Journal of Engineering Mechanics, Trans. ASCE, 2010, 136(10): Bayesian Network Enhanced with Structural Reliability Methods: Methodology

Transcription

1 Ths artcle appeared n: Journal of Engneerng Mechancs, Trans. ASCE, 200, 36(0: Bayesan Network Enhanced wth Structural Relablty Methods: Methodology Danel Straub & Armen Der Kureghan 2 Abstract We combne Bayesan networks (BNs and structural relablty methods (SRMs to create a new computatonal framework, termed enhanced Bayesan network (ebn, for relablty and rsk analyss of engneerng structures and nfrastructure. BNs are effcent n representng and evaluatng complex probablstc dependence structures, as present n nfrastructure and structural systems, and they facltate Bayesan updatng of the model when new nformaton becomes avalable. On the other hand, SRMs enable accurate assessment of probabltes of rare events represented by computatonally demandng, physcally-based models. By combnng the two methods, the ebn framework provdes a unfed and powerful tool for effcently computng probabltes of rare events n complex structural and nfrastructure systems n whch nformaton evolves n tme. Strateges for modelng and effcently analyzng the ebn are descrbed by way of several conceptual examples. The companon paper apples the ebn methodology to example structural and nfrastructure systems. Assocate Professor, Engneerng Rsk Analyss Group, Technsche Unverstät München, Arcsstr. 2, München, Germany. Emal: straub@tum.de 2 Tase Professor of Cvl Engneerng, Dept. of Cvl & Envronmental Engneerng, Unv. of Calforna, Berkeley, CA Emal: adk@ce.berkeley.edu Straub & Der Kureghan (200a /32

2 Introducton Structural relablty methods (SRMs have been developed and successfully appled n the engneerng communty to solve for the probablty of an event E that s gven through an ntegral of the form Pr E f d x x x x ( E ( The event E s defned as a doman E n the outcome space of random varables,,,, whch are specfed through ther jont probablty densty functon ( 2 n (PDF f ( x. In structural system relablty theory (Dtlevsen and Madsen, 996, E s defned n terms of a set of contnuous and contnuously dfferentable lmt-state functons g (x,,, m, n the form E where ( x mn max g ( x,, max g ( x 0 (2 kk C C K C k s an ndex set denotng the k-th cut set of the system. The problem s sad to be a component relablty problem when m ; a parallel-system relablty problem when K ; and a seres-system relablty problem when each cut set contans only one ndex. The form n ( corresponds to a general system relablty problem. Occasonally, the structural relablty problem s defned n the total-probablty form Pr( E pe ( x f ( x dx (3 x where p (x E s the condtonal probablty of event E gven x. Provded p (x E s contnuously dfferentable wth respect to x, one can easly show (Wen and Chen, 987 that the above form reduces to Equaton ( wth E ( x { x ( x 0}, where x φ s the outcome of a standard normal random varable. β( x [ p ( x] s the φ E condtonal relablty ndex gven x, wheren [ ] denotes the nverse of the standard normal cumulatve probablty functon. Ths form s applcable when random varables exogenous to also nfluence event E. Straub & Der Kureghan (200a 2/32

3 Methods for solvng ntegrals of the form n ( nclude the Frst- and Second-Order Relablty Methods (FORM and SORM and a varety of smulaton approaches, ncludng crude Monte Carlo, mportance samplng, drectonal samplng and subset smulaton. In the followng we wll refer to ths class of methods as SRMs. These methods are well-documented n a varety of textbooks and artcles, ncludng (Dtlevsen and Madsen, 996, (Rackwtz, 200 and (Der Kureghan, 2005, and are mplemented n a number of educatonal and commercal software (Ellngwood, Bayesan networks (BNs, also known as belef networks, are probablstc models that facltate effcent representaton of the dependence structure among random varables by graphcal means. BNs have been developed durng the past 25 years, mostly n the feld of artfcal ntellgence, for representng probablstc nformaton and reasonng (Russell and Norvg, They have found applcatons n many felds such as statstcal modelng, language processng, mage recognton and machne learnng, and have begun to be used n engneerng rsk analyss. Recent applcatons n ths feld are reported, e.g., n (Frs-Hansen, 2000; Faber et al., 2002; Frs-Hansen, 2004; Mahadevan and Rebba, 2005; Grêt-Regamey and Straub, 2006; Nshjma et al., 2009; Bens et al., Snce the BN methodology s lkely to be new to most readers of ths journal, ts essental elements are brefly ntroduced n a separate secton. As would be expected, both SRMs and BNs have advantages and lmtatons. SRMs are applcable to contnuous random varables wth a known jont dstrbuton. Any form of statstcal dependence can be handled. However, several SRMs, e.g., FORM and SORM, are not sutable for dscrete random varables. Furthermore, these methods are dffcult to apply for non-experts, partcularly n the context of nformaton updatng, and are not easly presentable n a graphcal form. They cannot generally be ncluded n automated algorthms that can run wthout an expert. On the other hand, the BN s hghly effectve for analyzng dscrete random varables and for nformaton updatng. It s also an effectve tool for decson-makng, and ts graphcal form provdes a concse representaton of statstcal dependence that can be understood also by non-experts. In ts dscrete form, the BN can be run by the lay engneer, even n an automated mode for near-real tme decson support. However, for contnuous random varables, the BN has practcal lmtatons on the type of dstrbutons and the form of statstcal dependence Straub & Der Kureghan (200a 3/32

4 that can be handled. Furthermore, t s not deally suted for computng small probabltes, whch s the specalty of SRMs. It follows that a combnaton of the two methods can potentally be a powerful tool for probablstc analyss and decson-makng. So far, few publcatons have consdered the use of BNs n cvl engneerng rsk analyss from a methodologcal vewpont. Frs-Hansen (Frs-Hansen, 2000 combnes example applcatons wth a dscusson of some of the methodologcal ssues nvolved, such as the dscretzaton of random varables. A SRM s used to compute the condtonal probablty tables of the BN, smlar to what s proposed n ths paper, yet wthout a formal framework. In (Frs-Hansen, 2004, an applcaton of BN to structural relablty problems s nvestgated, based on dscretzng all random varables. Addtonally, a number of authors dscuss the modelng of relablty problems usng BNs (Bobbo et al., 200; Mahadevan and Rebba, 2005; Langseth and Portnale, 2007; Nel et al., 2008, but these do not nclude structural relablty applcatons. The objectve of ths paper s to explore the possblty of combnng the SRM and BN nto an enhanced tool for probablstc analyss and decson-makng. Denoted enhanced Bayesan Network or ebn, the proposed tool s a BN whch has both dscrete and contnuous nodes wth arbtrary dstrbutons and nterdependences. We explore the rules and computatons that are necessary to reduce an ebn nto a reduced BN (rbn wth dscrete nodes only, for whch exstng exact methods of nference can be used. The reducton s performed through a process of elmnaton of contnuous nodes. We show that the requred computatons can be performed by an SRM through background analyses, whch can reman hdden to the user of the rbn. Varous alternatves for elmnatng contnuous nodes so as to optmally produce the rbn are explored. Fnally, modelng strateges that enable effcent computatons are presented and the lmtatons of the approach are dscussed. The applcaton of the methodology s shown n a companon paper, consderng relablty analyss of an ndvdual structural system and a system of structural systems. Straub & Der Kureghan (200a 4/32

