Enabling a Powerful Marine and Offshore Decision-Support Solution Through Bayesian Network Technique

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1 Rsk Analyss, Vol. 26, No. 3, 2006 DOI: /j x Enablng a Powerful Marne and Offshore Decson-Support Soluton Through Bayesan Network Technque A. G. Eleye-Datubo, 1 A. Wall, 1 A. Saajed, 1 and J. Wang 1 A powerful practcal soluton s by far the most desred output when makng decsons under the realm of uncertanty on any safety-crtcal marne or offshore unts and ther systems. Wth data and nformaton typcally beng obtaned ncrementally, adoptng Bayesan network (BN) s shown to realstcally deal wth the random uncertantes whle at the same tme makng rsk assessments easer to buld and to check. A well-matched methodology s proposed to formalze the reasonng n whch the focal mechansm of nference processng reles on the sound Bayes s rule/theorem that permts the logc. Expandng one or more nfluencng nodal parameters wth decson and utlty node(s) also yelds an nfluence dagram (ID). BN and ID feasblty s shown n a marne evacuaton scenaro and that of authorzed vessels to floatng, producton, storage, and offloadng collson, developed va a commercal computer tool. Senstvty analyss and valdaton of the produced results are also presented. 1. INTRODUCTION If all the nformaton that could be known about a martme hazardous event/stuaton were obtanable for ts rsk assessment, then the results of such studes that are accurately carred out would not be subject to uncertanty. Instead, data and nformaton are typcally obtaned ncrementally. Thus, the nherent uncertanty can be due to mperfect understandng of the doman, ncomplete knowledge of the state of the doman at the tme where a gven task s to be performed, randomness n the mechansms governng the behavor of the doman, or a combnaton of these. It s necessary then to model the assessment doman such that the probablstc measure of each event becomes more relable n lght of the new nformaton beng receved. In vew of ths, the doman that s represented can be put out n an ntutve vsual 1 Marne, Offshore & Transport Research Group, School of Engneerng, Lverpool John Moores Unversty, Byrom Street, Lverpool, L3 3AF, UK. Address correspondence to Jn Wang, Lverpool John Moores Unversty; tel: ; fax: ; j.wang@ ljmu.ac.uk. format as a Bayesan network (BN) model. The BN reasonng system can be vewed as the generalzaton of prepostonal logc and resoluton theorem provng that ncorporates the treatment of uncertanty for the structure of the complex argument. Probablty and Bayes s theory ensure that nferences based on the network are sound. As essental n a rsk-based marne communty, reasonng wth ncomplete knowledge s one of the fundamental features of human ntellgence. Competent expert and engneerng judgment (to compensate for any lack of mature data) ncorporated n a BN can ad n provdng ts sold knowledge base. The generc nature of ths technque means that t can be developed further and appled wdely n marne and offshore applcatons. Wth ths phlosophy n a logcal framework, adoptng BN to formalze reasonng about system dependablty wll make assessments easer to buld, check, and certanly update. The analogy of BN models can be further expanded/transformed to output nfluence dagrams (IDs) that are hghly ntutve n the decson-makng process. Such dagrams ad the vsblty of a large number of nteractng ssues and ther effects on the /06/ $22.00/1 C 2006 Socety for Rsk Analyss

