 − αi≤ P(Ai)
i = 1,...,n.
(2)
with Bi ⊆ Ai. The probability family induced by a
cloud [δ(x),π(x)] will be noted P[δ,π].
Moreover, the following property linking clouds and
possibility distributions has been shown by Dubois and
Prade [11]:
Proposition 1. A probability family P[δ,π]described
by the cloud (δ,π) is equivalent to the family Pπ∩P1−δ
described by the two possibility distributions π and 1−
δ.
3Probability intervals and random
sets
Most of the results presented in this section can be
found in [1, 6], where more details can be found. There
are two main approaches to build a belief function Bel
from a set Leof probability intervals.
The first one, explored in [13] by Lemmer and Kyburg,
considers probability intervals as a partial specification
of a belief function and consists to find a belief function
Bel1that extends the intervals s.t.
Bel1(xi) = liand Pl1(xi) = ui∀i
(3)
As shown in [13], finding such a belief function is pos-
sible iff the tree following conditions hold
?n
i=1li≤ 1 ≤?n
i=1ui
?
j?=ilj+ ui≤ 1 and
?
j?=iuj+ li≥ 1
∀ i
?n
i=1li+?n
i=1ui≥ 2
where the two first conditions correspond to non-
emptiness and reachability (which are always satisfied
by supposition). Lemmer and Kyburg also provide a
means to build one of the belief function satisfying
constraints (3). Let us note that with this method,
we have Bel1(A) ≥ P(A), which imply PBel1⊂ PLe.
Thus, The belief function Bel1is an inner approxima-
tion of the family PLe.
The second approach, extensively explored by De-
noeux [6], considers probability intervals Le as some
"most committed" information and try to find a
conservative belief function Bel2 s.t.
PLe(A) ∀A. Since there exist a lot of such belief func-
tions, Denoeux also proposes to find the belief function
Bel2(A) ≤
Bel2that maximizes a given specifity criterion, in or-
der to keep as much information as possible (in [6], this
criterion is the sum of belief degrees over events). Ob-
viously, with this approach, we have PLe⊂ PBel2, and
Bel2is this time an outer approximation of the family
PLe. Methods given in [6] insure that Bel2(xi) = li,
but it can be checked from the examples in [6] that
Pl2(xi) > ui, even if the three conditions required by
the first approach hold.
4Probability intervals and
(Generalized) P-boxes
Given a set L of probability intervals and a mean-
ingful ordering relation ≤Rbetween elements xi, one
can easily build a generalized p-box [F,F]L from L.
Given the consecutive sets Ai = (xi]R,∀xi ∈ X and
the ordering s.t. xi≤Rxj iff i < j, lower and upper
generalized cumulative distributions corresponding to
L are, respectively
FR(xi) = P(Ai) = max(?
FR(xi) = P(Ai) = min(?
xi∈Ailj,1 −?
xi∈Aiui,1 −?
xi/ ∈Aiuj)
xi/ ∈Aili)
(4)
Now, if we consider that this p-box is all the infor-
mation we have, one can easily find back probability
intervals L?from this information s.t.
P?(xi) = l?
i= max(0,P(Ai) − P(Ai−1))
?(xi) = u?
P
i= P(Ai) − P(Ai−1)
and we have the following proposition
Proposition 2. Given an initial set L of probability
intervals over a space X, and given the transforma-
tions
− − − − →
p − box
[FR(x),FR(x)]
Set L
− − − − − − →
IntervalSet L?
we have that PL⊆ PL?.
Proof. Looking at equations given above, we can easily
express [l?
i] in term of values li,uiof the original set
L, this gives us
i,u?
l?
i
= max(0,
?
xi∈Ai
li−
?
xi∈Ai−1
ui,
?
xi∈Ai
li+
?
xi?∈Ai−1
li− 1,
1 −
?
xi?∈Ai
ui−
?
xi∈Ai−1
ui,
?
xi?∈Ai−1
li−
?
xi?∈Ai
ui)

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= max(0,li+
?
xi∈Ai−1
(li− ui),
li+ (
?
X
?
