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On the Connection Between Probability Boxes and Possibility Measures

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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, in- cluding 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 W be a possibility space. A subset of W is called Probability boxes [8], or p-boxes for short, are pairs of an event. Denote the set of all events by Ã(W), and the lower and upper cumulative distribution functions, and set of all finitely additive probabilities on Ã(W) 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 Ã(W). With P, we been connected to info-gap theory [9], random sets [10], associate a lower probability P on fA : Ac 2 K g 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) = fP 2 : (8A 2 )(P(A) ≤ P(A))g: 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) = supfP(A) : P 2 M (P)g possibility measure can be interpreted as a p-box. Con- A ⊆ versely, we provide necessary and sufficient conditions for all W. 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 2 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. © 2011. The authors - Published by Atlantis Press 836 ∗ 1 Note that (0W−;x] = [0W;x]. Now, let W = W[f0W−g, and define F F H = f(x0;x1] [ (x2;x3] [···[ (x2n;x2n+1]: ∗ x0 ≺ x1 ≺ ··· ≺ x2n+1 in W g: 0 1 W Proposition 2. [13, Prop. 4] For any A 2 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 W , it holds that EF;F (A) = PF;F (A), where n+1 H 2.2. P-Boxes PF;F (A) = 1 − ∑ maxf0;F(x2k) − F(x2k−1)g; (1) k=0 In this section, we revise p-boxes defined on totally pre- ordered (not necessarily finite) spaces. For further de- with x−1 = 0W− and x2n+2 = 1W. tails, see [13]. Start with a totally preordered space (W;). So, is To calculate EF;F (A) for an arbitrary event A ⊆ W, 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 2 W 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) C2H ;A⊆C W: [x;y] = fz 2 W: x z yg For intervals, we immediately infer from Proposi- tion 2 and Eq. (2) that (‘i.p.’ stands for ‘immediate pre- (x;y) = fz 2 : x ≺ z ≺ yg W decessor’) and similarly for [x;y) and (x;y]. E ((x;y]) = F(y) − F(x) (3a) We assume that W has a smallest element 0W and a F;F largest element 1 —we can always add these two ele- W EF;F ([x;y]) = F(y) − F(x−) (3b) ments to the space W. ( 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 : W ! [0;1] for which F(1W) = 1. F(y−) − F(x) if y has an i.p. For each x 2 W, F(x) is interpreted as the probability ( F(y) − F(x−) y of [0 ;x]. if has no i.p. W EF;F ([x;y)) = The quotient set of W with respect to ' is denoted by F(y−) − F(x−) if y has an i.p. W= ': (3d) [x]' = fy 2 W: y ' xg for any x 2 W 1 for any x ≺ y in W, where F(y−) denotes supz≺y F(z) W= ' = f[x]' : x 2 Wg: and similarly for F(x−). If W= ' 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 quo- tient space W= ' for any z 2 W. We also have that [x]' of W= '. Definition 1. A probability box, or p-box, is a pair EF;F (fxg) = F(x) − F(x−) (4) (F;F) of cumulative distribution functions from W to [0;1] satisfying F ≤ F. for any x 2 W. A p-box is interpreted as a lower and an upper cumu- lative 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. f[0W;x]: x 2 Wg [ f(y;1W]: y 2 Wg Definition 3. A maxitive measure is an upper probabil- defined by ity P on Ã(W) satisfying P(A [ B) = maxfP(A);P(B)g for every A and B ⊆ W. PF;F ([0W;x]) = F(x) and PF;F ((y;1W]) = 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(W) = 1. we introduce an element 0W− such that: Possibility measures are a particular case of maxitive 0W− ≺ x for all x 2 W measures. F(0W−) = F(0W−) = F(0W−) = 0: 1 In case x = 0W, evidently, 0W− is the immediate predecessor. 837 Definition 5. A (normed) possibility distribution is a Here, A x means z x for all z 2 A, and similarly mapping p : W ! [0;1] satisfying supx2W p(x) = 1. A y ≺ A means y ≺ z for all z 2 A.
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