Possibility Theory and Statistical Reasoning

Didier Dubois
Institut de Recherche en Informatique de Toulouse
Université Paul Sabatier, 118 route de Narbonne
31062 Toulouse Cedex 4, France

May 15, 2006

Abstract

Numerical possibility distributions can encode special convex families of probability mea-
sures. The connection between possibility theory and probability theory is potentially fruitful
in the scope of statistical reasoning when uncertainty due to variability of observations should be
distinguished from uncertainty due to incomplete information. This paper proposes an overview
of numerical possibility theory. Its aim is to show that some notions in statistics are naturally
interpreted in the language of this theory. First, probabilistic inequalites (like Chebychev’s)
offer a natural setting for devising possibility distributions from poor probabilistic information.
Moreover, likelihood functions obey the laws of possibility theory when no prior probability is
available. Possibility distributions also generalize the notion of confidence or prediction inter-
vals, shedding some light on the role of the mode of asymmetric probability densities in the
derivation of maximally informative interval substitutes of probabilistic information. Finally,
the simulation of fuzzy sets comes down to selecting a probabilistic representation of a possibility
distribution, which coincides with the Shapley value of the corresponding consonant capacity.
This selection process is in agreement with Laplace indifference principle and is closely con-
nected with the mean interval of a fuzzy interval. It sheds light on the “defuzzification” process
in fuzzy set theory and provides a natural definition of a subjective possibility distribution that
sticks to the Bayesian framework of exchangeable bets. Potential applications to risk assessment
are pointed out.

1 Introduction

There is a continuing debate in the philosophy of probability between subjectivist and objectivist
views of uncertainty. Objectivists identify probabilities with limit frequencies and consider subjec-
tive belief as scientifically irrelevant. Conversely subjectivists consider that probability is tailored
to the measurement of belief, and that subjective knowledge should be used in statistical inference.
Both schools anyway agree on the fact that the only reasonable mathematical tool for uncertainty
modelling is the probability measure. Yet, the idea, put forward by Bayesian subjectivists, that
it is always possible to come up with a precise probability model, whatever the problem at hand
looks debatable. This claim can be challenged due to the simple fact that there are at least two

1
kinds of uncertain quantities: those which are subject to intrinsic variability (the height of adults
in a country), and those which are totally deterministic but anyway ill-known, either because they
pertain to the future (the date of the death of a living person), or just because of a lack of knowl-
edge (I may not know in this moment the precise age of the President). It is clear that the latter
cause of uncertainty is not “objective”, because when lack of knowledge is at stake, it is always
somebody’s knowledge, which may differ from somebody else’s knowledge. However, it is not clear
that incomplete knowledge should be modelled by the same tool as variability (a unique probability
distribution) [72, 42, 49]. One may argue with several prominent other scientists like Dempster [19]
or Walley [108] that the lack of knowledge is precisely reflected by the situation where the proba-
bility of events is ill-known, except maybe for a lower and an upper bound. Moreover one may also
have incomplete knowledge about the variability of a non-deterministic quantity if the observations
made were poor, or if only expert knowledge is available. This point of view may to some extent
reconcile subjectivists and objectivists: it agrees with subjectivists that human knowledge matters
in uncertainty judgements, but it concedes to objectivists that such knowledge is generally not rich
enough to allow for a full-fledged probabilistic modelling.

Possibility theory is one of the current uncertainty theories devoted to the handling of incomplete
information, more precisely it is the simplest one, mathematically. To a large extent, it is similar to
probability theory because it is based on set-functions. It differs from the latter by the use of a pair
of dual set functions called possibility and necessity measures [28] instead of only one. Besides, it is
not additive and makes sense on ordinal structures. The name “Theory of Possibility” was coined
by Zadeh [119]. In Zadeh’s view, possibility distributions were meant to provide a graded semantics
to natural language statements. However, possibility and necessity measures can also be the basis
of a full-fledged representation of partial belief that parallels probability. It can be seen either as
a coarse, non-numerical version of probability theory, or a framework for reasoning with extreme
probabilities (Spohn [102]), or yet a simple approach to reasoning with imprecise probabilities
[36]. The theory of large deviations in probability theory also handles set-functions that look like
possibility measures [90]. Formally, possibility theory refers to the study of maxitive and minitive
set-functions, respectively called possibility and necessity measures such that the possibility degree
of a disjunction of events is the maximum of the possibility degrees of events in the disjunction,
and the necessity degree of a conjunction of events is the minimum of the necessity degrees of
events in the conjunction. There are several branches of possibility theory, some being qualitative,
others being quantitative, all satisfying the maxitivity and minitivity properties. But the variants of
possibility theory differ for the conditioning operation. This survey focuses on numerical possibility
theory. In this form, it looks of interest in the scope of coping with imperfect statistical information,
especially non-Bayesian statistics relying on likelihood functions (Edwards, [46]), and confidence
or prediction intervals. Numerical possibility theory provides a simple representation of special
convex sets of probability functions in the sense of Walley [108], also a special case of Dempster’s
upper and lower probabilities [19], and belief functions of Shafer [96]. Despite its radical simplicity,
this framework is general enough to model various kinds of information items: numbers, intervals,
consonant (nested) random sets, as well as linguistic information, and uncertain formulae in logical
settings [39].

Just like probabilities being interpreted in different ways (e.g., frequentist view vs. subjective view),
possibility theory can support various interpretations. Hacking [63] pointed out that possibility can
be understood either as an objective notion (referring to properties of the physical world) or as
an epistemic one (referring to the state of knowledge of an agent). Basically there are four ideas

2
each of which can be conveyed by the word ’possibility’. First is the idea of feasibility, such as
ease of achievement, also referring to the solution to a problem, satisfying some constraints. At the
linguistic level, this meaning is at work in expressions such as “it is possible to solve this problem”.
Another notion of possibility is that of plausibility, referring to the propensity of events to occur.
At the grammatical level, this semantics is expressed by means of sentences such as “it is possible
that the train arrives on time”. Yet another view of possibility is logical and it refers to consistency
with available information. Namely, stating that a proposition is possible means that it does not
contradict this information. It is an all-or-nothing version of plausibility. The last semantics of
possibility is deontic, whereby possible means allowed, permitted by the law. In this paper, we
focus on the epistemic view of possibility, which also relies on the idea of logical consistency. In
this view, possibility measures refer to the idea of plausibility, while the dual necessity functions
attempt to quantify the idea of certainty. Plausibility is dually related to certainty, in the sense
that the certainty of an event reflects a lack of plausibility of its opposite. This is a striking
difference with probability which is self-dual. The expression It is not probable that “not A” is
equivalent to saying It is probable that A, while the statement It is not possible that “not A” is not
equivalent to saying It is possible that A. It has a stronger meaning, namely: It is necessary that
A. Conversely, asserting that it is possible that A does not entail anything about the possibility
nor the impossibility of “not A”. Hence we need a dual pair of possibility and necessity functions.

There are not so many extensive works on possibility theory. The notion of epistemic possibility
seems to appear during the 1940’s in the work of the English economist G. L. S. Shackle [95],
who called degree of potential surprise of an event its degree of impossibility, that is, the degree
of certainty of the opposite event. The first book on possibility theory based on Zadeh’s view [33]
emphasises the close links between possibility theory and fuzzy sets, and mainly deals with numer-
ical possibility and necessity measures. However it already points out some links with probability
and evidence theories. Klir and Folger [76] insist on the fact that possibility theory is a special
case of belief function theory, with again a numerical flavour. A more recent detailed survey [38]
distinguishes between quantitative and qualitative sides of the theory. Basic mathematical aspects
of qualitative possibility theory are studied at length by De Cooman [14]. More recently this author
has investigated numerical possibility theory as a special case of imprecise subjective probability
[16]. Qualitative possibility theory is also studied in detail by Halpern [64] in connection with other
theories of uncertainty. The paper by Dubois, Nguyen and Prade [39] provides an overview of the
links between fuzzy sets, probability and possibility theory. Uncertainty theories are also reviewed
in the recent book by Klir [75] with emphasis on the study on information measures.

This paper is devoted to a survey of results in quantitative possibility theory, pointing out connec-
tions to probability theory and statistics. It however centers on the author’s and close colleagues’
views of the topic and does not develop mathematical aspects. The next section provides the basic
equations of possibility theory. Section 3 shows how to interpret possibility distributions in the
probabilistic setting. Section 4 bridges the gap between possibility distributions and confidence
intervals. Section 5 envisages possibility theory from the standpoint of subjective probability and
the principle of Insufficient Reason. Section 6 discusses the possibilistic counterpart to mathemat-
ical expectation. The last section points out the potential of possibility theory for uncertainty
propagation in risk assessment.

3
2 Basics of Possibility Theory

The primitive object of possibility theory is the possibility distribution, which assigns to each element
u in a set U of alternatives a degree of possibility π(u) ∈ [0, 1] of being the correct description of a
state of affairs. This possibility distribution is a representation of what an agent knows about the
value of some quantity x ranging on U (not necessarily a random quantity). Function πx reflects the
more or less plausible values of the unknown quantity x. These values are assumed to be mutually
exclusive, since x takes on only one value (its true value). When πx (u) = 0 for some u, it means
that x = u is considered an impossible situation. When πx (u) = 1, it means that x = u is just
unsurprising, normal, usual, a much weaker statement than when probability is 1. Since one of the
elements of U is the true value, the condition
∃u, πx (u) = 1
is assumed for at least one value u ∈ U . This is the normalisation condition. It claims that at least
one value is viewed as totally possible. Indeed if ∀u ∈ U, πx (u) < 1, the representation would be
logically inconsistent because suggesting that all values in U are partially impossible for x. The
degree of consistency of a submormalized possibility distribution is cons(π) = supu∈U πx (u).

2.1 The Logical View

The simplest form of a possibility distribution on the set U is the characteristic function of a subset
E of U , i.e.,πx (u) = 1 if x ∈ E, 0 otherwise. It models the situation when all that is known about
x is that it cannot lie outside E. This type of possibility distribution is naturally obtained from
experts stating that a numerical quantity x lies between values a and b; then E is the interval [a, b].
This way of expressing knowledge is more natural than arbitrarily assigning a point-value u∗ to x
right away, because it allows for some imprecision.

Possibility as logical consistency has been put forward by Yager [113]. Consider a piece of incomplete
information stating that “x ∈ E” is known for sure. This piece of information is incomplete insofar
as E contains more than one element, i.e., the value of x can be any one (but only one) of the
elements in E. Given such a piece of information, a set-function ΠE is built from E by the following
procedure 
1, if A ∩ E 6= ∅ (x ∈ A and x ∈ E are consistent)
ΠE (A) = (1)
0, otherwise (A and E are mutually exclusive).
Clearly, ΠE (A) = 1 means that given that x ∈ E, x ∈ A is possible because the intersection
between set A and set E is not empty, while ΠE (A) = 0 means that x ∈ A is impossible knowing
that x ∈ E. It is easy to verify that such Boolean possibility functions ΠE satisfy the “maxitivity”
axiom :
ΠE (A ∪ B) = max(ΠE (A), ΠE (B)). (2)
Dually, a proposition is certain if and only if it logically derives from the available knowledge.
Hence necessity also conveys a logical meaning in terms of deduction, just as possibility is a matter
of logical consistency. The certainty of the event x ∈ A, knowing that x ∈ E, is then evaluated by
the following index, called necessity measure:

1, if E ⊆ A
NE (A) = (3)
0, otherwise.

