Comparing Alternative Methods for Inference in Multiply Sectioned Bayesian Networks

Transcription

1 From: FLAIRS-02 Proeedings. Copyright 2002, AAAI ( All rights reserved. Compring Alterntive Methods for Inferene in Multiply Setioned Byesin Networks Y. Xing Dept. Computing & Informtion Siene University of uelph, Ontrio, Cnd Astrt Multiply setioned Byesin networks (MSBNs) provide one frmework for gents to estimte the stte of domin. Existing methods for multi-gent inferene in MSBNs re sed on linked juntion forests (LJFs). The methods re extensions of messge pssing in juntion trees for inferene in singlegent Byesin networks (BNs). We onsider extending other inferene methods in single-gent BNs to multi-gent inferene in MSBNs. In prtiulr, we onsider distriuted versions of loop utset onditioning nd forwrd smpling. They re ompred with the LJF method in terms of off-line ompiltion, inter-gent messges during ommunition, onsistent lol inferene, nd preservtion of gent privy. Introdution Consider lrge unertin prolem domin populted y set of gents. The gents n e hrged with mny possile tsks depending on the nture of the pplition. One ommon tsk is to estimte wht is the true stte of the domin so tht they n t ordingly. Multiply setioned Byesin networks (MSBNs) (Xing 1996) provide one frmework to ondut suh tsk. An MSBN onsists of set of interrelted Byesin sunets eh of whih enodes n gent s knowledge on sudomin. Proilisti inferene n e performed in distriuted fshion while nswers to queries re ext with respet to proility theory. Existing methods for multi-gent inferene in MSBNs re extensions of lss of methods for inferene in singlegent Byesin networks (BNs): messge pssing in juntion trees (Jensen, Luritzen, & Olesen 1990; Shfer 1996; Mdsen & Jensen 1998). The linked juntion forest (LJF) method (Xing 1996) ompiles the sunet t eh gent into juntion tree (JT). Inter-gent messge pssing is performed through linkge tree etween pir of djent gents. The distriuted Shfer-Shenoy propgtion nd distriuted lzy propgtion (Xing & Jensen 1999) ompile the sunet t n gent into set of JTs, one for eh djent gent. The JT is then used for messge pssing with the gent. Inferene methods, other thn messge pssing in JTs, hve een proposed for resoning in single-gent BNs. In Copyright 2002, Amerin Assoition for Artifiil Intelligene ( All rights reserved. this work, we onsider extending two of them for multigent inferene in MSBNs: loop utset onditioning nd forwrd smpling. Weompre their performne with the LJF method with fous on gent utonomy nd gent privy. Setion introdues MSBNs nd the LJF inferene method. Setion presents distriuted loop utset onditioning with its properties nlyzed. Setion nlyzes distriuted forwrd smpling. MSBNs nd inferene with LJFs An MSBN (Xing 1996) M is olletion of Byesin sunets tht together defines BN. To ensure ext inferene, sunets re required to stisfy ertin onditions. First we introdue terminologies to desrie these onditions. Let i =(V i,e i )(i =0, 1) e two grphs (direted or undireted). 0 nd 1 re sid to e grph-onsistent if the sugrphs of 0 nd 1 spnned y V 0 V 1 re identil. iven onsistent grphs i =(V i,e i )(i =0, 1), the grph =(V 0 V 1,E 0 E 1 ) is lled the union of 0 nd 1, denoted y = 0 1.iven grph =(V,E), V 0 nd V 1 suh tht V 0 V 1 = V nd V 0 V 1, nd sugrphs i of spnned y V i (i =0, 1), wesy tht is setioned into 0 nd 1. The sunets in n MSBN must stisfy hypertree ondition: Definition 1 Let = (V,E) e onneted grph setioned into sugrphs { i = (V i,e i )}. Let the i s e orgnized s onneted tree Ψ where eh node is leled y i nd eh link etween k nd m is leled y the interfe V k V m suh tht for eh i nd j, V i V j is ontined in eh sugrph on the pth etween i nd j in Ψ. Then Ψ is hypertree over. Eh i is hypernode nd eh interfe is hyperlink. The interfe etween sunets in n MSBN must form d-sepset: Definition 2 Let e direted grph suh tht hypertree over exists. A node x ontined in more thn one sugrph with its prents π(x) in is d-sepnode if there exists one sugrph tht ontins π(x). Aninterfe I is d-sepset if every x I is d-sepnode. In multi-gent system, d-sepnode is shred y more thn one gent nd is lled puli node. A node internl to single gent is lled privte node. The struture of 534 FLAIRS 2002