2 696 Eleye-Datubo et al. decson. They can also offer the beneft of a robust practcal soluton that s requred for acheved safety at an affordable cost. Hence, the fnal scheme of the BN can gve a model n whch reasonng s justfed, whle t enables a powerful marne decson-support soluton that s easy to use, flexble, and approprate for the assessment task. 2. LITERATURE BRIEFING ON BN Untl 20 years ago, the ssue of orderng possble belefs, both for belef revson and for acton selecton, was seen as ncreasngly mportant and problematc, and at the same tme, dramatc new developments n computatonal probablty and decson theory drectly addressed perceved shortcomngs. The key development (Pearl, 1988) was the dscovery that a relatonshp could be establshed between a well-defned noton of condtonal ndependence n probablty theory and the absence of arcs n a drected acyclc graph (DAG). Ths relatonshp made t possble to express much of the structural nformaton n a doman ndependent of the detaled numerc nformaton, n a way that both smplfes knowledge acquston and reduces the computatonal complexty of reasonng. The resultng graphc models have come to be known as BNs. BNs are at the cuttng edge of expert systems research and development. Unlke the tradtonal rulebased approach to expert systems, they are able to replcate the essental features of plausble reasonng (reasonng under condtons of uncertanty) and combne the advantages of an ntutve vsual representaton wth a consstent, effcent, and mathematcal bass n Bayesan probablty. Crtcally, they are capable of retractng belef n a partcular case when the bass of that belef s explaned away by new evdence. Because of the development of propagaton algorthms (Laurtzen & Spegelhalter, 1988; Pearl, 1988; Russell & Norvg, 2003), followed by avalablty of easyto-use commercal software and growng number of creatve applcatons (Jensen, 1993; SERENE Consortum, 1999), BN has caught the sudden nterest of research n dfferent research felds snce the early 1990s. Perhaps the greatest testament to the usefulness of Bayesan problem-solvng technques s the wealth of practcal applcatons that have been developed snce then n areas of ntellgent decson, safety assessment, nformaton flterng, autonomous vehcle navgaton, weapons schedulng, medcal dagnoss, pattern recognton, and computer network dagnoss (Heckerman et al., 1995). Snce most real-lfe problems nvolve nherently uncertan relatonshps, BN s a technology wth huge potental for applcaton across many domans. IDs, whch further extend the noton of BNs by ncludng decson nodes and utlty nodes, have been used n human relablty assessment (Humphres, 1995) and decson makng on exploson protecton offshore (Bolsover & Wheeler, 1999). A good reference work for the computatonal method underlyng the mplementaton of them n Hugn s descrbed n (Jensen et al., 1994). The Hugn software (Jensen, 1993) enables a powerful rsk assessment soluton that s easy to use, flexble, and approprate for use on marne and offshore applcatons. Other renowned program packages for BN buldng and nfluencng nclude MSBNx (Kade et al., 2001), created at Mcrosoft Research, and Netca (Netca, 2002), the commercal program developed by Norsys Software Corp. 3. SEMANTICS OF A BN Fundamental to the dea of BNs s the concept of modularty, whereby a complex system s bult by combnng smpler parts of components that are related n a causal manner. A BN provdes factorzed representaton of a probablty model that explctly captures much of the structure typcal n human-engneered models. More generally, a BN s a DAG that encodes a condtonal probablty dstrbuton (CPD) at ts nodes on the bass of arcs receved. Therefore, by defnton: BN = DAG encoded wth CPD. The graphcal structure of a BN (.e., the DAG) depcts a qualtatve llustraton of the nteractons among the set of random (.e., chance) varables, such as hazardous events, that t models. Numercally, a BN represents the jont probablty dstrbuton (JPD) among the modeled varables. Ths dstrbuton s descrbed effcently, explorng probablstc ndependences among the modeled varables. Each node s descrbed by a probablty dstrbuton (PD) condtonal on ts drect predecessors that has ts values entered nto a condtonal probablty table (CPT),.e., a matrx of condtonal probabltes, assocated wth the node. The encoded nodes wth no predecessors are descrbed by pror PDs. Those wth predecessors are descrbed by posteror PDs.