4
Clearly the information x ∈ E logically entails x ∈ A when E is contained in A, so that certainty
applies to events that are logically entailed by the available evidence. It can be easily seen that
NE (A) = 1 − ΠE (Ac ), denoting Ac the complement of A. In other words, A is necessarily true if
and only if “not A” is impossible. Necessity measures satisfy the “minitivity” axiom

NE (A ∩ B) = min(NE (A), NE (B)). (4)

2.2 Graded possibility and necessity

However this binary representation is not entirely satisfactory. If the set E is too narrow, the piece
of information is not so reliable. One is then tempted to use wide uninformative sets or intervals
for the range of x. And sometimes, even the widest, safest set of possible values does not rule
out some residual possibility that the value of x lies outside it. So it is natural to use a graded
notion of possibility. Then, formally a possibility distribution π coincides with the membership
function µF of a fuzzy subset F of U after Zadeh [117]. For instance, a fuzzy interval is a fuzzy
set M of real numbers whose α-cuts Mα = {u, µM (u) ≥ α} for 0 < α ≤ 1, are nested intervals,
usually closed ones (see Dubois, Kerre et al. [26] for an extensive survey). If u and u0 are such that
πx (u) > πx (u0 ), u is considered to be a more plausible value than u0 . The possibility degree of an
event A, understood as a subset of U is then

Π(A) = sup πx (u). (5)

u∈A

It is computed on the basis of the most plausible values of x in A, neglecting other realisations.
The degree of necessity is then N (A) = 1 − Π(Ac ) = inf u∈A
/ 1 − πx (u) [28, 33].

A possibility distribution πx is at least as informative (we say specific) as another one πx0 if and only
if πx ≤ πx0 (see, e.g., Yager, [115] ). In particular, if ∀u ∈U, πx (u) = 1, πx contains no information at
all (since it expresses that any value in U is possible for x). The corresponding possibility measure
is then said to be vacuous and denoted Π> .

Remark. The possibility measure defined in (5) satisfies a strong form of maxitivity (2) for the
union of infinite families of sets. On infinite sets, axiom (2) alone does not imply the existence of a
possibility distribution satisfying (5). For instance, consider the natural integers, and a set function
assigning possibility 1 to infinite subsets of integers, possibility 0 to finite subsets. This function is
maxitive in the sense of (2) but does not fulfil (5) since π(n) = Π({n}) = 0, ∀n = 0, 1 . . ..

The possibilistic representation is capable of modelling several kinds of imprecise information within
a unique setting. It is more satisfactory to describe imprecise numerical information by means of
several intervals with various levels of confidence rather than a single interval. Possibility distri-
butions can be obtained by extracting predictions intervals from probability measures as shown
later on, or more simply by linear approximation between a core and a support provided by some
expert. A possibility distribution πx can more generally represent a finite family of nested confi-
dence subsets {A1 , A2 , . . . , Am } where Ai ⊂ Ai+1 , i = 1, . . . , m − 1. Each confidence subset Ai is
attached a positive confidence level λi . The links between the confidence levels λi ’s and the degrees
of possibility are defined by postulating λi is the degree of necessity (i.e. certainty) N (Ai ) of Ai .
It entails that λ1 ≤ . . . ≤ λm due to the monotonicity of the necessity function N . The possibility

5
distribution equivalent to the weighted family {(A1 , λ1 ), (A2 , λ2 ), . . . , (Am , λm )} is defined as the
least informative possibility distribution π that obeys the constraints λi = N (Ai ), i = 1, . . . , m. It
comes down to maximizing the degrees of possibility π(u) for all u ∈ U , subject to these constraints.
The solution is unique and is, ∀u,

1, if u ∈ A1
πx (u) = (6)
/ i 1 − λi , otherwise,
mini:u∈A

which also reads

πx (u) = min max(1 − λi , Ai (u)),
i=1,...,m

where Ai (·) is the characteristic function of Ai . The set of possibility values {π(u) : u ∈ U } is then
finite. This solution is the least committed one with respect to the available data, since by allowing
the greatest possibility degrees in agreement with the constraints, it defines the least restrictive
possibility distribution. Conversely, the family {(A1 , λ1 ), (A2 , λ2 ), . . . , (Am , λm )} of confidence in-
tervals can be reconstructed from the possibility distribution πx . Suppose that the set of possibility
values πx (u) is {α1 = 1, α2 ≥ α3 . . . ≥ αm } and let αm+1 = 0. Then

Ai = {u : πx (u) ≥ αi }; λi = 1 − αi+1 , ∀i = 1, . . . , m.

In particular, if λm = 1, then Am is the subset which for sure contains x; moreover, Am = U if no

strict subset of U surely includes x. This analysis extends to an infinite nested set of confidence
intervals. Especially if M is a fuzzy interval with a continuous membership function, then the corre-
sponding set of confidence intervals is {(Mα , 1 − α), 1 ≥ α > 0}. Denoting Mα (·) the characteristic
function of the cut Mα , it holds that N (Mα ) = 1 − α, and µM (u) = inf α∈(0,1] max(α, Mα (u))(=
inf{α ∈ (0, 1] : α > µM (u)}).

2.3 Conditioning in possibility theory

Conditioning in possibility theory has been studied as a counterpart to probabilistic conditioning.

However there is no longer a unique meaningful definition of conditioning, unlike in probability
theory. Moreover the main difference between numerical and qualitative possibility theories lies in
the conditioning process. The first notion of conditional possibility measure goes back to Hisdal
[69]. She introduced the set function Π(· | A) through the equality

∀B, B ∩ A 6= ∅, Π(A ∩ B) = min(Π(B | A), Π(A)). (7)

In order to overcome the existence of several solutions to this equation, the conditional possibility
measure can be defined, as proposed by Dubois and Prade [33], as the least specific solution to this
equation, that is, when Π(A) > 0 and B 6= ∅,
 
1, if Π(A ∩ B) = Π(A)
Π(B | A) = (8)
Π(A ∩ B), otherwise .

The only difference with conditional probability is that the renormalisation via division is changed
into a simple shift to 1 of the plausibility values of the most plausible elements in A. This form
of conditioning agrees with a purely ordinal view of possibility theory and makes sense in a finite
setting only. However, applied to infinite numerical settings, it creates discontinuities, and does

6
not preserve the infinite maxitivity axiom. Especially Π(B | A) = supu∈B π(u | A) may fail to hold
for non-compact events B [14]. The use of the product instead of minimum in the conditioning
equation (7) enables infinite maxitivity to be preserved through conditioning. In close agreement
with probability theory, it leads to
Π(A ∩ B)
Π(B | A) = (9)
Π(A)
provided that Π(A) 6= 0. Then N (B | A) = 1 − Π(B c | A). See De Baets et al. [13] for a complete
mathematical study of possibilistic conditioning, leading to the unicity of the product-based notion,
in the infinite setting. The possibilistic counterpart to Bayes theorem looks formally the same as
in probability theory:
Π(B | A) · Π(A) = Π(A | B) · Π(B). (10)
However the actual expression of Bayes theorem in possibility theory is different, due to the max-
itivity axiom and normalization. Consider the problem of testing hypothesis H against its com-
plement, upon observing evidence E. One must compute Π(H | E) in terms of Π(E | H), Π(E |
H c ), Π(H), Π(H c ) as follows:
Π(E | H) · Π(H)
Π(H | E) = min(1, ) (11)
Π(E | H c ) · Π(H c )
where max(Π(H), Π(H c )) = 1. Moreover one must separately compute Π(H c | E). It is obvious
that max(Π(H | E), Π(H c | E)) = 1. The merit of this counterpart to Bayes rule is that it
does not require prior information to be available. In case of lack of such prior information, the
uniform possibility Π(H) = Π(H c ) = 1 is used. Contrary to the probabilistic Bayesian approach,
uninformative possibilistic priors are truly uninformative and invariant under any form of rescaling.
It leads to compute
Π(E | H)
Π(H | E) = min(1, ) (12)
Π(E | H c )
and Π(H c | E) likewise. It comes down to comparing Π(E | H) and Π(E | H c ), which corresponds
to some existing practice in statistics, called likelihood ratio tests. Namely, the likelihood function
is renormalized via a proportional rescaling [46]; [2]. This approach has been successfully developed
for use in practical applications for instance by Lapointe and Bobée [81].

Yet another counterpart to Bayesian conditioning in quantitative possibility theory can be derived,
noticing that in probability theory, P (B | A) is an increasing function of both P (B ∩ A) and
x
P (Ac ∪ B). This function is exactly f (x, y) = x+1−y . Then the following expression becomes
natural ([37]))
Π(A ∩ B)
Π(B |f A) = . (13)
Π(A ∩ B) + N (A ∩ B c )
The dual conditional necessity is such that
N (A ∩ B)
N (B |f A) = 1 − Π(B c |f A) = . (14)
N (A ∩ B) + Π(A ∩ B c )
Interestingly, this conditioning also preserves consonance, and yields a possibility measure on the
conditioning set A with distribution having support A:
(
π(u)
max(π(u), N (A)+π(u) ), if u ∈ A
π(u |f A) = (15)
0, otherwise.

7
The |f - conditioning clearly leads to a loss of specificity on the conditioning set A. In particular, if
N (A) = 0 then πA is totally uninformative on A. On the contrary, the product-based conditioning
(9) operates a proportional rescaling of π on A, and corresponds to a form of information accumula-
tion similar to Bayesian conditioning. It suggests that the product-based conditioning corresponds
to a revision of the possibility distribution π into π 0 = Π(· | A), interpreting the conditioning event
A in a strong way as Π(Ac ) = 0 (Ac is impossible). On the other hand, the other conditioning
Π(B |f A) is hypothetical, it evaluates the plausibility of B in the context where A is true without
assuming that Ac is impossible. It corresponds to a form of contextual query-answering process
based on the available uncertain information, focusing it on the reference class A. In particular, if
π does not inform about A (N (A) = 0), then restricting to context A only produces uninformative
results, that is π(u |f A) is vacuous on A while π(u | A) is generally not so. A related topic is that
of independence in possibility theory, which is omitted here for the sake of brevity. See Dubois et
al. [39] for an extended discussion.

3 Relationship between probability and possibility theories

In quantitative theories on which we focus here, degrees of possibility are numbers that generally
stand for upper probability bounds. Of course the probabilistic view is only one among other
interpretive settings for possibility measures. Possibility degrees were introduced in terms of ease
of attainment and flexible constraints by Zadeh [119] and as an epistemic uncertainty notion by
Shackle [95], with little reference to probability and/or statistics. Levi [84, 85] was the first to
relate Shackle’s ideas to probability theory, in connection with Dempster’s attempts to rationalize
fiducial inferences [19]. Further on, Wang [112], Dubois and Prade [31] and others have developed
a frequentist view of possibility, which suggests a bridge between possibility theory and statistical
science. Possibility degrees then offer a simple approach to imprecise (set-valued) statistics, in
terms of upper bounds of frequency. The comparison between possibility and probability theories
is made easy by the parallelism of the constructs (i.e. the use of set-functions), which is not the case
when comparing fuzzy sets and probability. As a mathematical object, maxitive set functions have
been already studied by Shilkret, [99]. Possibility theory can also be viewed as a graded extension
of modal logic where the dual notions of possibility and necessity already exist for a long time, in an
all-or-nothing format. The notion “more possible than” was actually first modelled by David Lewis
[86], in the setting of modal logics of counterfactuals, by means of a complete preordering relation
among events satisfying some prescribed properties. This notion was independently rediscovered by
Dubois [22] in the setting of decision theory, in an attempt to propose counterparts to comparative
probability relations for possibility theory. Maxitive set-functions or equivalent notions have also
emerged as a key tool in various domains, such as belief revision (Spohn [102]), non monotonic
reasoning [6], game theory (the so-called unanimity games, Shapley [98]), imprecise probabilities
(Walley [109]), etc. Due to the ordinal nature of the basic axioms of possibility and necessity
functions, there is no enforced commitment to numerical possibility and necessity degrees. So there
are basically two kinds of possibility theories: quantitative and qualitative [38]. This survey deals
with quantitative possibility.

8
3.1 Imprecise Probability.

In the quantitative setting, we can interpret any pair of dual necessity/possibility functions [N, Π]
as upper and lower probabilities induced from specific convex sets of probability functions. Let π
be a possibility distribution inducing a pair of functions [N, Π]. We define the probability family
P(π) = {P, ∀A measurable, N (A) ≤ P (A)} = {P, ∀A measurable, P (A) ≤ Π(A)}. In this case,
supP ∈P(π) P (A) = Π(A) and inf P ∈P(π) P (A) = N (A) hold (see [15, 36]). In other words, the family
P(π) is entirely determined by the probability intervals it generates. Any probability measure
P ∈ P(π) is said to be consistent with the possibility distribution π.

