On the Connection Between Probability Boxes and Possibility Measures

EUSFLAT-LFA 2011 July 2011 Aix-les-Bains, France

On the connection between probability boxes and possibility measures

Matthias C. M. Troffaes1 Enrique Miranda2 Sebastien Destercke3

1Durham University, Durham, United Kingdom 2University of Oviedo, Oviedo, Spain 3UMR1208, French Agricultural Research Centre for International Development, Montpellier, France

Abstract [14, Section 4.6.6] and [16], and in much more detail in [13]. We explore the relationship between p-boxes on totally The paper is organised as follows: in Section 2, we preordered spaces and possibility measures. We start by give the basics of the behavioural theory of imprecise demonstrating that only those p-boxes who have 0–1- probabilities, and recall some facts about p-boxes and valued lower or upper cumulative distribution function possibility measures; in Section 3, we first determine can be possibility measures, and we derive expressions necessary and sufficient conditions for a p-box to be for their natural extension in this case. Next, we estab- maximum preserving, before determining in Section 4 lish necessary and sufficient conditions for a p-box to necessary and sufficient conditions for a p-box to be a be a possibility measure. Finally, we show that almost possibility measure; in Section 5, we show that almost every possibility measure can be modelled by a p-box. any possibility measure can be seen as particular p-box, Whence, any techniques for p-boxes can be readily ap- and that many p-boxes can be seen as a couple of possi- plied to possibility measures. We demonstrate this by bility measures; some special cases are detailed in Sec- deriving joint possibility measures from marginals, un- tion 6. Finally, in Section 7 we apply the work on mul- der varying assumptions of independence, using a tech- tivariate p-boxes from [13] to derive multivariate possi- nique known for p-boxes. bility measures from given marginals, and in Section 8 we give a number of additional comments and remarks. Keywords: Possibility measures, maxitive measures, p- Note that proofs are omitted for brevity, whence be- boxes, coherent upper previsions, natural extension. ware that results are presented in the order that they ap- pear most logical, and not in the order that they are most 1. Introduction easily proven. Possibility measures [1] are supremum preserving set 2. Preliminaries functions, and are widely applied in many fields, including data analysis [2], cased-based reasoning [3], and 2.1. Imprecise Probabilities psychology [4]. In this paper we are concerned with quantitative possibility theory [5], where degrees of pos- We briefly introduce imprecise probabilities; see [17, 18, sibility range in the unit interval. Their interpretation as 14, 19] for more details. upper probability [6, 7] fits our purpose best. Let Ω be a possibility space. A subset of Ω is called Probability boxes [8], or p-boxes for short, are pairs of an event. Denote the set of all events by ℘(Ω), and the lower and upper cumulative distribution functions, and set of all finitely additive probabilities on ℘(Ω) by P. are often used in risk and safety studies, in which cumu- An upper probability is any real-valued function P de- lative distributions play an essential role. P-boxes have fined on an arbitrary subset K of ℘(Ω). With P, we been connected to info-gap theory [9], random sets [10], associate a lower probability P on {A : Ac ∈ K } via and also, partly, to possibility measures [11]. P-boxes P(A) = 1 − P(Ac). Consider the set can be defined on arbitrary finite spaces [12], and, more (P) = {P ∈ : (∀A ∈ )(P(A) ≤ P(A))}. generally, even on arbitrary totally preordered spaces M P K [13]—we will use this extensively. The upper envelope E of M (P) is called the natural This paper aims to consolidate the connection be- extension [14, Thm. 3.4.1] of P: tween possibility measures and p-boxes, making as few assumptions as possible. We prove that almost every E(A) = sup{P(A) : P ∈ M (P)} possibility measure can be interpreted as a p-box. Con- A ⊆ versely, we provide necessary and sufficient conditions for all Ω. The corresponding lower probability is E E(A) = − E(Ac) E for a p-box to be a possibility measure. denoted by , so 1 . Clearly, is the (P) To study this connection, we use imprecise proba- lower envelope of M . P coherent bilities [14], of which both possibility measures and p- We say that is (see [14, Sec. 3.3.3]) when, A ∈ boxes are particular cases. Possibility measures are ex- for all K , P(A) = E(A). plored as imprecise probabilities in [6, 7, 15], and p- boxes are studied as imprecise probabilities briefly in P is called coherent whenever P is.

