Probability boxes on totally preordered spaces for multivariate modelling

Matthias C. M. T roﬀaes and Sebastien Desterck e. Probability boxes on totally pr e ordered spaces fo r multiv aria te mo delling. In t ernational Journal of Appr oximate R e asoning , 20 11. In press. htt p://d x.doi. org/10.1016/j.ijar.2011.02.001 PR OBABILITY BO XES ON TOT ALL Y PREORDERED SP A CES F OR MUL TIV ARIA TE MODELLING MA TTHIAS TR OFF AES AND SEBASTIEN DESTERCKE Abstract. A pair of low er and upp er cumulativ e distribution f unctions, also called probability box or p-b ox, is among the most popular models used in i m precise pr obability theory . They arise naturally in exper t elicitation, for instance in cases where b ounds are sp eciﬁed on the quan tiles of a random v ariable, or when quant iles ar e sp eciﬁed only at a ﬁnite num b er of p oints. Many practical and formal results concerning p-boxes already exist i n the l iterature. In this paper, we provide new eﬃcien t to ols to construct multiv ariate p-b ox es and develop algori thms to draw infer ences fr om them. F or this purp ose, we formali se and exten d the theory of p-b o xes using W all ey’s behavioural theory of i mprecise probabilities, and heavily r ely on its notion of natural exten sion and existing results about indep endence mo deling. In particular, we all o w p-b oxes to b e deﬁned on arbitrary totally preordered spaces, hence thereby also admitting multiv ariate p-b ox es via probability b ounds ov er any collection of nested sets. W e focus on the cases of independence (using the facto rization prop erty ), and of unknown dependence (using the F r ´ ec het bounds), and w e sho w that our approach extends the pr obabili stic arithmetic of Williamson and Downs. Two design pr oblems—a damp ed oscillator, and a river dike— demonstrate the pr actical feasibility of our r esults. 1. Introduction Imprecise probability [41] refers to uncertaint y mo dels applicable in situations where the av ailable infor mation do es no t allow us to single out a uniq ue probability measure for a ll random v ariables inv olved. Examples of such mo dels include 2- a nd n -mo notone c apacities [5], low er and upp er previs ions [43, 4 4, 41], b elief functions [38], credal sets [31], p ossibility and necessity measures [23, 6], interv al probabilities [4 2], and coherent risk measures [1, 1 5]. Unlik e cla ssical probability mo dels, which ar e describ ed by proba bility mea sures, impr e c ise probability mo dels require more complex mathematical to ols, such as non-linear functiona ls and non-additive measures [41]. It is therefore of interest to co nsider particula r imprecise probability mo dels that yield simple mathematica l descr iptions, po ssibly a t the exp ense of gener ality , but gaining e a se of use, e lic itation, and gra phical representation. One such mo del is co nsidered in this pap er: pairs of low er and upp er distribution functions, also called pr ob ability b oxes , or brieﬂy , p-b oxes [24, 25]. P -b oxes are o ften used in risk o r safety studies, in which cumulativ e distributions play an ess e n tial r ole. Man y theo retical prop erties and practical asp ects of p-boxes hav e alrea dy b een studied in the literatur e . Previous work includes pro babilistic arithmetic [4 5], which provides a very eﬃcient numerical framework for particular statistica l inferences with p-b oxes (and which we g eneralise in this pap er). In [2 6], p-b oxes a re co nnected to info-gap theory [2]. The relation b etw een p-b oxes a nd rando m sets was inv estigated in [3 0] and a pplied in [35 ]: man y r esults and techniques applica ble to rando m sets are also applicable to p-b oxes. Finally , a r e cent extensio n of p-b oxes to arbitr ary ﬁnite spaces [1 9] yields p otential applica tion of p-b oxes to a muc h mo re general set of problems , s uch as robust design ana lysis [2 0, 28], a nd sig nal pro cessing [21]. Key wor ds and phr ases. Low er prevision, p-box, m ultiv ariate, Choquet integral, F r´ ec het bounds, full component. 2 M. TROFF AES AND S. DESTE RCKE In this pap er, we study p-b oxes within the framework of the theory of coher ent low er pre- visions. Cohere nt low er previsio ns were introduced by Williams [4 3] as a generalis ation of de Finetti’s work [11], and were developed further by W alley [41]. Coheren t low er previsio ns gen- eralize many of the other imprecise probability mo dels in the literatur e, and are equiv a lent to closed co nv ex sets of ﬁnitely additive probability measures. Studying p-b oxes by means of low er previsio ns has at least t wo adv an tages :  Low er pr evisions can b e deﬁned on arbitrar y s pa ces, and thus enable p-boxes to b e used for mor e gener a l problems, a nd not just pr o blems concer ning the r eal line.  Low er previsions c ome with a powerful inference to ol, called natur al ext ension , whic h reﬂects the least-committal conseq uences o f a ny given ass essments. Natural extensio n generalise s many k nown extensions, including for insta nce Cho quet integration for 2- monotone measures. In this pap er, we study the natur a l extension of a p- box, a nd we derive a num ber of useful expres s ions for it. This leads to new numerical to ols that provide exact inferences on arbitrar y event s, and even on ar bitrary (b ounded) ra ndom quantities. F rom the point of v iew of coherent low er previsio ns, p-b oxes hav e already been s tudied br ieﬂy in [41 , Section 4.6.6] a nd [39]. Lower and upp er distribution functions asso ciated with a sequence of moments hav e also b een co nsidered [33]. As alr eady mentioned, [1 9] extended p-b oxes to ﬁnite tota lly preo rdered s paces. In this pap er, we extend p-b oxes fur ther to arbitra ry totally preo rdered spaces. Our generalis ation has many useful featur es that classica l p-boxes do not have:  W e enco mpass, in one sweep, p- boxes deﬁned on ﬁnite spaces, as well as (contin uous) p-b oxes on closed r eal interv a ls.  Perhaps even more imp or tantly , a s we do not impo se anti-symmetry on the or der ing, we ca n also handle pro duct spa ces by considering an appropria te total preo rder—for instance, one induced b y a metric—a nd thus a ls o admit multiv aria te non-ﬁnite p-b oxes, which ha ve no t b een consider ed befo re. Whence, we can sp ecify p-b oxes directly o n the pro duct space. Contrast this with the usua l multiv a riate approach to p-b oxes, s uch as probabilistic arithmetic [4 5], that consider one mar ginal p- b ox per dimension and dr aw inferences fro m a join t mo del built aro und some infor mation a b out v ariable dep endencies (of course , we can still do the sa me, and will do so in Section 7).  Our approach is also useful in elicitation, a s it allows uncertaint y to b e expr essed as probability bounds ov er any collection of (po ssibly m ultiv ariate) nested sets—also s ee [28, 18] for a discuss io n o f similarly constr ucted mo dels. Thus, unlike classica l p-b oxes, we are not r estricted to even ts of the type r8 , x s , even on the real line. Our approa ch is thus r ather diﬀerent, and far more gener a l, tha n the one usually cons idered for inferences with p-b oxes. Indeed, we ﬁrst deﬁne a join t p-box o ver so me multiv ar iate space of int eres t, either directly or by using ma rginal mo dels a nd a dependence mo del, after which we dr aw exact infer ences from this joint p-b ox using natura l extension. In contrast, usua l metho ds [36] such a s for instance proba bilistic ar ithmetic [45] start out with margina l p-b oxes each deﬁned on the r e a l line, a nd pr ovide to ols to make inferences for sp eciﬁc multiv ar iate even ts. The pap er is org a nised as follows: Section 2 provides a brief intro duction to the theory of coherent low er previs ions, used in the rest o f the pa p er. Section 3 then intro duce s and studies the p-b ox mo del from the p oint of view of low er previs io ns. Section 4 provides an express ion for the natura l extension o f a p-b ox to all even ts, via the partition top o logy induced b y the equiv alence classes of the pr eorder a nd a dditivity on full comp o ne nts. Section 5 studies the natural extens ion to a ll gambles, via the Cho q ue t in tegr a l. Section 6 studies an imp or tant sp ecia l case of p-b oxes who s e preor der is induced by a re al-v alued mapping , as this will usually b e the P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 3 most conv enient wa y to s p ecify a multiv aria te p-b ox. Section 7 discuss e s the constr uctio n of such multiv a riate p-b oxes from marginal coher ent low er previsio ns under arbitrar y dep endency mo dels. F or tw o impo rtant sp ecial cases—e pistemic indep endence, and completely unknown depe ndence (the F r´ echet case)—clo sed expres sions a r e derived. W e also derive pro babilistic arithmetic as a specia l c a se of our appr oach. Section 8 demonstra tes the theory with some examples. As a ﬁrst example, we infer b o unds on the exp ectation of the da mping ratio o f a damp ed harmonic oscillato r whose parameter s are describ ed by a multiv ariate p-b ox. As a s econd example, we der ive the exp ected overﬂo w height for a river dike, ag a in using a multiv aria te p-box. Finally , Section 9 ends with our main co nclusions and op en pro blems. 