3 Marne and Offshore Decson-Support Soluton Usng BN BAYESIAN INFERENCE MECHANISM Bayesan nference s a process by whch observatons of a real-world stuaton are used to update the random uncertanty about one or more varables characterzng aspects of that stuaton. It reles on the use of Bayes s rule/theorem (Bayes, 1763) as ts rule of nference, defnng the manner n whch uncertantes ought to change n lght of newly made observatons. Ths subjectve probablty theory s only part of the Bayesan nference mechansm. Together wth the applcable results of such probablty concepts as the product and sum rules, the concept of condtonal ndependence (Pearl, 1988), dependency separated or d-separated (Pearl, 1988), the technques of margnalzaton (Velldo & Lsboa, 2001), and the pattern of nference (Wellman & Henron, 1993; Laurtzen & Spegelhalter, 1988; Pearl, 1988), t provdes the basc tool for both Bayesan belef updatng and for treatng probablty as logc. In order to apply these tools, the pror probabltes and the lkelhood probabltes must be obtaned Bayes s Theorem/Rule In order to make probablty statements about the model parameters, the analyss must begn wth provdng ntal or pror probablty estmates for specfc outcomes or events of nterest. Then from sources such as a specal report, a database, a case study, etc., some addtonal nformaton (.e., data or evdence) about the event, or an entrely new event(s), s obtaned. In lght of ths new nformaton provdng new data belef, t s desrable to mprove the state of knowledge, and thus the pror probablty values are updated by calculatng revsed probabltes, referred to as the posteror probabltes (these probabltes provde the bass for acton). Bayes s theorem provdes a means for makng these probablty calculatons. Essentally, t s a relatonshp between condtonal and margnal probabltes, and s gven for two events, A and B, by Equaton (1). P(B A)P(A) P(A B) =. (1) P(B) Each term n Bayes s theorem has a conventonal name. The term P(A) s called the pror probablty of A. It s pror n the sense that t precedes any nformaton about B and ths s what causes all the arguments. P(A) s also the margnal (total) probablty of A. The term P(A B) s called the posteror probablty of A, gven B. It s posteror n the sense that t s derved from or entaled by the specfed value of B. The term P(B A), for a specfc value of B, s called the lkelhood functon for A, gven B and can also be wrtten as L(A B). The term P(B) sthepror or margnal (total) probablty of B, but also one that provdes evdence of nterest for the probablty update of A. Its nverse s usually regarded as a normalzng constant, α. Wth ths termnology, the theorem may be paraphrased as lkelhood pror posteror = P(A B) evdence = αl(a B)P(A). (2) Generally, for an event B wth states {b 1,..., b m }, the posteror probablty on the event A can be computed from the Bayes s rule as P(A b 1,...,b m ) = P(b 1,...,b m A)P(A). (3) P(b 1,...,b m ) The process of Bayes s theorem s repeated every tme new or addtonal nformaton becomes avalable, so that as Lndley (1970) puts t, today s posteror probablty s tomorrow s pror. Thus, as the number of peces of evdence ncreases, the dependence of the posteror on the orgnal estmated pror decreases The Lkelhood Functon The lkelhood prncple (Fsher, 1922; Edwards, 1992) states that all the relevant nformaton n the model s contaned n the lkelhood functon (whch s of fundamental mportance n the theory of Bayesan nference). Lkelhood s a soltary term used to represent such a functon and s one of several nformal synonyms for probablty ; so sometmes, P(B A) s called the lkelhood of A, gven B, and s denoted by L(A B). The reason for ths s that f, for example, a 1,..., a n are possble states of event A wth an effect on the event B n whch b s known, then P(b a ) s a measure of how lkely t s that a s the cause. Moreover, ths s a smple, compellng concept that has a host of good statstcal propertes and can be derved from the reasonng logc as well as by expert judgment. 5. STRUCTURAL EFFECTS ON THE INFERENCE PROCESSING One of the best features of BNs s that one can ncorporate new node(s) as the data become avalable. Thus, t follows that one effect can be a cause of

4 698 Eleye-Datubo et al. a new/another node and a cause can also be the effect of a new/another node. Owng to ths addtonal capablty of a BN model, t can consttute a descrpton of the probablstc relatonshps among the system s varables that amount to a factorzaton of the jont dstrbuton of all varables nto a seres of margnal and condtonal dstrbutons. Evdence propagaton may take place va a message-postng scheme Jont Probablty Dstrbuton A probablstc model may consst of a set of varables X = {X 1,X 2,..., X n }, whch explots condtonal ndependence to represent the JPD over X havng the product form (Pearl, 1988): P(x 1,...,x n ) = P(x 1 parent(x 1 )) P(x 2 parent(x 2 ))...P(x n parent(x n )) n = P(x parents(x )). (4) =1 P(x 1,x 2,..., x n ) gves the JPD and, lke the CPD, t s a table of values where one entry s made for each possble combnaton of values that ts varables can jontly take. The JPD for a problem captures the probablty nformaton of every possble combnaton of a set of varables, and ther states. Once a JPD has been defned for a problem, then t s possble, usng t along wth the axoms of probablty, to answer any probablstc query regardng any