F F H = {(x0,x1] ∪ (x2,x3] ∪ ··· ∪ (x2n,x2n+1]: ∗ x0 ≺ x1 ≺ ··· ≺ x2n+1 in Ω }.

0 1 Ω Proposition 2. [13, Prop. 4] For any A ∈ H , that is A = (x0,x1] ∪ (x2,x3] ∪ ··· ∪ (x2n,x2n+1] with x0 ≺ x1 ≺ ··· ≺ ∗ H Figure 1: Example of a p-box on [0,1]. x2n+1 in Ω , it holds that EF,F (A) = PF,F (A), where

n+1 H 2.2. P-Boxes PF,F (A) = 1 − ∑ max{0,F(x2k) − F(x2k−1)}, (1) k=0 In this section, we revise p-boxes defined on totally preordered (not necessarily finite) spaces. For further de- with x−1 = 0Ω− and x2n+2 = 1Ω. tails, see [13]. Start with a totally preordered space (Ω,). So, is To calculate EF,F (A) for an arbitrary event A ⊆ Ω, transitive, reflexive and any two elements are compara- H ∗ use the outer measure [14, Cor. 3.1.9] PF,F of the upper ble. As usual, we write x ≺ y for x y and x 6 y, x y probability PH defined in Eq. (1): for y ≺ x, and x ' y for x y and y x. For any two x, F,F y ∈ Ω exactly one of x ≺ y, x ' y, or x y holds. We H ∗ H also use the following common notation for intervals in EF,F (A) = PF,F (A) = inf PF,F (C). (2) C∈H ,A⊆C Ω: [x,y] = {z ∈ Ω: x z y} For intervals, we immediately infer from Proposi- tion 2 and Eq. (2) that (‘i.p.’ stands for ‘immediate pre- (x,y) = {z ∈ : x ≺ z ≺ y} Ω decessor’) and similarly for [x,y) and (x,y]. E ((x,y]) = F(y) − F(x) (3a) We assume that Ω has a smallest element 0Ω and a F,F largest element 1 —we can always add these two ele- Ω EF,F ([x,y]) = F(y) − F(x−) (3b) ments to the space Ω. ( A cumulative distribution function is a non- F(y) − F(x) if y has no i.p. EF,F ((x,y)) = (3c) decreasing map F : Ω → [0,1] for which F(1Ω) = 1. F(y−) − F(x) if y has an i.p. For each x ∈ Ω, F(x) is interpreted as the probability ( F(y) − F(x−) y of [0 ,x]. if has no i.p. Ω EF,F ([x,y)) = The quotient set of Ω with respect to ' is denoted by F(y−) − F(x−) if y has an i.p. Ω/ ': (3d)

[x]' = {y ∈ Ω: y ' x} for any x ∈ Ω 1 for any x ≺ y in Ω, where F(y−) denotes supz≺y F(z) Ω/ ' = {[x]' : x ∈ Ω}. and similarly for F(x−). If Ω/ ' is ﬁnite, then one can z− z Because F is non-decreasing, F is constant on elements think of as the immediate predecessor of in the quotient space Ω/ ' for any z ∈ Ω. We also have that [x]' of Ω/ '.

