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-SEPARATON: FOM THOMS TO LGOTMS

Dn Geige omas Va & Jdea ea

Cogive ysems Laborao Compr iene DepmnUvesty of Clfoia Los Ageles CA 9024ggesd

ABTACT

An fie lgor is dvelod ta idees alnddenis mpld by e toology of a Bays ss aimit ssfro te sodess d ometeess o -searato wth esc o obably tho he al-goh ime E I w E is numbrof edgs i te nwork

1. ITODUO

Bayesan newoks ecode optis of a poabiiy distbo sng drced acycic gaphs dags)Te sage is spad aong any disciplies suchas: Acia Inellgce [Peal 988] ecisonAnalysis [Howad d Mahson 8; hace9 Econo Wod 6 eeics Wg3 hilosohy lymo al. 987 ad Stats

ts atzen and Seglhalr 988 87A Baysn netwok is a pa D P wh D s adag ad is a obabiiy disbon cald heunderlying dsbuton ach node in Dcoesnds o a varable a set of ods coresond to o a st o vaabes X ad xdnots vaus da fm domain of alro he (coss pdct) doman of rescivly X X, XK ) caled a (condtinal ep) sm.

The mrtnce of d -searaion stems fm hefoowing teom [Vema and Peal 1988; Geigernd Pea88}.

t Pn = {P I (D is a Bayesannetwo en

Te "y sonness stes a weee J L )0 ods D , mst reese a -eedecy tat olds in eve udelyg - Te " art (cmlees) es at andeendency that is not deeed y dseaaoaot be haed y all distibutions in P and,

ece aot b eveaed y nonnmerc metods

HE E

n s ecton we eveop a lnea time gofo idetyig the se of nodes at ar d-searated frm y The soundness and comleteness f d -eparaon guaantees that te set ofvaables K corrsnig o e of nodes she maxima se of varales ha can deied asig indendent of give wout sogto numecal calcuaions. he oosed agotis of w kwra Sea-goim; it nds nodes eachale frm hugh active tail (by , hence the maxima set ofnodes satisfyg )D This task can eviewed as instance of a mo genera ak ofding a path in a dicted gaph for wch some

ci pai of links a restrctd not to aprouv. I h separation isviewed a a sccaion fo suc retictions foreampl two lins v v w ano aparconsecuvely n ative trl uness v L orha a descendent in L e following notations aemloyed: D = V, E) i a dctd graph (notnecesrly acyclic) whe V a set of nodes,E VxV is the set of dircted) ln and E is a list of pi of adj h not apar consecuvely F connoe fail) We saytat oded air of liks 1 is eg ffe 1 and hat a a ff evey par oadacent liks on it is egal We emhasize at by"pa we mean a dicted path, not a tal

We pose a simle aoi fo e foowing

pe Given a nte diected graph D (V ast !:E E and a set of nods J d anodes rachale frm via a legal path in D ego nd pr a t moc ofhose d Eve [7.

Inp: A ree grh D V set of le- rs d

Ott bel e e c e sbee w "ee t

i hble m J v a lel h.

ew ne s V n j J , he lnk e em w1 Ll s n l j E wih R . bel le ne w uene

:= I

i Fn l ule lk v w e le ne v lbele tu v w a e lk s

eh ln w n S w +I e re ne w wR

v := + 1, G Se

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h

Bayesin network D V )nd wo ijoi sets of noes

J

ndL. lit of incomg in (in-Ji) foeh no V no {a I (J L, a)D}

i) Conu he ollowing tbe:{e if h esenent[v - fle ohew

i nr de gp whee = vv

iii i goh nd te se o nodes whih hve egl ph fo , whee of ( v i egl ff * nd eithe i hedtohed eo l ue o 2) n htohe node onhe l uw n D

iK

Re K)

Th ctess of i ago i talishd by oowig gumet

mm 4: For vy od J a i hble om J vi legl l i i thre iv ph by m J a in

Fo u L n x0 i 0-x a) aci il b L i D did phx0-x a) i a D' nd vi . (W v liminad e a J u L ochal conin; h il x- x a) i o o o-cv au b ou dniio, JL d {a m disjit Tm 5 h ud by h go ixcly {a (J ) }

h et ntt n Sp ii) U d hb v legal pah i u, b em

in all d o in

J L l J vi

i

L) n D Hw, L 0 ods J u L d a i o achb om J (by i pah by L) ho K V - J xac t {a aD

Nt, w ow th h comxiy o is w z th arim s

. Th t i im a f: Ini-iay ark al fZ ih e oo iomi ls o that hn to ps nd o on. Th w e id mo o, h i

qu 0 ros h o p qu cottin l f a nod h il h link h aa m v i oih -lt d tt cmptly expily th tology D Th p l quis0 I ) p Uing h wo lis h tsk dg g te ( gh q olyot tme; i = v lad npndng upon do - D whh h dedt h ak o h oli o v o he v o o lc Ts o opo p encod lk s d H S qu o mo a 0 i hi iho up bund sming I VI) the e gorm

Th algohm c alo myd tef hethe s mn D l a ax ta dpad gn L ad bsee t hod ax- In t i ask, go sghly ivd K ha b te L [1988 ve

y d o lgoim o h akThi agoihm coi o ollog F g b eg D hih n e od i J u Ld oing i id ks Scond fo td gap cad stpng nt o ik o D a

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t

t tt

t

t

A Bayesian etwork D VE)and wo dsjon es o odes JadL.