5 2 Bayesan networks We lmt ourselves to ntroducng the most mportant concepts of the BN as requred for the remander of the paper. For an extensve ntroducton to BN, we refer to the standard textbook of Jensen and Nelsen (Jensen and Nelsen, 2007 and the revew paper on the applcaton of BN for engneerng relablty applcatons by Langseth and Portnale (Langseth and Portnale, Broader ntroductons to BN and related concepts of probablstc knowledge representaton and reasonng are provded by (Pearl, 988 and (Russell and Norvg, BNs are probablstc models based on drected acyclc graphs. They represent a probablty measure ( z over the outcome space of a set of random varables Z ( Z,..., Z N. Each varable Z can be defned n a dscrete and fnte outcome space (dscrete random varable or a contnuous outcome space (contnuous random varable. For dscrete random varables, π( z Pr( Z z p( z s the jont probablty mass N functon (PMF. For contnuous random varables, π( z Pr( Z z / z f ( z s the jont PDF. The sze of the jont outcome space of Z for whch ( z must be defned ncreases exponentally wth the number of varables, but the BN enables an effcent modelng by factorng the jont probablty dstrbuton nto condtonal (local dstrbutons for each varable gven ts parents. A smple BN wth fve varables s llustrated n Fgure, where Z s a parent of Z 3 and Z 4, and Z 2 s a parent of Z 4 and Z 5. Z Z 2 Z 3 Z 4 Z 5 Fgure. A smple Bayesan network. The jont probablty measure for ths network s gven as z, z2, z3, z4, z5 z z2 z3 z z4 z, z2 z5 z2 z (4 Straub & Der Kureghan (200a 5/32

6 whch can be wrtten n the compact and general form N z paz z (5 where pa( Z denotes the set of parents of Z. In addton, let ch( Z denote the chldren of Z and sp( Z the spouses of Z. The spouses are all varables that share a chld wth Z but are not chldren of Z. For example, n the BN shown n Fgure, pa( Z s the empty set, ch( Z { Z3, Z4} and sp( Z { Z2}. The parents, chldren and spouses of a varable together form the Markov blanket of that varable, blz ( paz ( chz ( spz ( (6 The Markov blanket s an mportant concept, snce, for gven values of bl( Z, Z s statstcally ndependent of all other varables,.e., z z z blz (7 where z denotes realzatons of all varables other than Z. Ths ndependence relaton follows drectly from the d-separaton rules formulated by Pearl (Pearl, 988, whch represent the ndependence assumptons encoded n the graphcal structure of the BN. When evdence (nformaton s avalable on a set of varables n the BN, the dstrbutons of all remanng varables are updated usng Bayes rule. The rules of d-separaton help to dentfy those varables whose probablty measure change upon the evdence. As an example, f the evdence { Z5 e5} s avalable on the BN n Fgure, the margnal probablty measures on the varables Z 2 and Z 4 wll change, whereas those of varables Z and Z 3 wll not change because they are d-separated from Z 5. As an example, ( z e s computed as 4 5 Straub & Der Kureghan (200a 6/32

7 z e z, e e5 4, Z z z z2 e5 z2 z z Z 2 2 e5 z2z2 Z2 e5 z2z2 z4 z, z2 z Z 2 Z e5 z2z2 Z2 (8 Equaton (8 s for the case of dscrete random varables; for contnuous random varables the summaton operatons must be replaced wth ntegraton operatons. From the numerator n the last lne of Equaton (8, we can deduce the prncple behnd exact nference algorthms, whch are avalable for solvng BNs wth dscrete random varables (Laurtzen and Spegelhalter, 988; Shenoy and Shafer, 990; Zhang and Poole, 996; Dechter, 996; Jensen and Nelsen, In essence, these algorthms am to fnd an optmal orderng of the summaton operatons requrng the lowest CPU tme and/or storage capacty. Avalable algorthms perform dfferently dependng on the applcaton. It has been shown that fndng an optmal orderng s an NP-complete problem (Cooper, 990. Ths concept from computer scence ndcates that t s not promsng to search for an algorthm that fnds the optmal orderng n all cases. Several algorthms have been mplemented n a number of commercally or freely avalable software (see Murphy Murphy, 200 for an overvew. Detaled knowledge of these algorthms s not necessary for ths paper. To apprecate the dffculty of the problem, observe that the computaton n Equaton (8 requres multplcaton operatons n the jont outcome space of Z, Z 2 and Z 4. In the language of (Jensen and Nelsen, 2007, the varables Z, Z 2 and Z 4 form the largest clque for ths BN, thus necesstatng the computaton of the potental n the doman of these three varables, potentals beng tables of condtonal probabltes. Snce the sze of the potental ncreases exponentally wth the number of varables nvolved, the sze of the largest potental to handle s crtcal for the performance of the algorthm. Consder a network consstng of one chld node wth 20 parent nodes. General nference wll requre manpulatng probabltes n the jont space of all these varables (.e., the largest clque Straub & Der Kureghan (200a 7/32