Definition 1. A probability box, or p-box, is a pair EF,F ({x}) = F(x) − F(x−) (4) (F,F) of cumulative distribution functions from Ω to [0,1] satisfying F ≤ F. for any x ∈ Ω. A p-box is interpreted as a lower and an upper cumulative distribution function (see Fig. 1), or more specifi- 2.3. Possibility and Maxitive Measures cally, as an upper probability P on the set of events F,F Brevity is stipulated; see [1, 5, 6, 7] for details. {[0Ω,x]: x ∈ Ω} ∪ {(y,1Ω]: y ∈ Ω} Definition 3. A maxitive measure is an upper probabil- defined by ity P on ℘(Ω) satisfying P(A ∪ B) = max{P(A),P(B)} for every A and B ⊆ Ω. PF,F ([0Ω,x]) = F(x) and PF,F ((y,1Ω]) = 1 − F(y). Proposition 4. [16, Def. 3.22, Thm. 3.46] A maxitive We denote by EF,F the natural extension of PF,F to all events. To simplify the expression for natural extension, measure P is coherent whenever P(/0) = 0 and P(Ω) = 1. we introduce an element 0Ω− such that: Possibility measures are a particular case of maxitive 0Ω− ≺ x for all x ∈ Ω measures.

F(0Ω−) = F(0Ω−) = F(0Ω−) = 0. 1 In case x = 0Ω, evidently, 0Ω− is the immediate predecessor.

837 Definition 5. A (normed) possibility distribution is a Here, A x means z x for all z ∈ A, and similarly mapping π : Ω → [0,1] satisfying supx∈Ω π(x) = 1. A y ≺ A means y ≺ z for all z ∈ A. Thus, /0 x and y ≺ /0 possibility distribution π induces a possibility measure for all x and y. Π on ℘(Ω), given by: An important special case is summarized in the following corollary: Π(A) = supπ(x) for all A ⊆ Ω. x∈A Corollary 8. Let (F,F) be a p-box with 0–1-valued F, and let B = {x ∈ Ω∗ : F(x) = 0}. If Ω/ ' is order com- If we write EΠ for the conjugate lower probability of plete, then, for any A ⊆ Ω, the upper probability Π, then: EF,F (A) = minF(supA ∩ [0Ω,y]). c y∈Bc EΠ(A) = 1 − Π(A ) = 1 − sup π(x). c x∈A If, in addition, Bc has a minimum, then A possibility measure is maxitive, but not all maxitive E (A) = F(supA ∩ [0 ,minBc]). measures are possibility measures. F,F Ω As an imprecise probability model, possibility mea- If, in addition, F = I , then [1Ω]' sures are not as expressive as for instance p-boxes. This poor expressive power is also illustrated by the fact that, EF,F (A) = F(supA). (5) for any event A: Note that Eq. (5) is essentially due to [7, paragraph preceeding Theorem 11]—they work with chains and Π(A) < 1 =⇒ EΠ(A) = 0, and multivalued mappings, whereas we work with total pre- E (A) > 0 =⇒ Π(A) = 1, Π orders. The case of order complete Ω/ ' is applicable meaning that every event has a trivial probability bound for instance when Ω is a finite space, or when it is an on at least one side. Their main attraction is that cal- interval of real numbers. culations with them are very easy: to find the upper (or In case of 0–1-valued F, we arrive at the following lower) probability of any event, a simple supremum suf- expression: fices. Proposition 9. Let (F,F) be a p-box with 0–1-valued F, In the following sections, we characterize the circum- and let C = {x ∈ Ω∗ : F(x) = 0}. Then, for any A ⊆ Ω, stances under which a possibility measure Π is the natural extension of some p-box (F,F). In order to do so, EF,F (A) = 1 − max sup F(y). x∈C ∗ we first characterise the conditions under which a p-box y∈Ω : y≺A∩(x,1Ω] induces a maxitive measure. Again, we summarize an important special case in the following corollary: 3. P-boxes as Maxitive Measures Corollary 10. Let (F,F) be a p-box with 0–1-valued ∗ 3.1. Necessary and Sufficient Condition for F, and let C = {x ∈ Ω : F(x) = 0}. If Ω/ ' is order Maxitivity complete, and C has a maximum, then, for any A ⊆ Ω, 1 − F(infA ∩Cc) if A ∩Cc has Let us begin by characterising under which conditions  the natural extension of a p-box is maxitive:  no minimum EF,F (A) = c c 1 − F((minA ∩C )−) if A ∩C has Theorem 6. The natural extension E of a p-box  F,F  a minimum. (F,F) is maximum preserving if and only if at least one of F or F is 0–1-valued. If, in addition, F = 1, then ( Hence, for the purposes of this paper we can restrict 1 − F(infA) if A has no minimum our attention to p-boxes (F,F) where at least one of F EF,F (A) = or F is 0–1-valued. Next, we provide expressions for the 1 − F(minA−) if A has a minimum. natural extensions of such p-boxes, and determine under which conditions these are possibility measures. 4. P-Boxes as Possibility Measures. In this section, we identify when p-boxes coincide ex- 3.2. Natural Extension of Maxitive P-Boxes actly with a possibility measure. All results in this sec- In case of 0–1-valued F, we arrive at the following ex- tion rely on the following trivial, yet important, lemma: pression: Lemma 11. For a p-box (F,F) there is a possibility measure Π such that E = Π if and only if Proposition 7. Let (F,F) be a p-box with 0–1-valued F, F,F and let B = {x ∈ Ω∗ : F(x) = 0}. Then, for any A ⊆ Ω, EF,F (A) = supEF,F ({x}) for all A ⊆ Ω x∈A EF,F (A) = min inf F(x). y∈Bc x∈Ω∗ : A∩[0 ,y]x Ω and in such a case, π(x) = EF,F ({x}).