A of ncomg lns (n-l) foeah oe .

A e o oe K were {I J L }

( Con e ollowng bl:

-{ o dscedn[vl- false otherwise

() Con a gap E')wee =E { _ E}

() Usng agorm 1 nd e se o al nodes Kw ave a ega pa o wep o s ( , _ w s lega w and ee 1 s eaoead neo e w ad deede[] e o 2 s no a adtoead noe oe w ad Z

(v K = V -

Reu K).

e corcness of is aohm is esabshed bythe followng arent.

Lem 4: or eve node J u s achable from J va a leal tal n D ff ere is an ac-tve pat by L fm J to in D

For a J u and J f (0 )is a active tal by n D en he directed pa 0 a ega n n se esa. (We have eiinated e case J u L ortecnical convenence te tral ( ) s notacve no noactve cause, by ou dentn J and } mst dsjoint

horm 5 he set red by he agor sexaty a J L a }

22

Proof he setK constcted in Step (ii containsal nodes achable fm J va a legal path in D .Thus by lemma contains al nodes not nJ u that a eachable rom J via an acve alby n D Howeve, (J hods iff J L d is not reachabe fm J by act b L efo, V ' u sexactly the set {I L

Nxt we show tat the complexity of e ao s 0 () we anayze he agoi step bystep. he t step s lemented as follos: nially ark al nodes of wth te Foow te o k o h o h ps dthen o thei parents and so on his way ea ls exaned at most once hnce e entie step equs 0 ( E ) oraons The second step rqishe cotcion of lst for each node ha speciesall he lnks that mana fro n D outlst).The nlst and the outlst completely d expictys e olog o This sep ao equs (IEI steps Using the two lis the task of ndinga leal par in step ii) of aotn 2 quires oyconstat me; f = u s laled ndepenng upon he dtion of v in wheer is o has a dscendent in Z ee s o he outlst o v, or lns of he nls o, or a selected Thus a constant nube ooperaons pe encounteed lnk is peod

ence Step (i) ues no o tan 0 ( )oeation whh is therefore the up boud assuin E I VI fo h ent ao

e ave agorth can also epoyed tovefy wheher a scic stateent on dag Siply nd e set a o l nodestat a dsepaated rom gven and obsee a L holds in i K n fact oris ta ago can sihy be ipoved byforcin tenaton once e condtion K K mas ben detecd Lautzen at a [988] have re-

prpsed ane ar r skheir aoi consiss of he foowing steps.ist o a dag D' by eovn om D a nodeswhich not cesto of any node n J u u and emving ther incdent lnks) Second fo anundrected graph G caed the oa gah, ysrippig e dctionalt of the ln of D and

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coectng y two ode at ha a coo dn D whch s or has a descendent in . ird,they show tha hold (undirctd)ps eween a in G are ntecepe y L

he complety of the moral gaph goth i beca a gap G ma upto I VI inks Hence, chekng separation in G q ep Tus, ur lgtm sa modae impovement as it ony eus O(IEIss. T g s sgnican ainl n spase gaphre I I = IV We ot t f mmmb f en f each ode s oune by constant thn t o aots achie he sasymc beavi .e linea in E . On oehand, when e task is to nd nodes d separated

by

L(not eey alan a gen inen

dence) then a bte foce appcaon o the morgraph algortm reues I steps cause foreach node not in u L the agohm mus contcta new mora grap. Hnce fo is tas, ou lgothm ofers a condeabe impome

The inerence engne Bayes netoks haslso en sed fo decsion naysis; an aalyst con-sls an expe ect inomation abot a decsonpoblem fomulates the apppae network andten b an automated sequence o gaphical and p-bablisc mapato a opmal decision s ob

taned [Howard and Mateson 198; Omsted 1984Shachter 1988] hen such a newor is constrctedi is mtant to deene e nfoaon neededto answe a gien uery P x I x L here {} L a abtra se of odes the netwok becausesome nodes might coain no rleat infoaon tothe decsion pobem ad elcting their numericalpaamete s a wase of efo hachte 988] As-suming that each node X stos e conditional ds-tbuton P I x( the tak s to entfy he setM f noes that must b consuld i e r ocompting P x x or ateativey the set of