2. Preliminaries W e s ta rt with a brief intro duction to the imprecis e probability mo dels that we shall use in this pap er. W e r efer to [4 ], [43] and [41] for mor e details . Se e also [3 2] for a brief summary of the theor y . Let Ω b e the p o ssibility s pace. A subset of Ω is calle d an event . A gamble o n Ω is a b ounded real-v alued function on Ω. The set of a ll ga m bles on Ω is denoted b y L p Ω q , or simply by L if the pos sibility s pa ce is clea r fr om the context. A particular t yp e of gam ble is the indic ator of an even t A , which is the gamble that tak es the v alue 1 on e le men ts of A and the v alue 0 elsewhere, and is denoted by I A , or simply by A if no confusion with A as even t is p ossible. A lower pr evision P is a r eal-v alued functional de ﬁned on an a rbitrary subset K of L , a nd is in terpreted as follows: for any gamble f in K a nd a ny ǫ ¡ 0, the transa ction f  P p f q  ǫ is acceptable to the sub ject who has assess ed this low er previs ion. Hence, lo wer pr evisions summarize a sub ject’s supremum buying price s for a collection of gambles, a nd it can be ar g ued that in this wa y they mo del a sub ject’s belief ab out the true state x in Ω (se e [41] for a more in-depth explanation). A lo wer previsio n deﬁned on a s et of indica tors of even ts is usually called a lower pr ob ability . By P , we denote the conjugate u pp er pr evision o f P : for every f such that  f P K , P p f q   P p f q . The upp er previsio n P p f q can b e interpreted a s a sub ject’s inﬁmum selling price for f . A low er pre v ision on K is called c oher ent (see [43, p. 5 ] and [41, pp. 7 3–75, Sec. 2.5]) when for all p in N , all f 0 , f 1 , . . . , f p in K a nd all non-negative real num b ers λ 0 , λ 1 , . . . , λ p , sup x P Ω  p ¸ i  1 λ i p f i p x q  P p f i qq  λ 0 p f 0 p x q  P p f 0 qq  ¥ 0 . A low er prevision on the set L of all g ambles is co herent if and only if (see [4 3, p. 11, Sec. 1.2.2] and [4 1, p. 75, Sec. 2.5.5 ]) (C1) P p f q ¥ inf f , (C2) P p λf q  λP p f q , a nd (C3) P p f  g q ¥ P p f q  P p g q for all ga mbles f , g and all non-neg ative r eal num b ers λ . A functional on L sa tisfying P p f q ¥ inf f and P p f  g q  P p f q  P p g q for a ny pair o f ga mbles f and g is called a line ar pr evisi on on L [41, p. 88, Sec. 2 .4 .8], a nd the set of all linear previsions on L is denoted by P . A linear prevision is the expe ctation op erator with resp ect to its restriction to e vents, whic h is a ﬁnitely additive probability . An alter native c hara cterisation of coherence via linear pre v isions g o es as follows. Let P be a low er prevision on K and let M p P q denote the set of all linea r previsions on L that dominate P on K : M p P q  t Q P P : p f P K qp Q p f q ¥ P p f qqu 4 M. TROFF AES AND S. DESTE RCKE Then P is coherent if and only if P agrees with the low er env elop e o f M p P q on K , that is, if and only if P p f q  min Q P M p P q Q p f q for a ll f P K (see [43, p. 18] a nd [41, p. 138, Sec. 3.3.3]). A c o nsequence of this is that a low er env elop e of co herent low er pre visions is again a coherent low er prevision. Given a co herent low er prevision P on K , its natur al extension [41, Chapter 3] to a la rger set K 1 of g ambles ( K 1  K ) is the p oint wise s mallest coher ent lower previs ion on K 1 that ag rees with P on K . Be cause it is the po int wise smallest, the natura l extension is the most conser v ativ e (or least-committal) coher e nt extension, and thereby r eﬂects the minimal b ehavioural consequences of P on K 1 . T aking natural extension is transitive [39, p. 98, Cor. 4.9]: if E 1 is the natural extension of P to K 1 and E 2 is the na tur al extension of E 1 to K 2  K 1 , then E 2 is also the natural extension of P to K 2 . Hence, if we know the natur a l extensio n of P to the s e t of a ll g ambles then we also know the natura l extension of P to any set o f ga m bles that includes K . The natural extensio n to all g a mbles is usually denoted by E . It holds that E p f q  min Q P M p P q Q p f q for an y f P L [41, p. 1 3 6, Sec. 3.4.1]. A particular cla ss o f coherent low er previs ions of interest in this pap er are c ompletely monotone lower pr evisi ons [10, 9]. A low er previsio n P deﬁned on a lattice o f gambles K , i.e., a set of gambles clos ed under po int-wise maximum and point-wise minim um, is called n -mo notone if for all p P N , p ¤ n , a nd all f , f 1 , . . . , f p in K : ¸ I t 1 ,...,p u p 1 q | I | P  f ^ © i P I f i  ¥ 0 . A lower pr evision which is n - monotone for all n P N is called c ompletely m onotone . 3. P-Boxes In this s ection, we introduce the formalism of p-b oxes deﬁned on totally pr e ordered (not necessarily ﬁnite) spaces, as well as the ﬁeld of even ts H , whic h will be instrumen tal to study the natural ex tension of p-b oxes. Hence, in contrast to the work by [24], we do no t r e s trict p-b oxes to interv als on the real line. Let p Ω , ¨ q be a total pr eorder: so ¨ is trans itive and reﬂexive a nd any tw o elements ar e comparable. W e write x  y for x ¨ y and x  y , x ¡ y for y  x , a nd x  y for x ¨ y and y ¨ x . F o r any tw o x , y P Ω exactly one of x  y , x  y , or x ¡ y holds. W e also use the following common notatio n for in terv als in Ω : r x, y s  t z P Ω : x ¨ z ¨ y u p x, y q  t z P Ω : x  z  y u and similar ly for r x, y q and p x, y s . F or simplicity , w e assume that Ω ha s a smalles t elemen t 0 Ω and a larges t e le ment 1 Ω . This is no essential assumption, since we can alwa ys add these tw o elements to the space Ω. A cumulative distribution fun ction is a ma pping F : Ω Ñ r 0 , 1 s which is non-decr easing a nd satisﬁes moreover F p 1 Ω q  1. F or ea ch x P Ω, we interpret F p x q as the proba bilit y of the interv al r 0 Ω , x s . Note that we do not impose F p 0 Ω q  0, so we a llow t 0 Ω u to car r y non-zero mass, whic h happ ens commonly if Ω is ﬁnite. Also no te tha t distribution functions are not ass umed to b e right-con tinuous—this would ma ke no sens e since we hav e no top ology deﬁned y et on Ω—but even if there is a top ology for Ω, we make no contin uity assumptions. P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 5 By Ω {  we denote the quo tient set of Ω with resp ect to the equiv alence relation  induced by ¨ , that is: r x s   t y P Ω : y  x u for any x P Ω Ω {   tr x s  : x P Ω u Because F is non-decr easing, F is constant on elements r x s  of Ω {  . Deﬁnition 1. A pr ob ability b ox , or p-b ox , is a pair p F , F q of cumulativ e distributio n functions from Ω to r 0 , 1 s satisfying F ¤ F . Note that some deﬁnitions, such as in [19], diﬀer in tha t they use the co arsest pr eorder for which F and F a re no n-decreas ing , eﬀectively imp o sing that x  y if and only if F p x q  F p y q and F p x q  F p y q . This pap er follows [17], only imp o sing that F and F a re no n-decreas ing, so x  y implies F p x q  F p y q and F p x q  F p y q , but not the other way around, thereby admitting more pr eorders, p os sibly leading to tighter b ounds (see Example 6 further o n). A p-b ox is interpreted as a lower and an upp er cumulativ e distribution function. In W alley’s framework, this mea ns that a p-b ox is interpreted as a low er probability P F ,F on the set o f even ts K  t r 0 Ω , x s : x P Ω u Y tp y , 1 Ω s : y P Ω u by P F ,F pr 0 Ω , x sq  F p x q and P F ,F pp y , 1 Ω sq  1  F p y q . In the pa rticular ca se o f p-b oxes on r a, b s  R it w as mentioned b y [41, Section 4 .6.6] and prov en by [39, p. 9 3 , Thm. 3.59] tha t P F ,F is coher ent. Mor e generally , it is straightforw ard to show that p-b oxes on an a rbitrary totally preordered space p Ω , ¨ q are coher ent as well. 1 W e denote by E F ,F the natural extension of P F ,F to all gambles. W e study this natural extensio n extensively further on. When F  F , we s ay that p F , F q is pr e cise , and we denote the corr esp onding low er prevision on K b y P F and its natural extension to L by E F (with F :  F  F ). A few examples of p-b oxes on r 0 , 1 s are illustrated in Fig 1. Given a p-b ox, w e can co nsider the set o f distribution functions that lie b etw een F and F , Φ p F , F q  F : F ¤ F ¤ F ( . W e can e asily express the natural extension E F ,F of P F ,F in terms of Φ p F , F q : E F ,F is the lower env elop e of the natural extensions of the cumulative distribution functions 2 F that lie b etw een F and F : E F ,F p f q  inf F P Φ p F ,F q E F p f q (1) for all gambles f o n Ω. A similar re s ult for p-b oxes on r 0 , 1 s can b e found in [41, Section 4.6 .6] and [3 3, Theor em 2]. T o see why this holds, note that P F ,F is the low er env elop e of the s et M p P F ,F q of linear previsions dominating P F ,F , b ecause P F ,F is a coherent lower proba bilit y . Each of the linear previs io ns Q in M p P F ,F q has a cumulativ e distribution function F Q , and it is easy to see that F Q P Φ p F , F q . Conv erse ly , any linear prevision whos e cumulativ e distribution function F be lo ngs to Φ p F , F q m ust b elong to M p P F ,F q . Therefor e, the set M p P F ,F q coincides with the set of a ll linear pr evisions who se cum ulative distribution function b elongs to Φ p F , F q , which establis hes Eq. (1). W e shall use this equa tion rep eatedly in subsequent pro ofs. This 1 The pro of is vir tually identical to the one given in [ 39, p. 93, Thm. 3.59]. 2 The natural extension of a cumulativ e distribution function F is sim ply understo o d to b e the natural extension of the precise p-box p F , F q . 6 M. TROFF AES AND S. DESTE RCKE Ω 1 0 1 F F Ω 1 0 1 F x F Ω 1 0 1 F :  F  F Ω 1 0 1 F F Figure 1. Examples of p-b oxes on r 0 , 1 s . allows us moreover to give a sens itivit y analy sis interpretation to p-b oxes: we can alwa ys regard them as a mo del for the impre c ise knowledge of a cumulativ e distribution function. W e end this sectio n with a tr ivial, yet very useful, a pproximation theorem: Theorem 2. L et P b e any c oher ent lower pr evision deﬁne d on L . The le ast c onservative p-b ox p F , F q on p Ω , ¨ q whose natu ra l ex tension is dominate d by P is given by F p x q  P pr 0 Ω , x sqq F p x q  P pr 0 Ω , x sq for al l x P Ω . Pr o of. O bviously , the na tur al extensio n E F ,F of the p-b ox, with F and F as ab ov e, is dominated by P , b ecause P is an extension of P F ,F (see fo r instance [39, p. 98 , Prop. 4.7 ]). An y o ther p-b ox p G, G q whose natur al extension is domina ted by P must satisfy: G p x q  P G,G pr 0 Ω , x sq  E G,G pr 0 Ω , x sq ¤ P pr 0 Ω , x sq  F p x q G p x q  1  P G,G pp x, 1 sq  E G ,G pr 0 Ω , x sq ¥ P pr 0 Ω , x sq  F p x q so p F , F q is indeed the lea st conse r v ativ e one.  4. Na tural Extension to All Events The rema inder of this pap er is devoted to the na tur al extension E F ,F of P F ,F , and to v ar ious conv enient expressions for it. W e star t by g iving the form of the natura l ex tension o n the ﬁeld of even ts gene r ated by K . 