838 We say that a p-box (F,F) is a possibility measure Theorem 13. For every possibility measure Π on Ω with when the conditions of Lemma 11 are satisﬁed. possibility distribution π such that π(Ω) = {π(x): x ∈ Taking into account that every possibility measure is Ω} is compact, there is a preorder on Ω and an upper maxitive, we deduce from Theorem 6 that, for a p-box cumulative distribution function F such that the p-box (F = I ,F) is a possibility measure with possibility to be a possibility measure, it is necessary that at least [1Ω]' one of F or F is 0–1-valued. The next corollary char- distribution π. In fact, one may take the preorder to acterizes the additional regularity conditions to arrive at be the one induced by π (so x y whenever π(x) ≤ π(y)) necessity and sufﬁciency: and F = π.

Corollary 12. Assume that Ω/ ' is order complete and The representing p-box is not necessarily unique: let (F,F) be a p-box. Then (F,F) is a possibility mea- Example 14. Let = {x ,x } and let be the possibil- sure if and only if either Ω 1 2 Π ity measure determined by the possibility distribution (L1) F is 0–1-valued, π(x1) = 0.5 π(x2) = 1. (L2) F(x) = F(x−) for all x ∈ Ω that have no immediate predecessor, and By Theorem 13, this possibility measure can be obtained if we consider the order x ≺ x and the p-box (F ,F ) (L3) {x ∈ Ω∗ : F(x) = 1} has a minimum, 1 2 1 1 given by or F1(x1) = 0 F1(x2) = 1 (U1) F is 0–1-valued, F1(x1) = 0.5 F1(x2) = 1. (U2) F(x) = F(x+) for all x ∈ Ω that have no immediate successor, and However, we also obtain it if we consider the order x2 ≺ x1 and the p-box (F2,F2) given by (U3) {x ∈ Ω∗ : F(x) = 0} has a maximum. F = I F2(x1) = 1 F2(x2) = 0.5 Note that, in case [1Ω]' , condition (L2) is essen- F (x ) = 1 F (x ) = 1. tially due to [7, Observation 9]. Also note that, for EF,F 2 1 2 2 to be a possibility measure, the conditions are still nec- Indeed, by Corollary 12, E is a possibility measure. essary even when Ω/ ' is not order complete: the proof F2,F2 in this direction does not require order completeness. By Eq. (4), its associated possibility distribution is As a special case, when Ω/ ' is ﬁnite, EF,F is a pos- E (x ) = F(x ) − F(x −) = 1 sibility measure with possibility distribution F2,F2 2 2 2