8 nvolves all 2 varables. Even f each varable has only two states, the potental to handle already contans 2 2 entres! Exact nference algorthms exst also for two specal cases of so-called hybrd BNs, whch nvolve both contnuous and dscrete random varables. The frst case s BNs wth nodes that are defned as Gaussan random varables, whose means are lnear functons of ther parents. The applcaton of such BNs s rather lmted, n partcular snce the contnuous nodes must not have any dscrete chldren. The second case s BNs whose nodes are defned as mxtures of truncated exponentals (MTE. Such MTEs can be thought of as an extenson of dscrete random varables, whereby the probablty densty wthn each nterval s approxmated by a lnear combnaton of exponental functons nstead of a constant (Langseth et al., As an alternatve to exact nference, approxmate nference algorthms usng smulaton technques have been developed, of whch Markov Chan Monte Carlo (MCMC methods (Glks et al., 996; Beck and Au, 2002 have become especally popular. Combnatons of exact and approxmate nference algorthms are also avalable (see, e.g., Murphy, We wll brefly return to dscuss approxmate nference, but otherwse restrct ths paper to exact nference n BNs wth dscrete random varables. 3 Framework for an enhanced Bayesan Network 3. Defntons We defne as enhanced Bayesan networks (ebns a subclass of BNs that have the followng propertes: a The BN has nodes that are defned n a fnte sample space (dscrete nodes and nodes that represent vectors of contnuous random varables (contnuous nodes. b The states of each dscrete node that s a chld of at least one contnuous node are defned as domans n the outcome space of ts parents, n whch case the node s determnstc, or are defned by a PMF that s parameterzed by the parent nodes, n whch case the node s random. Straub & Der Kureghan (200a 8/32

9 More formally, let us defne a set of dscrete random varables Y { Y,, Y ny } and a set of vectors of (not necessarly ndependent contnuous random varables {,, }. The complete set of nodes n the network s denoted Z {, Y }. A n dscrete varable Y has a set of possble states ( k y, where s the ndex for the varable and k s the ndex of the state, and a condtonal PMF p y pa Y Y y pa Y. To enhance readablty, we omt the state ndex ( k ( k [ ( ] Pr[ ( ] k whenever there s no ambguty. A contnuous node wth ndex represents a vector of contnuous random varables (,,,, and s characterzed by the jont n condtonal PDF f[ x pa( ]. The BN s then characterzed by the combned measure YY p yx f x p y pa Y f pa x (9 When the dscrete node Y has exclusvely contnuous parents wth realzatons denoted by x pa Y, the condtonal PMF p[ y pa( Y ] s defned accordng to Equatons ( or (3. ( Thus, each state ( k y ( k s defned through a doman x ( pa( Y. If the node Y addtonally has dscrete parents y pa Y (, ths functon must be defned for each combnaton of the states of y pa Y separately and the correspondng notaton of the ( k doman s x, k ( pa pa( Y (, wheren k pa denotes the jont state of the dscrete parents of Y. As an example, y may descrbe the operatonal modes of a mechancal system and pa( Y ( k x represent the models descrbng the system performance for dfferent, k ( pa pa( Y modes k pa ; or y pa Y may be a varable representng whether or not a wave hts the deck ( ( k of an offshore platform, and x represent the approprate structural models for each case., k ( pa pa( Y All dscrete nodes n the ebn are defned as sngle varables. Ths s no lmtaton, snce for dscrete varables t s straghtforward to transform the jont space of several varables nto the space of a sngle varable. As an example, f varable has two states 0 and, Straub & Der Kureghan (200a 9/32

10 and varable has two states 0 and, the jont space of the two varables can be represented by a sngle varable wth four dscrete states 00, 0, 0 and. 3.2 Inference problem and soluton strategy The nference problem consdered here s that of determnng p( y j y e, where Y j are the random varables of nterest and Ye y e s the avalable evdence. j Y and Y e are both subsets of Y, the set of dscrete random varables. Ths s a restrcton of the general case, n that all varables of nterest are dscrete random varables and all evdence s on dscrete random varables. However, for many applcatons these restrctons are not crtcal. As we wll show, equvalent dscrete varables can be ntroduced for any contnuous varable of nterest. Furthermore, as descrbed later, certan types of evdence on contnuous random varables can be handled by ntroducng a bnary dscrete random varable wth one of ts states correspondng to the observed evdence. We propose to solve nference problems of the type descrbed above through a two-step procedure. The frst step s the determnaton of p( y, the jont PMF of the dscrete varables Y. p( y s obtaned through elmnaton of the contnuous nodes n the ebn and s represented by a BN tself. The resultng reduced BN s referred to as rbn. In the second step, snce the rbn conssts only of dscrete nodes, p y y s evaluated from p( y by use of exstng algorthms for exact nference. ( j e The motvaton for transformng the ebn nto a rbn consstng only of dscrete nodes s the avalablty of exact and easy-to-use nference algorthms for BNs wth dscrete nodes, and the possblty to utlze well establshed SRMs. An alternatve strategy that s not nvestgated here s the use of approxmate algorthms that can perform nference drectly n hybrd BNs,.e., BNs wth both contnuous and dscrete nodes (Langseth et al., In the companon paper (Straub and Der Kureghan, 200 we use smulaton-based algorthms for hybrd BNs to check our models, but t s noted that such methods are ether computatonally neffcent n the general case (e.g., n the case of rejecton or lkelhood samplng, or they have unknown rates of convergence (e.g., n the case of MCMC, makng them dffcult to employ n automated algorthms. By automated Straub & Der Kureghan (200a 0/32

11 algorthms we mean algorthms that can be ncluded n software and then be effcently appled by engneers and other experts who are not specalsts n the underlyng probablstc modelng and algorthms. Automated algorthms are partcularly relevant for near-real-tme decson support systems. The prncple advantage of the proposed computaton strategy s that only the development of the rbn s tme-consumng and requres specalst knowledge n probablty and relablty analyss. By mplementng the resultng rbn n software that allows exact nference, the resultng model can be appled by non-specalsts. Furthermore, the rbn s easly extended to a decson graph, thus allowng drect decson optmzaton. We beleve that the development of such automated algorthms wll advance the dssemnaton of probablstc methods n practce for a varety of complex engneerng decson problems under uncertanty. 3.3 Determnng the rbn In order to establsh the rbn, the contnuous nodes are removed from the network. An algorthm for elmnaton of nodes n an nfluence dagram, whch ncludes the BN as a specal case, s descrbed n (Shachter, 988, Shachter, Node elmnaton algorthm Followng (Shachter, 986, we frst defne as barren nodes all random varables wthout chldren that do not receve any evdence. If s a barren node n an ebn, we can smply remove t together wth the lnks drectng to t, wthout changng p( y. Second, consder theorem 2 from (Shachter, 986 that descrbes the condtons for reversng a drected lnk (arc: Gven that there s an arc (,j between chance nodes and j, but no other drected (,j-path n a regular nfluence dagram, arc (,j can be replaced by arc (j,. Afterward, both nodes nhert each other's condtonal predecessors. Accordng to ths theorem, a drected lnk between two nodes (from node to node j can be reversed f there s no other path n the same drecton (otherwse the new network would become cyclc, by addng drected lnks from all parents of node to node j and from all parents of node j to node. Straub & Der Kureghan (200a /32