4.1. Extension to the Field Generated by the Domain. Let H be the ﬁeld of even ts generated by the domain K o f the p-b ox, i.e., even ts of the t yp e r 0 Ω , x 1 s Y p x 2 , x 3 s Y    Y p x 2 n , x 2 n  1 s P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 7 for x 1  x 2  x 3      x 2 n  1 in Ω (if n is 0 we simply take this expression to b e r 0 Ω , x 1 s ) and p x 2 , x 3 s Y    Y p x 2 n , x 2 n  1 s for x 2  x 3      x 2 n  1 in Ω. Clea rly , these even ts form a ﬁeld: the union and intersection of any t wo even ts in H is aga in in H , and the complement of any ev ent in H also is aga in in H . T o simplify the descriptio n of this ﬁeld, and the expression of natural extension, we in tro duce an elemen t 0 Ω  such that: 3 0 Ω   x for all x P Ω F p 0 Ω q  F p 0 Ω q  F p 0 Ω q  0 In par ticular, p 0 Ω  , x s  r 0 Ω , x s . If we let Ω   Ω Y t 0 Ω u , then H  tp x 0 , x 1 s Y p x 2 , x 3 s Y    Y p x 2 n , x 2 n  1 s : x 0  x 1      x 2 n  1 in Ω  u . (2) Since the pro cedure of natural extension is transitive, in order to calculate the natural exten- sion of P F ,F to a ll gambles, we shall ﬁrst consider the extension from K to H , then the natural extension fr o m H to the s e t of a ll even ts, and ﬁnally the natural ex tension fro m the s e t o f all even ts to the se t of a ll gambles. In the case o f a prec ise p-box, P F has a unique ex tension to a ﬁnitely additive probability measure o n H . Prop ositi o n 3 . E F r estricte d to H is a ﬁnitely additive pr ob ability me asur e. Mor e over, for any A P H , that is A  p x 0 , x 1 s Y p x 2 , x 3 s Y    Y p x 2 n , x 2 n  1 s with x 0  x 1      x 2 n  1 in Ω  , it holds that E F p A q  n ¸ k  0 p F p x 2 k  1 q  F p x 2 k qq (3) Pr o of. Be c ause E F pr 0 Ω , x sq  P F pr 0 Ω , x sq  F p x q and E F pr 0 Ω , x sq  1  E F pp x, 1 Ω sq  1  P F pp x, 1 Ω sq  F p x q for all x , E F is linear on the linea r s pa ce s pa nned by tr 0 Ω , x s : x P Ω u [39, p. 102 , Prop. 4.18 ]. This linear spac e includes H , which proves that E F is additive on H , and consequently it is a ﬁnitely a dditiv e probability measure o n H . The ex pressions for E F p A q follow immediately .  W e now extend Prop os ition 3 to p-boxes. Prop ositi o n 4. F or any A P H , that is A  p x 0 , x 1 s Y p x 2 , x 3 s Y    Y p x 2 n , x 2 n  1 s with x 0  x 1      x 2 n  1 in Ω  , it holds that E F ,F p A q  P H F ,F p A q , wher e P H F ,F p A q  n ¸ k  0 max t 0 , F p x 2 k  1 q  F p x 2 k qu . (4) Pr o of. W e ﬁrst show tha t E F ,F p A q ¤ P H F ,F p A q . Cons ider a cumulativ e distribution function F in Φ p F , F q which satisﬁes F p x 2 k q  F p x 2 k q F p x 2 k  1 q  max t F p x 2 k  1 q , F p x 2 k qu 3 The cunning reader notes that one could av oid i n tro ducing 0 Ω  b y imposing F p 0 Ω q  F p 0 Ω q  0. How ev er, this leads to an apparen t loss of generality when linking p-boxes to other uncertain t y models, and a sli gh tly more complicated equation for the natural extension—therefore, for our purp ose, it is simpl er to stick to the m ost general formulation. ( F p 1 Ω q  F p 1 Ω q  1 follows from coherence, so nothing is lost there.) 8 M. TROFF AES AND S. DESTE RCKE for all k  0 , . . . , n . Note that 0 ¤ F p x 0 q ¤    ¤ F p x 2 n  1 q ¤ 1, so there is a cum ulative distribution function s a tisfying the ab ov e equalities. Secondly , note that for each k  0 , . . . , 2 n  1, F satisﬁes F p x k q ¤ F p x k q ¤ F p x k q . Hence, there is indeed a cumulativ e distribution function F in Φ p F , F q satisfying the ab ove equa lities . By E q. (3), E F p A q  n ¸ k  0 max t F p x 2 k  1 q , F p x 2 k qu  F p x 2 k q  n ¸ k  0 max t 0 , F p x 2 k  1 q  F p x 2 k qu  P H F ,F p A q with re s p e c t to F . Using E q. (1), we deduce that P H F ,F p A q ¥ E F ,F p A q . Next, we s how that E F ,F p A q ¥ P H F ,F p A q . Let F b e any cum ulative distribution function in Φ p F , F q . Then, E F pp x 2 k , x 2 k  1 sq  F p x 2 k  1 q  F p x 2 k q ¥ max t 0 , F p x 2 k  1 q  F p x 2 k qu since F p x 2 k  1 q ¥ F p x 2 k  1 q and  F p x 2 k q ¥  F p x 2 k q . But, b ecaus e E F is a ﬁnitely additive probability measure on H (Prop os ition 3), P H F ,F p A q ¤ E F p A q . This holds for any F in Φ p F , F q , and hence Eq. (1) implies that P H F ,F p A q ¤ E F ,F p A q .  Note that it is po ssible to der ive a close d expressio n fo r the upp er prev ision E F ,F as well, similar to E q. (4), howev er tha t expre s sion is not as e a sy to work with. In pr actice, to ca lc ulate E F ,F p A q for an even t A P H , it is by far easies t ﬁrs t to calculate E F ,F p A c q using Eq. (4) (observe that a lso A c P H ), and then to a pply the conjuga cy relation: E F ,F p A q  1  E F ,F p A c q . The lo wer pr obability P F ,F on K deter mined by the p-b ox usually do es not hav e a unique coherent extension to the ﬁeld H (unless F  F ), as shown in the fo llowing example. Example 5 . Consider the distribution functions F and F given by F p x q  x for x P r 0 , 1 s and F p x q  0 if x ¤ 0 . 5, F p x q  2 p x  0 . 5 q if x ¥ 0 . 5. F rom Pro p osition 3, bo th P F and P F hav e a unique extension to the ﬁeld H . Let us deﬁne P 1 on H by P 1 p A q :  min t P F p A q , P F p A qu for all A . The n P 1 is a coherent low er prevision on H , and it is not diﬃcult to show that P 1  P F ,F p A q for any A P K . Now, giv en the in terv al p 0 . 5 , 0 . 6 s , w e deduce from P rop osition 3 that P 1 pp 0 . 5 , 0 . 6 sq  min t P F pp 0 . 5 , 0 . 6 sq , P F pp 0 . 5 , 0 . 6 squ  min t F p 0 . 6 q  F p 0 . 5 q , F p 0 . 6 q  F p 0 . 5 qu  min t 0 . 2 , 0 . 1 u  0 . 1 . How ever, it follows from Prop ositio n 4 that E F ,F pp 0 . 5 , 0 . 6 sq  max t 0 , F p 0 . 6 q  F p 0 . 5 qu  max t 0 , 0 . 2  0 . 5 u  0. 4.2. Inner M easure. The inner measure P H F ,F  of the coherent lower pro bability P H F ,F deﬁned in E q. (4) coincides w ith E F ,F on all even ts [41, Cor. 3.1 .9, p. 127 ]: E F ,F p A q  P H F ,F  p A q  sup C P H ,C  A P H F ,F p C q , (5 ) The next example demonstr ates that the choice o f the preorder ¨ can hav e a signiﬁcant impact, even for the sa me F and F . P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 9 Example 6 . T ake Ω  t 0 , 1 , 2 , 3 , 4 u and consider : F p x q  F p x q  # 0 if x  2 , 1 otherwise . Consider the total pr eorder ¨ 1 deﬁned b y 0  1 1  1 2  1 3  1 4 a nd the usual total ordering ¨ 2 deﬁned by 0  2 1  2 2  2 3  2 4. With ¨ 1 , we hav e for a ny ev ent A  Ω that E F ,F p A q  # 1 if t 2 , 3 , 4 u  A, 0 otherwis e . using Eqs. (4) and (5). How ever, with ¨ 2 , E F ,F p A q  # 1 if 2 P A, 0 otherwise . F or ease o f nota tion, fr om now onw ards, we denote E F ,F by E when no confusio n a bo ut the cum ulative distribution functions determining the p-b ox can a rise. In principle, the pr oblem o f natural extension to all event s is solved: simply ca lculate the inner measure as in Eq. (5), using Eq. (4) to ca lculate P H F ,F p C q for elements C in H . How ever, the inner measure still involv es calculating a supr emum , which may b e non-obvious. What we show next is tha t Eq. (4) ca n be extended to arbitra ry ev ents, by ﬁr st taking the top olog ical int erio r with resp ect to a very simple top olo gy , follow ed by a (p ossibly inﬁnite) sum over the so-called full comp o nents of this interior. 4.3. The Pa rtition T op ology . Consider the p artition top olo gy on Ω g enerated by the equiv a- lence c lasses r x s  , that is, the top olo g y generated by τ :  tr x s  : x P Ω u . This top olo gy is very s imila r to the discrete top o lo gy , except tha t it is not Hausdor ﬀ, unles s ¨ is anti-symmetric: if x  y but x  y then x and y cannot be top ologic ally separated, since every neighbor ho o d of x is also a neighborho o d o f y a nd vice versa. If ¨ is a nti-symmetric (for example, the usual or dering o n the rea ls is), then the pa rtition top ology reduces to the discrete top ology , that is, every set is clop en (clo sed and op en). The open sets in this top o logy are a ll unio ns of eq uiv alence clas ses (or , subs e ts o f Ω {  , if you like). Hence, in this top olo gy , every op en set is a lso close d. In particular , every interv a l in p Ω , ¨ q is clop en. The top ologica l interior of a set A is g iven b y the union of a ll equiv alence classe s contained in A : int p A q  ¤ tr x s  : r x s   A u (6) and the top ologic a l c lo sure is given b y the union o f all equiv alence classe s which intersect with A : cl p A q  ¤ tr x s  : r x s  X A  Hu . (7) Lemma 7. F or any subset A of Ω , E p A q  E p int p A qq and E p A q  E p cl p A qq . Pr o of. Clea rly E p int p A qq ¤ E p A q bec ause int p A q  A . If we ca n also s how that E p int p A qq ¥ E p A q the desir ed result is e s tablished. Consider an ele men t C of the ﬁeld H which is included in A . Since C is in particular an op en set in the partitio n top ology , it is a subset o f int p A q . The mo notonicity of E implies that E p C q ¤ E p int p A qq . Co nsequently , E p A q  P H F ,F  p A q  sup C P H ,C  A P H F ,F p C q  sup C P H ,C  A E p C q ¤ E p int p A qq . 10 M. TROFF AES AND S. DESTE RCKE The equality E p A q  E p cl p A qq now follows fr om the fact that cl p A q  p int p A c qq c and E p A q  1  E p A c q  4.4. Additivity on F ull Com p onents. Next, w e determine a co ns tructive expres s ion o f the natural ex tension E on the clop en subsets of Ω. Deﬁnition 8. [3 7, § 4.4] A set S  Ω is called ful l if r a, b s  S for any a ¨ b in S . What do these full sets lo o k like? Obviously , any full s et is clo p en, as it m ust b e a union of equiv alence classes. Lemma 9. Every fu l l set is clop en. Pr o of. O bserve that r a, a s  r a s  , and apply Deﬁnition 8.  Under an additional completeness a ssumption, the full se ts are precisely the interv a ls. Lemma 10 . If Ω {  is or der c omplete, that is, if every s ubset of Ω {  has a supr emu m (minimal upp er b ound) and inﬁmum (max imal lower b ound), then every ful l set is an int erval, that is, it c an b e written as r x, y s , r x, y q , p x, y s , or p x, y q , for some x , y in Ω . Pr o of. Co nsider a full set S in Ω. Since, b y Lemma 9, S is clope n, we may co nsider S as a subset of Ω {  . So r x s   inf S and r y s   sup S exist, b y the o rder co mpletenes s o f Ω {  . Consider the case x P S and y R S . Apply the deﬁnitions of inf and sup to establish that z P S if and only if x ¨ z  y . The other thr ee c a ses are prov en similarly .  