2 n MSBN is multiply setioned DA (MSDA) with hypertree orgniztion: Definition 3 A hypertree MSDA = i i, where eh i is DA, is onneted DA suh tht (1) there exists hypertree Ψ over, nd (2) eh hyperlink in Ψ is d-sepset. An MSBN is then defined s follows, where potentil over set of vriles is non-negtive distriution of t lest one positive prmeter, nd uniform potentil onsists of 1 s only. Definition 4 An MSBN M is triplet (V,,P). V = i V i is the domin where eh V i is set of vriles, lled sudomin. = i i ( hypertree MSDA) is the struture where nodes of eh DA i re leled y elements of V i. Let x e vrile nd π(x) e ll prents of x in. Foreh x, extly one of its ourrenes (in i ontining {x} π(x))isssigned P (x π(x)), nd eh ourrene in other DAs is ssigned uniform potentil. P = i P i is the joint proility distriution (jpd), where eh P i is the produt of the potentils ssoited with nodes in i.atriplet S i =(V i, i,p i ) is lled sunet of M. Two sunets S i nd S j re sid to e djent if i nd j re djent. An MSBN supports knowledge representtion of oopertive multi-gent system, where eh sunet enodes the prtil knowledge of n gent on the domin. We denote y A i the gent whose knowledge is enoded in sunet S i. Eh A i n only oserve lolly. One multi-gent MSBN is onstruted, gents my perform proilisti inferene y omputing the query P (x e), where x is ny vrile within the sudomin of n gent, nd e denotes the oservtions mde y ll gents. The key omputtion is to propgte the impt of oservtions to ll gents, whih we term s ommunition. Itisperformed y inter-gent messge pssing. Hene ommunition requires system-wide inter-gent messge pssing. Asgents re utonomous, onstnt system-wide messge pssing is either unville or undesirle. Most of the time, eh gent A i omputes the query P (x e i, e i ), where e i is the lol oservtions mde y A i nd e i is the oservtions mde y other gents up to the ltest ommunition. Note tht e i is not expliitly ville to A i nd only its impt is propgted to A i. This omputtion is termed lol inferene. Eh sunet my e multiply onneted. Multiple undireted pths my lso exist ross different sunets. To filitte ext inferene with messge pssing, the LJF method ompiles eh sunet into JT, lled lol JT, nd onverts eh d-sepset into JT, lled linkge tree. Lol inferene is performed y messge pssing in the lol JT s for single-gent BNs. Inter-gent messge pssing during ommunition is performed using the linkge trees. A ommunition requires pssing O(2 g) inter-gent messges, where g is the numer of gents. The size of the messge is liner on the numer of lusters in the linkge tree nd is exponentil on the rdinlity of the lrgest luster. During onstrution of n MSBN, whether puli node hs ny prent or hild in sunet (ut not how mny or wht they re) needs to e reveled. The onditionl proility distriution of puli node my e negotited mong relevnt gents to pool the diverse expertise together. Other thn these, onstrution of n MSBN, its ompiltion into n LJF, nd ommunition using LJF revel no dditionl informtion regrding the internls of eh gent. In the following setions, we extend two ommon inferene methods in single gent BNs to multi-gent MSBNs. Bsed on gent utonomy nd privy, we ssume (1) tht onstnt system-wide messge pssing is not ville, (2) tht the d-sepnodes re puli,ut ll other vriles in eh sunet nd their ssoited distriutions re privte to orresponding gent, nd (3) tht only peer-to-peer oordintion is ville. Loop utset onditioning Single-gent oriented Cutset onditioning (Perl 1988) onverts multiply onneted BN into multiple tree-strutured BNs nd performs the λ π messge pssing in eh of them. The key step of the method is to hypothetilly oserve set C of nodes, the loop utset, sotht ll loops re roken. The posterior distriution of vrile x given oservtion e 1 is omputed y P (x e 1 )= P (x,e 1 )P ( e 1 ), (1) where =( 1,..., n ) is ny onfigurtion of C. For eh, P (x,e 1 ) is otined y λ π in orresponding treestrutured BN. P ( e 1 ) n e lulted s P ( e 1 )=onst P (e 1 )P (), (2) where P (e 1 ) is otined y λ π. To otin P () in Eq. (2), n nestrl ordering of vriles is used to ompute the following ftors (Suermondt & Cooper 1991) whose produt is P (): P ( 1 ),P( 2 1 ),..., P ( n 1,..., n 1 ). (3) If the oservtions onsist of m vlues e 1,..., e m, the ove n e generlized to ompute in sequene to otin P (e 1 ),P(e 2,e 1 ),..., P (e m,e 1,..., e m 1 ) (4) P ( e 1 ),P( e 1,e 2 ),..., P ( e 1,..., e m ), (5) nd ompute in sequene P (x,e 1 ),P(x,e 1,e 2 ),..., P (x,e 1,..., e m ) (6) to otin P (x e 1 ),P(x e 1,e 2 ),..., P (x e 1,..., e m ). (7) Inferene in MSBNs y distriuted utset onditioning In n MSBN, loops n exist oth within lol DAs nd ross multiple lol DAs. Finding loop utset C tht n rek ll loops requires distriuted serh. It n e performed using vrition of TestAyliity (Xing 1998), FLAIRS

3 polynomil lgorithm for verifying yliity of the struture of n MSBN. A simple vrition works s follows: Strt y reursively mrking terminl nodes (with no more thn one djent node) in eh gent. After ll suh nodes hve een mrked, mrk til-to-til or hed-to-til node s in ny gent nd dd s to C y propgtion s there is no entrlized ontrol. At lest one loop is now ut open. Mrk reursively new terminl nodes until no more. Repet the ove until ll nodes re mrked. See (Xing 1998) for detils on multi-gent node mrking. In Figure 1, C = {f,h,j, o} is loop utset. Note tht f is privte to A 1, h is privte to A 2, o is privte to A 0, nd j is shred y A 2 nd A 0. f e d 1 n l m Figure 1: The struture of trivil MSBN. After suh utset is found, the multiply onneted DA union needs to e onverted to O(2 C ) tree-strutured DA unions, one for eh onfigurtion of C, nd distriuted λ π messge pssing needs to e performed in eh of them 1. We onsider the omputtion of sequenes (3) through (7) where the oservtions e 1,..., e m re those otined sine the lst ommunition. Note tht e 1,..., e m s well s vriles in C re distriuted mong gents. First, onsider the omputtion of sequene (3). This omputtion needs to e performed following n nestrl ordering of vriles in the domin. The ordering n e defined through nother simple vrition of TestAyliity (Xing 1998), desried s follows: Reursively mrk root nodes in ll g gents in multiple rounds. At most one node per gent n e mrked t ith round nd the node is given n index etween i nd i+g 1. The indexes then define n nestrl ordering. Figure 2 shows n exmple. The ville indexes for A 0 re 0, 3, 6,... nd those for A 1 re 1, 4, 7,... Inround 0 A 0, y ooperting with A 2, reognizes tht the puli node j is root nd indexes it with 0. A 1 indexes f with 1. A 2 is notified y A 0 with the index of j, ut otherwise it hs no root node to index. In round 1, A 0 indexes l with 3 nd A 1 indexes with 4. The proess ontinues until ll nodes re indexed. Using the nestrl ordering, sequene (3) n e otined y extending the method of Suermondt nd Cooper (Suermondt & Cooper 1991). For the exmple in Figure 2 nd the loop utset C = {f,h,j, o}, the sequene P (f),p(j f),p(o f,j),p(h f,j,o) n e omputed. As there is no entrlized ontrol, to oordinte the omputtion so tht it follows the nestrl ordering is quite involved: 1 Equivlently, one ould proess the sme DA union O(2 C ) times one for eh distint. Itismtter of implementtion. 