The pairs (interval Ai , necessity weight λi ) suggested above can be interpreted as stating that the
probability P (Ai ) is at least equal to λi where Ai is a measurable set (like an interval containing
the value of interest). These intervals can thus be obtained in terms of fractiles of a probability
distribution. We define the corresponding probability family as follows: P = {P, ∀Ai , λi ≤ P (Ai )}.
If the sets Ai are nested (A1 ⊂ A2 ⊂ . . . ⊂ An , as can be expected for a family of confidence
intervals), then P generates upper and lower probabilities of events that coincide with possibility
and necessity measures induced by the possibility distribution (6) (see Dubois and Prade [36],
De Cooman and Aeyels [15]) for details). Recently, Neumaier [89] has refined the possibilistic
representation of imprecise probabilities using so-called ‘clouds”, which come down to considering
the set of probability measures consistent with two possibility distributions. Walley and de Cooman
[110] show that fuzzy sets representing linguistic information can be captured in the imprecise
probability setting.

In a totally different context, well-known notions of probability theory such as probabilistic in-
equalities can be interpreted in the setting of possibility theory as well. Let x be the quantity to
be estimated and suppose all is known is its mean value x∗ and its standard deviation σ. Cheby-
chev inequality defines bracketing approximations of symmetric intervals around x∗ for unknown
probability distributions, namely it can be proved that for any probability measure P having these
characteristics,
1
P (X ∈ [x∗ − aσ, x∗ + aσ]) ≥ 1 − 2 for a ≥ 1. (16)
a
The preceding inequality suggests a simple method to build distribution-free possibilistic approxi-
mations of probability distributions [24], letting π(x∗ − aσ) = π(x∗ + aσ) = min(1, a12 ) for a > 0.
It is easy to check that P ∈ P(π). Due to the nested structure of intervals involved in probabilistic
inequalities, one may consider that such classical results can be couched in the terminology of possi-
bility theory and yield convex sets of probability functions representable by possibility distributions.
Of course such a representation based on Chebychev inequality is rather weakly informative.

Viewing possibility degrees as upper bounds of probabilities leads to the justification of the |f -
conditionalization of possibility measures. This view of conditioning can be called Bayesian possi-
bilistic conditioning (Dubois and Prade [37]; Walley, [109]) in accordance with imprecise probabili-
ties since Π(B |f A) = sup{P (B | A) : P (A) > 0, P ≤ Π}. Such Bayesian possibilistic conditioning
contrasts with product conditioning (9) which always supplies more specific results than the above.
See De Cooman [16] for a detailed study of this form of conditioning and Walley and De Cooman
[111] for yet other forms of conditioning in possibility theory.

9
3.2 Random Sets

From a mathematical point of view, the information modelled by πx in (6), induced by the family of
confidence sets {(A1 , λ1 ), (A2 , λ2 ), . . . , (Am , λm )} can also be viewed as a nested random set. Indeed
letting νi = λi − λi−1 , ∀i = 1, . . . , m + 1 (assuming the conventions: λ0 = 0; λm+1 = 1; Am+1 = U ):
X
∀u, πx (u) = νi . (17)
i:u∈Ai

The sum of weights ν1 , . . . , νm+1 is 1. Hence the possibility distribution is the one-point cover-
age function of a random set [29], which can be viewed as a consonant belief function [97]. This
view lays bare both partial ignorance (reflected by the size of the Ai ’s) and uncertainty (the νi ’s)
contained in the information. And νi is the probability that the source supplies exactly Ai as a
faithful representation of the available knowledge about x (it is not the probability that x belongs
to Ai ). Equation (17) still makes sense when the set of Ai ’s is not nested. This would be the
case if the random set {(A1 , ν1 ), (A2 , ν2 ), . . . , (Am+1 , νm+1 )} models a set of imprecise sensor ob-
servations along with the attached frequencies. In this case, and contrary to the nested situation,
the probability weights νi can no longer be recovered from π using (17). However, this equation
supplies an approximate representation of the data [34]. The random set view of possibility theory
is developed in more details in (Joslyn [73], Gebhardt and Kruse, [55, 56, 57]). Note that the
product-based conditioning of possibility measures formally coincides with Dempster rule of con-
ditioning, specialised to possibility measures, i.e., consonant plausibility measures of Shafer [96].
It comes down to intersecting all sets Ai with the conditioning set A, assigning the weight νi to
non-empty intersections Ai ∩ A. Then, these weights are renormalized so that their sum be one.

A continuous possibility distribution on the real line can be defined by a probability measure on
the unit interval (for instance the uniformly distributed one) and a multivalued mapping from (0, 1]
to R, defining a family of nested intervals, following Dempster [19]. The probability measure is
carried over from the unit interval to the real line via the multiple-valued mapping that assigns
to each level α ∈ (0, 1] an α-cut of the fuzzy interval restricting an ill-known random quantity of
interest (see Dubois and Prade [32], Heilpern [66], for instance). The possibility distribution is then
viewed as a random interval I and π(u) = P (u ∈ I) the probability that the value u belongs to a
realization of I, as per equation (17).

3.3 Likelihood Functions

When prior probabilities are lacking, likelihood functions can be interpreted as possibility distri-
butions, by default. Suppose a probabilistic model delivering the probability P (u | y) of observing
x = u, when a parameter y ∈ V , is fixed. Conversely, when x = u∗ is observed, the probability
P (u∗ | v) is understood as the likelihood of y = v, in the sense that the greater P (u∗ | v) the more
y = v is plausible. It is easy to see that ∀A ⊆ V

min P (u∗ | v) ≤ P (u∗ | A) ≤ max P (u∗ | v).

v∈A v∈A

It is clear that the upper bound of the probability P (u | A) is a possibility measure (see Dubois
et al. [27]). Besides insofar as P (u∗ | A) is genuinely viewed as the likelihood of the proposition

10
y ∈ A, it is clear that whenever A ⊂ B, the proposition y ∈ B should be at least as likely as the
proposition y ∈ A. So by default it should be that P (u∗ | A) ≥ maxv∈A P (u∗ | v) as well [11].
So, in the absence of prior information it is legitimate to define π(v) = P (u∗ | v), i.e. to interpret
likelihood functions as possibility distributions. In the past, various people suggested to interpret
fuzzy set membership degrees µF (v) as the probability P (0 F 0 | v) of calling an element v an 0 F 0
[70, 105].

The lower bound minv∈A P (u∗ | v) for P (u∗ | A) can be interpreted as a so-called “degree of
guaranteed possibility” [43] ∆(A) = minv∈A π(v), which models the idea of evidential support of all
realizations of event A. Viewing P (u∗ | v) as a degree of evidential support of v when observing u∗ ,
∆(A) is clearly the minimal degree of evidential support for y ∈ A, that evaluates to what extent
all values in the set A are supported by evidence.

In general maxv P (u∗ | v) 6= 1 since in this approach, the normalisation with respect to v is not
warranted. And in the continuous case P (u∗ | v) may be greater than 1. Yet, it is natural to assume
that maxv P (u∗ | v) = 1. It corresponds to existing practice in statistics whereby the likelihood
function is renormalized via a proportional rescaling (Edwards [46]). Besides, a value y = v can
be ruled out when P (u∗ | v) = 0, while it is only highly plausible if P (u∗ | v) is maximal. If a
value v of parameter y must be selected, it is legitimate to choose y = v ∗ such that P (u∗ | v)
is maximal. Hence possibilistic reasoning accounts for the principle of maximum likelihood. For
instance, in the binomial experiment, observing x = u∗ in the form of k heads and n − k tails
for a coin where probability of heads is y = v, leads to consider the probability (= likelihood)
P (u∗ | v) = v k · (1 − v)n−k , as the degree of possibility of y = v and the choice y = nk for the optimal
parameter value is based on maximizing this degree of possibility.

The product-based conditioning rule (9) can also be justified in the setting of imprecise probability,
and maximum likelihood reasoning, noticing that, if event A, occurs one may interpret the prob-
ability degree P (A) as the (second-order) degree of possibility of probability function P ∈ P(π),
say π 0 (P ) in the face of the occurrence of A (the likelihood that P is the model governing the
occurrence of A). Applying the maximum likelihood reasoning comes down to restricting P(π) to
the set {P ≤ Π, P (A) = Π(A)} of probability functions maximizing the probability of occurrence
of the observation A. It is clear that Π(B | A) = sup{P (B | A), P (A) = Π(A) > 0, P ≤ Π} [42].

4 From confidence sets to possibility distributions

It is usual in statistics to summarize probabilistic information about the value of a model param-
eter θ by means of an interval containing this parameter with a degree of confidence attached to
it, in other words a confidence interval. Such estimation methods assume the probability measure
Px (· | θ) governing an observed quantity x to have a certain parameterized shape. The estimated pa-
rameter θ is expressed in terms of the observed data using a mathematical expression (an empirical
estimator), and the distribution Pθ of the empirical evaluation of the parameter is then computed
from the distribution of the measured quantity. Finally an interval [a, b] such that Pθ (θ ∈ [a, b]) ≥ α
is extracted, where α is a prescribed confidence level. For instance, θ is the mean value of Px , and
Pθ is the distribution of the empirical mean θ∗ = x1 +...+x
n
n
, based on n observations.

11
The practical application of this otherwise well-founded estimation method is often ad hoc. First,
the choice of the confidence threshold is arbitrary (it is usually considered equal to 0.95, with no
clear reasons). Second, the construction of the confidence interval often consists in locating α2 - and
(1 − α2 )-fractiles, which sounds appropriate only if the distribution Pθ is symmetric. It is debatable
because in the non-symmetric case (for instance using a χ2 function for the estimation of variance)
some excluded values may have higher density than other values lying in the obtained interval.
Casting this problem in the setting of possibility theory may tackle both caveats. Namely, first,
a possibility distribution can be constructed whose α-cuts are the confidence intervals of degree
1 − α, and then such intervals can be located in a principled way, requesting that the possibility
distribution be as specific as possible, thus preserving informativeness.

To simplify the framework, we consider the problem of determining prediction intervals, which
consists in finding an interval containing, with a certain confidence level, the value of a variable x
having a known distribution. The question is dual to the confidence interval problem, in the sense
that in the latter the parameter θ is fixed, and the interval containing it changes with the chosen
data sample, while, in the following, the interval is prescribed and the value it brackets is random.

4.1 Basic principles for probability-possibility transformations

Let P be a probability measure on a finite set U , obtained from statistical data. Let E be any
subset of U . Define a possibility distribution πE on U by letting

1, if u ∈ E
πE (u) = (18)
1 − P (E), otherwise,

It is obvious to check that by construction ΠE (A) ≥ P (A), ∀A ⊆ U . Then the information repre-
sented by P can be approximated by the statement : “x ∈ E with confidence at least P (E)”, which
is precisely encoded by πE . Note that if the set E is chosen such that P (E) = 0, πE is the vacuous
possibility distribution. If E is too large, πE is likewise not informative. In order for the statement
“x ∈ E with confidence P (E)” not to be trivial, P (E) must be high enough, and E must be narrow
enough. Since these two criteria are obviously antagonistic, one may either choose a confidence
threshold α and minimise the cardinality of E such that P (E) ≥ α, or conversely fix the cardinality
of E and maximize P (E). Doing so for each value of confidence or each value of cardinality of the
set is equivalent to finding a sequence {(E1 , α1 ), (E2 , α2 ), . . . , (Ek , αk )} of smallest prediction sets.
If we let the probability measure be defined by p(ui ) = pi and assume Pi p1 > . . . > pn > pn+1 = 0,
then it is obvious that the smallest set E with probability P (E) ≥ j=1 pj is E = {u1 , u2 , . . . , ui }.
Similarly, {u1 , u2 , . . . , ui } is also the set of cardinality i having maximal probability. So the set of
most informative prediction sets is nested and is
i
X
{({u1 }, p1 ), ({u1 , u2 }, p1 + p2 ), . . . , ({u1 , . . . , ui }, pj ) . . .}.
j=1

It comes down to turning a probability distribution P into a possibility distribution π P of the

following form ([29, 18]):
Xn
πiP = pj , ∀i = 1, . . . , n, (19)
j=i

12
denoting π P (ui ) = πiP . This possibility distribution can be called the optimal fuzzy prediction
set induced by P . Note that π P is a kind of cumulative distribution function, with respect to the
ordering on U defined by the probability values. Readers familiar with mathematical social sciences
will notice it is a so-called Lorentz curve (see for instance Moulin[88]). It can be proved that this
transformation of a probability measure into a possibility measure satisfies three basic principles
(Dubois et al, [41]):

1. Possibility-probability consistency: one should select a possibility distribution π consistent

with p, i.e., such that P ∈ P(π).