Note that Ω {  can b e made or der co mplete via the Dedekind completion [37, § 4.34]. Deﬁnition 11. [3 7, § 4.4] Given a clo p en set A  Ω a nd a n element x of A , the ful l c omp onent C p x, A q of x in A is the lar gest full set S which satisﬁes x P S  A . Lemma 12. The ful l c omp onent s of any clop en set A form a p artition of A . Pr o of. This is easily s hown using a similar result fo r total or de r s given in [37, 4.4(a)].  In the following theorem, we prove that the natural extension E is additive o n full compo nents. Recall that the sum of a (p ossibly inﬁnite) family p x λ q λ P Λ of non-nega tive real num bers is deﬁned as ¸ λ P Λ x λ  sup L  Λ L ﬁ nite ¸ λ P L x λ If the outcome of the ab ove sum is a ﬁnite num ber, at most c o untably many of the x λ ’s ar e non-zero [3 7, 10 .40]. Theorem 13. L et B b e a clop en su bset of Ω . L et p B λ q λ P Λ b e the ful l c omp onents of B , and let p C λ q λ P Λ 1 b e the ful l c omp onents of B c . Then E p B q  ¸ λ P Λ E p B λ q E p B q  1  ¸ λ P Λ 1 E p C λ q Pr o of. Since p B λ q λ P Λ is a partition of B , and b ecause the functional E is monotone and sup er- additive, E p B q  E pY λ P Λ B λ q ¥ E pY λ P L B λ q ¥ ¸ λ P L E p B λ q for every ﬁnite subset L o f Λ, and c onsequently E p B q ¥ ° λ P Λ E p B λ q . W e are left to show that also E p B q ¤ ° λ P Λ E p B λ q . P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 11 Let ǫ ¡ 0. By Eq. (5) , the inner meas ur e coincides with the natural extension, so there is a C in H s uch that C  B and E p B q ¤ E p C q  ǫ . F rom Eq. (2), we can write C  p x 0 , x 1 s Y p x 2 , x 3 s Y    Y p x 2 n , x 2 n  1 s for some x 0  x 1      x 2 n  1 in Ω  . Let us denote C k :  p x 2 k , x 2 k  1 s for k  0, . . . , n . It is easily es tablished that C 0 , . . . , C n are the full comp onents of C . Applying Pr op osition 4 twice, and using the fact that each full comp onent C k of C must b e a subset of exactly o ne full comp onent B λ of B (this is a co nsequence of C  B ), we ﬁnd that E p C q  n ¸ k  0 E p C k q  ¸ λ P Λ E pY k : C k  B λ C k q (note that the union is H for those λ where no C k  B λ —so only a ﬁnite n umber of terms can be non-z ero in the la tter sum) and co nsequently E p B q ¤ E p C q  ǫ  ¸ λ P Λ E pY k : C k  B λ C k q  ǫ ¤ ¸ λ P Λ E p B λ q  ǫ, taking into account the monotonicity of E . As this holds for any ǫ ¡ 0, we arrive at the desir ed inequality . The expr ession for E simply follows from the conjugacy rela tion E p B q  1  E p B c q , once noted tha t B c is c lo p en as well.  In o ther words, the natural ex tens ion E of a p-b ox is arbitr ari ly additive on ful l c omp onents . In particula r, interestingly , it is σ -a dditive on full comp onents (but obviously not a dditive, let alone σ -additive, on a r bitrary even ts). Example 14 . Additivity on full comp onents is not s uﬃcient for a low er pr obability to b e equiv- alent to a p-b ox, even in the ﬁnite case. F or example, consider Ω  t 1 , 2 , 3 u with the usual ordering, so H  ℘ p Ω q . Let P b e the low er pro bability deﬁned by P pt 1 uq  P pt 2 uq  P pt 3 uq  0 . 1 It can b e c heck ed that P is coherent, and that the natural extensio n E o f P is the low er env elop e of the probability mas s functions p 0 . 8 , 0 . 1 , 0 . 1 q , p 0 . 1 , 0 . 8 , 0 . 1 q , and p 0 . 1 , 0 . 1 , 0 . 8 q . Mo reov er, E is additive on full comp onents, beca use E pt 1 , 3 uq  E pt 1 uq  E pt 3 uq (every other subset of Ω is already full). How ever, E pt 2 uq  max t 0 , E pt 1 , 2 uq  E pt 1 uqu bec ause E pt 1 , 2 uq  0 . 2 and E pt 1 uq  0 . 8. This shows that E violates Pr op osition 4 and as a consequence it is not the natural extension to even ts of a p-b ox. 4.5. Summary. Let us summarize all results s o far, and expla in how, in practice, E p A q and E p A q of an arbitra ry even t A can b e calc ulated. Prop ositio n 4 g av e the natural extensio n to the ﬁeld H ; we are now in a p ositio n to generalize it to all even ts, at leas t when Ω {  is o rder complete. Indeed, consider an ar bitr ary even t A . By L e mma 7, it suﬃces to calculate the natura l extension of int p A q or cl p A q . Ca lculating the in terior o r closur e with re sp ect to the partition top ology will usually b e trivia l (see examples further on), and moreover, the top ologica l in terior or closure of a s et is alwa ys clop en, so now we only need to know the natural extension of clop en sets. Now, by Theo r em 1 3, it follows that we only need to ca lc ulate the natural extension o f the full comp onents p B λ q λ P Λ of in t p A q or the full comp onents p C λ q λ P Λ of cl p A q c  int p A c q —note that 12 M. TROFF AES AND S. DESTE RCKE each of these full compo nents is a lso clop en. Also, ﬁnding the full compo nents will o ften be a trivial o pe r ation—commonly , there will only b e a few. But, by Lemma 10, if, in additio n, Ω {  is or der complete, then each full comp onent is an int erv al. And for interv als , we immediately infer from Prop os ition 4 and Eq. (5) that: E pp x, y sq  max t 0 , F p y q  F p x qu (8a) E pp x, y qq  max t 0 , F p y q  F p x qu (8b) E pr x, y sq  # max t 0 , F p y q  F p x qu if x has no immediate predecessor max t 0 , F p y q  F p x qu if x has a n immediate predecessor (8c) E pr x, y qq  # max t 0 , F p y q  F p x qu if x has no immediate predecess or max t 0 , F p y q  F p x qu if x has an immediate predecess or (8d) for any x  y in Ω, 4 where F p y q denotes sup z  y F p z q and s imilarly fo r F p x q . The equalities hold b eca use, if x  y in Ω, and x  is a n immedia te predec e s sor of x , then r x, y s  p x  , y s and r x, y q  p x  , y q . Reca ll also that F p 0 Ω q  F p 0 Ω q  0 by conv ention. If Ω {  is ﬁnite, then one can think of z  as the immediate predecessor of z in the quotient space Ω {  for an y z P Ω; In other w or ds, we have a s imple constructive means of calculating the natural extension of any ev ent. 4.6. Sp ecial Cases. The ab ov e equa tions hold for any p Ω , ¨ q with order complete quotient space. In most cas e s in practice, either  Ω {  is ﬁnite, or  Ω {  is connected meaning that for any tw o elements x  y in Ω there is a z in Ω such that x  z  y , 5 (this is the case for instance when Ω is a closed interv al in R and ¨ is the usua l order ing of rea ls) Moreov er, if Ω {  is c o nnected, then F will commonly satisfy F p y q  F p y q for all y in Ω. F or example, in case Ω is a closed interv al in R , this happ ens precisely when F p 0 q  0 a nd F is left-contin uous in the usua l sense. Obviously , if Ω {  is ﬁnite, then every element of Ω ha s a n immediate pre decessor (remember, we take the immediate predece s sor of 0 Ω to be 0 Ω  ), and if Ω {  is connected, then no element except 0 Ω has an immediate pr edecessor . By Lemma 7, Theor em 13, and Eqs . (8), we conclude: Corollary 15. If Ω {  is ﬁ nite, then every ful l set B is of the form r a, b s , and E p A q  ¸ λ P Λ max t 0 , F p b λ q  F p a λ qu E p A q  1  ¸ λ P Λ 1 max t 0 , F p b 1 λ q  F p a 1 λ qu wher e pr a λ , b λ sq λ P Λ ar e the ful l c omp onent s of int p A q , and pr a 1 λ , b 1 λ sq λ P Λ 1 ar e the ful l c omp onent s of int p A c q  cl p A q c . 4 In case x  0 Ω , eviden tly , 0 Ω  is the immediate pr edecessor. 5 This terminology stems from the fact that, in this case, Ω {  is connected with resp ect to the order topology [37, § 15.46(6)]. P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 13 Corollary 16 . If Ω {  is or der c omplete and c onne cte d, and F p y q  F p y q for al l y in Ω , then E p A q  ¸ λ P Λ max t 0 , F p sup B λ q  F p inf B λ qu E p A q  1  ¸ λ P Λ 1 max t 0 , F p sup C λ q  F p inf C λ qu wher e p B λ q λ P Λ ar e the ful l c omp onents of in t p A q and p C λ q λ P Λ 1 ar e the ful l c omp onents of int p A c q  cl p A q c . Beware of F p 0 Ω q  F p 0 Ω q  0 in the la st c orollar y . 4.7. Example. Let’s inv estiga te a particular type o f p- boxes on the unit s quare r 0 , 1 s 2 . Fir st, we must sp ecify a preorder on Ω. A natural yet naive wa y of doing so is, for instance, saying that p x 1 , y 1 q ¨ p x 2 , y 2 q whenever x 1  y 1 ¤ x 2  y 2 Consider a p- b ox p F , F q on pr 0 , 1 s 2 , ¨ q . Since F is required to b e non-dec reasing with r esp ect to ¨ , it follows that F p x, y q is consta nt on elements of r 0 , 1 s 2 {  , whic h mea ns that F p x 1 , y 1 q  F p x 2 , y 2 q whenever x 1  y 1  x 2  y 2 . Thus, we may think of F p x, y q as a function of a single v ariable z  x  y , a nd we write F p z q . Similarly , we write F p z q . Our deﬁnition o f ¨ means that o ur p-b ox sp eciﬁes b ounds on the pr obability of right-angled triangles (restricted to r 0 , 1 s 2 ) whos e hypothenuses are or thogonal to the dia g onal: F p z q ¤ p ptp x, y q P r 0 , 1 s 2 : x  y ¤ z uq ¤ F p z q (9) Observe that the p-b ox is given directly on the tw o-dimensional pro duct space, without the need to deﬁne marginal p-b oxes fo r each dimensio n. The bas e τ for our partition top olo gy is g iven by τ  ttp x, y q P r 0 , 1 s 2 : x  y  z u : z P r 0 , 2 su F or example, the top olog ical interior of a rectangle A  r a, b s  r c, d s is e mpty , unless a  c  0 or b  d  1, beca use in all other cases, no element of τ is a s ubset of A . In the cases wher e a  c  0 and min t b, d u  1, or ma x t a, c u ¡ 0 and b  d  1 (if a  c  0 and b  d  1 then the interior is Ω), r esp ectively , w e have: int pr 0 , b s  r 0 , d sq  tp x, y q P r 0 , 1 s 2 : x  y ¤ min t b, d uu int pr a, 1 s  r c, 1 sq  tp x, y q P r 0 , 1 s 2 : x  y ¥ 1  max t a, c uu Consequently , E p A q  0 for all rec tangles A , except fo r E pr 0 , b s  r 0 , d sq  F p min t b, d uq E pr a, 1 s  r c, 1 sq  1  F p 1  max t a, c uq Fig. 2 illustr ates the situation. So, for the purp ose o f making inferences ab out the lower proba- bilit y o f even ts that a re rectang le s , the ordering ¨ was o bviously po or ly chosen. In g eneral, one should cho ose ¨ in a way that Ω {  c ontains go o d appr oximations for al l events of inter est. F or example, in the ca s e of r e c tangles, we could for instance discr etize r 0 , 1 s 2 int o smalle r squares, and imp ose so me ordering on thes e squa r es. Of cours e , it may not b e entirely obvious how to interpret the low er and upp er cum ulative distribution functions on s uch discretized space , since ther e is no natural ordering on such discretization. Another strategy would b e to start from a reference p o int (e.g., an elicited mo dal v alue) and then to choose the ordering ¨ such that in terv als corresp o nd to co ncent ric r egions of in terests around the refer ence p oint. Aga in, all of this is po ssible b eca use our theory concerns p-b oxes o n ar bitrary tota lly preor de r ed spaces, and is not limited to the real line with its natural o rdering. Mo re r ealistic e x amples in which such concentric regions are used are given in Section 8. 