12 The process of elmnatng node lnks from barren node. Then, n the ebn proceeds by frst reversng all drected to ch(, the chldren of, untl ch( s the empty set and s a, together wth all the lnks pontng to t, can smply be removed. Fgure 2 shows an example of the process. The order of the reversng operatons can be chosen freely as long as t s ensured that the resultng network s acyclc at any stage; n Fgure 2, the lnk from to Y 6 cannot be reversed frst, as ths would lead to the cycle Y Y Y. Later, we wll show that the order of the reversng operatons can nfluence the form of the rbn, and we wll address the optmal orderng. It s noted that upon elmnaton of the contnuous nodes, dscrete nodes that were defned as determnstc functons of ther parents (through domans encapsulatng the uncertanty n ther contnuous parent nodes. become random nodes, reverse (,Y 5 reverse (,Y 6 remove Y 4 Y 4 Y 4 Y 4 Y 5 Y 6 Y 5 Y 6 Y 5 Y 6 Y 5 Y 6 Y 7 Y 7 Y 7 Y 7 Fgure 2. Illustraton of an enhanced Bayesan network and a lnk reversal sequence for removal of node to arrve at the rbn Computng the condtonal probablty tables (potentals of the rbn For the gven structure of the rbn, as determned through the node elmnaton algorthm, t s necessary to compute the condtonal probablty tables (potentals of the varables. Here we show that these computatons can be performed through a SRM. To dstngush between the ebn and the rbn structure, let pa( Y denote the parents of varable Y n the ebn and let pa ( Y denote ts parents n the rbn. Straub & Der Kureghan (200a 2/32

13 Let Y C denote all dscrete varables that are chldren of at least one contnuous varable n the ebn, YC { Y[ ch( ch( n ]}, and let Y NC denote all remanng dscrete varables. It follows from the node elmnaton algorthm that all varables n Y NC wll have the same parents n the rbn as n the orgnal ebn, and therefore the same condtonal probablty tables. Ths s also evdent when formulatng the jont PMF of all dscrete nodes for the general case: p y p y x f x dxdx n x n nx x YY nx p y pa Y f pa x dx dx p y pa Y p y pa Y f pa x dx dx n n x YY x NC YYC (0 Snce the parents of the varables n Y NC do not nclude any, the condtonal probablty terms for the Y NC are taken outsde the ntegral n Equaton (0. It follows that py pa( Y py pa( Y, YY NC ( and the remanng condtonal probablty terms must correspond to the ntegral n the last lne of Equaton (0: p y pa( Y p y pa( Y f x pa( dx dx (2 x n n YY x C YYC The rght-hand sde of Equaton (2 corresponds to the general formulaton n Equaton (3. Therefore, we can use a SRM to solve for the jont probablty of the Y C. In the case where all varables Y C are defned as domans n the space of ther contnuous parents, we can reformulate Equaton (2 to read YY C p ( k y pay f x y x x ( x YY C ( k, k pa P x dx (3 Straub & Der Kureghan (200a 3/32

14 where f x y f [ x pa( ] ( P, wth Y P denotng the set of dscrete parents to any of the contnuous varables, YP { Y[ pa( pa( n ]}. Note that Equaton (3 has the form of Equaton (. If the probabltes n the rbn are computed drectly usng ths equaton, then the problem must be solved for all combnatons of the states of the varables Y C, Y and pa( Y. Ths number can become prohbtvely large P C n general ebn models. However, under certan crcumstances, t s possble to take advantage of the ndependence assumptons encoded n the rbn to lmt the number of SRM calculatons. Ths s the case f t s possble to reformulate the ntegral n Equaton (3 nto the form f xy dx f x y dx P l Pl l x( x l xll( xl ( k x x l l, kpa l YY Cl (4 In Equaton (4, the contnuous random varables are separated nto groups l, wth correspondng dscrete chldren Y Cl and dscrete parents Y Pl, such that the SRM calculatons can be performed separately for these groups. The total number of SRM calculatons s then gven by the product of the number of states of Y Cl, Y Pl and pa( Y Cl, summed over all groups l. In the followng secton, we show how groups can be dentfed for whch the decomposton n Equaton (4 holds. 3.4 The Markov envelope In the precedng secton, a method was presented for reducng the ebn to a rbn by means of a SRM. It was postulated that the number of SRM calculatons can be reduced by dentfyng groups of contnuous random varables l for whch such calculatons can be performed separately. In ths secton we show that these groups are unquely defned by the graphcal structure of the ebn, and that the mnmum number of SRM calculatons requred, therefore, follows drectly from the graphcal structure of the ebn and the number of states of ts dscrete varables. Straub & Der Kureghan (200a 4/32

15 Frst, we observe that f a contnuous node j has no other contnuous node n ts Markov blanket, bl( j as defned n Equaton (6, then ths node can be treated separately n the process of establshng the rbn. We can show ths by consderng the general formulaton for the rbn, Equaton (0. The rbn was obtaned by ntegratng out the contnuous varables. If a contnuous node j has no other contnuous node n ts Markov blanket, then pa( j as well as sp( Y do not contan any other contnuous node. Furthermore, no contnuous node can have can separate the ntegraton over j j as ts parent. For these reasons, we j from the ntegraton over the other contnuous varables n (0. We can also separate the correspondng terms of the rbn on the lefthand sde of (2 and wrte a separate equaton for all terms nvolvng j : py pa( Y p y pa( Y f x pa( dx (5 j j j j Ych( j Ych( j Snce pa ( Y can only nclude varables that appear on the rght-hand sde of (5, all varables n pa ( Y are part of the Markov blanket of j. Ths proves that we can treat together wth bl( as a separate ebn when establshng the rbn. It also follows j that t s mmateral f the lnks from j contnuous varables, whch are outsde the Markov blanket of j are reversed before or after the lnks from other j. Second, t follows from the node elmnaton algorthm that: a all dscrete nodes n ch( wll have all parents of as parents n the rbn; b the dscrete node n ch( whose lnk from s reversed last wll have all other dscrete varables n ch( as well as all dscrete varables n sp( as parents,.e., the node wll have as parents all other dscrete varables n bl( ; c f bl( ncludes other contnuous varables, then, after removal of, these wll have all varables of bl( n ther respectve Markov blankets. As a consequence, after elmnaton of a second varable j, one node n ch( j wll have all other nodes that were part of the Markov blankets of and j n Straub & Der Kureghan (200a 5/32