2. Ordinal faithfulness : the chosen possibility distribution should preserve the ordering of ele-
mentary events, namely, π(u) > π(u0 ) if and only if p(u) > p(u0 ). We cannot require the same
condition for events since the possibility ordering is generally coarser than the probability
ordering.

3. Informativity: The information content of π should be maximized so as to preserve as much

from P as possible. It means finding the most specific possibility distribution in P(π). In the
case of a statistically induced probability distribution, the rationale of preserving as much
information as possible is natural.

It is clear that π P satisfies all above three requirements and is the unique such possibility distri-
bution. Note that the prediction sets organize around the mode of the distribution. The mode is
indeed the most frequent value and is the most natural characteristics for an expert. When some
elements in U are equiprobable the unicity of the possibility transform remains, but equation (19)
must be applied to the well-ordered partition U1 , . . . , Uk of U induced by p, obtained by grouping
equiprobable elements (elements in Ei have the same probability which is greater than the one of
elements in Ei+1 ). Then equipossibility on each set Ui of the partition is assumed. For instance,
the uniform probability is transformed into the uniform (vacuous) possibility distribution.

However it makes sense to relax the ordinal faithfulness condition into a weak form: p(u) > p(u0 )
implies π(u) > π(u0 ). Doing so, the most specific possibility transform is no longer unique. For
instance, if p1 = . . . = pn = n1 then selecting any linear ordering of elements and applying (19)
yields a most specific possibility distribution consistent with P . More generally if U1 , . . . , Uk is the
well-ordered partition of U induced by p, then the most specific possibility distributions consistent
with p are given by (19) applied to any linear ordering of U coherent with U1 , . . . , Uk (using an
arbitrarily ranking of elements within each Ei ).

The above approach has been recently extended by Masson and Denoeux [87] to the case when
the empirical probability values pi are parameters of a multinomial distribution, themselves esti-
mated by means of confidence intervals. The problem is then to define a most specific possibility
distribution covering all probability functions satisfying the constraints induced by the confidence
intervals.

13
4.2 Alternative approaches to probability-possibility transforms

The idea that some consistency exists between possibilistic and probabilistic representations of
uncertainty was suggested by Zadeh [119]. He defined the degree of consistency P between a possibility
distribution π and a probability measure P as follows: Cons(P, π) = i=1,n πi · pi . It is the
probability of the fuzzy event whose membership function is π. However Zadeh also described
the consistency principle between possibility and probability in an informal way, whereby what is
probable should be possible. Dubois and Prade [28] translated this requirement via the inequality
Π(A) ≥ P (A) that founds the interpretation of possibility measures as upper probability bounds.
There are two basic approaches to possibility/probability transformations. We presented one in
the previous section 4.1, the other one is due to Klir [74, 58]. Klir’s approach relies on a principle
of information invariance, while the other one, described above, is based on optimizing information
content [41]. Both respect a form of probability-possibility consistency. Klir tries to relate the
notion of possibilistic specificity and the notion of probabilistic entropy. The entropy of a probability
measure P is defined by
Xn
H(P ) = − pj · log pj . (20)
j=1

In Klir’s view, the transformation should be based on three assumptions :

1. A scaling assumption that forces each value πi to be a function of pi (where p1 ≥ p2 ≥ . . . ≥ pn ,

that can be ratio-scale, interval scale, Log-interval scale transformations, etc.

2. An uncertainty invariance assumption according to which the entropy H(p) should be nu-
merically equal to some measure E(π) of the information contained in the transform π of
p.

3. Transformations should satisfy the consistency condition π(u) ≥ p(u), ∀u.

The information measure E(π) can be the logarithmic imprecision index of Higashi and Klir [68]
n
X
E(π) = (πi − πi+1 ).Log2 i, (21)
i=1

or the measure of total uncertainty as the sum of two heterogeneous terms estimating imprecision
and discord respectively (after Klir and Ramer [78]). The uncertainty invariance equation E(π) =
H(p), along with a scaling transformation assumption (e.g., π(x) = αp(x) + β, ∀x), reduces the
problem of computing π from p to that of solving an algebraic equation with one or two unknowns.

Klir’s assumptions are debatable. First, the scaling assumption leads to assume that π(u) is a
function of p(u) only. This pointwiseness assumption may conflict with the probability/possibility
consistency principle that requires Π ≥ P for all events. See Dubois and Prade ([28], pp. 258-
259) for an example of such a violation. Then, the nice link between possibility and probability,
casting possibility measures in the setting of upper and lower probabilities cannot be maintained.
The second and the most questionable prerequisite assumes that possibilistic and probabilistic
information measures are commensurate. The basic idea is that the choice between possibility and
probability is a mere matter of translation between languages “neither of which is weaker or stronger

14
than the other” (quoting Klir and Parviz, [77]). It means that entropy and imprecision capture the
same facet of uncertainty, albeit in different guises. The alternative approach recalled in section
4.1 does not make this assumption. Nevertheless Klir was to some extent right when claiming some
similarity between entropy of probability measures and specificity of possibility distributions. In a
recent paper [25], the following result is indeed established :

Theorem 1 Suppose two probability measures P and Q such that π P is strictly less specific than
π Q , then H(P ) > H(Q).

It is easy to check that if π P is strictly less specific than π Q , then P is (informally) less peaked
than Q (to use a terminology due to Birnbaum [7]), so that one would expect that the entropy
of P should be higher than that of Q. Besides, viewing π P as a kind of cumulative distribution,
the comparison of probability measures via the specificity ordering of their possibility transforms is
akin to a form of stochastic dominance called “majorization” in Hardy et al. [65]’s famous book on
inequalities where a general result encompassing the above theorem is proved. This result lays bare
a partial ordering between probability measures that seem to underlie many indices of dispersion
(entropy, but Gini index as well and so on) as explained in [25], a result that actually dates back
to the book of Hardy et al..

4.3 The continuous case

The problem of finding the best prediction interval for a random variable or best confidence intervals
for a parameter does not seem to have received much attention, except for symmetric distributions.
If the length L of the interval is prescribed, it is natural to maximise its probability. Nevertheless,
it is not difficult to prove that, for a probability measure with a continuous unimodal density p, the
optimal prediction interval of length L, i.e., the interval with maximal probability is IL = [aL , aL +L]
where aL is selected such that p(aL ) = p(aL + L). It is a cut {u, p(u) ≥ β} of the density, for a
suitable value of a threshold β. This interval has degree of confidence P (IL ) (often taken as 0.95)
[41]. Conversely, the most informative prediction interval at a fixed level of confidence, say α is
also of this form (choosing β such that P ({u, p(u) ≥ β}) = α. Clearly, in the limit, when the
length L vanishes, IL reduces to the mode of the distribution,viewed as the “most frequent value”.
It confirms that the mode of the distribution is the focus point of prediction intervals (not the
mean), although such best prediction intervals are not symmetric around the mode. This result
extends to multimodal density functions, but the resulting prediction set may consist of several
disjoint intervals, since the best prediction set is of the form {u, p(u) ≥ β} for a suitable value of a
threshold β[24]. It also extends to multidimensional universes [91].

Moving the threshold β from 0 to the height of the density, a probability measure having a unimodal
probability density p with strictly monotonic sides can be transformed into a possibility distribution
whose cuts are the prediction intervals of p with various confidence levels. The most specific
possibility distribution π consistent with p, and ordinally equivalent to it, is obtained, such that
[41]:
∀L > 0, π(aL ) = π(aL + L) = 1 − P (IL ).
Hence the α-cut of the optimal (most specific) π is the (1 - α)- prediction interval of p. These
prediction intervals are nested around the mode of p. Going from objective probability to possibility

15
thus means adopting a representation of uncertainty in terms of confidence intervals. Dubois,
Mauris et al., [24] have found more results along this line for symmetric densities. Noticeably, each
side of the optimal possibilistic transform is convex and there is no derivative at the mode of π
because of the presence of a kink. Hence given a probability density on a bounded interval [a, b],
the symmetric triangular fuzzy number whose core is the mode of p and the support is [a, b] is an
upper approximation of p (in the style of, but much better than Chebychev inequality) regardless of
its shape. In the case of a uniform distribution on [a, b], any triangular fuzzy number with support
[a, b] provides a most specific upper approximation. These results and more recent ones (Baudrit
et al. [3]) justify the use of triangular fuzzy numbers as fuzzy counterparts to uniform probability
distributions, and model free approximations of probability functions with bounded support. This
setting is also relevant for modeling sensor measurements [83].

A related problem is the selection of a modal value from a finite set of numerical observations
{x1 ≤ . . . ≤ xK }. Informally the modal value makes sense if a sufficiently high proportion of close
observations exist. It is clear that such a modal value does not always exist, as in the case of the
uniform probability. Dubois et al. [40] formalized this criterion as finding the interval length L for
which the difference between the empirical probability value P (IL ) for the most likely interval of
L
length L, and the (uniform) probability Lmax is maximal, where Lmax = xK − x1 is the length of
the empirical support of the data. If L is very small or if it is very close to Lmax , this difference is
very small. The obtained interval achieves a trade-off between specificity (the length L of IL ) and
frequency.

5 Possibility theory and subjective probability

So far, we have considered possibility measures in the scope of modeling information stemming
from statistical evidence. There is a subjectivist side to possibility theory. Indeed, possibility
distributions can arguably be advocated as a more natural representation of human uncertain
knowledge than probability distributions. The representation of subjective uncertain evidence by
a single probability function relies on interpreting probabilities as betting rates, in the setting of
exchangeable bets. The probability P (A) of an event is interpreted as the price of a lottery ticket
an agent is ready to pay to a banker provided that this agent receives one money unit if event
A occurs. A fair price is enforced, namely if the banker finds the buying price offered by the
agent too small, they must exchange their roles. The additivity axiom is enforced using the Dutch
book argument, namely the agent loses money each time the buying price proposal is such that
P (A) + P (Ac ) 6= 1. Then the uncertain knowledge of an agent must always be represented by a
single probability distribution [82], however ignorant this agent may be.

However it is clear that degrees of probability obtained in the betting scheme will depend on the
partition of alternatives among which to bet. In fact, two uniform probability distributions on two
different frames of discernment representing the same problem may be incompatible with each other
(Shafer [96]). Besides, if ignorance means not being able to tell if one contingent event is more or
less probable than any other contingent event, then uniform probabilities cannot account for this
postulate because, unless the frame of discernment is binary, even assuming a uniform probability,
some contingent event will have a probability higher than another [42]. Worse, if an agent proposes
a uniform probability as expressing his beliefs, say on facets of a die, it is not possible to know if

16
this probability is the result of sheer ignorance, or if the agent really knows that the underlying
process is random.