( EF ,F (x1) = F(x1) − F(x1−) = 0.5, F(x) if x min{y: F(y) = 1} 2 2 π(x) = 0 otherwise. i.e., π, as with the given ordering, x2− = 0Ω− and x1− = x2. when F is 0–1-valued, and Moreover, a p-box may be maximum preserving even ( 1 − F(x−) if F(x) = 1 if neither the lower nor the upper distribution function is π(x) = vacuous, as we have shown in Corollary 12. 0 otherwise. Also, there are possibility measures which cannot be when F is 0–1-valued. represented as p-boxes when π(Ω) is not compact: We now characterise when, conversely, possibility Example 15. Let Ω = [0,1], and consider the possibil- measures can be represented as p-boxes. We shall see ity distribution given by π(x) = (1 + 2x)/8 if x < 0.5, that this is possible for almost all of them. π(0.5) = 0.4 and π(x) = x if x > 0.5; note that π(Ω) = [0.125,0.25) ∪ {0.4} ∪ (0.5,1] is not compact. The or- 5. From Possibility Measures to P-Boxes dering induced by π is the usual ordering on [0,1]. Let Π be the possibility measure induced by π. We show 5.1. Possibility Measures as Specific P-Boxes that there is no p-box (F,F) on ([0,1],), regardless of the ordering on [0,1], such that E = Π. [11] already discuss the link between possibility mea- F,F sures and p-boxes defined on the real line with the usual By Corollary 12, if EF,F = Π, then at least one of F ordering. They show that any possibility measure can be or F is 0-1–valued. Assume first that F is 0-1–valued. approximated by a p-box, however at the expense of los- By Eq. (4), EF,F ({x}) = F(x)−F(x−) = π(x). Because ing some information. We substantially strengthen their π(x) > 0 for all x, it must be that F(x−) = 0 for all x, result, and even reverse it: we prove that any possibility so F = π. Because F is non-decreasing, x y if and measure with compact range can be exactly represented only if F(x) ≤ F(y); in other words, can only be the by a p-box with vacuous lower cumulative distribution usual ordering on [0,1] for (F,F) to be a p-box. Hence, F = I F = I{1}. function, that is, with [1Ω]' . In other words, generally speaking, possibility measures are a special case of For (F,F) to induce the possibility measure Π, we p-boxes on totally preordered spaces. know from Corollary 12 that F(x) = F(x−) for every

839 x that has no immediate predecessor, that is, for every Example 17. Consider x ≺ y ∈ Ω. The distance between x > 0. But, F(0.5) = 0.4 6= 0.25 = F(0.5−). E and its approximation min{E ,E } on the inter- F,F πF πF Similarly, if F would be 0–1-valued, then we deduce val (x,y] is given by from Eq. (4) that F(x) = 1 for every x, again because min{E ((x,y]),E ((x,y])} − E((x,y]) π(x) > 0 for all x. Therefore, F(x−) = 1−π(x) for all x. πF πF But, because F is non-decreasing, one can easily show = min{F(y),1 − F(x)} − (F(y) − F(x)) that can only be the inverse of the usual ordering on = min{F(x),1 − F(y)}. [0,1] for (F,F) to be a p-box. Now, for (F,F) to induce the possibility measure Π, Therefore, the approximation will be close to the exact we know from Corollary 12 that F(x) = F(x+) for every value only when either F(x) is close to zero or F(y) is x that has no immediate successor in with respect to , close to one. that is, for every x ≺ 0, or equivalently, for every x > 0. In general, the set M (Π1) ∩ M (Π2) associated with Whence, two possibility distributions π1 and π2 is not a p-box. In- deed, it may be empty (take for instance Ω = {1,2} and F(x) = F(x+) = inf F(y−) = 1 − supπ(y) the possibility measures determined by the distributions y x y 0. This leads to a contradiction: by the deﬁ- sult in another uncertainty model, such as a cloud [20]. nition of π, we have on the one hand, In particular, when M (Π1) ∩ M (Π2) is non-empty, the resulting coherent upper probability is in general not F(0.5−) = sup (1 − supπ(y)) = 0.5 a possibility measure. An interesting open problem is x>0.5 y