16 the ebn as parents. More generally, consder a set of contnuous nodes dentfed as follows: Start wth a sngle contnuous node and put t nto M M, whch s ; add all contnuous nodes that are part of the Markov blanket of the frst node; add all contnuous nodes that are part of the Markov blankets of the addtonal nodes; and so on. We then defne as a Markov envelope the aggregaton of all varables (dscrete and contnuous that are part of the Markov blankets of all varables n,.e., { bl( }. Ths M M concept s llustrated n Fgure 3. It follows from the above consderatons that one dscrete varable n each of the Markov envelopes wll have all other dscrete varables n the envelope as parents n the rbn. Furthermore, the contnuous random varables wthn a Markov envelope form the mnmum groups holds. l for whch the equalty n Equaton (4 envelope 2 3 envelope 2 Y 4 Y 5 4 Y 6 Fgure 3. Illustraton of the prncple of envelopes of Markov blankets of contnuous varables. and 3 are both n bl( 2, thus the envelope contans bl(, bl( 2 and bl( 3, whereas bl( 4 contans no other contnuous varables and forms an ndvdual envelope. The fact that for one node Y n each Markov envelope pa ( Y wll nclude all other dscrete varables n the envelope has fundamental mplcatons for the resultng rbn. Independent of the orderng of lnk reversals, the szes of these envelopes determne the number of SRM computatons, snce the potental of one node n each envelope wll nclude all other dscrete varables n the envelope. (The number of entres n the table s n m, wth n beng the number of dscrete varables n the envelope and m beng the number of states of the -th varable. Furthermore, the maxmum sze of these envelopes represents a lower lmt to the maxmum clque sze n the rbn. The mportance of ths Straub & Der Kureghan (200a 6/32

17 observaton stems from the fact that the maxmum clque sze s a crucal parameter for the computatonal speed n performng exact nference n BNs (Dechter, 996. It follows from the above analyss that, to ensure computatonal feasblty of the rbn, the number of dscrete varables n any Markov envelope n the ebn must be lmted. The maxmum feasble number depends on m, but even n the extreme case of m 2 for all, the Markov envelope should not contan more than 5-20 dscrete varables. In a later secton, we dscuss modelng strateges to deal wth ths problem. 3.5 Illustraton The dervaton of the rbn s llustrated on the example depcted n Fgure 2. Snce ths ebn contans only one node wth contnuous varables, there exsts only one Markov envelope, consstng of the varables and Y3 Y6. Frst, we demonstrate how the rbn shown on the far rght of Fgure 2 can be derved from the ebn through algebrac manpulatons. The jont probablty measure for ths ebn s wrtten as,, 7 x x 2 3, x 3 5 4, x 6 5, x 7 5 p y y f p y p y y p y y y p y y f y p y y p y y p y y (6 The rbn s obtaned through ntegraton over the doman of,:,,,, p y y p y y x f x dx 7 7,,, p y p y y p y y y p y y p y y f x y p y y x p y y x dx (7 wth,,,, f x y p y y x p y y x dx p y y y x f x y dx p y, y y, y (8 and nsertng n (7 we obtan Straub & Der Kureghan (200a 7/32

18 ,, 7 2 3, , 6 3, p y y p y p y y p y y y p y y p y y y y p y y (9 Ths formulaton corresponds to the rbn obtaned n Fgure 2 by the node elmnaton algorthm, snce p y, y y, y p( y y, y, y p( y y,. The condtonal ( y4 probablty tables of varables Y 5 and Y 6 n the resultng network must be computed. In accordance wth (3, ( k5 ( k6 ( k3 ( k4 ( k3,, p y y y y f x y dx ( ( k5 ( k6 x { y5, k ( x 4 y 6, k ( x 5 } ( k5 Here, ( x s the doman that defnes the event Y5 k5 n the space of gven that Y 4 4 Y5, k4 ( k6 k, and ( x s defned accordngly. Equaton (20 can be solved usng a Y6, k5 system SRM. The ndvdual potentals for Y 5 and Y 6 are then obtaned smply by 5 3, 4 5, 6 3, 4 p y y y p y y y y (2 p y y, y, y Y p y5, y6 y3, y4 (22 p y y, y The dervaton of the rbn for ths example, therefore, requres solvng m m m ( m system structural relablty problems (wth m beng the number of states of Y. 3.6 Obtanng an optmal rbn from a gven ebn Although the mnmum number of requred SRM calculatons to produce the rbn s determned by the structure of the ebn, t s possble to obtan dfferent rbns for a gven ebn dependng on the selected order of lnk reversals and elmnaton of contnuous nodes. Ths s demonstrated by the example n Fgure 4. In ths secton we brefly dscuss the optmal orderng of the lnk reversal and node elmnaton actons. Straub & Der Kureghan (200a 8/32

19 a reverse (, reverse (, and remove b reverse (, reverse (, and remove Fgure 4. The nfluence of the order of lnk reversals on the fnal rbn: Orderng (a leads to an addtonal lnk n the rbn. The crteron for the optmalty of the rbn depends on the envsoned applcaton. Commonly, ether the rbn leadng to the lowest requred CPU tme or the one leadng to the mnmum storage requrement s consdered as optmal. In other nstances, computatonal or storage ssues may not be as mportant as havng a rbn that has lnks wth logcal (causal nterpretaton. In most cases, however, these crtera wll concde. For example, n Fgure 4, the orderng (b leads to a rbn that s optmal accordng to all the above crtera. It has been shown n a precedng secton that each Markov envelope can be consdered ndvdually n the elmnaton algorthm. Therefore, only the orderngs of lnk reversals and node elmnatons wthn the Markov envelopes are relevant for the optmalty of the rbn. Snce the number of dscrete varables wthn a Markov envelope must necessarly be lmted, the number of combnatons of lnk reversal orders to consder wll generally be relatvely small. In that case, the analyst may be able to determne the optmal rbn through nspecton of the ebn graph. Based on observatons made earler, one can state that the lnk reversal should generally start wth the lnks gong to nodes wth the fewest parents, f the goal s to have the rbn wth a mnmal number of lnks. Exceptons to ths rule, however, may occur. It s noted that the modelng choces made n establshng the ebn are more decsve for computatonal performance than the order of lnk reversals n establshng the rbn. Ths Straub & Der Kureghan (200a 9/32