Several scholars, and noticeably Walley [108] challenged the unique probability view, and more
precisely the exchangeable bet postulate. They admit the idea that an agent will not accept to buy
a lottery ticket pertaining to the occurrence of A beyond a certain maximal price, nor to sell it
under another higher price. The former buying price is interpreted as the lower probability P∗ (A),
and the upper probability P ∗ (A) is viewed as the latter selling price. Possibility measures can be
interpreted in this way if their distribution is normalized. The possibilistic axiom P∗ (A ∩ B) =
min(P∗ (A), P∗ (B)) indicates an agent is not ready to put more money on A and on B than he would
put on their conjunction. It indicates a cautious behavior. In particular, the agent would request
free lottery tickets for all elementary events except at most one, namely the one with maximal
possibility if unique (because N ({u}) = 0, ∀u ∈ U whenever π1 (u1 ) = π2 (u2 ) = 1 for u1 6= u2 ).
This subjectivist view of possibility measures was first proposed by Robin Giles [60] in 1982, and
developed by De Cooman and Aeyels [15].

The above subjectivist framework presupposes that decisions will be made on the basis of imprecise
probabilistic information. Another view, whose main proponent is Smets [100], contends that
one should distinguish between a credal level whereby an agent entertains beliefs and incomplete
information is explicitly accounted for (in terms of belief functions according to Smets, but it could
be imprecise probabilities likewise), and a so-called “pignistic” level where the agent should stick to
classical decision theory when making decisions. Hence, this agent should use a unique probability
distribution when computing expected utilities of such decisions. Under the latter view the question
solved by Smets is how to derive a unique prior probability from a belief function. This approach
is recalled here and related to Laplace Insufficient Reason principle. An interesting question is
the converse problem : how to reconstruct the credal level from a prior probability distribution
supplied by an expert.

5.1 A generalized Insufficient Reason principle

Laplace proposed that if elementary events are equally possible, they should be equally probable.
This is the principle of Insufficient Reason that justifies, on behalf of respecting the symmetries
of problems, the use of uniform probabilities over the set of possible states. Suppose there is a
non-uniform possibility distribution on the set of states representing an agent knowledge. How to
derive a probabilistic representation of this knowledge, where this dissymmetry would be reflected
through the betting rates of the agent? Clearly, changing a possibility distribution into a probability
distribution increases the informational content of the considered representation. Moreover, the
probability distribution should respect the exiting symmetries in the possibility distribution, in
agreement with Laplace principle.

A solution to this problem can be found by considering the more general case where the available
information is represented by a random set R = {(A1 , ν1 ), (A2 , ν2 ), . . . , (Am , νm )}. A generalised
Laplacean indifference principle is then easily formulated: the weights pi bearing on the focal sets
are then uniformly distributed on the elements of these sets. So, each focal set Ai is changed
uniform probability measure P i over Ai , and the resulting probability P P is of the form
into a P
PP = m i
i=1 νi · P . This transformation, already proposed by Dubois and Prade [29] comes down

17
to a stochastic selection process that first selects a focal set according to the distribution ν, and
then picks an element at random in the focal set Ai according to Laplace principle. The rationale
behind this transformation is to minimize arbitrariness by preserving the symmetry properties of
the representation.

Since a possibility distribution can be viewed as a nested random set, the pignistic transformation
applies: the transformation of π yields a probability pπ defined by [30]:
X πj − πj+1
pπ ({ui }) = , (22)
j
j=i,...,n

where π({ui }) = πi and π1 ≥ . . . ≥ πn ≥ πn+1 = 0. This probability is also the gravity center of
the set P = {P | ∀A, P (A) ≤ Π(A)} of probability distributions dominated by Π [41]. Hence it can
be viewed as applying the Insufficient Reason Principle to the set of probabilities P(π), equipping
it with a uniformly distributed meta-probability, and then selecting the mean value.

This transformation coincides with the so-called pignistic transformation of belief functions (Smets
[100]) as it really fits with the way a human would bet if possessing information in the form
of a random set. Smets provides an axiomatic derivation of this pignistic probability: the basic
assumptions are anonymity (permuting the elements of U should not affect the result) and linearity
(the pignistic probability of a convex combination of random sets is the corresponding convex sum of
pignistic probabilities derived from each random set). The pignistic probability also coincides with
the Shapley value in game theory [98], where a cooperative game can be viewed as a non additive
set function assigning a degree of strength to each coalition of agents. The obtained probability
measure is then a depiction of the overall strength of each agent. Smets proposed similar axioms
as Shapley.

5.2 A Bayesian approach to subjective possibility

If we stick to the Bayesian methodology of eliciting fair betting rates from the agent, but reject
the credo that degrees of beliefs coincide with these betting rates, it follows that the subjective
probability distribution supplied by an agent is only a trace of this agent’s beliefs. While his beliefs
can be more faithfully represented by a set of probabilities, the agent is forced to be additive by the
postulates of exchangeable bets. Noticeably, the agent provides a uniform probability distribution
whether (s)he knows nothing about the concerned phenomenon, or if (s)he knows the concerned
phenomenon is purely random. In the Transferable Belief Model [101], the agent provides a pignistic
probability induced by a belief function. Then, given a subjective probability, the problem consists
in reconstructing the underlying belief function.

There are clearly several belief functions corresponding to a given pignistic probability. It is in
agreement with intuition to consider the least informative among those. It means adopting a
pessimistic view on the agent’s knowledge. This is in contrast with the case of objective probability
distributions where the available information is of statistical nature and should be preserved. Here,
the available information being provided by an agent, it is not supposed to be as precise. One
way of proceeding consists in comparing contour functions in terms of the specificity ordering of
possibility distributions. Dubois et al. [44] proved that the least informative random set with a

18
prescribed pignistic probability is unique and consonant. It is based on a possibility distribution
π sub , previously suggested in [30] with a totally different rationale:
X
π sub (ui ) = min(pj , pi ). (23)
j=1,n

More precisely, let F(p) be the set of random sets R with pignistic probability p. Let πR be the
possibility distribution induced by R using the one-point coverage equation (17). Define R1 to be
at least as informative a random set as R2 whenever πR1 ≤ πR2 . Then the least informative R in
F(p) is precisely the consonant one such that πR = π sub . Note that the pignistic transformation is a
bijection between possibility and probability distributions. Equation (23) is also the transformation
converse to eqn. (22). The subjective possibility distribution is less specific than the optimal fuzzy
prediction interval (19), as expected, that is π sub > πp , generally. By construction, π sub is a
subjective possibility distribution. Its merit is not to assume human knowledge is precise, like in
the subjective probability school. The transformation (23) was first proposed in [30] for objective
probability, interpreting the empirical necessity of an event as summing the excess of probabilities
of realizations of this event with respect to the probability of the most likely realization of the
opposite event.

6 Fuzzy intervals and possibilistic expectations

A fuzzy interval [26] is a possibility distribution on the real line whose cuts are (generally closed)
intervals. One interesting question is whether one can extract from such fuzzy intervals the kind of
information useful in practice, that statisticians derive from probability distributions: cumulative
distributions, mean values, variance, for instance. This section summarizes some known results on
these issues.

6.1 Possibilistic cumulative distributions

Consider a fuzzy interval M with membership function µM viewed as a possibility distribution. The
core of M is an interval [m∗ , m∗ ] = {u, µM (u) = 1}. The upper cumulative distribution function of
the fuzzy interval M is F ∗ (a) = ΠM ((−∞, a]) = sup{µM (x) : x ≤ a}, hence :

∗ µM (a), if a ≤ m∗
∀a ∈ R, F (a) = (24)
1, otherwise.

Similarly, the lower distribution function F∗ (a) = NM ((−∞, a]) = 1 − ΠM ((a, +∞)) = inf{1 −
µM (x) : x > a} such that:

if a < m∗

0,
∀a ∈ R, F∗ (a) = (25)
1 − limx→a+ µM (x), otherwise.

The upper distribution function F ∗ matches the increasing part of the membership function of M .
The lower distribution function F∗ reflects the decreasing part of the membership function of M of
which it is the fuzzy complement. In the imprecise probability view, M encodes a set of probability

19
measures P(µM ). The upper and lower distribution functions are limits of distribution functions in
this set [32]. However, the set of probability measures whose cumulative distributions lie between
F∗ and F ∗ is a superset of P(µM ) as indicated in [32] (see also [3]).

Ferson and Ginzburg [52] call a p-box a pair of cumulative distribution functions (F , F ) with F ≤ F .
It can be viewed as a generalized interval as well. The above definitions shows that a fuzzy interval
induces a p-box. But such generated p-boxes are less informative than the possibility distributions
they are computed from. The point is that in the p-box view the two cumulative distributions are
in some sense independent. They correspond to two random variables x− and x+ > x− defining
the random interval [x− , x+ ] with possibly independent end-points (See Heilpern [66, 67]; Gil [59]).
Note that the intersection of all such generated intervals should not be empty so as to ensure the
normalization of M . On the contrary, in the random set view, nested cuts Mλ = [m∗λ , m∗λ ] are
generated as a whole (hence a clear dependence between endpoints). Variables x− and x+ then
depend on a single parameter λ such that [x− (λ), x+ (λ)] = [m∗λ , m∗λ ]. In the p-box view, intervals
of the form [x− (α), x+ (β)] are generated for independent choices of α and β.

6.2 Possibilistic integrals

Possibility and necessity measures are very special cases of Choquet capacities that encode families
of probabilities. The natural notion of integral in this framework is Choquet integral. A capacity
is a monotone set-function σ (if A ⊆ B, then σ(A) ≤ σ(B)), defined on an algebra A of subsets
of U , with σ(∅) = 0. The Choquet integral of a bounded function φ from a set U to the positive
reals, with respect to a Choquet capacity (or fuzzy measure) σ, is defined as follows:
Z ∞
Chσ (φ) = σ({u, φ(u) ≥ α})dα, (26)
0

provided that the cutset {u, φ(u) ≥ α} is in the algebra A. See Denneberg [20] for a mathematical
introduction. When σ = P , a probability measure, it reduces to a Lebesgue integral. When σ = Π,
a possibility measure, and φ = µF is the membership function of a fuzzy set F , also called a fuzzy
event, it reads Z 1 Z 1
ChΠ (F ) = Π(Fα )dα = sup{µF (u) : π(u) ≥ α}dα. (27)
0 0 u

In order to see that this identity holds, it suffices to Rnotice (with De Cooman [16]) that the possibility
1
degree of an event A can be expressed as Π(A) = 0 Π> (A ∩ πβ )dβ where πβ is the β-cut of π and
R 1>Ris1 the vacuous possibility measure (Π> (A) = 1 if A 6= ∅ and 0 otherwise). Then ChΠ (F ) reads
Π
0 0 RΠ> (Fα ∩ πβ )dαdβ. It is allowed to commute the two integrals and the result follows, noticing
1
that 0 Π> (Fα ∩ πβ )dβ = sup{µF (u) : π(u) ≥ α}.

Equation (27) is a definition of the possibility of fuzzy events different from Zadeh’s (namely
Π(F ) = supu∈U min(π(u), µF (u))) because the maxitivity of ChΠ (F ) w.r.t. F is not preserved
here. That is, ChΠ (F ∪ G) 6= max(ChΠ (F ), ChΠ (G)) when µF ∪G = max(µF , µG ). The Choquet
integral w.r.t. a necessity measure can be similarly defined, and it yields :
Z 1 Z 1
ChN (F ) = N (Fα )dα = inf {µF (u) : π(u) ≥ α}dα. (28)
0 0 u

20
In order to see it, we use the duality between necessity and possibility, N (Fα ) = 1 − Π((Fα )c ) and
denoting F c the fuzzy complement of µF as µF c = 1 − µF we use the fact that (Fα )c = {u, µF c (u) >
1 − α}. Plugging these identities into eqn. (27) yields the above expression of the Choquet integral
of F w.r.t. a necessity measure. Moreover, the interval [ChN (F ), ChΠ (F )] encloses all expectations
of F w.r.t. all probabilities in P(π), boundaries being attained.