14 M. TROFF AES AND S. DESTE RCKE x  y ¤ 0 . 5 0 . 5 ¤ x  y ¤ 1 . 2 x  y ¥ 1 . 2 b d x  y ¤ min t b, d u a c x  y ¥ 1  max t a, c u Figure 2. Shap e of interv a ls induced b y ¨ , and calculation of the top ologica l interior. 5. Na tural Extension to All Gambles Next, w e esta blish that p-b oxes ar e completely mono to ne, and that therefor e their natura l extension to all gambles can b e expressed a s a Cho q uet integral. W e further simplify the c alcu- lation o f this Cho quet in tegral via the low er and upper oscillation of ga m bles with resp ect to the partition top o lo gy introduced earlie r . 5.1. Complete Monoto nicit y. As shown in [33, Section 3.1], the natural extens io n E F of a distribution function F on r 0 , 1 s is co mpletely monotone. It is fairly easy to genera lise this r esult to distribution functions on an a rbitrary tota lly preorder ed space Ω. 6 How ever, given this, and Eq. (1), we canno t immediately deduce the complete mo notonicity of E F ,F , b ecause the lower env elop e of a set of completely monotone lower pr e visions is not necessar ily completely monotone . W e prove next that such a n en velope is indeed completely monotone in the ca se of p-boxes. This is an improv ement with resp ect to prev ious results [19], where the rela tio n betw een p-boxes and complete monoto nicit y w as established for ﬁnite spaces. Let P H F ,F denote the restriction of E F ,F to H , given by Prop ositio n 4. Theorem 17. P H F ,F is a c ompletely monotone c oher ent lower pr ob abili ty. Pr o of. Clea rly , P H F ,F is coherent as it is the natural extension to H of a coherent low er probability P F ,F [41, p. 123 , 3.1.2(a)]. T o prov e that it is completely monotone, we must establish that for all p P N , 2 ¤ p ¤ n , a nd all A 1 , . . . , A p in H : P H F ,F  p ¤ i  1 A i  ¥ ¸ H I t 1 ,...,p u p 1 q | I | 1 P H F ,F  £ i P I A i  . (10) F or any p P N , 2 ¤ p ¤ n , and an y A 1 , . . . , A p in H , consider the ﬁnite ﬁeld g e ne r ated by A 1 , . . . , A p . Let Q denote the res tr iction of P H F ,F to this ﬁnite ﬁeld. By [19, Sec. 3], Q is completely monotone on this ﬁnite ﬁeld. In particular, Eq. (10) is satisﬁed. But this means that Eq. (10) is satisﬁed for all p P N , 2 ¤ p ¤ n , and all A 1 , . . . , A p in H , which establishes the theo rem.  6 Indeed, by [9, Thm. 5 & Thm. 9], the natural extension of any ﬁnitely additive probability on a ﬁeld is completely monotone. P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 15 5.2. Cho quet In tegral Representation. Complete monotonicity allows us to characterise the natural ex tension on all gambles, as we sho w in the following theorem: Theorem 18 . The natur al extension E of P F ,F is given by the Cho quet inte gr al E p f q  inf f  » sup f inf f E pt f ¥ t uq d t for every gamble f . Mor e over, E is c ompletely monotone on al l gambles. Similarly, E p f q  inf f  » sup f inf f E pt f ¥ t uq d t. Pr o of. Immediate from Theorem 17 once o bs erved that E  P H F ,F  on all even ts, and [9, Theo- rems 8 and 9]. The latter tw o theor ems state that:  Given a coherent n -monotone ( n ¥ 2) lower pro bability P deﬁned on a ﬁeld (her e, H ), its na tural extensio n to a ll g ambles is given by its Cho quet integral p C q ³  d P   If a coherent low er probability P deﬁned on a ﬁeld is n -monotone ( n ¥ 2), then its natural ex tension to all g ambles is n -monoto ne.  5.3. Lo w er and Upp er Oscill ation. By Lemma 7, to turn T heo rem 18 in a n eﬀective algo- rithm, we must calculate int pt f ¥ t uq for every t . F or tunately , there is a very simple wa y to do this. F or any gamble f on Ω and any top olog ic al base τ , deﬁne its lower oscil lation a s the gamble osc p f qp x q  sup C P τ : x P C inf y P C f p y q F or the partition top o logy which we introduced ear lier, this s impliﬁes to osc p f qp x q  inf y Pr x s  f p y q (11) The upp er os c illation is: osc p f qp x q   osc p f qp x q  sup y Pr x s  f p y q (12) F or a subset A o f Ω, we deduce from the ab ov e deﬁnition and from E q . (6) that the lower oscillation of I A is I int p A q , so the low er oscillation is the natural genera lisation of the topolo gical int erio r to gambles. Similarly , w e see from Eq. (7) that the upp er oscillatio n of I A is I cl p A q . Prop ositi o n 19. F or any gamble f on Ω , int pt f ¥ t uq  t osc p f q ¥ t u (13a) cl pt f ¥ t uq  t osc p f q ¥ t u (13b) so, in p articular, E p f q  inf osc p f q  » sup osc p f q inf osc p f q E pt osc p f q ¥ t uq d t  E p osc p f qq (14a) E p f q  inf osc p f q  » sup osc p f q inf osc p f q E pt osc p f q ¥ t uq d t  E p osc p f qq (14b) 16 M. TROFF AES AND S. DESTE RCKE Pr o of. E qs. (13) are easily establis hed using the deﬁnitions of interior and closur e, a nd low er and upper o scillation. F or exa mple, x P int pt f ¥ t uq if and only if there is a C in τ such that x P C a nd C  t f ¥ t u But, for our choice of τ , necessarily C  r x s  if x P C P τ , so the a b ove ho lds if and o nly if  y P r x s  : f p y q ¥ t And this holds if and only if osc p f qp x q ¥ t where we used the deﬁntion of osc p f q . Let us prov e Eqs. (14). It follows from Eq. (11) that f ¥ osc p f q , and as a co nsequence E p f q ¥ E p osc p f qq . W e are left to prov e that E p f q ¤ E p osc p f qq . Indeed, using Lemma 7 , and Eqs. (13), E p f q  inf f  » sup f inf f E pt f ¥ t uq d t  inf f  » sup f inf f E pt osc p f q ¥ t uq d t and since obviously , by Eq. (11), inf f  inf osc p f q ,  inf osc p f q  » sup f inf osc p f q E pt osc p f q ¥ t uq d t and since sup f ¥ sup osc p f q , using the usual prop er ties of the Riemann integral,  inf osc p f q  » sup osc p f q inf osc p f q E pt osc p f q ¥ t uq d t  » sup f sup osc p f q E pt osc p f q ¥ t uq d t Now use the fact that t osc p f q ¥ t u  H for t P p sup osc p f q , sup f s , so the las t term is zero.  Concluding, to calculate the na tural extensio n of any gamble, in pra c tice , we must simply determine the full comp onents of the cut sets o f its low er or upp er oscillation, a nd calculate a simple Riema nn integral of a monotonic function. Examples will b e given in Section 8 . 6. P-Boxes Whose Preorders are Induced by a Real-V alued Function In practice , the most conv enient wa y to sp ecify a preor der ¨ on Ω such that Ω {  is or der complete and co nnected is by means of a bounded rea l-v alued function Z : Ω Ñ R . F or instance, in the example in Section 4.7, we used Z p x, y q  x  y . Also see [2] and [28]. Let us assume fr o m now onw ards that Z is a s urjective mapping fro m Ω to r 0 , 1 s . F or a ny x and y in Ω, deﬁne x ¨ y whenever Z p x q ¤ Z p y q . Because Z is surjective, Ω {  is o rder complete a nd co nnected. In par ticular, Ω ha s a smallest and larg est element , for which Z p 0 Ω q  0 a nd Z p 1 Ω q  1. Mo r eov er, we can think of a ny cumulativ e distribution function on p Ω , ¨ q as a function o ver a s ingle v ariable z P r 0 , 1 s . Conseq ue ntly , we can think of any p-b ox on p Ω , ¨ q as a p-b ox on pr 0 , 1 s , ¤q . In particular, for a ny subset I of r 0 , 1 s we write E p I q for E p Z  1 p I qq . F o r example, for a , b in r 0 , 1 s , and A  Z  1 pp a, b sq  Ω, we hav e that E p A q  E pp a, b sq  max t 0 , F p a q  F p b qu P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 17 by Prop ositio n 4. Similar expr essions for other types of int erv als follow from Eqs. (8 ). The top olo g ical in terio r a nd closure ca n be related to the so-ca lled lower and upp er inverse of Z  1 . Indeed, consider the m ulti-v alued ma pping Γ :  Z  1 : r 0 , 1 s Ñ ℘ p Ω q . Be cause for every x in Ω, it holds that r x s   Γ p Z p x qq , it follows that, for any subset A of Ω, int p A q  Γ p Γ  p A qq , and cl p A q  Γ p Γ  p A qq , where Γ  and Γ  denote the low er and upp er inv ers e of Γ resp ectively , that is Γ  p A q  t z P r 0 , 1 s : Γ p z q  A u , and Γ  p A q  t z P r 0 , 1 s : Γ p z q X A  Hu (see fo r instance [16]). 7 Theorem 20 . L et A b e an arbitr ary subset of Ω . Then E p A q  ¸ λ P Λ E p I λ q E p A q  1  ¸ λ P Λ 1 E p J λ q wher e p I λ q λ P Λ ar e the ful l c omp onents of Z p int p A qq  Γ  p A q and p J λ q λ P Λ 1 ar e the ful l c omp onents of Z p int p A c qq  Z p cl p A q c q  Γ  p A c q . If, in addition, F is left-c ontinuous as a fu n ction of z P r 0 , 1 s and F p 0 q  0 , then E p A q  ¸ λ P Λ max t 0 , F p sup I λ q  F p inf I λ qu E p A q  1  ¸ λ P Λ 1 max t 0 , F p sup J λ q  F p inf J λ qu Pr o of. Indeed, by Cor o llary 16, E p A q  E p int p A qq  ¸ λ P Λ E p B λ q where p B λ q λ P Λ are the full comp onents of int p A q . So , the result is established if we ca n show that p Z  1 p I λ qq λ P Λ are the full comp onents o f int p A q . Obviously , since p I λ q λ P Λ partitions Z p int p A qq , it follows that p Z  1 p I λ qq λ P Λ partitions ¤ λ P Λ Z  1 p I λ q  Z  1 p Z p int p A qqq  int p A q where the latter eq uality follows fro m the fact that int p A q is clop en, i.e ., is a union o f eq uiv alence classes. W e are left to prove that each set Z  1 p I λ q is a full comp onent. Clearly , Z  1 p I λ q is full: for any t wo x and y in Z  1 p I λ q , it holds that r x, y s  t v P Ω : Z p x q ¤ Z p v q ¤ Z p y qu  Z  1 pr Z p x q , Z p y qsq  Z  1 p I λ q where we used the fact that r Z p x q , Z p y qs  I λ in the last s tep. Consider any x P Z  1 p I λ q . The desired result is establis hed if we can show that Z  1 p I λ q is the lar gest full set S which satisﬁes x P S  int p A q . Suppo se S is larger , that is, S is full, x P S  int p A q , and Z  1 p I λ q ( S . Since b oth sets ar e clop en, it m ust be that there is so me y P S such that r y s  X Z  1 p I λ q  H . But this implies that I λ  Z p Z  1 p I λ qq ( Z p S q 7 W e follow the terminology in [34] and [ 22], among others. Beware that Γ  and Γ  are sometimes called upp er and lower inv erse instead [40, 3], or str ong and we ak inv erse [14]. 