Hence, EF,F coincides with Π in neither case. 6. Natural Extension of 0–1-Valued P-Boxes Another way of relating possibility measures and p- boxes goes via random sets (see for instance [10] and From Proposition 7, we can derive an expression for the [12]). natural extension of a 0–1-valued p-box: Proposition 18. Let (F,F) be a p-box where F = 5.2. P-boxes as Conjunction of Possibility Measures ICc ,F = IBc for some C ⊆ B ⊆ Ω. Then for any A ⊆ Ω,  In [12], where p-boxes are studied on ﬁnite spaces, it is 0 if there are x ∈ C and y ∈ Bc s.t.  shown that a p-box can be interpreted as the conjunction c c EF,F (A) = A ∩C x,A ∩ B ∩C = /0,y ≺ A ∩ B of two possibility measures, in the sense that (P ) M F,F 1 otherwise. is the intersection of two sets of additive probabilities induced by two possibility measures. The next propo- Moreover, Corollary 12 allows us to determine when sition extends this result to arbitrary totally preordered this p-box is a possibility measure: spaces. Proposition 19. Assume that Ω/ ' is order complete. c c Proposition 16. Let (F,F) be a p-box such that (F,IΩ) Let (F,F) be a p-box where F = IC ,F = IB for some and (I ,F) are possibility measures (see Corol- C ⊆ B ⊆ Ω. The following statements are equivalent: [1Ω]' lary 12). Then, (F,F) is the intersection of two pos- 1. (F,F) is a possibility measure. sibility measures deﬁned by the distributions 2. Bc has a minimum and C has a maximum. πF (x) = F(x) and πF (x) = 1 − F(x−), In particular, we can characterise under which conditions a precise p-box, i.e., one where F = F := F, in- in the sense that M (P ) = M (Π ) ∩ M (Π ). F,F πF πF duces a possibility measure. The natural extension of This suggests a simple way (already mentioned precise p-boxes on the unit interval was considered in in [12]) to conservatively approximate EF,F by using the [21, Section 3.1]. From Theorem 6, the natural exten- two possibility distributions: it holds that sion of F can only be a possibility measure when F is 0–1-valued. If we apply Propositions 18 and 19 with max{E (A),E (A)} ≤ E (A) B = C we obtain the following: πF πF F,F Corollary 20. Let (F,F) be a precise p-box where F = ≤ EF,F (A) ≤ min{Eπ (A),EπF (A)}. F F is 0–1-valued, and let B = {x ∈ Ω∗ : F(x) = 0}. Then, This approximation is computationally attractive, as it for every subset A of Ω, allows us to use the supremum preserving properties of  c possibility measures. However, as next example shows, 0 if there are x ∈ B,y ∈ B  c the approximation will usually be very conservative, and EF,F (A) = s.t. A ∩ B x and y ≺ A ∩ B hence not likely to be helpful. 1 otherwise.

840 If moreover Ω/ ' is order complete, then (F,F) is a This result is a consequence of [13, Lemma 22], possibility measure if and only if B has a maximum and which gives the least conservative p-box that domi- Bc has a minimum. nates the natural extension of the combination of some marginal p-boxes by . Here we are considering the par- As a consequence, we deduce that a precise 0–1- ticular case where the marginal p-boxes represent possi- valued p-box never induces a possibility measure in case bility measures, and approximating of the corresponding (Ω,) = ([0,1],≤), except if F = I[0,1]. p-box by a possibility measure.