of the varables. Ths ncludes ther value gven addtonal evdence, that s, ther posteror probabltes, although the space, and consequently, tme complexty requred n representng and manpulatng the JPD s exponental n the number of varables consdered (D Ambroso, 1999). For example, the JPD requred to represent a system wth 20 bnary values would have 2 20 (1,048,576) values. Ths causes a problem n the elctaton, storage, and manpulaton of these values, thus makng the use of JPDs unfeasble for any practcal use. Fortunately, when modelng most real systems, advantage s taken of any nherent structure the system has by modelng the system as a graph (D Ambroso, 1999). In the general case, a JPD over a set of varables, X = {X 1,X 2,..., X n }, can be defned recursvely usng the product rule (Equaton (5)): P(X 1, X 2,...,X n ) = P(X 1 X 2,...,X n )P(X 2,...,X n ) = P(X 1 X 2,...,X n )P(X 2 X 3,...,X n )P(X 3,...,X n ) = P(X 1 X 2,...,X n ) P(X 2 X 3,...,X n ) P(X n 1 X n )P(X n ). (5) Ths factorzaton property of JPDs s referred to as the chan rule of probabltes and s one that allows any orderng of varables n the factorzaton. Such a rule s especally sgnfcant for BNs, because t provdes a means of calculatng the full JPD from condtonal probabltes, whch s what a BN stores. For example, the JPD for three events, A, B, and C, can be expressed more compactly as: P(A B, C)P(B, C) = P(A, B, C) = P(B A, C)P(A, C). (6) Then, n applyng Equaton (5), Bayes s theorem specfes the probablty of an event A, gven the condton that an event B and an event C both occur (B A C) as: P(A B, C) = 5.2. Belef Probablty Update P(B A, C)P(A C). (7) P(B C) Evdence s new nformaton about a random varable that causes a change about ts PD. Newly avalable evdence s brought about when a partcular state of an event happens. The effect of such new evdence wll certanly propagate throughout the network and thereby cause the posteror probabltes of other events to teratvely be recalculated. Ths s achevable by message postng along the edges (Pearl, 1988). Therefore, ntroducng the noton of evdence s mperatve n the reasonng wth BN. Nonetheless, t s worth notng that the real power and generalzaton of BN s that entered evdence propagates n both drectons, even though the graph s drected. Suppose there s an nterest n a gven event C (referred to as the query varable) havng a jont probablty P(c), over C. Before any evdence becomes avalable, the propagaton process conssts of calculatng the margnal probabltes P(C = c ), or smple P(c ), for each C. Now, suppose some evdence has become avalable to the event C. In ths stuaton, the propagaton process conssts of calculatng the condtonal probabltes P(C = c ε = e), or smple P(c e),

5 Marne and Offshore Decson-Support Soluton Usng BN 699 where ε s a set of evdental nodes wth known values ε = e. The newly avalable evdence, ε, can be decomposed nto two subsets: ε +, the subset of ε that can be accessed from C though ts parents (top-down),.e., propagates n the drecton of the arcs. ε, the subset of ε that can be accessed from C though ts chldren (bottom-up),.e., propagates aganst the drecton of the arcs. For the probablty of C = c, gven that e = e + for a parent and e = e for a chld: P(c e) = P ( c e, e + ) P ( e = c, e + ) ( P c e + ) P ( e e + ). (8) Snce C d-separates ε from ε + (.e., ε ε +, where stands for d-separaton), condtonal ndependence can be used to smplfy the frst term n the numerator and then 1/P(e e + ) can be treated as a normalzng constant, α, so that: P(c e) = α P ( e ) ( ) c P c e +. (9) Accordng to the Bayes s theorem conventonal nterpretaton (Equaton (2)), posteror s pror scaled by lkelhood and normalzed by evdence (so (posterors) = 1), thus Equaton (9) can be rewrtten as where P(c e) = αλ (c )π (c ), (10) λ (c ) represents P(e c ), a message passed onto c as lkelhood evdence; and π (c ) represents P(c e + ), a message passed onto c as pror evdence. To compute the functons λ (c ) and π (c ), suppose a typcal node C has parents B = {B 1,..., B m } and chldren A = {A 1,..., A n } (see Fg. 1). The evdence ε + can be parttoned nto m dsjont components, one for each parent of C : ε + = { ε + B 1 C,...,ε + B m C }, (11) where the evdence ε + B j C s the subset of ε + contaned n the B j -sde of the lnk B j C. Smlarly, the evdence ε can be parttoned nto n dsjont components, that s: ε { ε A 1 C,...,ε } A n C, (12) where the evdence ε B j C s the subset of ε contaned n the A j -sde of the lnk A j C. Then, gven an nstantaton of b = {b 1,..., b m } of the parents of C, π (c ) can be computed (.e., topdown propagaton) va a recursve soluton (Pearl, 1986; Castllo et al., 1997). Lkewse, gven an nstantaton of a = {a 1,..., a n } of the chldren of C, λ (c ) can be computed (.e., bottom-up propagaton). The CPTs of the events never change by enterng new evdence; only the new-fangled/belef probablty n each of ts possble states s determned by the belef probablty n the states of the nodes to whch t s drectly connected. The algorthm smultaneously updates belef for all the nodes, causng them to become posteror probabltes, untl the network reaches equlbrum. In other words, the JPD of the varables changes each tme new nformaton s learnt about the observable varables. Such calculatons for the propagaton of probabltes n a BN are usually Fg. 1. Evdence propagaton va message postng.