6.3 Expectations of fuzzy intervals

The simplest non-fuzzy substitute of the fuzzy interval M is its core, or its mode when its core is a
singleton (in Dubois and Prade’s early works [28], what is called mean value of a fuzzy number is
actually its mode). Under the random set interpretation of a fuzzy interval, upper and lower mean
values of M in the sense of Dempster [19], can be defined, i.e., E∗ (M ) and E ∗ (M ), respectively,
such that [32, 66]:
Z 1
E∗ (M ) = inf Mλ dλ; (29)
0
Z 1
E ∗ (M ) = sup Mλ dλ. (30)
0
Note that these expressions are Choquet integrals of the identity function with respect to the
possibility and the necessity measures induced by M . The mean interval of a fuzzy interval M
is defined as E(M ) = [E∗ (M ), E ∗ (M )]. It is thus the interval containing the mean values of all
random variables compatible with M (i.e., P ∈ P(µM )). It is also the Aumann integral of the α-cut
mapping : α ∈ (0, 1] −→ Mα , as recently proved by Ralescu [92].

That the mean value of a fuzzy interval be an interval seems to be intuitively satisfactory. Partic-
ularly the mean interval of a (regular) interval [a, b] is this interval itself. The same mean interval
obtains in the random set view and the imprecise probability view of fuzzy intervals, and is also
the one we get by considering the cumulative distribution of the p-box induced by M . The upper
and lower mean values are additive with respect to the fuzzy addition, since they satisfy, for u.s.c.
fuzzy intervals [32, 66]:

E∗ (M + N ) = E∗ (M ) + E∗ (N ); (31)
∗
E (M + N ) = E ∗ (M ) + E ∗ (N ), (32)

where µM +N (z) = supx min(µM (x), µN (z − x)). This property is a consequence of the additivity
of Choquet integral for the sum of comonotonic functions.

6.4 The Mean Interval and Defuzzification.

Finding a scalar representative value of a fuzzy interval is often called defuzzification in the liter-
ature of fuzzy control (See Yager and Filev [116], and Van Leekwijk and Kerre [107] for extensive
overviews). Various proposals exist:

• the mean of maxima (MOM), which is the middle point in the core of the fuzzy interval M ,

21
 • the center of gravity. This is the center of gravity of the support of M , weighted by the
membership grade.

• the center of area (median): This is the point of the support of M that equally divides the
area under the membership function.

The MOM sounds natural as a representative of a fuzzy interval M in the scope of possibility
theory where values of highest possibility are considered as default plausible values. This is in
the particular case when the maximum is unique. However, the MOM clearly does not exploit all
the information contained in M since it neglects the membership function. Yager and Filev [116]
present a general methodology for extracting characteristic values from fuzzy intervals. They show
that all methods come down to a possibility-probability transformation followed by the extraction of
characteristic value such as a mean value. Note that the MOM, the center of gravity and the center
of area come down to renormalizing the fuzzy interval as a probability distribution and computing
its mode, expected value or its median, respectively. These approaches are ad hoc. Moreover, the
renormalization technique (dividing the membership function by its surface) is itself arbitrary since
the obtained probability may fail to belong to P(µM ), the set of probabilities dominated by the
possibility measure attached to M [28]. In view of the quantitative possibility setting, it seems that
the most natural defuzzication proposal is the middle point of the mean interval [114]
1
E∗ (M ) + E ∗ (M )
Z
(inf Mλ + sup Mλ )
E(M ) = dλ = . (33)
0 2 2

Only the mean interval accounts for the specific possibilistic nature of the fuzzy interval. The
choice of the middle point expresses a neutral attitude of the user and extends the MOM to an
average mean of cut midpoints. Other choices are possible, for instance using a weighted average of
E∗ (M ) and E ∗ (M ). Fullér and colleagues [9, 53] consider introducing a weighting function on [0, 1]
in order to account for unequal importance of cuts when computing upper and lower expectations.

E(M ) has a natural interpretation in terms of simulation of a “fuzzy variable”. Chanas and
Nowakowski [10] investigate this problem in greater detail. Namely, consider the two step random
generator which selects a cut at random (by choosing λ ∈ (0, 1]), and a number in the cut Mλ . The
corresponding random quantity is x(α, λ) = α · inf Mλ + (1 − α) · sup Mλ . The mean value of this
random variable is E(M ) and its distribution is PM with density [41] :
Z 1
Mλ (x)
pM (x) = dλ. (34)
0 sup Mλ − inf Mλ

The probability distribution PM is in fact, the center of gravity of P(µM ). It corresponds to the
pignistic transformation of M , obtained by considering cuts as uniformly distributed probabilities.
The mean value E(M ) is linear in the sense of fuzzy addition and scalar multiplication [94, 54].

6.5 The variance of a fuzzy interval

The notion of variance has been extended to fuzzy random variables [79], but little work exists on
the variance of a fuzzy interval. Fuller and colleagues [9, 53] propose a definition as follows:

22
 1
sup Mλ − inf Mλ 2
Z
V (M ) = ( ) f (λ)dλ, (35)
0 2

where f is a weight function. However appealing this definition may sound, it lacks proper inter-
pretation in the setting of imprecise probability. In fact, the very idea of a variance of a possibility
distribution is somewhat problematic. A possibility distribution expresses information incomplete-
ness, and does not so much account for variability. The variance of a constant but ill-known
quantity makes little sense. The amount of incompleteness is then well-reflected by the area under
the possibility distribution, which is a natural characteristics of a fuzzy interval. Other indices
of information already mentioned in section 4.2 are variants of this simpler index. Additionally,
it is clear that the expression of V (M ) depends upon the area under the possibility distribution
(suffices to let f (λ) = 1). So it is not clear that the above definition qualifies as a variance: the
wider a probability density, the higher the variability of the random variable, but the wider a fuzzy
interval, the more imprecise.

However if the possibility distribution π = µM stands for subjective knowledge about an ill-known
random quantity, then it is interesting to compute the range V (M ) of variances of probability
functions consistent with M , namely

V (M ) = {variance(P ), P ∈ P(µM )}. (36)

Viewing a fuzzy interval as a set of cuts, determining V (M ) is closely related to the variance of a
finite set of interval-valued data, for which results and computational methods exist [50]. In the
nested case, which applies to fuzzy intervals, the determination of the upper bound of the variance
interval is a NP-hard problem, while the lower bound is trivially 0. The upper bound of V (M )
can be called the potential variance of M , and it is still an open problem to find a closed form
expression for it in the general case, let alone its comparison with the definition (35) by Fuller.
In the symmetric case, Dubois et al. [23] proved that the potential variance of M is precisely
given by (35) when f (λ) = 1. We can conjecture that this result holds in the general case, which
would suggest that the amount of imprecision in the knowledge of a random variable reflects its
maximal potential range of variability. Also of interest is to relate the potential variance to the
scalar variance of a fuzzy random variable, introduced by Koerner [79].

7 Uncertainty propagation with possibility distributions

A very important issue for applications of possibility theory is that of propagating uncertainty
through mathematical models of processes. Traditionally this problem was addressed as one of
computing the output distribution of functions of random variables using Monte-Carlo methods
typically. Monte-Carlo methods applied to the calculation of functions of random variables cannot
account for all types of uncertainty propagation [48]. The necessity of different tools for telling
variability from partial ignorance leads to investigate the potential of possibility theory in addressing
this issue, since this theory is tailored to the modeling of incomplete information. Suppose there
are k < n random variables (X1 , . . . , Xk ) and n − k possibilistic quantities (Xk+1 , . . . , Xn ). The
problem, already outlined by Kaufmann and Gupta [62] is to compute the available information on
a function f (X1 , . . . , Xn ).

23
First assume that only two possibilistic quantities are present (k = 0, n = 2). The extension
principle of Zadeh [118] proposes a solution to the propagation problem in the setting of possibility
theory [33]. Consider a two place function f . If a joint possibility relating two ill-known quantities
x1 and x2 is separable, i.e., π = min(π1 , π2 ), then the possibility distribution πf of f (x1 , x2 ) is
sup{min(π1 (u1 ), π2 (u2 )) : f (u1 , u2 ) = v}, if f −1 ({v}) 6= ∅;

πf (v) = (37)
0, otherwise .

This proposal lays bare a basic issue that has to be solved prior to performing the propagation
step: how to represent joint possibility distributions and what type of independence is involved?
In the purely possibilistic framework, the choice π = min(π1 , π2 ) for the joint distribution reflects
a principle of minimal specificity: the largest, least committed joint possibility distribution whose
projections are π1 and π2 is π = min(π1 , π2 ). So the above calculation presupposes nothing on
the possible dependence between quantities x and y. Moreover, the extension principle applied to
non-fuzzy possibility distributions (valued on {0, 1}) reduces to standard interval calculations.

In the setting of random sets [57], the partial lack of information gives birth two possible levels of
dependence (that can hardly be told apart using single probability distributions): on top of the
possible dependence between variables, one must consider the possible dependence between the
sources of information. The above notion of joint possibility distributions relies on a dependence
assumption between sources, but no dependence assumption between variables is made. It presup-
poses that if the first source delivers a cut (A1 )λ of π1 then the other one delivers (A2 )λ for the
same value of λ. Two nested random sets with associated one-point coverage functions π1 and π2
then produce a nested random set with one-point coverage function min(π1 , π2 ). In other words, it
comes down to working with confidence intervals having the same levels of confidence. For instance,
if π1 and π2 are supplied by the same expert, such a dependence assumption between confidence
levels looks natural.

On the contrary, the assumption that quantities x1 and x2 are independently observed will not lead
to a nested random set, since cuts (A1 )λ and (A1 )ρ are compatible, for λ 6= ρ. Hence the set of
joint observations is not equivalent to a joint possibility distribution [35].

Let πi be associated with the nested random interval with mass assignment function νi for i = 1, 2.
Let ν(A, B) be the joint mass assignment whose projections are νi for i = 1, 2. That is:
P
ν1 (A) = PB ν(A, B);
(38)
ν2 (B) = A ν(A, B).

Note that since ν(A, B) is assigned to the Cartesian product A × B, there is still no dependence
assumption made between the variables x1 and x2 . Assuming independence between the sources
of information leads to define a joint random set describing (x1 , x2 ) by means of Dempster rule
of combination of belief functions, that is, ν(A, B) = ν1 (A) · ν2 (B). The random set induced on
f (x1 , x2 ) has mass function νf defined by
X
νf (C) = ν1 (A) · ν2 (B), (39)
A,B:f (A,B)=C

where f (A, B) = C is obtained by interval analysis. The one-point coverage function of νf (equa-
tion 17) can be directly expressed in terms of π1 and π2 by the sup-product extension principle

24
(changing minimum into product in (37) [34]. The random set setting for computing with pos-
sibility distributions thus encompasses both fuzzy interval and random variable computation. In
practice the above calculation can be carried out on continuous possibility distributions using a
Monte-Carlo method that selects α-cuts of π1 and π2 , and interval analysis on selected cuts.

A total absence of knowledge about dependence between sources may also be assumed. If the joint
mass function ν(A, B) is unknown, a more heavy computation scheme can be invoked. Namely, for
any event C of interest it is possible to compute probability bounds induced by the only knowledge
of the marginal distributions
P π1 and π2 . For instance the lower probability bound can be obtained by
minimizing P∗ (C) = A,B:f (A,B)⊆C P ν(A, B) under the constraints (38), and the upper probability
bound by maximizing P ∗ (C) = A,B:f (A,B)∩C6=∅ ν(A, B) under the same constraints. The obtained
bounds are the most conservative one may think of [4].

When k variables are probabilistic, other quantities being possibilistic, one may perform Monte-
Carlo sampling of random variables (X1 , . . . , Xk ) and fuzzy interval analysis on possibilistic quan-
tities (Xk+1 , . . . , Xn ) [47, 61]. This presupposes that random variables and possibilistic quantities
are independently informed, random variables being mutually independent and possibilistic vari-
ables depending on a single source of information(for instance several sensors and a human expert,
respectively). Then f (X1 , . . . , Xn ) is a fuzzy random variable for which average upper and lower
cumulative distributions can be derived[5]. Such kinds of hybrid calculations were implemented
by Ferson[47], Guyonnet et al. [61] in the framework of risk assessment for pollution studies, and
geology [1]. However the above schemes making other kinds of assumptions between possibilistic
variables can be accommodated in a straightforward way to the hybrid probability/possibility sit-
uation. The above outlined propagation methods should be articulated in a more precise way with
other techniques which propagate imprecise probabilities in risk assessment models [51, 71]. Note
that casting the above calculations in the setting of imprecise probabilities enables dependence
assumptions to be made between marginal probabilities dominated by π1 and π2 , while the random
set approach basically accounts for (in)dependence assumptions between the sources of information.
The study of independence when both variability and lack of knowledge are present is not yet fully
understood (see for instance Couso et al. [12]).