nodes ta can b assied arbiay ndioa dtibtions without aectng te quantity P x x .he euie et a b idef th separation citeion e epsent e paaetes of dsbuin P a a du pa P onode hs is clealy a lgitate repsentatonolyg w hfa f e eey

ode o n aP lx P c garded a pantof . F dummy nodes at ard sepaated om by psent aabes tat a

condionay depennt o J given L ad so inoaon stod in ese nodes can be noredhus e nfomation ud to copP x I x resdes in e set of dmmy odes whichae no d sepaad fo J gien ooer hecompeteness of dsepaaton fuer implies at M mnma o node i M c exempte fompcessng on purely topogic gouds ie.wtout considering the numercal aues o e pobabtes noe) The agothm below saizes ese consderaos:

t p Bayesn network, e f

a

p f M ha c uf m P j

C b gg dmm d v' f v v ddg l

() U ah 2 t d -pad J b L

L M h f dm d h idd

We conclde with an xampe Consie henetwork D of Figu 3 a a que P 3.

Fg

Th computaton of P quires on to multplhe aces P31 and 1 d o ov

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te vues of X These two matrices stod athe dmmy nodes ' and wich a te y dummy ndes not -separated fm node 3 (given hs agorim 3 reveals te fact hat e paramete reprsentd by node 2 d 4'

(P(x1), P (4 ae nt nded fr the cmputatin f P (3). Note however at knowing evaue o X4 might inence te computation ofP (x ) cause X and X cold deendent evale of X 2 n te her hand never afects thiscompttin cause X 2 s indendent o X hiseample shows at he qestion of wheer a vaiable inuences e comptan f a query and equesion f wheher e pareters stod wih atvarable inence e same quer ar tw diffentquesions. Agm 3 by presenting paraete dummy vaiables slves the ater prblem by

trasfing it t an instance of the fer

Shachter wa e st to address he pblem fidenifying ilevant parameters [Shachter 188Or foan provides several advantages Firstwe distingish tween sensitivity t variable instantiations and senitivity t paameer vales and thealg we provide cn tailrd t slve eierne of tese prblems Shacter's algr hdeste second poblem and thefo i des n vea te independencies a are implied b e tplogy f e dag o eample in ig Shachtersagtm would coecy cnclde hat nodes

2ad

4 bth cntain no relevant iation fr e comptatin of P(). Yet X2 is indepndent f Xwhile X4 and X might be dendent a distnctinnot addssed in Shachters agoitm Second rmed is mprised of two comonents 1 declarative charcterization of he independencies encodedin he network ie e d separain e crionad 2 pdrl implementation of the criteiondened in hs appoach facilitas a clear pofof e validity and maimality of e gaphica cteron independent of e detils f the algrhmfolowed by profs f he algms coctness

ad optimaity ( it reqires y EI) steps). Shacters taent the caracterizaton f teneeded pamete is insepable from he aghm

3) Sacer aso cosders deesc arabes wcw Geger a al 9

hece it is harde to establsh prfs f cocnesand maimaity

ACKNOEDGEMT

We hak Eli Gai fr his help in devepng agori 1 and to Aaria a and Rss Shahter ormany smating discssins.

REERES Evn. 1979 Graph Agothm, Compe Sciee Pe.

G rsnal unican 988.

D. ige & . Pel Agust 988 The gi o CauaModls Pro. of he 4h Wokop on Uri

AI t Pal, Mnta pp. 36-47

D Geie, T S. V d J PeaL 89 dniyn ideden Bayein netwos", ehni e , U Cogniiv Sym arao eption.

C Glymour R Schenes P Spies d K el 1987 D s-

cvrng Caual re Nw Yrk: caemi Pre.R. A H & J E an. 98 Ience Diag

Pps d Applns of D ylo Pa, C Saegc Decios Grop

auin D. Spegltr 989 mptwit prabiii o grapical stus d he api

cato to expe systms. Ra S o BSL zn AP Dwid, BN Lrs HG Lee Ocb 1988 Indndnc is f Ded Ma Field, Tnica Rr R 8832 Aarg

sitnt alg Dm.

S l 983 eesng d Soing iinPoblem PD. is, ES Dept trd er

Pl. 88 Pobab Resonng n ege sem:Nwor of Pa nfe San Mae C M

g un.

RD Scte 988 Pobbilic eren d ee Di

gr" O Rsrh ol. 36 58904.Q. mi. Jue 98 enc Diagr for aia

odg hical re#1 depamen o tat

t eit Wick Cveny, Egd

T & . l. Augu 988 Casal Newo Sem-

tic d Epesienes eeg of he

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Workhop on Uetai A!, Pl Mo p35359

H Wo 9. Eom Ml Buildig At:

Nort-Hod ublhng CoS Wrig 934. 'T Metho of Pat Cofc" A MaJh

Sis. Vo. p 6

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