7. Constructing Multivariate Possibility Measures 7.2. Natural Extension: The Fréchet Case from Marginals n The natural extension i=1Pi of P1,..., Pn is the upper In [13], multivariate p-boxes were constructed from envelope of all joint (finitely additive) probability mea- marginals. We next apply this construction together with sures whose marginal distributions are compatible with the p-box representation of possibility measures, given the given marginal upper probabilities. So, the model is by Theorem 13, to build a joint possibility measure from completely vacuous (that is, it makes no assumptions) some given marginals. As particular cases, we consider about the dependence structure, as it includes all possi- the joint, ble forms of dependence. See [24, p. 120, §3.1] for a rigorous definition. In this paper, we only need the fol- (i) either without any assumptions about dependence lowing equality, which is one of the Fréchet bounds (see or independence between variables, that is, using for instance [14, p. 122, §3.1.1]): the Fréchet-Hoeffding bounds [22], ! n n (ii) or assuming epistemic independence between all n i=1Pi Ai = minPi(Ai) ∏ i= variables, which allows us to use the factorization i=1 1 property [23]. for all A1 ⊆ X1,..., An ⊆ Xn. Let us consider n variables X1,..., Xn, and assume Theorem 22. The possibility distribution that for each variable Xi we are given a possibility measure Πi with corresponding possibility distribution πi on n π(x) = maxπi(xi) (6) Xi. We assume that the range of all marginal possibil- i=1 ity distributions is [0,1]; in particular, Theorem 13 ap- plies, and each marginal can be represented by a p-box induces the least conservative upper cumulative distri- on ( , ), with vacuous F , and F = . Remember bution function on (Ω,) that dominates the natural ex- Xi i i i πi n tension Πi of Π ,..., Πn. that the preorder i is the one induced by πi. i=1 1 In other words, if we consider the marginal credal sets 7.1. Multivariate Possibility Measures M (Π1),..., M (Πn) and consider the set M of all the The construction in [13, Sect. 7] employs the following finitely additive probabilities on Ω whose Xi-marginals belong to M (Πi) for i = 1,...,n, then M is included in mapping Z, which induces a preorder on Ω = X1 × the credal set of the possibility distribution π as defined ··· × Xn: n in Eq. (6). Z(x1,...,xn) = maxπi(xi). i=1 Since it is based on very mild assumptions, it is With this choice of Z, we can easily find the possibility not surprising that the possibility distribution given by measure which represents the joint as accurately as pos- Eq. (6) is very uninformative (that is, very close to a sible, under any rule of combination of coherent lower vacuous model where π(x) = 1 for every x): we shall probabilities: have π(x) = 1 as soon as πi(xi) = 1 for some i, even if π j(x j) = 0 for every j 6= i. In particular, if one of the Lemma 21. Let be any rule of combination of coher- marginal possibility distributions is vacuous, then so is ent upper probabilities, mapping the marginals P1,..., π. This also shows that the corresponding possibility Jn Pn to a joint coherent upper probability i=1 Pi on all measure Π may not have Π1,..., Πn as its marginals. events. If there is a continuous function u for which 7.3. Independent Natural Extension n n ! K Pi ∏Ai = u(P1(A1),...,Pn(An)) In contrast, we can consider joint models which satisfy i=1 i=1 the property of epistemic independence between the dif- ferent X1,..., Xn. This property means, roughly speak- for all A1 ⊆ X1,...,An ⊆ Xn, then the possibility distribution π defined by ing, that the conditional models that we can derive from the joint coincide with the marginals. The most conser- π(x) = u(Z(x),...,Z(x)) vative of these models is called the independent natural n extension ⊗i=1Pi of P1,..., Pn. See [23] for a rigorous induces the least conservative upper cumulative distri- definition and properties, and [15] for a study of joint bution function on (Ω,) that dominates the combina- possibility measures that satisfy epistemic independence Jn tion i=1 Πi of Π1,..., Πn. in the case of two variables. In this paper, we only need