If the user of such methods is interested by the risk of violating some threshold for the output
value, upper and lower cumulative distributions can be derived. This mode of presentation of
the results lays bare the distinction between lack of knowledge (the distance between the upper
and lower cumulative distributions) and variability (the slopes of the cumulative distributions
influenced by the variances of the probabilistic variables). These two dimensions would be mixed
up in the variance of the output if all inputs were represented by single probability distributions.
Besides note that upper and lower cumulative distributions only partially account for the actual
uncertain output (contrary to the classical probability case). So more work is needed to propose
simple representations of the results that may help the user exploit this information. The proper
information to be extracted clearly depends on the question of interest (for instance determining
the likelihood that the output value lies between two bounds cannot be addressed by means of the
upper and lower cumulative distributions).

25
8 Conclusion

Quantitative possibility theory seems to be a promising framework for probabilistic reasoning under
incomplete information. This is because some families of probability measures can be encoded
by possibility distributions. The simplicity of possibility distributions make them attractive for
practical applications of imprecise probabilities, and more generally for the representation of poor
probabilistic information. Besides, cognitive studies for the empirical evaluation of possibility
theory have recently appeared [93]. Their experiments suggest “that human experts might behave
in a way that is closer to possibilistic predictions than probabilistic ones”. The cognitive validation
of possibility theory is clearly an important issue for a better understanding of when possibility
theory is most appropriate.

The connection between possibility and probability in the setting of imprecise probabilities makes it
possible to envisage a unified approach for uncertainty propagation with heterogeneous information,
some sources providing regular statistics, other ones subjective information under the form of
ill-defined probability densities [71]. Other applications of possibilistic representations of poor
probabilistic knowledge are in measurement [83].

Moreover a reassessment of non-Bayesian statistical methods in the light of possibility theory seems
to be promising. This paper only hinted to that direction, focusing on some important concepts
in statistics, such as confidence intervals and the maximum likelihood principle. However much
work remains to be done. In the case of descriptive statistics, a basic issue is the handling of set-
valued or fuzzy data: how to extend existing techniques ranging from the calculation of empirical
mean, variances and correlation indices [21], to more elaborated data analysis methods such as
clustering, regression analysis [45], principal component analysis and the like. Besides, possibility
theory can offer alternative representations of classical data sets, for instance possibilistic clustering
[80, 106] where class-membership is graded, fuzzy regression analysis where a fuzzy-valued affine
function is used [103, 104]. Concerning inferential statistics, the use of possibility theory suggests
substituting a single probabilistic model with a concise representation of a set of probabilistic
models among which the user is not forced to choose if information is lacking. The case of scarce
empirical evidence is especially worth studying. For instance, Masson and Denoeux [87] consider
multinomial distributions induced by a limited number of observations from which possibly large
confidence intervals on probabilities of realizations are obtained. These intervals delimit a set of
potential probabilistic models to be encompassed by means of a single possibility distribution. Since
confidence intervals are selected using a threshold, doing away with this threshold leads to a higher
order possibility distribution on probabilistic models, similar to De Cooman’s approach to fuzzy
probability intervals[17]. Lastly, the representation of high-dimensional possibility measures can be
envisaged using the possibilistic counterpart to Bayesian networks [8].

Acknowledgements This paper is based on previous works carried out with several people, espe-
cially Henri Prade, Gilles Mauris, Philippe Smets, and more recent works with Cedric Baudrit and
Dominique Guyonnet.

26
References

[1] Bardossy G., Fodor J., 2004. Evaluation of Uncertainties and Risks in Geology, Springer,
Berlin.

[2] Barnett V., 1973. Comparative Statistical Inference, J. Wiley, New York.

[3] Baudrit C., Dubois D., Fargier H., 2004. Practical representation of incomplete probabilis-
tic information. In: M. Lopez-Diaz et al. (Eds) Soft Methodology and Random Information
Systems (Proc. 2nd International Conference on Soft Methods in Probability and Statistics,
Oviedo), Springer, Berlin, 149-156. To appear in Comput. Stat. & Data Anal.

[4] Baudrit. C., Dubois D., 2005. Comparing Methods for Joint Objective and Subjective Un-
certainty Propagation with an example in a risk assessment, Proc. 4th Int. Symposium on
Imprecise Probabilities and Their Applications Pittsburgh, USA, 2005, 31-40.

[5] Baudrit. C., Guyonnet D., Dubois D., 2005. Post-processing the hybrid method for addressing
uncertainty in risk assessments, J. Environmental Engineering, 131,1750-1754.

[6] Benferhat S., Dubois D. and Prade H., 1997. Nonmonotonic reasoning, conditional objects and
possibility theory, Artificial Intelligence, 92, 259-276.

[7] Birnbaum Z. W., 1948. On random variables with comparable peakedness, Annals of Mathe-
matical Statistics, 19, 76-81.

[8] Borgelt C., and Kruse R., 2002. Learning from imprecise data: possibilistic graphical models
Computational Statistics & Data Analysis, 38, 449-463.

[9] Carlsson C., and Fuller R., 2001. On possibilistic mean value and variance of fuzzy numbers
Fuzzy Sets and Systems, 122, 315-326.

[10] Chanas S. and Nowakowski M., 1988. Single value simulation of fuzzy variable, Fuzzy Sets and
Systems, 25, 43-57.

[11] Coletti G. and Scozzafava R., 2003. Coherent conditional probability as a measure of un-
certainty of the relevant conditioning events, Proc. ECSQARU03, Aalborg, LNAI vol. 2711,
Springer Verlag, Berlin, 407-418.

[12] I. Couso, S. Moral, and P. Walley. A survey of concepts of independence for imprecise proba-
bilities Risk Decision and Policy 5 165–181, 2000.

[13] De Baets B., Tsiporkova E. and Mesiar R., 1999. Conditioning in possibility with strict order
norms, Fuzzy Sets and Systems, 106, 221-229.

[14] De Cooman G., 1997. Possibility theory Part I: Measure- and integral-theoretics groundwork;
Part II: Conditional possibility; Part III: Possibilistic independence, Int. J. of General Systems,
25(4), 291-371.

[15] De Cooman G., Aeyels D., 1999. Supremum-preserving upper probabilities. Information Sci-
ences, 118, 173 -212.

[16] De Cooman G., 2001. Integration and conditioning in numerical possibility theory. Annals of
Mathematics and AI, 32, 87-123

27
[17] De Cooman G., 2005. A behavioural model for vague probability assessments Fuzzy Sets and
Systems, 154, 305-358.

[18] Delgado M. and Moral S., 1987. On the concept of possibility-probability consistency, Fuzzy
Sets and Systems, 21, 311-318.

[19] Dempster A. P., 1967. Upper and lower probabilities induced by a multivalued mapping, Ann.
Math. Stat., 38, 325-339.

[20] Denneberg D., 1994. Nonadditive Measure and Integral, Kluwer Academic, Dordrecht, The
Netherlands.

[21] Denoeux T., Masson M.-H., Hbert P.-A., 2005, Nonparametric rank-based statistics and sig-
nificance tests for fuzzy data Fuzzy Sets and Systems, 153, 1-28.

[22] Dubois D., 1986. Belief structures, possibility theory and decomposable confidence measures
on finite sets, Computers and Artificial Intelligence (Bratislava), 5, 403-416.

[23] Dubois D., Fargier H., Fortin J., 2005. The empirical variance of a set of fuzzy intervals. Proc.
of the IEEE Int. Conf. on Fuzzy Systems, Reno, Nevada, IEEE Press, p. 885-890.

[24] Dubois D., Foulloy L., Mauris G., Prade H., 2004. Possibility/probability transformations,
triangular fuzzy sets, and probabilistic inequalities. Reliable Computing, 10, 273-297.

[25] Dubois D. Huellermeier E., 2005. A notion of comparative probabilistic entropy based on the
possibilistic specificity ordering. In L. Godo (Ed.) Proc. of the Europ. Conf. ECSQARU’05,
Barcelona, LNAI 3571, Springer, Berlin, 848-859.

[26] Dubois D., Kerre E., Mesiar R., Prade H., 2000. Fuzzy interval analysis. In: D. Dubois H. Prade
(Eds), Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series, Kluwer, Dordrecht,
483-581.

[27] Dubois D., Moral S. and Prade H., 1997. A semantics for possibility theory based on likelihoods,
J. of Mathematical Analysis and Applications, 205, 359-380.

[28] Dubois D. and Prade H., 1980. Fuzzy Sets and Systems: Theory and Applications, Academic
Press, New York.

[29] Dubois and Prade H., 1982. On several representations of an uncertain body of evidence, In
M.M. Gupta, and E. Sanchez (Eds) Fuzzy Information and Decision Processes, North-Holland,
Amsterdam, 167-181.

[30] Dubois D. and Prade H., 1983. Unfair coins and necessity measures: towards a possibilistic
interpretation of histograms. Fuzzy Sets and Systems, 10, 15-20.

[31] Dubois D. and Prade H., 1986. Fuzzy sets and statistical data, Europ. J. Operations Research,
25, 345-356.

[32] Dubois D. and Prade H., 1987. The mean value of a fuzzy number, Fuzzy Sets and Systems,
24, 279-300.

[33] Dubois D. and Prade H., 1988. Possibility Theory, Plenum Press, New York.

28
[34] Dubois D. and Prade H., 1990. Consonant approximations of belief functions. Int. J. Approx-
imate Reasoning, 4, 419-449.

[35] Dubois D. and Prade H., 1991. Random sets and fuzzy interval analysis, Fuzzy Sets and
Systems, 42, 87-101.

[36] Dubois D. and Prade H., 1992. When upper probabilities are possibility measures, Fuzzy Sets
and Systems, 49, 65-74.

[37] Dubois D. and Prade H., 1997. Bayesian conditioning in possibility theory, Fuzzy Sets and
Systems, 92, 223-240.

[38] Dubois D. and Prade H., 1998. Possibility theory: Qualitative and quantitative aspects, In
Gabbay D.M. and Smets P., (eds.) Handbook of Defeasible Reasoning and Uncertainty Man-
agement Systems Vol. 1, Kluwer Academic Publ., Dordrecht, 169-226.

[39] Dubois D., Nguyen H. T., Prade H., 2000. Possibility theory, probability and fuzzy sets:
misunderstandings, bridges and gaps. In: D. Dubois H. Prade (Eds), Fundamentals of Fuzzy
Sets, The Handbooks of Fuzzy Sets Series, Kluwer, Dordrecht, 343-438.

[40] Dubois D., Prade H., Rannou E., 1998. An improved method for finding typical values. Proc. of
7th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-based
Systems (IPMU’98), Paris. Editions EDK, Paris, pp. 1830–1837.

[41] Dubois D., Prade H. and Sandri S., 1993. On possibility/probability transformations. In: Fuzzy
Logic. State of the Art, In: R. Lowen, M. Roubens (eds.), Kluwer Acad. Publ., Dordrecht,
103-112.

[42] Dubois D., Prade H. and Smets P., 1996. Representing partial ignorance, IEEE Trans. on
Systems, Man and Cybernetics, 26, 361-377.

[43] Dubois, D., Prade, H., Smets, P., 2001. Not impossible vs. guaranted possible in fusion and
revision. Proc. of the 6th.European Conference (ESCQARU 2001), Toulouse, Springer-Verlag,
LNAI 2143, 522-531

[44] Dubois D. Prade H., Smets, P., 2003. A definition of subjective possibility. Badania Operacyjne
I Decyzje (Wroclaw University, Poland) #4, 7-22.