X\xi
li− 1),
1 −
ui
li+
?
xi?∈Ai
(li− ui))
and, given that ui ≥ li and that the set L is reach-
able and non-empty, we have that l?
procedure can be followed for the bounds u?
have PL⊆ PL?. We also have l?
in degenerate cases of precise information or complete
ignorance.
i≤ li. The same
i, and we
i= li;u?
i= ui∀i only
Not surprisingly, this proposition shows that trans-
forming probability intervals into p-boxes implies a
loss of information. Let us note that the same argu-
ment holds when one wants to transform a generalized
p-box [F,F] into probability intervals, and then get
back a generalized p-box [F?,F
using equations (4) (i.e. we have P[F,F]⊂ P[F?,F?]).
Example 1. Let us take the same four probability in-
tervals as in the example given in [14], summarized in
the following tabular
?] from these intervals
x1
0.10
0.28
x2
0.34
0.56
x3
0.25
0.46
x4
0
0.08
li
ui
if we consider the order R s.t. xi≤Rxjiff i ≤ j, this
gives us the following generalized p-box
FF
A1= {x1}
A2= {x1,x2}
A3= {x1,x2,x3}
A4= X
0.10
0.46
0.92
1
0.28
0.75
1
1
and if we want to get back probability intervals from
this sole generalized p-box, we obtain
x1
0.10
0.28
x2
0.18
0.65
x3
0.17
0.54
x4
0
0.08
l?
u?
i
i
which is clearly less informative than the first proba-
bility intervals.
5Probability intervals and possibility
distributions
The problem of transforming a probability distribution
into a quantitative possibility distribution has been
adressed by many authors (see [10] for an extended
discussion about the links between probabilities and
quantitative possibility theory). In this paper, we will
follow the same line as Masson and Denoeux in [14],
where authors study the problem of transforming in-
terval probabilities into a possibility distribution.
Concerning the transformation of a precise probability
into a possibility distribution, a first consistency prin-
ciple was informally stated by Zadeh [18] as: what is
probable should be possible. It was later translated by
Dubois and Prade [12] as the mathematical constraint
P(A) ≤ Π(A)
∀A ⊆ X
and the possibility measure Π is said to dominate
the probability measure P. The transformation of a
probability into a possibility then consists of choos-
ing a possibility measure amongst those dominating
P. Dubois and prade [12] then proposed to add the
following constraint
p(xi) ≤ p(xj) ↔ π(xi) ≤ π(xj)
and to choose the least specific possibility distribution
(π?is more specific than π if π?≤ π ∀x) respecting
these two constraints. They showed that the solution
exists and is unique.Let us consider the order on
probability masses s.t.
p(x1) ≤ p(x2) ≤ ... ≤ p(xj) ≤ ... ≤ p(xn)
Dubois and Prade’s transformation can then be for-
mulated as
π(xi) =
i
?
j=1
p(xj)
When working with a set L = {[li,ui],i = 1,...,n} of
probability intervals, the order induced on probability
masses is no longer complete, and the partial order is
reduced to
p(xi) ≤ p(xj) ↔ ui≤ lj
and two probabilities p(xi),p(xj) are incomparable if
intervals [li,ui],[lj,uj] intersect in some way. Let us
note M this partial order and C the set of its linear ex-
tensions (a linear extension Cl∈ C is a complete order
that is compatible with the partial order M). Given
this partial order, Masson and Denoeux [14] propose
the following procedure to transform the set of proba-
bility intervals into a possibility distribution:
1. For each order Cl∈ C and each element xi, solve
π(xi)Cl=max
p(x1),...,p(xn)
?
(j)≤σ−1
σ−1
ll
(i)
p(xj)
(5)

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under the constraints



?
k=1,...,np(xk) = 1
lk≤ pk≤ uk
p(xσl(1)) ≤ p(xσl(2)) ≤ ... ≤ p(xσl(n))
where σl is the permutation of p(xk) associated
with the linear extension Cl
2. Take the distribution dominating all distributions
π(xi)Cls.t.