841 the following equality for the independent natural exten- of the joint are outer approximations of the original sion: marginals, and will in general not coincide with the original marginals. n n ! n O Pi ∏Ai = ∏Pi(Ai) (7) i=1 i=1 i=1 8. Conclusions for all A1 ⊆ X1,..., An ⊆ Xn. Both possibility measures and p-boxes can be seen as Theorem 23. The possibility distribution coherent upper probabilities. We used this framework n to study the relationship between possibility measures n π(x) = maxπi(xi) (8) and p-boxes. Following [13], we allowed p-boxes to be i=1 defined on arbitrary totally preordered spaces, whence induces the least conservative upper cumulative distri- including p-boxes on finite spaces, on real intervals, and bution function on (Ω,) that dominates the indepen- even multivariate ones. n We began by considering the more general case of dent natural extension ⊗i=1Πi of Π1,..., Πn. maxitive measures, and proved that a necessary and Note that the possibility measure determined Eq. (8) sufficient condition for a p-box to be maxitive is that is only an outer approximation of the independent natu- at least one of the cumulative distribution functions of ral extension: as shown in [15, Sec. 6], there is no least the p-box must be 0–1 valued. Moreover, we deter- conservative possibility measure that corresponds to the mined the natural extension of a p-box in those cases and independent natural extension of possibility measures. gave a necessary and sufficient condition for the p-box One interesting consequence of Theorem 23 is that to be supremum-preserving, i.e., a possibility measure. we can also use the possibility distribution determined As special cases, we also studied p-boxes where both by Eq. (8) under some other independence conditions the lower and the upper distribution functions are 0–1– for imprecise probabilities, such as strong or Kuznetsov valued, and in particular precise 0–1 valued p-boxes. independence [25, 26]. This is because the joint model Secondly, we showed that almost every possibility they determine also satisfies the factorization condition measure can be represented as a p-box. Hence, in gen- given by Eq. (7), and as a consequence the least conser- eral, p-boxes are more expressive than possibility mea- vative upper cumulative distribution that dominates the sures, while still keeping a relatively simple representa- strong or the Kuznetsov product of the marginal possi- tion and calculus [13], unlike many other models, such bility measures is also determined by Eq. (8). See [23] as for instance lower previsions and credal sets, which for more information on the relationship between factor- typically require far more advanced techniques, such as ization and independence. linear programming. We do not consider the minimum rule and the product Finally, we considered the multivariate case in more rule n n detail, by deriving a joint possibility measure from given minπi(xi) and πi(xi), marginals using the p-box representation established in i=1 ∏ i=1 this paper and results from [13]. as their relation with the theory of coherent lower pre- In conclusion, we established new connections be- visions is still unclear. However, we can compare the tween both models, strengthening known results from above approximation with the following outer approxi- literature, and allowing many results from possibility mation given by [27, Proposition 1]: theory to be embedded into the theory of p-boxes, and n vice versa. n π(x) = min(1 − (1 − πi(xi)) ). (9) As future lines of research, we point out the generali- i=1 sation of a number of properties of possibility measures The above equation is an outer approximation in case of to p-boxes, such as the connection with fuzzy logic [1] random set independence, which is slightly more con- or the representation by means of graphical structures servative than the independent natural extension [28, [29], and the study of the connection of p-boxes with Sec. 4], so in particular, it is also an outer approximation other uncertainty models, such as clouds and random of the independent natural extension. Essentially, each sets. n distribution πi is transformed into 1 − (1 − πi) before applying the minimum rule. It can be expressed more Acknowledgements simply as n n 1 − max(1 − πi(xi)) . i=1 Work partially supported by projects TIN2008-06796- C04-01 and MTM2010-17844 and by a doctoral grant If for instance π(xi) = 1 for at least one i, then this formula provides a more informative (i.e., lower) upper from the IRSN. bound than Theorem 23. On the other hand, when all π(xi) are, say, less than 1/2, then Theorem 23 does bet- References ter. Finally, note that neither Eq. (8) nor Eq. (9) are proper [1] L. A. Zadeh. Fuzzy sets as a basis for a theory of joints, in the sense that, in both cases, the marginals possibility. Fuzzy Sets and Systems, 1:3–28, 1978.

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