0 o j k j k g h 2 i 0 f,1 1 l,3 h,26,4,4 j,0 j,0,7 m,21 i,29,12,12 k,20 g,23 d,10 n,15 2 e,13 o,18 k,20 Figure 2: Anestrl ordering where the index of eh node is shown eside the lel. First, for eh onfigurtion rrnged in the given nestrl ordering ( 1,..., n ), the gent with i must instntite the orresponding vrile ording to the order, sine messges need to e pssed mong gents fter the instntition if i+1 is ontined in distint gent. Seond, messge pssing must e performed prtilly sine prt of the DA union still ontins loops due to utset vriles yet to e instntited. For instne (Figure 2), fter instntiting vrile f, A 1 n only propgte its impt to nd e ut not eyond sine the rest of the DA union still ontins loops. Third, the messge pssing my involve eh gent multiple times nd hene tivities of gents must e refully oordinted. For exmple, fter A 0 instntited j, its impt needs to e propgted to l nd lolly, then to, d nd in A 1, nd finlly k to n nd o within A 0 gin. The sequene (3) needs to e omputed for eh onfigurtion nd the results need to e propgted to t lest one gent to ompute P (C). The gent n then send P (C) to every other gent for their lol usge. Next, onsider the omputtion of sequene (4). iven, sequene (4) n e otined y m messge propgtions in eh orresponding DA union. Sine e 1,..., e m re distriuted nd the sequene needs to e omputed in order, gents must oordinte the omputtion. After the first propgtion over the system, the gent with e 1 otins P (e 1 ). Itthen enters e 1, followed y the seond propgtion over the system. The gent with e 2 otins P (e 2,e 1 ), nd the proess ontinues. Note tht results for sequene (4) re distriuted. Sequene (6) n e otined through the sme proess with eh gent seleting its lol x. From sequene (4), sequene (5) n e otined similrly to Eq. (2). Sine the results for sequene (4) re distriuted, this omputtion needs to e oordinted. The gent with P (e 1 ) omputes P ( e 1 ) through Eq. (2). Note tht to derive the normlizing onstnt, the omputtion nnot e performed until sequene (4) hs een omputed for eh. It then sends P ( e 1 ) to other gents. The gent with P (e 2,e 1 ) will then ompute P ( e 1,e 2 ) nd sends it. The proess ontinues then t the next gent. From the results of sequenes (5) nd (6), eh gent will e le to ompute sequene (7). Note tht lthough the omputtion for sequenes (4) through (7) n e performed in ny order of e 1,..., e m, the order must e greed nd followed onsistently y ll gents for ll four sequenes. Lol evidentil inferene nnot e performed when the system-wide messge pssing is sent. For instne, A 0 nnot perform lol utset onditioning using its sunet 536 FLAIRS 2002

4 only, sine the dependene through sudomins in other gents nnot e ounted for. Between ommunitions, pproximte lol inferene using only the lol sunet is possile, ut the posteriors otined is not ext, nd n e signifintly different from wht will e otined fter glol ommunition. We summrize s follows: 1. Distriuted serh for loop utset nd nestrl ordering n e performed off-line. The rest of the omputtion must e performed on-line sine the omputtion depends on the oservtions e 1,..., e m. Note tht P (C) must e omputed on-line. 2. The omputtion of P (C) for eh requires O( C ) rounds of system-wide messge pssing. If omputtions for ll s re performed sequentilly, O( C 2 C ) rounds of messge pssing re needed. To redue inter-gent messge pssing, the O(2 C ) messges, one for eh, my e thed, mking O( C ) rounds of messge pssing suffiient with eh messge O(2 C ) times long. The omputtion of sequenes (4) through (7) requires one round of system-wide messge pssing for eh element in sequenes (4) nd (6). Hene O(m) rounds of messge pssing re needed, with messge thing. Overll, O( C + m) rounds of messge pssing re needed. 