20 s because the form of the ebn determnes the number of SRM calculatons and presents a lower bound on the maxmum clque sze of the rbn. For ths reason, emphass should be placed on establshng a computatonally effcent ebn model by focusng on the szes of ts Markov envelopes. Ths s dscussed n a later secton on modelng strateges. 3.7 Evdence and nference on contnuous varables A major motvaton for the use of a BN s ts capablty for Bayesan updatng when evdence, such as measurement results, montorng data or observatons of performances of structures, becomes avalable. Incluson of evdence on any set of varables n the rbn s supported by the avalable exact nference algorthms for dscrete-varable BNs. Therefore, n the ebn approach, potental evdence on contnuous varables should somehow be represented n terms of dscrete varables so they reman present n the rbn. To ths end, when establshng the ebn, t s necessary to antcpate the type of evdences that may become avalable on contnuous varables and ntroduce correspondng dscrete varables. Potental evdence for a group of contnuous varables e may be descrbed by a set of domans e, ( xe,,...,me, n whch the varables e mght be observed to fall. Let p e Pr[ e e, ( xe]. Then a dscrete varable Y e s ntroduced as a chld of e wth m states that are defned through the domans ( x. The evdence ( x } e, e { e e, e s thus represented n the rbn as evdence { Y e } on Y e. Note that the probabltes p are of the form n Equaton ( and are easly computed by an SRM. It s possble to envson evdence of zero probablty. An event of the form { h( 0}, where h denotes a determnstc functon, has zero probablty when varables. In ths case all SRM computatons that nvolve e are contnuous e must be performed condtonal on the zero-probablty event. SRM enables such computatons through surface ntegraton (Schall et al., 988 or by relablty senstvty analyss (Madsen, 987. However, t mght often be easer and more practcal to represent such zeroprobablty observatons wth domans of small probablty. For example, for the event e Straub & Der Kureghan (200a 20/32

21 mentoned above, one may consder the doman ( x {0 h( x h} wth h a suffcently small value. If observatons of such a functon over an nterval are antcpated, then the nterval needs to be dscretzed and correspondng domans ntroduced so that the method descrbed n the precedng paragraph apples. Snce nference (updatng can be made only on varables that are present n the rbn, the outcome space of contnuous varable on whch nference s desred must be dscretzed. Methods for such dscretzaton are descrbed next. e e e 3.8 Dscretzaton of contnuous random varables Dscretzaton of random varables n the ebn (or more precsely, dscretzaton of the outcome spaces of contnuous random varables may be necessary for two reasons. Frst, f we are nterested n the posteror dstrbuton of a contnuous random varable, ts outcome space should be dscretzed to allow nference on the varable n the rbn. Second, an effcent strategy to reduce the sze of the Markov envelopes s to selectvely dscretze contnuous random varables, as we wll demonstrate n the next secton. Surprsngly, lttle lterature s avalable on dscretzaton of contnuous random varables n the context of engneerng rsk analyss, gven that the approach s commonly used n engneerng practce. Some consderatons can be found n (Frs-Hansen, 2000; Nel et al., 2008; Straub, There s need for a formalzed approach, whch, however, s beyond the scope of ths paper. We lmt ourselves to proposng an approach that s drectly based on the ebn and whch s accurate as well as practcal for many applcatons. Consder dscretzaton of the outcome space of a contnuous random varable wth condtonal cumulatve dstrbuton functon F [ x pa( ]. In the ebn, we replace by two random varables, a dscrete varable Y and a contnuous varable chld of Y. Y nherts all parent varables of chldren of, whle, whch s a becomes the parent to all the. Ths s llustrated n Fgure 5. The outcome space of Y conssts of states, denoted by y ( k, k m. These correspond to mutually exclusve, collectvely m Straub & Der Kureghan (200a 2/32

22 exhaustve ntervals n the outcome space of the orgnal varable of s dentcal to the outcome space of.. The outcome space Orgnal model: dscretzed: Y 2 Y 4 2 Fgure 5. Dscretzaton of a random varable by replacng t wth Y and. The proposed approach to dscretzaton makes drect use of the ebn framework. By mantanng a contnuous random varable the condtonal dstrbutons of the chldren of varable wth parents (the parents of varable n the ebn, t s not necessary to redefne ; t suffces to replace the condtonng. We must, however, determne the condtonal PMF of Y gven ts and the dstrbuton of s later elmnated n the process of establshng the rbn. The condtonal PMF of Y s obtaned as condtoned on Y. The contnuous ( k p y pa ( F x ( ( k pa F x k pa (23 n whch x k state k of Y. If and x k are the lower and upper boundares of the nterval correspondng to s a determnstc functon of ts parents, h[ pa( ], then the states of Y can be defned drectly as domans n the space of pa( : ( k [ pa( x ] { x h[ pa( x ] 0} { h[ pa( x ] x 0}. If k k has no parent, then the condtonal dstrbuton of gven Y y s obtaned as ( k Straub & Der Kureghan (200a 22/32

23 0, x xk F x F x F x y x x x ( k k, k k F x k F x k, x k x (24 In ths case, the dscretzaton does not ntroduce any approxmaton, snce the margnal dstrbuton of margnal dstrbuton of s dentcal to that of. However, f has parents, then the s not generally known and an approxmaton s requred. A straghtforward choce s the unform dstrbuton wthn each dscretzed nterval n whch case 0, x xk x x F x y x x x, xk x ( k k, k k xk xk x -, k x k (25 The unform assumpton s not sutable f the dscretzaton nterval s bounded only on one sde, as t occurs for the ntervals n the extreme tals of unbounded dstrbutons. In such cases, a dfferent dstrbuton must be selected. For example, for the unbounded nterval ( x, one may select the exponental dstrbuton (Straub, 2009 k F 0, k ( k x y exp x x k, xk x x x (26 where must be selected by the analyst to reflect the antcpated rate of decay of the tal of the margnal dstrbuton. Even when ( k ( has parents n the ebn, ts margnal dstrbuton mght be known and F x y can be determned from (24. In such cases, the dscretzaton stll entals an approxmaton, snce cannot fully reflect the dstrbuton of wthn one nterval of Y for gven values of pa(. In other words, even though the margnal dstrbuton of Straub & Der Kureghan (200a 23/32