π(xi) = max
Cl∈Cπ(xi)Cl
∀i
(6)
this procedure insures that the resulting possibility
distribution π will dominate every probability distri-
bution contained in PL. We thus have PL⊂ Pπ. Let
us note that this transformation can produce an im-
portant loss of information.
Example 2. Taking back probability intervals of ex-
ample 1, we have here three possible linear extensions
Cl∈ C
C1
C2
C3
=(L4,L1,L3,L2)
(L4,L1,L2,L3)
(L4,L3,L1,L2)
=
=
using the above method on these three orders gives the
following distribution (see [14] for more details)
x1
0.64
x2
1
x3
1
x4
0.08
π
Let us note that, if this transformation produces the
most specific possibility distribution dominating a pre-
cise probability distribution (i.e.
?ui= 1), this is no longer the case when the probabil-
ity distribution is imprecisely known through intervals
(ui ?= li). To see this, let us first recall that an up-
per generalized R-cumulative distribution FRcan be
seen as a possibility distribution dominating a prob-
ability measure, since it is a maxitive measure (i.e.
we have maxx∈AFR(x) ≥ Pr(A),∀A ⊆ X). Any R-
cumulative distribution dominating a family PL in-
duced by probability intervals is thus also a possibility
measure dominating this family. Now, in our example,
let us consider the following order <Rbetween the four
elements: x4<Rx1<Rx3<Rx2. From this order,
we can build the following R-cumulative distribution:
ui = li ∀i and
x1
0.36
x2
1
x3
0.66
x4
0.08
FR= πR
which still dominates PLand is more specific than the
distribution built by the method given in [14]. This
shows that Masson and Denoeux’s method tends to
give conservative bounds, and thus can result in an
important loss of information, which seems hard to
justify. This important loss is due to the fact that Mas-
son and Denoeux do not consider any specific ordering
relation R between the elements of X, an assumption
that is made in the counter-example given above.
6Probability intervals and clouds
Let L be a set of probability intervals and PLthe asso-
ciated family. In this section, we introduce a method,
inspired from [14], which transforms intervals L into a
cloud [δ,π]L.
From property 1, we know that families described
by clouds are equivalent to the intersection of two
families described by possibility distributions.
other property of clouds is that a discrete thin cloud,
up to the right transformation, can represent a pre-
cise probability distribution.
a0 = 0 < a1 < ... < an = 1.
π2(xi) = 1 − ai−1 = 1 − δ(xi). Now let us consider
the cloud [δ = 1 − π2,π = π1]. It has been proved
by Dubois and Prade [11] that the probability family
P[δ,π]induced by this cloud contains a unique proba-
bility measure P s.t. p(xi) = ai− ai−1∀i = 1,...,n.
An-
Let π1(xi) = ai s.t.
Consider then
Given this property, two requirements that should, in
our opinion, follow any transformation of probability
intervals into clouds are the following:
• If intervals L describe a precise probability distri-
bution, then the transformation should result in
the corresponding thin cloud.
• The family P[δ,π]Lshould be an outer approxima-
tion of PL(i.e. PL⊂ P[δ,π]L)
Let us consider the distribution π built with the
method described in section 5 as the upper distribu-
tion of the cloud. By reversing the inequality under
the summand in equation (5), we can build another
distribution πδin the following way:
1. For each order Cl∈ C and each element xi, solve
πCl
δ(xi)= max
p(x1),...,p(xn)
?
(i)≤σ−1
σ−1
ll
(j)
p(xj)
=1 −
min
p(x1),...,p(xn)
?
(j)<σ−1
σ−1
ll
(i)
p(xj)
=1 − δCl(xi)
with the same constraints as in section 5
2. Take the distribution dominating all distributions
πCl
δ(xi)
πδ(xi) = 1 − δ(xi) = max
Cl∈CπCl
δ(xi)
∀i
(7)

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And [δ,π] is actually a cloud (δ < π, since δ(xi) is
a minimum computed over less intervals L(xi) than
π(xi), which is moreover a maximum). It can be eas-
ily checked that if L is reduced to a precise probability
distribution, then the above method results in the thin
cloud corresponding to this distribution. We also have
that PL⊂ Pπδand PL⊂ Pπby construction, thus we
have PL⊂ P[δ,π]L= Pπδ∩ Pπ, and our two require-
ments are satisfied.