6 718 Eleye-Datubo et al. Fg. 36. Collson- FPSO mpact probablty set to 100% n shuttle tanker loss whle full. Fg. 37. Addton of evdence and resultng events from the Collson- FPSO stuaton. probablty value of each hghlghted node s ncreased (note from Fg. 38). Those that have sgnfcantly ncreased by a wder margn are especally the Spll/Release node and the Human Injury node. The Ignton node has remaned the same n probablty value, snce t s only a pece of evdence for exploson and fre outbreak, and not a resultng ncdent of the collson to the FPSO n ths scenaro. Wth the Exploson node set to a falure of 100% blast durng a 100% mpact on collson wth the FPSO, the probablty of 96.66% ndcates a hgh amount n certanty for structural damage to happen (Fg. 39). The same can be sad for the Human Injury node, whch now has a probablty value of 84.26%. As such, a great deal of attenton wll have to be pad to ncreasng safety for these represented nodes. Thus, the rsk analyst and decsonmakers mght fnd t approprate to consder modelng out an ID for exploson. Other such peces of typcal evdence as the human element (wth states such as error and nterventon ), weather condton (wth sea states of calm, harsh, adverse, and severe ) (see Fg. 40), electrcal/electronc aspects, etc., can be made nto new nodes and added to dversfy the range of the BN applcablty n ths scenaro. The scenaro settngs for ths case study can enable a domnant decson n a marne and offshore rsk assessment study. Nonetheless, as extensons to the scenaro network may le n the dscrepancy of the rsk analyst and decsonmakers, the author has chosen to keep the network to an acceptable sze. It s best, however, that the rsk analyst s aware, n tacklng a scenaro effectvely, of beng twsted n the complextes that very large BNs brng.

7 Marne and Offshore Decson-Support Soluton Usng BN 719 Fg. 38. Stuaton for resultng events from Collson- FPSO mpact probablty set to 100%. Fg. 39. Stuaton for resultng events wth Collson- FPSO and exploson falure set to 100%. Fg. 40. Some added typcal evdence for a shuttle tanker loss of poston.