18 M. TROFF AES AND S. DESTE RCKE bec ause Z p y q belo ngs to Z p S q but not to Z p Z  1 p I λ qq . But, this would mean that I λ is not a full comp onent of Z p int p A qq —a contradiction. So, Z  1 p I λ q m ust b e the largest full set S which satisﬁes x P S  int p A q .  Regarding gambles, no te that the low er os c illation is constant on equiv alence clas ses (this follows immediately from its deﬁnition). Hence, we ma y consider osc p f q for a gamble f on Ω as a function of z P r 0 , 1 s , and we can us e Prop o sition 19 to write: Prop ositi o n 21. F or any gamble f on Ω , E p f q  inf osc p f q  » sup osc p f q inf osc p f q E pt z : osc p f qp z q ¥ t uq d t E p f q  inf osc p f q  » sup osc p f q inf osc p f q E pt z : osc p f qp z q ¥ t uq d t 7. Constructing Mul tiv aria te P-Boxes from Marginals In this section, we construct a m ultiv ariate p-box fr om marginal coherent low er previs ions under ar bitrary rules of combination. As s p ecia l c ases, we derive expressions for the join t, (i) either without an y as sumptions a bo ut dep endence or indep e ndence betw een v ariables, that is, using the F r´ ec het-Ho eﬀding b ounds [29], (ii) or as s uming epistemic indep endence b etw een all v ariables, that is, us ing the factor ization prop erty [7]. W e also der ive Williamson and Downs’s [45] probabilistic ar ithmetic as a sp ecial case o f our framework. Spec iﬁc a lly , co nsider n v ar iables X 1 , . . . , X n assuming v alues in X 1 , . . . , X n , and assume that marginal low er prev isions P 1 , . . . , P n , are given for each v ariable—e ach of these could b e the natural extension of a p-b ox, although we do not r equire this. So , each P i is a cohere nt low er prevision on L p X i q . 7.1. Multiv ariate P-Bo xes. The ﬁrst step in constructing our mu ltiv ariate p-b ox is to deﬁne a mapping Z to induce a pr eorder ¨ on Ω  X 1      X n . The follo wing ch oice w ork s p erfectly for our purp ose: Z p x 1 , . . . , x n q  n max i  1 Z i p x i q where each Z i is a s urjective mapping from X i to r 0 , 1 s and hence, also induces an mar ginal preorder ¨ i on X i . Each P i can b e approximated b y a p- b ox p F i , F i q on p X i , ¨ i q , deﬁned by F i p z q  P i p Z  1 i pr 0 , z sqq F i p z q  P i p Z  1 i pr 0 , z sqq This approximation is the b es t p ossible one, by Theor em 2. Beware that even thoug h diﬀerent c hoices o f Z i may induce the sa me total preo rder ¨ i , they might lead to a diﬀerent total preorder ¨ induced by Z . Whence, our joint to tal preor der ¨ is not uniquely deter mined by ¨ i . Roughly sp eaking, the Z i sp ecify how the marginal preor ders ¨ i scale r elative to one another. As we s ha ll see, this eﬀectively means that our choice of Z i aﬀects the precis ion o f our inferences: a go o d choice will ensure that a ny even t of interest can be well approximated by elements of Ω {  . Of course, nothing pre vents us, at least in theory , to co nsider the s et of all Z i which induce some given marginal total pr eorders ¨ i , and whence to work with a set of p-b oxes. In some cases, this may result in quite complica ted calcula tions. How ever, in Section 7.4, we will s ee an example where this approa ch is feasible. An ywa y , with this choice of Z , w e can eas ily ﬁnd the p-b ox whic h repre sents the joint as accurately a s p ossible, under a ny rule of combination of coherent lo wer previsions: P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 19 Lemma 22. Consider any ru le of c ombination d of c oher ent lower and upp er pr evisio ns, map- ping the mar ginals P 1 , . . . , P n to a joint c oh er ent lower pr evisi on Ä n i  1 P i on al l gambles. Supp ose ther e ar e functions ℓ and u for which: n ä i  1 P i  n ¹ i  1 A i   ℓ p P 1 p A 1 q , . . . , P n p A n qq and n ä i  1 P i  n ¹ i  1 A i   u p P 1 p A 1 q , . . . , P n p A n qq , for al l A 1  X 1 , . . . , A n  X n . Then, the c ouple p F , F q deﬁne d by F p z q  ℓ p F 1 p z q , . . . , F n p z qq F p z q  u p F 1 p z q , . . . , F n p z qq is the le ast c onservative p-b ox on p Ω , ¨ q whose natur al extension E F ,F is dominate d by the c ombinatio n Ä n i  1 P i of P 1 , . . . , P n . Pr o of. By Theorem 2, the leas t conser v ativ e p-b ox on p Ω , ¨ q whose natura l extensio n is domi- nated by the joint P  Ä n i  1 P i is g iven b y F p z q  P p Z  1 pr 0 , z sqq F p z q  P p Z  1 pr 0 , z sqq Now, observe tha t the set Z  1 pr 0 , z sq is a pro duct o f margina l interv als : Z  1 pr 0 , z sq  tp x 1 , . . . , x n q P Ω : n max i  1 Z i p x i q ¤ z u  tp x 1 , . . . , x n q P Ω : p i  1 , . . . , n qp Z i p x i q ¤ z qu  n ¹ i  1 t x i P X i : Z i p x i q ¤ z u  n ¹ i  1 Z  1 i pr 0 , z sq . The des ired equa lities follow immediately .  7.2. Natural Extension: The F r´ ec het Case. The natur al ex tension ⊠ n i  1 P i of P 1 , . . . , P n is the lo wer env elop e of a ll joint distr ibutio ns (or, linear previsions) whose marginal dis tr ibutions (or, mar ginal linear previsions) a re compatible with the given marginal lower previsions. So, the mo del is co mpletely v acuous about the dep endence structure, a s it inc ludes all p os sible forms of dep endence. W e refer to for ins tance [8, p. 120, § 3.1] for a r igorous deﬁnition. In this pap er, we only need the following equalities, which ar e known as the F r´ ec het b ounds [2 7] (see for instance [46, p. 13 1] for a mor e recent discussion): n ò i  1 P i  n ¹ i  1 A i   max # 0 , 1  n  n ¸ i  1 P i p A i q + and (15a) n ò i  1 P i  n ¹ i  1 A i   n min i  1 P i p A i q (15b) for all A 1  X 1 , . . . , A n  X n . Theorem 23 . The p-b ox p F , F q deﬁne d by F p z q  max # 0 , 1  n  n ¸ i  1 F i p z q + F p z q  n min i  1 F i p z q is the le ast c onservative p-b ox on p Ω , ¨ q whose natu r al extension E F ,F is dominate d by the natu r al extension ⊠ n i  1 P i of P 1 , . . . , P n . Pr o of. Immediate, by Lemma 22 and Eqs. (1 5).  20 M. TROFF AES AND S. DESTE RCKE The next exa mple shows that, even when P i are p-b oxes, the joint p-b ox will in gene r al only be an outer approximation (although the closest one that is a p- box) o f the joint lower previsio n. Example 24 . Cons ider t wo v ariables X and Y with domain X  t x 1 , x 2 u , with x 1  x 2 , and Y  t y 1 , y 2 u , with y 1  y 2 . Consider F 1 p x 1 q  0 . 4 , F 1 p x 1 q  0 . 6 , F 1 p x 2 q  F 1 p x 2 q  1 , F 2 p y 1 q  0 . 2 , F 2 p y 1 q  0 . 3 , F 2 p y 2 q  F 2 p y 2 q  1 . Let P 1 be the natural extension o f p F 1 , F 1 q , and let P 2 be the natural extension o f p F 2 , F 2 q . Consider the even ts A  t x 1 u  Y and B  X  t y 2 u . W riting P for ⊠ n i  1 P i , we hav e that P p A q  P 1 pt x 1 uq  max t 0 , F 1 p x 1 q  F 1 p x 1 qu  0 . 4 P p B q  P 2 pt y 2 uq  max t 0 , F 2 p y 2 q  F 2 p y 1 qu  0 . 7 whence, P p A Y B q  max t P p A q , P p B qu  0 . 7 , P p A X B q  max t 0 , 1  2  P p A q  P p B qu  0 . 1 . But this mea ns that P is not even 2 -monotone, b ecause P p A Y B q  P p A X B q  P p A q  P p B q . Therefore , P cannot b e represe nt ed by a p- box, as p-b oxes a re completely monoto ne by Theorem 1 7. 7.3. Indep endent Natural Extension. In c o ntrast, the indep endent natura l extension b n i  1 P i of P 1 , . . . , P n mo dels e pis temic indep endence b etw een X 1 , . . . , X n . W e refer to [7] fo r a rigor ous deﬁnition a nd prop erties . In this pap er, we only need the following equalities: n â i  1 P i  n ¹ i  1 A i   n ¹ i  1 P i p A i q and (16a) n â i  1 P i  n ¹ i  1 A i   n ¹ i  1 P i p A i q (16b) for all A 1  X 1 , . . . , A n  X n . Theorem 25. The p-b ox p F , F q deﬁne d by F p z q  n ¹ i  1 F i p z q F p z q  n ¹ i  1 F i p z q is the le ast c onservative p-b ox on p Ω , ¨ q whose natur al extension E F ,F is dominate d by the indep e dent natur al extens ion b n i  1 P i of P 1 , . . . , P n . Pr o of. Immediate, by Lemma 22 and Eqs. (1 6).  Again, in gener a l, the joint p-b ox will only b e an outer approximation o f the actual joint low er prevision. Example 26 . Again, consider t wo v ariables X a nd Y with domain X  t x 1 , x 2 u , with x 1  x 2 , and Y  t y 1 , y 2 u , with y 1  y 2 . Consider F 1 p x 1 q  0 . 4 , F 1 p x 1 q  0 . 6 , F 1 p x 2 q  F 1 p x 2 q  1 , F 2 p y 1 q  0 . 3 , F 2 p y 1 q  0 . 5 , F 2 p y 2 q  F 2 p y 2 q  1 . P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 21 As b efore, let P 1 be the natura l extension of p F 1 , F 1 q , a nd let P 2 be the natura l extension of p F 2 , F 2 q . Consider the event s A  tp x 1 , y 1 q , p x 1 , y 2 qu and B  tp x 1 , y 2 q , p x 2 , y 1 qu . W riting P for b n i  1 P i , we hav e that P p A q  P 1 pt x 1 uq  0 . 4 P p B q ¥ 0 . 4 where the last ineq uality follo ws from the fact that all pro bability mass functions p which dom- inate P m ust satisfy p p x 1 | y 2 q ¥ P pt x 1 uq  0 . 4 and p p x 2 | y 1 q ¥ P pt x 2 uq  0 . 4, whence p p B q  p p x 1 | y 2 q p p y 2 q  p p x 2 | y 1 q p p y 1 q ¥ 0 . 4 p p p y 1 q  p p y 2 qq  0 . 4 for all p whic h dominate P . Because P is the low er en velope of all such p , the desir ed inequalit y follows. 8 Also, beca use of the factor ization pr op erty of the independent natura l extension, P p A Y B q  1  P ptp x 2 , y 2 quq  1  P 1 pt x 2 uq P 2 pt y 2 uq  1  0 . 6  0 . 7  0 . 58 P p A X B q  P ptp x 1 , y 2 quq  P 1 pt x 1 uq P 2 pt y 2 uq  0 . 4  0 . 5  0 . 2 . Again, this means that P cannot b e repres ent ed by a p- box, as it v iolates 2-mo no tonicity . 