24 s correctly represented, the statstcal dependence between approxmately represented n the dscretzaton. and ts parents s only Wth the presented approach, all dscrete chldren of the orgnal contnuous varable become chldren of, and, therefore, are part of the same Markov envelope. It can be desrable to have them n separate Markov envelopes, n order to reduce the number of SRM computatons and to lmt the complexty of the resultng rbn. Ths can be acheved by ntroducng a separate contnuous random varable j for each chld of, as llustrated n Fgure 6. Here, all j are defned through dentcal dstrbutons condtonal on Y as gven n Eqs. (24 to (26. Ths approach to dscretzaton ntroduces an addtonal approxmaton n the model: In realty, the j should be dentcal, but ths model only consders that they are n the same nterval as specfed by Y. Ths approxmaton does not affect the model of the margnal dstrbutons, but t leads to an underestmaton of the statstcal dependence among the chldren of Y. Orgnal model: dscretzed: Y a b c 2 Y 4 2 Y 4 Fgure 6. Alternatve dscretzaton of a random varable, separatng the chldren of. 4 Modelng strateges The proposed soluton strategy for the ebn has two computatonal bottlenecks: The number of SRM computatons necessary to determne the condtonal probablty tables, and the sze of the largest clque n the resultng rbn. The number of SRM computatons Straub & Der Kureghan (200a 24/32

25 ncreases exponentally wth the number of dscrete varables n each Markov envelope. The maxmum clque sze has as a mnmum value the number of dscrete varables n each Markov envelope, but t also depends on the dependence structure of the dscrete varables. As a consequence, to ensure computablty of the rbn, the szes of the Markov envelopes n the ebn must be lmted. In ths secton, we dscuss strateges for dong ths. a Dscretzaton of contnuous random varables Markov envelopes are separated when contnuous varables are not drectly lnked and f they are not connected through common chldren, as llustrated n Fgure 3. Therefore, an effcent strategy to reduce the sze of Markov envelopes, and thus the number of SRM calculatons and the complexty of the rbn, s to selectvely dscretze contnuous random varables. Ths strategy s partcularly effectve n herarchcal ebn structures, as shown n Fgure 7. If each Y,,...,5, has m dscrete states, then the evaluaton of the rbn for the orgnal model (a requres m 4 ( m system SRM calculatons, whereas model (b, wheren the contnuous varable 0 has been replaced wth the dscrete varable Y 0 and correspondng contnuous varables 0a,, 0e, requres only 5( m component SRM calculatons. If the components are dentcally defned (e.g., fve structural components of smlar type, then the number of component SRM calculatons s further reduced to ( m. An addtonal example of how dscretzaton reduces the sze of Markov envelopes s demonstrated n Fgure 8 for a dynamc ebn. Here, dscretzaton of the varables connectng the dfferent slces (correspondng to dfferent nstances of tme or space leads to a much smpler rbn structure. To ths end, the contnuous random varable replaced by the dscrete Y a and correspondng contnuous a and s b. The reducton n the number of requred SRM calculatons s dentcal to that of the example n Fgure 7. Straub & Der Kureghan (200a 25/32

26 a Orgnal model: 0 ebn rbn Y 4 Y 5 Y 4 Y 5 b Dscretzng 0 : ebn rbn Y 0 Y 0 0a 0b 0c 0d 0e Y 4 Y 5 Y 4 Y 5 Fgure 7. Dscretzng the common parent varable reduces the sze of the Markov envelopes. a Orgnal model: ebn rbn Y 4 Y 5 Y 4 Y 5 b Dscretzng - 5 : b 2b 3b 4b ebn rbn a a a Y 4a Y 5a a a a Y 4a Y 5a a 2a 3a 4a 5a Y 4 Y Y 4 Y 5 Fgure 8. Dscretzng the nterconnectng varables to reduce the sze of the Markov envelopes n the dynamc ebn. Straub & Der Kureghan (200a 26/32

27 b Causal and explct modelng Because the rules for encodng the dependence structure n the BN graph are derved from causal reasonng (Pearl, 988, causal modelng of the relatons among varables generally leads to the lowest number of lnks n a BN. Fgure 9 shows the classcal example of two test outcomes, of a system. If the test outcomes are modeled condtonally on the state of the system Y 0, as n the causal model on the left, then t can reasonably be assumed that s statstcally ndependent of gven Y 0. On the other hand, f the PMF of the system state s defned condtonal on the test outcomes (the socalled dagnostc model on the rght, then and Y 2 become statstcally dependent. (Ths becomes evdent from applyng the lnk reversal algorthm to the causal network. a Causal model: b Dagnostc model: Y 0 Y 0 Fgure 9. Causal versus dagnostc modelng n the ebn. Y 0 s system state and and are two outcomes of tests on Y 0. Addtonally, the complexty of the ebn and the resultng rbn can often be reduced by ncludng varables explctly as separate nodes n the ebn. Consder the example of a set of 5 equ-correlated random varables. In a drect representaton of these varables the last varable would have all other varables as parents (smlar to the rbn n Fgure 7a. However, equ-correlaton s typcally caused by a common nfluencng factor. If such a factor s explctly ncluded n the model as a common parent node, then the dependence among the varables can be fully represented by a smple network structure, smlar to the rbn n Fgure 7b. Ths structure can represent equ-correlaton among varables -Y 5 through the common factor Y 0. Straub & Der Kureghan (200a 27/32

28 c Mantanng causalty n the rbn To reduce the complexty of the rbn and to smplfy the SRM calculatons, t s often benefcal to mantan causalty n the rbn, even though ths mght requre dscretzng addtonal varables. An example s shown n Fgure 0a, n whch - 5 represent random varables nfluencng the system/component performance Y 0. Dscrete varables Y a and Y 2a are outcomes of tests performed on and 2. If the contnuous varables and 2 are dscretzed, as shown n Fgure 0b, the resultng rbn mantans causalty. If the states of Y 0 are defned by sngle lmt-state functons, t s then suffcent to calculate p( y 0 through component SRM calculatons to obtan the rbn. In the orgnal model, Fgure 0a, t s necessary to compute the jont PMF of Y 0, Y a and Y 2a wll necesstate system SRM calculatons., whch a Orgnal model: ebn rbn a a a a Y 0 Y 0 b Dscretzng and 2 : ebn rbn a a a a a 2a b 2b Y 0 Y 0 Fgure 0. By mantanng causalty n the rbn, system SRM calculatons mght be avoded. Straub & Der Kureghan (200a 28/32