[45] D’Urso P., 2003. Linear regression analysis for fuzzy/crisp input and fuzzy/crisp output
data.Computational Statistics & Data Analysis, 42, 47-72.

[46] Edwards W. F., 1972. Likelihood, Cambridge University Press, Cambridge, U.K.

[47] Ferson S., Ginzburg L.R., 1995. Hybrid Arithmetic. Proceedings of ISUMA-NAFIPS’95, IEEE
Computer Society Press, 619-623.

[48] Ferson S., 1996. What Monte Carlo methods cannot do. Human and Ecology Risk Assessment,
2, 990-1007.

[49] Ferson S., Ginzburg L.R., 1996. Different methods are needed to propagate ignorance and
variability. Reliability Engineering and Systems Safety, 54, 133-144.

[50] Ferson S., Ginzburg L., Kreinovich V., Longpre L., and Aviles M., 2002. Computing variance
for interval data is NP-hard. ACM SIGACT News, 33, 108-118.

29
[51] Ferson, S., Berleant, D., Regan, H.M., 2004. Equivalence of methods for uncertainty propa-
gation of real-valued random variables. International Journal of Approximate Reasoning, 36,
1-30

[52] Ferson S., Ginzburg L., Akcakaya R. 2006. Whereof one cannot speak: when input distributions
are unknown. Risk Analysis, to appear.

[53] Fullér R., and Majlender P., 2003. On weighted possibilistic mean and variance of fuzzy num-
bersFuzzy Sets and Systems, 136,363-374.

[54] Fortemps P. and Roubens M., 1996. Ranking and defuzzification methods based on area com-
pensation, Fuzzy Sets and Systems, 82, 319-330.

[55] Gebhardt J. and Kruse R., 1993. The context model: an integrating view of vagueness and
uncertainty. Int. J. Approximate Reasoning, 9, 283-314.

[56] Gebhardt J. and Kruse R., 1994. A new approach to semantic aspects of possibilistic rea-
soning. In M. Clarke et al. (Eds.) Symbolic and Quantitative Approaches to Reasoning and
Uncertainty, Lecture Notes in Computer Sciences Vol. 747, Springer Verlag, 151-160.

[57] Gebhardt J. and Kruse R., 1994. On an information compression view of possibility theory.
Proc 3rd IEEE Int. Conference on Fuzzy Systems. Orlando,Fl., 1285-1288.

[58] Geer J.F. and Klir G.J., 1992. A mathematical analysis of information-preserving transfor-
mations between probabilistic and possibilistic formulations of uncertainty, Int. J. of General
Systems, 20, 143-176.

[59] Gil M. A., 1992. A note on the connection between fuzzy numbers and random intervals,
Statistics and Probability Lett., 13, 311-319.

[60] Giles R., 1982. Foundations for a theory of possibility, Fuzzy Information and Decision Pro-
cesses (Gupta M.M. and Sanchez E., eds.), North-Holland, 183-195.

[61] Guyonnet D., Bourgine B., Dubois D., Fargier H., Cume B., Chiles J.P., 2003 Hybrid approach
for addressing uncertainty in risk assessments. Journal of Environmental Engineering, 126,
68-78.

[62] Kaufmann A. and Gupta M. M., 1985. Introduction to Fuzzy Arithmetic - Theory and Appli-
cations, Van Nostrand Reinhold, New York.

[63] Hacking I., 1975. All kinds of possibility, Philosophical Review, 84, 321-347.

[64] Halpern J., 2004. Reasoning about Uncertainty MIT Press, Cambridge, Mass.

[65] Hardy, G.H. Littlewood, J.E., Polya, G. Inequalities, Cambridge University Press, Cambridge
UK, 1952.

[66] Heilpern S., 1992. The expected value of a fuzzy number, Fuzzy Sets and Systems, 47, 81-87.

[67] Heilpern S., 1997. Representation and application of fuzzy numbers, Fuzzy Sets and Systems,
91, 259-268.

[68] Higashi H. and Klir G., 1982. Measures of uncertainty and information based on possibility
distributions, Int. J. General Systems, 8, 43-58.

30
[69] Hisdal E., 1978. Conditional possibilities independence and noninteraction, Fuzzy Sets and
Systems, 1, 283-297.

[70] Hisdal E., 1988. Are grades of membership probabilities? Fuzzy Sets and Systems, 25, 325-348.

[71] Helton J.C. Oberkampf W.L., Eds., 2004. Alternative Representations of Uncertainty, Relia-
bility Engineering and Systems Safety, vol. 85, Elsevier, 369 p.

[72] Hoffman F. O. and Hammonds J. S., 1994. Propagation of Uncertainty in Risk Assessments:
The Need to Distinguish Between Uncertainty Due to Lack of Knowledge and Uncertainty
Due to Variability, Risk Analysis, vol. 14, pp. 707-712.

[73] Joslyn C., 1997. Measurement of possibilistic histograms from interval data, Int. J. of General
Systems, 26(1-2), 9-33.

[74] Klir G.J., 1990. A principle of uncertainty and information invariance, Int. J. of General
Systems, 17, 249-275.

[75] Klir G.J., 2006. Uncertainty and Information. Foundations of Generalized Information Theory.
J. Wiley.

[76] Klir G.J. and Folger T., 1988. Fuzzy Sets, Uncertainty and Information, Prentice Hall, Engle-
wood Cliffs, NJ.

[77] Klir G.J. and Parviz B., 1992. Probability-possibility transformations: A comparison, Int. J.
of General Systems, 21, 291-310.

[78] Klir G.J., Ramer A., 1990. Uncertainty in the Dempster-Shafer theory : a critical re-
examination. Int. J. of General Systems, 18(2), 155-166.

[79] Koerner R., 1997. On the variance of fuzzy random variable, Fuzzy Set and Systems, 92, 83-93.

[80] Krishnapuram R., Keller J., 1993. A possibilistic approach to clustering, IEEE Trans. Fuzzy
Systems (1) 98-110.

[81] Lapointe S. Bobee B., 2000. Revision of possibility distributions: A Bayesian inference pattern,
Fuzzy Sets and Systems, 116, 119-140

[82] Lindley D.V., 1982. Scoring rules and the inevitability of probability, Int. Statist. Rev., 50,
1-26.

[83] Mauris G., Lasserre V., Foulloy L., 2001. A fuzzy appproach for the expression of uncertainty
in measurement. Int. J. Measurement, 29, 165-177.

[84] Levi I., 1973. Gambling with Truth, The MIT Press, Cambridge, Mass.

[85] Levi I., 1980. Potential surprize: its role in inference and decision-making. In: Applications of
Inductive logic In L.J. Cohen, M. Hesse, (Eds), Oxford UP, UK.

[86] Lewis D. L., 1979. Counterfactuals and comparative possibility, In : Ifs, Harper W. L., Stal-
naker R. and Pearce G., (eds.), D. Reidel, Dordrecht, 57-86.

[87] Masson, M. Denoeux T., 2006. Inferring a possibility distribution from empirical data, Fuzzy
sets and Systems, 157, 319-340.

31
[88] H. Moulin. Axioms of Cooperative Decision Making. Cambridge University Press, Cambridge,
MA, 1988.

[89] Neumaier A., 2004. Clouds, fuzzy sets and probability intervals. Reliable Computing, 10, 249-
272.

[90] Nguyen, H. T., Bouchon-Meunier, B., 2003. Random sets and large deviations principle as a
foundation for possibility measures, Soft Computing, 8:61-70.

[91] Nunez Garcia J., Kutalik Z. Cho K.-H and Wolkenhauer O., 2003. Level sets and minimum
volume sets of probability density functions. Int. J. Approximate Reasoning, 34, 25-48.

[92] Ralescu D., 2002. Average level of a fuzzy set. In C. Bertoluzza, M. A. Gil, and D. A. Ralescu,
(Eds), Statistical Modeling, Analysis and Management of Fuzzy data, Springer, Heidelberg,
119–126.

[93] Raufaste E., Da Silva Neves R., Mariné C., 2003. Testing the descriptive validity of possibility
theory in human judgements of uncertainty. Artificial Intelligence, 148: 197-218.

[94] Saade J. J. and Schwarzlander H., 1992. Ordering fuzzy sets over the real line: An approach
based on decision making under uncertainty, Fuzzy Sets and Systems, 50, 237-246.

[95] Shackle G. L.S., 1961. Decision, Order and Time in Human Affairs, (2nd edition), Cambridge
University Press, UK.

[96] Shafer G., 1976. A Mathematical Theory of Evidence, Princeton University Press, Princeton.

[97] Shafer G., 1987. Belief functions and possibility measures, In Bezdek J.C., (Ed.) Analysis of
Fuzzy Information Vol. I: Mathematics and Logic, CRC Press, Boca Raton, FL, 51-84.

[98] Shapley S., 1953. A value for n-person games. In Kuhn and Tucker, eds., Contributions to the
Theory of Games, II, Princeton University Press, 307-317.

[99] Shilkret N., 1971. Maxitive measure and integration, Indag. Math., 33, 109-116.

[100] Smets P., 1990. Constructing the pignistic probability function in a context of uncertainty,
In Henrion M. et al., (Eds.) Uncertainty in Artificial Intelligence, vol. 5, North-Holland, Am-
sterdam, 29-39.

[101] Smets P. and Kennes R., 1994. The transferable belief model, Artificial Intelligence, 66, 191-
234.

[102] Spohn W., 1988. Ordinal conditional functions: A dynamic theory of epistemic states, In
:Harper W. and Skyrms B., (Eds.) Causation in Decision, Belief Change and Statistics, 105-
134.

[103] Tanaka, H., Uejima, S. and Asai, K., 1982. Linear regression analysis with fuzzy model. IEEE
Trans. Systems Man Cybernet. 12, pp. 903-907.

[104] Tanaka, H., Lee H. 1999. Exponential possibility regression analysis by identification method
of possibilistic coefficients Fuzzy Sets and Systems, 106, s 155-165.

[105] Thomas S.F., 1995. Fuzziness and Probability, ACG Press, Wichita, Kansas.

32
[106] Timm H., Borgelt C., Doering C., Kruse, R., 2004. An extension to possibilistic fuzzy cluster
analysis. Fuzzy Sets and Systems, 147, 3-16.

[107] Van Leekwijk W. and Kerre E., 1999. Defuzzification: criteria and classification. Fuzzy Sets
and Systems, 118, 159-178.

[108] Walley P., 1991. Statistical Reasoning with Imprecise Probabilities, Chapman and Hall.

[109] Walley P., 1996. Measures of uncertainty in expert systems, Artificial Intelligence, 83, 1-58.

[110] Walley P. and de Cooman G., 1999. A behavioural model for linguistic uncertainty, Informa-
tion Sciences, 134, 1-37.

[111] Walley P. and de Cooman G., 1999. Coherence of rules for defining conditional possibility,
International Journal of Approximate Reasoning, 21, 63-107

[112] Wang P.Z., 1983. From the fuzzy statistics to the falling random subsets, In Wang P.P., (Ed.)
Advances in Fuzzy Sets, Possibility Theory and Applications, Plenum Press, New York, 81-96.

[113] Yager R.R., 1980. A foundation for a theory of possibility, J. Cybernetics, 10, 177-204.

[114] Yager R. R., 1981. A procedure for ordering fuzzy subsets of the unit interval, Information
Sciences, 24, 143-161.

[115] Yager R.R., 1992. On the specificity of a possibility distribution, Fuzzy Sets and Systems, 50,
279-292.

[116] Yager R. R. and Filev D., 1993. On the issue of defuzzification and selection based on a fuzzy
set, Fuzzy Sets and Systems, 55, 255-271.

[117] Zadeh L.A., 1965. Fuzzy sets, Information and Control, 8, 338-353.

[118] Zadeh L. A., 1975. The concept of a linguistic variable and its application to approximate
reasoning, Information Sciences, Part I: 8, 199-249; Part II: 8, 301-357; Part III: 9, 43-80.

[119] Zadeh L. A., 1978. Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems,
1, 3-28.

33