7.4. Sp ecial Case: Probabili stic Arithmetic. Let Y  X 1  X 2 with X 1 and X 2 real- v alued ra ndom v ariables. O ne can also consider substractio n, mu ltiplication, and division, but for simplicity , we stick to addition—the other three cas es follow along a lmost identical lines. Probabilis tic arithmetic [46] deals with the problem of estimating P Y pr8 , y sq  F Y p y q and P Y pr8 , y sq  F Y p y q for any y P R under the a ssumptions that  the uncertaint y o n X 1 and X 2 is given by p-b oxes p F 1 , F 1 q and p F 2 , F 2 q , with ¨ 1 and ¨ 2 the natur al or dering of real num be r s, 9 and  the dep endence structure is completely unknown (F r´ ec het ca se). Using the F r´ echet b o unds, Williamson a nd Do wns [45] provide explic it formulae for the diﬀeren t arithmetic op erations , thus pro viding v ery eﬃcien t algor ithms to make inferences from marginal p-b oxes. Let us show, for the particular case of addition, that their results ar e ca ptured by o ur joint p-b ox prop osed in Theo rem 23. O ther cases, no t tre ated he r e to sav e s pace, follow fr o m almost ident ical r e a soning. 10 The low er cumulative distribution function r esulting from proba bilistic arithmetic is, for any y P R F X 1  X 2 p y q  sup x 1 ,x 2 : x 1  x 2  y max t 0 , F 1 p x 1 q  F 2 p x 2 q  1 u . (17) Without m uch loss of genera lit y , and for o ur convenience, assume that b oth X 1 and X 2 lie in a bo unded interv al r a, b s . Let Z 1 and Z 2 be any s urjective maps r a, b s Ñ r 0 , 1 s which induce the usual ordering o n r 0 , 1 s . Some pr o p erties of Z 1 and Z 2 immediately follow: b oth ar e contin uous and strictly increasing , and s o ar e their in verses—we rely on this in a bit. T o a pply Theorem 23, we consider the total preorder ¨ on Ω  r a, b s 2 induced b y Z p x 1 , x 2 q  max t Z 1 p x 1 q , Z 2 p x 2 qu . Figure 3 illus trates the even t 11 t X 1  X 2 ¤ y u , with y P r 2 a, 2 b s , as well as the larg e st in terv al Z  1 pr 0 , z sq included in it. Recall that Z  1 pr 0 , z sq  Z  1 1 pr 0 , z sq  Z  1 2 pr 0 , z sq  r 0 , Z  1 1 p z qs  r 0 , Z  1 2 p z qs . 8 By linear programm ing, it can actually be shown that P p B q  0 . 4. 9 F or substraction and di vi sion, ¨ 2 is the r ev erse natural or dering. 10 Note that X 1 and X 2 are assumed to be p ositive in case of m ultiplication and divi s ion. 11 By t X 1  X 2 ¤ y u we mean tp x 1 , x 2 q P r 0 , 1 s 2 : x 1  x 2 ¤ y u . 22 M. TROFF AES AND S. DESTE RCKE y y Z  1 2 p z q Z  1 1 p z q t X 1  X 2 ¤ y u Z  1 pr 0 , z sq Figure 3. The even t t X 1  X 2 ¤ y u , and the la rgest interv a l Z  1 pr 0 , z sq in- cluded in it. Whence, for z such that Z  1 1 p z q  Z  1 2 p z q  y , we achiev e the lar gest interv al Z  1 pr 0 , z sq which is still included in t X 1  X 2 ¤ y u . There is alwa ys a unique such z bec a use also Z  1 1  Z  1 2 is contin uous and s trictly increa sing. Recall that, by Theorem 23 (now witho ut sho rtcuts in notation), F p Z  1 p z qq  max 0 , F 1 p Z  1 1 p z qq  F 2 p Z  1 2 p z qq  1 ( F p Z  1 p z qq  n min i  1 t F 1 p Z  1 1 p z qq , F 2 p Z  1 2 p z qqu is the least conse r v ativ e p-b ox o n p Ω , ¨ q whose natural e xtension is dominated by the na tural extension P 1 ⊠ P 2 of P 1 and P 2 . Also, as we hav e just s hown, Z  1 pr 0 , z sq , for our choice of z , is the top o logical interior of t X 1  X 2 ¤ y u . Whence, by Theorem 20, we ﬁnd that E F ,F pt X 1  X 2 ¤ y uq  E F ,F p Z  1 pr 0 , z sqq  F p Z  1 p z qq  max t 0 , F 1 p Z  1 1 p z qq  F 2 p Z  1 2 p z qq  1 u where we remember that z is chosen such that Z  1 1 p z q  Z  1 2 p z q  y . But, this holds for every v alid choice of functions Z 1 and Z 2 , whence P 1 ⊠ P 2 pt X 1  X 2 ¤ y uq ¥ sup x 1 ,x 2 : x 1  x 2  y max t 0 , F 1 p x 1 q  F 2 p x 2 q  1 u which indeed coincides with Eq. (1 7). Similar ar guments hold for the upp er cum ulative distri- bution functions, and other ar ithmetic op er ations. In conclusion, probabilistic arithmetic cons titutes a very sp eciﬁc cas e of our approa ch. 8. Examples In this section, w e investigate t wo diﬀerent ex amples in which p-b oxes are us ed to mo del uncertaint y ar ound so me parameters. The ﬁrst example concerns a damp ed harmonic oscillator, i.e., a classical engineering toy example. The sec o nd ex ample concerns the ev aluation of a river dike heigh t, an imp ortant issue in regio ns sub ject to p o ten tial ﬂo o ds. 8.1. Damp ed Harmoni c Oscil l ator. Co nsider a simple damp ed harmonic oscillator, w ith damping co eﬃcient c ¡ 0, spring constant k ¡ 0, and mass m ¡ 0. The damping r atio ζ p c, k q  c 2 ? k m P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 23 c k 2 1 rp c, k qs  osc p ζ qp z q osc p ζ qp z q Rectangular equiv alence cla ss. c k 2 1 rp c, k qs  Elliptical equiv alence c la ss. Figure 4. Diﬀerent po ssible eq uiv alence c lasses. determines how quickly the o scillator returns to its equilibrium state— ζ p c, k q  1 means fastest conv ergence. Supp o se the e ngineering desig n has alrea dy bee n completed, so the optimal v alues for c  and k  hav e been determined, such that ζ p c  , k  q  1. Without loss of generality , we choo se the units for mass, time, and length, such that m  k   ζ p c  , k  q  1 (so c   2). Of cour se, the a ctual v alues for c and k will diﬀer from their design v a lues, and uncertaint y m ust b e taken into acc o unt. Let us calculate the low er and upp er exp ectation o f ζ p c, k q , g iven that o ur uncerta int y a b out c and k is describ ed b y a p-b ox. First, w e must s pec ify a preo rder. F or this problem, it seems fairly natura l to ha ve b ounds on the quantiles of the distance b etw een the actual v alues p c, k q and the design v alues p 2 , 1 q . This comes down to for insta nce the following ch oice for Z : Z p c, k q  max t| c  2 | , 2 | k  1 |u F or simplicity , we only consider the region Z p c, k q ¤ 1. This mea ns that we are cer tain that c P r 1 , 3 s and k P r 0 . 5 , 1 . 5 s —if necessa ry , Z can b e rescaled to accomo da te lar ger or smaller regions. Note that w e ha ve taken a supremum nor m as distance. This simpliﬁes the calculations below, but of cours e, o ne might as well take the Euclidian norm, or any other reas onable distance function, at the exp ense of slightly more complicated calculations a nd dep endency mo delling (see Fig. 4). Equiv alence classe s rp c, k qs  are edg es of rectangles with vertices p 2  Z p c, k q , 1  Z p c, k q{ 2 q . What is a p-b ox for the preor der ¨ induced by Z ? A p-b ox p F , F q sp eciﬁes low er and upper bo unds fo r the probability of concentric rectang les around the design p oint p 2 , 1 q : F p z q ¤ p ptp c, k q : Z p c, k q ¤ z uq ¤ F p z q So, eﬀectiv ely , our p-b ox sp eciﬁes c o ncentric predictio n regions for the uncertain parameters c and k . 24 M. TROFF AES AND S. DESTE RCKE t 0 1 ? 6 1 3 ? 2 osc p ζ qp z q osc p ζ qp z q 1 Figure 5. The low er oscillation osc p ζ qp z q and upp er oscillatio n osc p ζ qp z q . t  1 0 1 z p t q 1 1 ? 6 3 ? 2 Figure 6. The function z p t q which determines the cut s e ts. W e can now calculate the lo wer and upp er ex pe c tation o f ζ p c, k q . First, we calculate the low er oscillation osc p ζ q and upp er oscillatio n osc p ζ q (see Fig. 5): osc p ζ qp z q  inf p c,k q : Z p c,k q z ζ p c, k q  2  z 2 a p 1  z { 2 q osc p ζ qp z q  sup p c,k q : Z p c,k q z ζ p c, k q  2  z 2 a p 1  z { 2 q Next, we ﬁnd the full comp onents of the even ts L t  t z P r 0 , 1 s : osc p ζ qp z q ¥ t u  # z P r 0 , 1 s : 2  z 2 a p 1  z { 2 q ¥ t + U t  t z P r 0 , 1 s : osc p ζ qp z q ¥ t u  # z P r 0 , 1 s : 2  z 2 a p 1  z { 2 q ¥ t + for all t P r 0 , 0 . 5 s . F ortunately , osc p ζ q is decr easing as function of z , and osc p ζ q is incre asing, and hence L t  r 0 , ℓ t s and U t  r u t , 1 s , with (see Fig . 6) ℓ t  2  t p t  a t 2  8 q :  z p t q with t P r 1 ? 6 , 1 s u t   2  t p t  a t 2  8 q :   z p t q with t P r 1 , 3 ? 2 s Note that the given b ounds for t arise fro m the minim um and maximum o f osc p f q and osc p f q . Concluding, by Prop o sition 21, when F p z q  F p z q for all z P r 0 , 1 s and F p 0 q  0, E p ζ q  1 ? 6  » 1 1 ? 6 F p z p t qq d t E p ζ q  1  » 3 { ? 2 1  1  F p z p t qq  d t P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 25 symbol name unit h ov erﬂow height of the river m q river ﬂo w r ate m 3 s  1 b river w idth m k Strickler co eﬃcient m 1 { 3 s  1 u upriver water lev el m d do wnriver water le vel m ℓ length of river stretch m T able 1. Meaning o f the v ariables used in Eq. (18) Int eres ting ly , b oth the low er and upp er exp ectation of ζ are determined by the low er cumu- lative distribution function only . Hence, in this pr oblem, we actually do not need to e licit the upper cumulative distribution function. 12 F or example, if the exp ert says that c and k a re indep endent , and the marginal low er cum u- lative distribution functions a re unifor m on c P r 1 , 2 s and k P r 0 . 5 , 1 . 5 s , so F 1 p z q  F 2 p z q  z , with preo r ders induced by Z 1 p c q  | c  2 | and Z 2 p k q  2 | k  1 | , then, by Theorem 25, b e c ause Z  max t Z 1 , Z 2 u , it follows that F p z q  z 2 , and E p ζ q  1 ? 6  » 1 1 ? 6 z p t q 2 d t  0 . 584 E p ζ q  1  » 3 { ? 2 1  1  z p t q 2  d t  1 . 664 8.2. River Dike Height Esti m ation. W e aim to estimate the minimal req uired dike height along a g iven stretch of river, using a simpliﬁed mo del that is used by the EDF (the F rench int egr ated ener gy op erator ) to mak e initial ev aluations [1 3]. Althoug h this mo del is quite simple, it provides a realistic industrial application. Skipping technical details, the mo del res ults in the following relationship: h p q , k , u, d q  $ ' & ' %  q k ? u  d ℓ b  3 5 if q ¥ 0 0 otherwise . (18) The mea ning of the v a riables is summarised in T able 1. F or the particular case under study , the river width is b  300 m and the river leng th is ℓ  6400 m . The remaining parameter s are uncertain. Exp ert asses sment leads to the following distributions:  The max imal ﬂow rate q has a Gum b el distribution 13 with loca tion para meter µ  1335 m 3 s  1 and scale pa rameter β  716 m 3 s  1 . F or calculatio ns, it is easier to w or k with symmetr ic distributions. Therefore, w e intro duce a v ariable r s atisfying q  µ  β ln p ln p r qq . 