3. Lol inferene nnot e performed extly. 4. At lest the numer of nodes in the loop utset nd the numer of vriles oserved system-wide must e reveled. Prtil informtion regrding the nestrl ordering of domin vriles is lso reveled. Inferene in MSBNs y distriuted forwrd smpling Aording to forwrd smpling (Henrion 1988), simultion must e performed in n nestrl ordering. Tht is, the vlue of hild node is determined fter the vlues of ll its prents hve een determined. The ordering n e otined y distriuted serh similr to tht for distriuted utset onditioning. In n MSBN, the prents of node my e loted in n djent gent. To simulte the vlue of the node, n inter-gent messge must e pssed. Consider the following senrio: An gent A 0 ontins vrile x ut its prents π(x) re ontined only in nother gent A 1. Thus, to determine the vlue of x, A 0 needs to wit until A 1 sends the vlues of π(x) to it. Furthermore, A 1 ontins vrile y ut its prents π(y) re ontined only in A 0, nd x is n nestor of y. Hene A 1 nnot determine the vlues for ll vriles it shres with A 0 nd s result nnot send them in one th. Insted, A 1 must wit until A 0 sends the vlues of π(y). Figure 3 illustrtes the sitution, where A 0 must wit for the vlue of from A 1, nd A 1 must wit for the vlue of from A 0. Sine the DA union of n MSBN is yli, dedlok is not possile. However, if direted pth rosses the interfe etween two gents k times, then messges must e pssed k nd forth k times etween the two gents efore vlues for ll vriles on the pth re simulted. Note tht direted pth my pss ross multiple gent interfes. Hene during simultion of eh se, the numer of messges etween eh pir of djent x y 0 1 Figure 3: Two djent lol DAs in n MSBN. gents re upper-ounded y the numer of d-sepnodes etween them. This implies tht O(g d) inter-gent messges re needed to simulte one se, where g is the numer of gents nd d is the rdinlity of the mximum d-sepset. If the ses re generted one y one, the ove mentioned ost for inter-gent messge pssing will e multiplied y the smple size. To redue this ost, inter-gent messges n e thed. For exmple, A 1 my generte K vlues for nd pss ll of them to A 0 nd lter reeives K vlues for eh of nd. The prie to e pid is tht ll K vlues for eh vrile must e sved in some wy until the omptiility of eh se with oservtions is resolved s disussed elow nd the ontriutions of the K ses to the posteriors re ounted. When the ses re generted one y one, the genertion of se n e terminted erly s soon s one gent found the prtil se to e inomptile with its lol oservtions. Beuse thed smpling is intended to redue inter-gent messge pssing, suh finding y n gent nnot e ommunited to other gents immeditely. After smple of ses is simulted, it is neessry to determine whih ses re omptile with the oservtions. This n e hieved y letting eh gent lel eh se tht is inomptile with its lol oservtions. All suh lels must then e pssed mong ll gents to weed out eh se tht is leled s inomptile y ny gent. Sine prents of vrile my not e ontined in n gent, lol inferene in the sene of system-wide messge pssing nnot e performed equivlently. An lterntive is tht, t the end of eh ommunition, eh gent reords down the posterior distriution of eh shred vrile tht is root lolly, nd use the lol sunet thus otined for lol inferene. For the exmple in Figure 3, A 1 my reord down P ( e) nd P ( e), where e stnds for the oservtions mde y ll gents efore the lst ommunition. After new lol oservtions re mde, A 1 n then perform lol inferene with lol forwrd smpling. The result, however, is not equivlent to wht would e otined with system-wide messge pssing, even when there is no oservtion other thn tht from A 0, sine the dependene etween nd through the loops in A 0 is not ounted for. We summerize s follows: 1. Distriuted serh for n nestrl ordering is needed. 2. With messge thing, O(d g) inter-gent messges re needed. The length of eh messge is in the order O(K d), where K is the smple size. Both vlues for shred vriles nd omptiility lels for ses need FLAIRS