Example 3. Again, we consider the probability in-
tervals given in example 1. From the three orders Ci
given above, we can obtain the following πδ
Ci
1
2
3
max
πδ(x1)
1
1
0.75
1
πδ(x2)
0.16
0.9
0.5
0.9
πδ(x3)
0.63
0.46
1
1
πδ(x4)
1
1
1
1
and, finally, the following cloud
x1
0.64
0
x2
1
0.1
x3
1
0
x4
0.08
0
π
δ
which, in this case, is only a little more informative
than the upper distribution taken alone (indeed, the
sole added constraint is that p(x2) ≤ 0.9).
As in the previous section, this method can be criti-
cized upon the ground that an important amount of
information is lost. Instead, one could take, for ex-
ample, the cloud associated to the generalized p-box
induced by the order x4<R x1<R x3<R x2 (since
generalized p-boxes are a particular case of clouds).
This would give the following distributions:
x1
0.36
0.1
0
x2
1
1
0.44
x3
0.66
0.44
0.1
x4
0.08
0
0
FR= πR
FR
δR
Where δR is FRafter a simple shift of values (the
necessity of this transformation, already emphasized
in [11], arises from the fact that sets Aiare strong α-
cuts, while Biare simple α-cuts). The cloud [δR,πR]
is obviously more specific than the first cloud while we
still have PL⊂ P[δR,πR].
7Conclusions
Probability intervals are very convenient to model un-
certainty, and can be encountered in various situations.
In this paper, we study how they can be transformed
into other popular models of imprecise probabilities.
Except for one, every method recalled or proposed here
give outer approximation of the family PL. This corre-
sponds to a cautious view, since there is no additional
information present in the model resulting from the
transformation (but some information can be lost in
the transformation process).
If probability intervals are reduced to precise probabil-
ity distributions, every proposed transformation result
in the model corresponding to this precise probability,
except for possibility distributions, which are the only
model studied here that can’t be seen as a generaliza-
tion of classical probabilities.
Interestingly enough, most (some subclasses of clouds
are not) of the studied representations in this paper are
special cases of random sets, allowing one to use all the
rich mathematical background of this theory as well
as many computational simplifications (e.g. easiness
to compute Choquet Integral).
Each of the models presented here has its own practical
interest in term of expressiveness or tractability. The
natural continuation of the work initiated in this paper
is to extend the study made by De Campos et al. [1]
to every model studied here. What becomes of ran-
dom sets, possibility distributions, generalized p-boxes
and clouds after fusion, marginalization, conditioning
or propagation? Is it still the same kind of represen-
tation after having applied these mathematical tools,
and under which assumptions? To which extends are
these representations informative? Can they easily be
elicited or integrated? If many results already exist
for random sets and possibility distributions, few have
been derived for generalized p-boxes or clouds, due to
the fact that these two latter representations have only
been recently proposed.
In term of behavioral interpretation of imprecise prob-
abilities [17], conditions of non-emptiness and reacha-
bility respectively correspond to avoiding sure loss and
to coherence of lower previsions. It is also interesting
to note that, if we see probability intervals as a con-
straint satisfaction problem (CSP) [4], non-emptiness
and reachability correspond to the notions of existence
of a solution and of bounds consistency. Another inter-
esting work should be to formalize links between CSP
and imprecise probabilities, with the aim to study to
which extent technics used in CSP can be used to solve
practical problems commonly encountered when deal-
ing with imprecise probabilities. To our knowledge,
this work largely remains to be done, although CSP
technics are already used to solve practical problems
related to (imprecise) probabilistic reasoning, most of
them being related to (credal) bayesian networks (see,
for example [3]).
Acknowledgement
This paper has been supported by a grant from the

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Institut de Radioprotection et de Sûreté Nucléaire
(IRSN). Scientific responsibility rests with the authors.
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