12 Of course this will not alwa ys b e the case—it just happens to b e so for this example. 13 The Gumbel distribution models the maximum of an exp onen tially dis tributed sample, and is used in extreme v alue theory [12] to model r ar e even ts such as ﬂoo ds. 26 M. TROFF AES AND S. DESTE RCKE p p k P r 30  15 z , 30  15 z sq k 30 p p k q 15 45 Figure 7. Deriv ation of the p- box for a tria ngular distribution. If r is unifor mly distributed o ver r 0 , 1 s , then q has the Gumbel distribution with locatio n parameter µ and s cale par ameter β . So, after transforma tio n, h p r , k , u, d q  $ ' & ' %  µ  β ln p ln p r qq k ? u  d ℓ b  3 5 if µ  β ln p ln p r qq ¥ 0 0 otherwise .  The Strickler co eﬃcient k has a symmetric triangular distr ibution ov er the in terv al r 15 m 1 { 3 s  1 , 45 m 1 { 3 s  1 s (with mo de at k   30 m 1 { 3 s  1 ).  There is als o uncertaint y ab out the w ater levels u and d , beca use sedimentary conditions are hard to characterise. Me a sured v alues are u   55 m and d   50 m , with measure- men t error deﬁnitely less than 1 m . These are als o mo delled by symmetric triangular distributions, on r 54 m, 56 m s and r 49 m, 51 m s resp ectively . Again, a na tur al ch oice for Z is the distance b etw een the exp ected v alues p r   1 { 2 , k   30 , u   55 , d   50 q and the actual v alues p r , k , u, d q : Z p r , k , u, d q  max t 2 | r  1 { 2 | , | k  30 |{ 15 , | u  55 | , | d  50 |u , The scale of the distances has b een chosen s uch that Z p r , k , u, d q ¤ 1 for all p oints of interest. Equiv alence classe s rp r , k , u, d qs  are b o r ders of 4 -dimentional boxes with vertices pp 1  z q{ 2 , 30  15 z , 55  z , 50  z q where z  Z p r , k , u, d q . The marg inal p-b oxes are, for r : F 1 p z q  F 1 p z q  p p 2 | r  1 { 2 | ¤ z q  p p r P rp 1  z q{ 2 , p 1  z q{ 2 sq  z bec ause r is uniformly distributed ov er r 0 , 1 s . F or k , we ha ve: F 2 p z q  F 2 p z q  p p| k  30 |{ 15 ¤ z q  p p k P r 30  15 z , 30  15 z sq  1  p 1  z q 2 (see Fig. 7). Simila r ly , for u and d , it is easily veriﬁed that: F 3 p z q  F 3 p z q  F 4 p z q  F 4 p z q  1  p 1  z q 2 The low er oscillation osc p h q and upp er oscillation osc p h q can b e c a lculated along the same lines as in the previo us example: osc p h qp z q  inf p r,k,u,d q : Z p r,k,u,d q z h p r , k , u, d q  o p z q osc p h qp z q  sup p r,k,u,d q : Z p r,k,u,d q z h p r , k , u, d q  o p z q P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 27 z 0 o p z q  1 1 5 10 Figure 8. The function o p z q which determines the low er and upper oscillatio n, and the cut sets. with o p z q  $ ' & ' %  µ  β ln p ln pp 1  z q{ 2 qq p 30  15 z q ? 5  2 z ℓ b  3 5 if µ  β ln p ln pp 1  z q{ 2 qq ¥ 0 0 otherwise . The function o p z q is depicted in Fig. 8: it is incre asing, with o p 1 q  0 (this is not immediately clear from the picture, but at higher s cale, it becomes appare nt ), o p 0 q  3 . 032, and o p 1 q  8 . Again, osc p h qp z q and osc p h qp z q are decre asing and incr easing in z , resp ectively . Hence the full comp onents of the even ts L t  t z P r 0 , 1 s : osc p h qp z q ¥ t u  t z P r 0 , 1 s : o p z q ¥ t u U t  t z P r 0 , 1 s : osc p h qp z q ¥ t u  t z P r 0 , 1 s : o p z q ¥ t u are o f the form L t  r 0 , ℓ t s and U t  r u t , 1 s again, with ℓ t   o  1 p t q for t ¤ o p 0 q u t  o  1 p t q for t ¥ o p 0 q As in the previous example, we do not nee d to elicit the upp er cumulativ e distr ibutions, and only the low er ones need to b e given. With unknown dependence, using Theore m 23, we hav e F p z q  max t 0 ,  3  z  3 p 1  p 1  z q 2 qu and w henc e E p h q  » o p 0 q 0 F p o  1 p t qq d t  1 . 515 E p h q  o p 0 q  » 8 o p 0 q  1  F p o  1 p t qq  d t  6 . 423 Therefore, to be o n the safe side, w e should c o nsider av erage overﬂo wing heights o f at least 6 . 5 m . F or c o mparison, us ing traditional methods instead of p- boxes, h has exp ectation 3 . 2 m , a ssuming independenc e b etw een all v ariables—this lies b etw een the low er a nd upper exp ectation that we just calc ula ted, a s exp ected. The interv al is o bviously m uch wider:  bec ause we have reduced a mu ltiv ariate problem to a univ ariate one, whence, leading to imprecision due to the diﬀerence b etw een low er and upp er o scillation,  and beca use we have no t made any a ssumption of indep endence, whence, leading to imprecision due to w eaker assumptions. Realistically , the decision maker may desir e a dike height t such that the upper pr obability of disaster E pt h ¥ t uq is le s s than a g iven threshold. It is easily veriﬁed that: E pt h ¥ t uq  E pt osc p h q ¥ t uq  1  F p o  1 p t qq 28 M. TROFF AES AND S. DESTE RCKE F or instance, for E pt h ¥ t uq  0 . 01, we need t to be 10 . 725 m . F or compariso n, using traditional metho ds instea d of p-boxes, t needs to b e abo ut 9 m , assuming independence b etw een all v ariables. In both examples, analytica l calculations are relatively simple due to the monotonicity of the target function with resp ect to the uncertain v ariables. Of co urse, this may not b e the case in general. 9. Conclusions P-b oxes ar e one of the most interesting imprecise pr obability mo dels from an op er a tional point of v ie w, b ecause they a re simply characterised by a low er and an upp er cumulativ e distribution function. In this pap er , for the purp os e of multiv aria te mo delling, we studied inferences (low er and upp er exp ectations in par ticular) from p-b oxes o n arbitra ry totally preo rdered s paces. F or this pur po se, we r e presented p-boxes as coherent low er previs ions, and studied their natur al extension. W e used a n as general as poss ible model by considering p-boxes whose low er and upp er cum ulative distribution functions a re deﬁned on a totally preordered space. Thereby , we extended the theory of p- boxes from ﬁnite to inﬁnite sets , and from total orders to tota l preorde r s. This allow ed us unify p- boxes on ﬁnite spaces and on interv als of r eals num b ers, and to ex tend the theory to the multiv a riate ca se. One very interesting r esult of this pap er is a practical mea ns of calculating the natural ex- tension of a p-b ox in this general setting. W e pr oved that the natura l extensio n of a p-b ox is ar bitrarily a dditive on full comp onents o f clop en sets with resp ect to the partition top ology induced by equiv alence class es of the underlying preor der (Theorem 13, Co rollar ie s 15 and 16). W e also prov ed that the natural extensio n is c ompletely mo notone, and therefore has a Cho- quet in tegra l representation (Theorem 18). Cons equently , to ca lculate the natural extension, we prov ed that it suﬃces to ca lculate the full comp o nents o f the cut sets of the lower osc illation, follow ed b y a simple Riemann in tegra l (Pro po sition 19). As a s pe c ia l case , we studied p-b oxes whose pr eorders are induced by a rea l-v alued mapping. Such p-b oxes are particular ly attractive, as they allow to build o r elicit a multiv ar iate uncertaint y mo del a t o nce. They corre sp ond to low er and upp er proba bilistic b ounds g iven over nested regions that can take arbitrary shap es. Consequently , we provided a new to ol to combine margina l p- b oxes into a joint p-b ox, under ar- bitrary rules of com bination, ther eby allowing any type o f dep endency mo delling (Lemma 2 2). As examples, w e consider ed tw o extreme cases: a ssuming nothing about dep endence (Theor e m 23), and ass uming epistemic indep endence (Theor em 25). Similar formulas a r e eas ily derived for any other rule of combination. Mor eov er, Williamson a nd Downs’s [45] proba bilistic a rithmetic obtains a s a sp ecia l case of our approa ch. W e demonstrated our metho dolog y on inference ab out a da mpe d harmonic oscilla to r, and on a river dike assess men t, showing that calcula tions ar e generally s tr aightforw ard. Of cour se, many op en proble ms rega r ding p- boxes remain to b e inv estigated. F or instance, even though there need not b e any relatio n b etw een the preorder and the dep endency mo de l— bec ause one can, in theo ry at least, a lwa ys cons truct a mult iv ariate p-b ox from marg inals for any dependency mo del and any preorder —some combin ations o bviously lea d to mor e imprecisio n than others. Our choice led to simple mathematical expressions, but is perhaps not the b est one po ssible in terms of precision. Ca n the dep e ndency mo de l infor m the ch oice of preo rder, to arrive at tighter b ounds? Also, the connection of p-b oxes with other uncertaint y mo dels, such as p oss ibilit y mea sures and clouds, deserves further inv estiga tion. P-BO XES ON TOT ALL Y PREORDERED SP A CES FOR MUL TIV ARIA TE M ODELLING 29 Ackno wledgements W ork par tially supp orted by a do ctor a l grant fro m the IRSN. W e are par ticula rly grateful to Enrique Miranda for the ma ny very fr uitful discussions, extremely useful sugges tions, and v arious con tributions to this pap er. W e also thank Gert de Co oman and Didier Dubois for their help with a very early dr aft o f this pap er. Finally , we thank b oth r eviewers, who se cons tructive comments helped improving the presentation of the pap er substa ntially . References [1] Ph. Artzner, F. Del baen, J.-M . Eb er, and D. Heath. Coheren t measures of risk. Mathematic al Financ e , 9:203–228, 1999. [2] Y. Ben-Haim . Info-gap de cision the ory: de cisions under sever e unc e rtainty . Academic Pr ess, London, 2006. [3] C. Berge. 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Durham University, Dept. of M a thema tical S ciences, Science Labora tories, South Ro ad, Durham DH1 3LE, United Kingdom E-mail addr ess : matthias.trof faes@gmail.com CIRAD, UMR1 208, 2 place P. Viala, F-34060 Montpellier cedex 1, France E-mail addr ess : sebastien.des tercke@cirad.fr

Probability boxes on totally preordered spaces for multivariate modelling

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