29 d Dvorcng of varables In some cases t s possble to reduce the number of parents to a varable by a dvorcng strategy (Jensen and Nelsen, Ths strategy s useful when the jont nfluence of a number of parent varables can be represented by a sngle ntermedate varable. As an example consder a case where the structural performance Y 0 s a functon of four dscrete varables -Y 4 and a number of contnuous varables. Suppose the falure state s descrbed by the lmt-state functon g y a y a y a y h(, where x h( x s any functon and a - a 3 are determnstc coeffcents. In ths case, the number of parents to Y 0 can be reduced to three by ntroducng the new varable Y5 a a2y2 ay 3 3, as llustrated n Fgure. dvorcng Y 54 Y 5 Y 54 Y 0 Y 0 Fgure. Illustraton of the dvorcng strategy (here, parents Y 4 and are dvorced from the remanng dscrete parents of Y 0. As a fnal remark, we note that the ebn approach explots condtonal ndependence among random varables n a probablstc model. Therefore, the approach does not present advantages for modelng problems that do not exhbt such ndependence. Consder a dscretzed random feld represented by a vector of dependent varables. Such a vector s not Markovan and cannot be represented by a herarchcal BN structure as n Fgure 7b. If observatons are avalable at n locatons modeled through n dscrete varables -Y n, one varable n the resultng rbn wll have all other n varables as parents. Such a model can become computatonally mpractcal even for a moderate number of observed varables. (Remember that the condtonal probablty table of the last node wll have m n entres, wth m beng the number of dscrete states of each varable Straub & Der Kureghan (200a 29/32

30 Y. Ths fundamental nablty to effcently model complex dependence structures among dscrete random varables that are not characterzed by causal relatons s the man lmtaton of the ebn approach. Although deas such as usng prncpal component analyss to reduce the dependence structure have been explored (Straub et al., 2008, further work s requred to address ths problem wthn the ebn framework. Luckly, most models used n cvl engneerng nvolve causal relatons among the varables. 5 Summary and Conclusons Ths paper presents the dea of usng structural relablty methods (SRMs for solvng enhanced Bayesan networks (ebns, whch nclude both contnuous and dscrete random varables whose states can be descrbed by sets of lmt state functons. The proposed approach employs a node elmnaton algorthm, whch removes the contnuous nodes to arrve at a reduced BN, the rbn, whch only has dscrete nodes and can be solved by exstng exact nference algorthms. SRMs are used to compute the condtonal probablty tables of the rbn as component or system relablty problems. To evaluate the number of SRM calculatons and the complexty of the resultng rbn the concept of Markov envelopes s ntroduced, and t s shown that the computatonal requrements are determned by the szes of the Markov envelopes. A number of modelng strateges are descrbed to reduce the szes of the Markov envelopes, and thereby reduce the number of requred SRM calculatons and the maxmum clque sze of the rbn. These strateges nclude selectvely dscretzng contnuous varables, mantanng causal relatons between the nodes, and dvorcng parent nodes. Addtonally, the problems of enterng evdence on contnuous varables and nference on contnuous varables are addressed. An alternatve for dealng wth the ebn that has not been consdered here s to perform approxmate nference drectly on the ebn, e.g., by means of MCMC. The potental advantage of such an approach les n the ablty to handle more general forms of dependence among the varables, ncludng those arsng from random felds. However, t s mportant to realze that approxmate nference has ts own lmtatons, n partcular when the nterest s n computng probabltes of rare events descrbed by lmt-state functons, and when the nterest s n near-real-tme nference and decson analyss. Straub & Der Kureghan (200a 30/32

31 The theoretcal framework presented n ths paper has many potental applcatons, especally for decson support n near-real-tme under uncertan and evolvng nformaton. Example areas of such applcaton nclude early warnng systems, emergency response and recovery plannng for natural hazards, and the optmzaton of nspecton, montorng and repar actons n nfrastructure systems. The companon paper presents two such applcatons related to structural and nfrastructure systems. References Beck, J. L., and S.-K. Au (2002, Bayesan Updatng of Structural Models and Relablty usng Markov Chan Monte Carlo Smulaton, Journal of Engneerng Mechancs, Trans. ASCE, 28(4, Bens, M. T., D. Straub, P. Frs-Hansen, and A. Der Kureghan (2009, Modelng nfrastructure system performance usng BN, n Proc. ICOSSAR'09, Osaka, Japan. Bobbo, A., L. Portnale, M. Mnchno, and E. Cancamerla (200, Improvng the analyss of dependable systems by mappng fault trees nto Bayesan networks, Relablty Engneerng & System Safety, 7(3, Cooper, G. F. (990, The computatonal complexty of probablstc nference usng Bayesan belef networks, Artfcal Intellgence, 42(2-3, Dechter, R. (996, Bucket elmnaton: A unfyng framework for probablstc nference, n Twelthth Conference on Uncertanty n Artfcal Intellgence, Portland, Oregon. Der Kureghan, A. (2005, Frst- and second-order relablty methods. Chapter 4, n Engneerng desgn relablty handbook, edted by E. Nkolads et al., CRC Press, Boca Raton, FL. Dtlevsen, O., and H. O. Madsen (996, Structural Relablty Methods, John Wley & Sons. Ellngwood, B. R. (2006, Structural safety specal ssue: General-purpose software for structural relablty analyss, Structural Safety, 28(-2, 2. Faber, M. H., I. B. Kroon, E. Kragh, D. Bayly, and P. Decosemaeker (2002, Rsk Assessment of Decommssonng Optons Usng Bayesan Networks, Journal of Offshore Mechancs and Arctc Engneerng, 24(4, Frs-Hansen, A. (2000, Bayesan Networks as a Decson Support Tool n Marne Applcatons, PhD thess, DTU, Lyngby, Denmark. Frs-Hansen, P. (2004, Structurng of complex systems usng Bayesan network., n Proceedngs Workshop on Relablty Analyss of Complex Systems, Techncal Unversty of Denmark, Lyngby. Glks, W. R., S. Rchardson, and D. J. Spegelhalter (996, Markov chan Monte Carlo n practce, Chapman & Hall, London. Grêt-Regamey, A., and D. Straub (2006, Spatally explct avalanche rsk assessment lnkng Bayesan networks to a GIS, Natural Hazards and Earth System Scences, 6(6, Jensen, F. V., and T. D. Nelsen (2007, Bayesan Networks and Decson Graphs. Informaton Scence and Statstcs, Sprnger, New York, NY. Langseth, H., T. D. Nelsen, R. Rumí, and A. Salmerón (2009, Inference n hybrd Bayesan networks, Relablty Engneerng and System Safety, 94(0, Straub & Der Kureghan (200a 3/32