5 to e trnsmitted etween gents. In omprison, ommunition using LJF psses O(2g) inter-gent messges, whih requires d/2 times less inter-gent messge pssing. 3. Lol inferene in the sene of system-wide messge pssing does not onverge to the orret posteriors in generl. 4. Prtil informtion regrding the nestrl ordering of domin vriles is reveled. Conlusion In this work, we ompre three lterntive methods for inferene in multi-gent MSBNs long four dimensions: the omplexity of off-line ompiltion, the mount of intergent messges during ommunition, the support of onsistent lol inferene, nd the privte informtion reveled. The LJF method requires off-line ompiltion of the LJF. Distriuted utset onditioning (DCC) requires off-line serh for loop utset nd n nestrl ordering. Distriuted forwrd smpling (DFS) requires off-line serh of n nestrl ordering. The mount of off-line ompiltion ompres s LJF > DCC >DFS. With g gents, ommunition y LJF n e performed with O(2g) inter-gent messges. The order is O(( C + m) g) for DCC where C is the utset nd m is the numer of oserved vriles. Note tht C is upper-ounded y the numer of loops in the MSDA. DFS psses O(d g) intergent messges where d is the rdinlity of the mximum d-sepset. Commonly, we hve C + m>d. The mount of inter-gent messges during ommunition ompres s Aknowledgements This work is supported y Reserh rnt OP from NSERC of Cnd. Referenes Henrion, M Propgting unertinty in Byesin networks y proilisti logi smpling. In Lemmer, J., nd Knl, L., eds., Unertinty in Artifiil Intelligene 2. Elsevier Siene Pulishers Jensen, F.; Luritzen, S.; nd Olesen, K Byesin updting in usl proilisti networks y lol omputtions. Computtionl Sttistis Qurterly (4): Mdsen, A., nd Jensen, F Lzy propgtion in juntion trees. In Pro. 14th Conf. on Unertinty in Artifiil Intelligene. Perl, J Proilisti Resoning in Intelligent Systems: Networks of Plusile Inferene. Morgn Kufmnn. Shfer, Proilisti Expert Systems. Soiety for Industril nd Applied Mthemtis, Phildelphi. Suermondt, J., nd Cooper, Initiliztion for the method of onditioning in Byesin elief networks. Artifiil Intelligene 50: Xing, Y., nd Jensen, F Inferene in multiply setioned Byesin networks with extended Shfer-Shenoy nd lzy propgtion. In Pro. 15th Conf. on Unertinty in Artifiil Intelligene, Xing, Y A proilisti frmework for oopertive multigent distriuted interprettion nd optimiztion of ommunition. Artifiil Intelligene 87(1-2): Xing, Y Verifition of dg strutures in oopertive elief network sed multi-gent systems. Networks 31: DCC > DFS > LJF. In this omprison, we hve foused on the numer of inter-gent messges (vs. the length of eh messge). This is sed on the ssumption tht eh messge hs ost funtion αc 1 + C 2, where C 1 is the onnetion ost etween given pir of gents, C 2 is the ost depending on the length of the messge, nd α quntifies how undesirle it is to pss messges etween gents frequently. It is ssumed tht αc 1 is identil ross methods (whih is vlid) nd is muh lrger thn C 2 no mtter whih method is used (whih is resonle pproximtion). LJF is the only method mong the lterntives tht supports onsistent lol inferene. Lol inferene using the other methods my led to errors of more or less ritrry size. The LJF method revels no privte informtion. DCC revels the size of utset nd the numer of oserved vriles, s well s prtil informtion on nestrl ordering. DFS revels prtil informtion on nestrl ordering. This nlysis nd omprison not only provide insight to the issues involved in multi-gent proilisti resoning, ut lso serve to those who implement suh inferene systems s guide out the pros nd ons of lterntives. 538 FLAIRS 2002