8 720 Eleye-Datubo et al. 9. BENEFITS AND LIMITATIONS OF BNs In BNs, each representaton possesses partcular advantages and dsadvantages that make t more, or less, sutable for ts ntended purpose. These have been recognzed and thus outlned n Sectons 9.1 and 9.2, respectvely Strengths of BNs The Bayesan framework offers several advantages over alternatve modelng approaches. The most mportant of these advantages are: It provdes ntutve vsual representaton wth a sound mathematcal bass n Bayesan probablty that translates nto a genune cause and effect relatonshp. Beng probablstc n ts approach, t facltates a meanngful communcaton of uncertanty. It s consstent wth the rsk assessment paradgm, and allows decsons to be made based on expected values. It s capable of combnng dverse data, expert judgment, and emprcal data. By ncorporatng expert judgment, the method s not paralyzed by a lack of observatonal data. It allows easy updatng of predcton and nference n a statstcally rgorous manner when observatons of model varables are made. Deletng or addng new nformaton does not also requre the whole network to be revsed. The assessment endponts are chosen so that they are of vtal nterest to stakeholders and decsonmakers, and can be easly conceved n terms of utlty for use n formal decson analyss. These partcular advantages offered by BN make t very useful n stuatons where uncertanty s unavodable Bayesan methods provde a mechansm to model the uncertanty. Thus, such methods can also be used where normal optmzaton and decsonmakng technques are dffcult to apply Dffcultes of Usng BNs In spte of ther remarkable power and potental to address nferental processes, there are some nherent lmtatons and labltes to BNs. These drawbacks nclude the followng: They cannot easly ncorporate unobserved varables, owng to the fact that the sze of the nternal CPT for a chld node can very quckly become qute large. There s computatonal complexty/dffculty (fllng n of detals of numercal recpe, computer tme, convergence montorng), whch s exponental n the number of nodes. These complex models wth large numbers of parameters, whch are often referred to as nonparametrc (NP), become NP-hard n complexty as they approach general multply-connected networks. Lkelhood functons are not always solvable analytcally (rather, heurstcs are extensvely used n practce). The complexty of nference s usually assocated wth large probablstc dependences recorded durng nference. However, a large model s preferable to a smaller one only f t provdes a suffcently large mprovement of ft to offset the penalty for ts addtonal complexty. 10. CONCLUDING REMARKS A BN could be used to model the components that affect rsk and how they nteract. Besdes, the graphcal nature of a BN makes the model ntutve for users to understand. The process of performng Bayesan updatng nvolves selectng a pror PD, calculatng the normalzng constant, formulatng the lkelhood functon, and then calculatng the posteror PD. The lkelhood functon ncorporates the objectve nformaton, whle the pror dstrbuton can nclude subjectve nformaton known about the dstrbutons of the model parameters. Therefore, the posteror dstrbuton ncorporates both the objectve and the subjectve nformaton nto the dstrbutons of the model parameters. Hence, BNs are well suted for modelng martme safety-crtcal systems predcton and rsk analyss. The methodology that has been proposed uses BNs to combne evdence from dfferent nformaton sources for a quanttatve assessment of a generc scenaro. A program tool, such as Hugn, can allow the model user to adjust the probabltes of states of nodes based on observed nformaton. The software can also propagate ths change through the network, and update the condtonal probabltes at each node based on the new nformaton. As shown n both the shp evacuaton and the authorzed vessels to FPSO nstallaton collson scenaro case studes, by usng BNs and a tool such as Hugn, t s possble to show all the mplcatons and results of a complex Bayesan argument based on the underlyng Bayes s theorem. Ths theorem s the

9 Marne and Offshore Decson-Support Soluton Usng BN 721 fundamental prncple governng the process of logcal Bayesan nference that determnes what conclusons can be made wth a degree of confdence based on the totalty of relevant evdence avalable. The probablstc predctons gve stakeholders and decsonmakers a realstc apprasal of the chances of achevng desred outcomes. The results from the case studes, as well as other renowned state-of-theart research work, do ndcate that BNs gve a sound and transparent approach to modelng marne operatonal rsk. Thus, BN s an ntegratve model that can be used effectvely wthn the exstng decsonmakng process. BNs can also be expanded to form IDs, whch permts rapd development of a practcal decson model. The value of IDs as a communcaton tool has been confrmed. Ther use s hghly ntutve and they provde a compact alternatve to decson trees such that, durng revew, persons who are not rsk analysts are able to nterpret the dagrams and propose mprovements to the decson model. ACKNOWLEDGMENTS Ths research s partally supported by the UK Engneerng and Physcal Scences Research Councl (EPSRC) through Grant No. GR/S85504 and the Insttute of Marne Engneerng, Scence and Technology (IMarEST) through a Stanley Gray Fellowshp. REFERENCES Bayes, T. (1763). An essay towards solvng a problem n the doctrne of chances. Phlosophcal Transactons of the Royal Socety of London, 53, Bolsover, A. J., & Wheeler, M. (1999). Decson-makng to treat an exploson hazard. In Proceedngs of the 8th Annual Conference on Safety on Offshore Installatons, November. London, UK: ERA Technology. Castllo, E., Gutérrez, J. M., & Had, A. S. (1997). Expert Systems and Probablstc Network Models. New York: Sprnger-Verlag. D Ambroso, B. (1999). Inference n Bayesan networks. 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