A Simple Method for Identifying Compelled Edges in DAGs

Transcription

1 A Smpl Mto or Intyn Compll Es n DAGs S.K.M. Won n D. Wu Dprtmnt o Computr Sn Unvrsty o Rn Rn Ssktwn Cn S4S 0A2 Eml: {won, nwu}@s.urn. Astrt Intyn ompll s s mportnt n lrnn t strutur (.., t DAG) o Bysn ntwork. A rpl mto (Ckrn 1995) ws propos to solv ts prolm. In ts ppr, w sow tt jont prolty struton n y Bysn ntwork n unquly rtrz y ts ntrns torzton. Bs on su n lr rtrzton, w sust smpl lortm to nty t ompll s o Bysn ntwork strutur. 1. Introuton Bysn ntworks v n wly us or mnn unrtnty usn prolty (Prl 1988). A Bysn ntwork onssts o rpl strutur, rt yl rp (DAG) n st o ontonl prolty strutons (CPDs). Ts two omponnts n jont prolty struton. T ontonl npnns sts y jont prolty struton n rprsnt y rnt DAGs. T noton o n quvln lss o DAGs ws stu y (Vrm & Prl 1990; Ckrn 1995). An quvln lss o DAGs onssts o ll t Bysn ntworks w n t sm jont prolty struton ut r n tr rsptv DAGs. Altou rnt DAGs n t sm quvln lss v rnt rpl struturs, rtn rt s rtn tr rtonlty n ll tos quvlnt DAGs. Ts s r rrr to s ompll s. Itsusul to nty ts ompll s n lrnn Bysn ntwork rom t. Ckrn (Ckrn 1995) propos rpl lortm or ntyn su s. In ts ppr, w sow tt jont prolty struton n y Bysn ntwork n unquly xprss s prout o mrnls v y notr prout o mrnls. W rr to su rtrzton o jont prolty struton s n ntrns torzton. All Bysn ntwork struturs vn t sm ntrns torzton lon to t sm quvln lss. Bs on ts lr lsston o Bysn ntworks, w sust smpl mto to nty ll t ompll s n vndag. Copyrt 2003, Amrn Assoton or Artl Intlln ( All rts rsrv. T ppr s ornz s ollows. W rst rvw t noton o quvlnt DAGs n Ston 2. Our lr rtrzton o jont prolty struton n y Bysn ntwork s suss n Ston 3. In Ston 4, w sr our lortm or ntyn ompll s o Bysn DAG. T onluson s vn n Ston Equvlnt DAGs In t ppr, w us uppr s lttrs to rprsnt st o vrls n lowr s lttrs to rprsnt snl vrl unlss otrws mnton. Lt U not nt st o srt vrls. W us p(u) to not jont prolty struton (jp) ovr U. Wll p(v ), V U, t mrnl (struton) o p(u) n p(x Y ) t ontonl prolty struton (CPD), or smply ontonl. By XY, wr X U, Y U,wmn X Y. Smlrly, y, wr U, U, wmn {} {}. Aorn to t nton o ontonl prolty, p(x Y )= p(xy ), wnvr p(y ) > 0, p(y ) w tus sy tt n t ov xprsson, t nomntor p(y ) s sor y t numrtor p(xy ) to yl ontonl p(x Y ). Itsnot tt p(w ) n sor y p(v ) to yl p(v W W ) n only W V. A Bysn ntwork (BN) n ovr st U = { 1, 2,..., n } o vrls onssts o two omponnts: () rt yl rp (DAG) D. E vrtx n D rprsnts vrl U. T prnts o t vrtx r not p( ). T rpl strutur o t DAG nos st o ontonl npnny normton; () quntton o D. E vrl n D s qunt wt ontonl prolty p( p( )). Ts two omponnts n (nu) jont prolty struton (jp) ovr U s ollows: n p(u) = p( p( )). (1) =1 W ll t quton n (1) t Bysn torzton. It s wort mntonn tt y lookn t t tors n t ov Bysn torzton, t DAG n rwn y rtn n rrow rom no n p( ) to t no. A v strutur n DAG D s n orr trpl o nos 516 FLAIRS 2003

2 (,, ) su tt (1) D ontns s n, n (2) n r not rtly onnt n D. T sklton o DAG s t unrt rp otn y roppn t rtonlty o vry n t DAG. Sn BN s lt y ts rpl strutur, nmly, t DAG, W tus wll us t trm BN or DAG ntrnly no onuson rss. It s n not (Vrm & Prl 1990) tt DAGs o rnt BNs my n t sm jp. Dnton 1 (Ckrn 1995) Two DAGs D n D r quvlnt or vry BN nu y D, tr xsts notr BN nu y D su tt ot nu BNs ns t sm jp, n v vrs. W us D 1 D 2 to not tt DAG D 1 s quvlnt to DAG D 2,orquvlntly, BN1 BN2. Grpl rtr (Vrm & Prl 1990; Ckrn 1995) v n propos to trmn wtr two vn DAGs r quvlnt or not. Torm 1 (Vrm & Prl 1990) Two DAGs r quvlnt n only ty v t sm sklton n t sm v-struturs. It s not r to s tt t rlton nus n quvln rlton. Lt {D} not ll t DAGs n ovr xst U o vrls. Nturlly, t rtrsts o t quvln rlton n us to roup t DAGs n {D} nto rnt quvln lsss. It s not tt sn quvlnt DAGs n t sm jp, ty rtnly no t sm st o ontonl npnny normton. Tror, ll t DAGs n t sm quvln lss no t sm ontonl npnny normton. () () Fur 1: Four quvlnt DAGs omprs n quvln lss. Exmpl 1 Consr t our DAGs sown n Fur 1. By Torm 1, ty r quvlnt to otr. Morovr, no otr DAGs n ovr t sm st o vrls r quvlnt to ny o tm, tror, ty omprs n quvlnt lss. Gvn n quvln lss o DAGs, t s not tt som s rtn tr rtonlty n ll t quvlnt DAGsnt quvln lss. Dnton 2 An x y n DAG D s ompll, or ny D D, x y s lso n D. Otrws, t s rvrsl. () (v) Exmpl 2 It n vr tt n t DAGs sown n F 1, t s,,,,,, n ppr n ll o t quvlnt DAGs, tror, ty r ompll s. On t otr n, t, or xmpl, s rvrsl. It s n suss n (Ckrn 1995) tt t ompll s n DAG v prtulr mportn n lrnn t strutur o Bysn ntworks rom osrv t. Morovr, n lortm ws vlop to nty ll ompll s n vn DAG. T lortm rst ns totl orrn ovr ll t s n t v DAG n tn ns out ll t ompll s s on t totl orrn. Howvr, t prsntton o t lortm n t proo o ts orrtnss r rtr omplt n r to omprn. For tls o t susson o t mportn on ntyn ompll s n t lortm, rrs r rrr to (Ckrn 1995). 3. Alr Rprsntton o Equvlnt DAGs In ts ston, w rst rly rvw rnt rpl rprsnttons o quvln lss o DAGs, w tn v novl lr rprsntton o quvlnt DAGs y stuyn t orm o Bysn torzton. Ts lr rprsntton o quvlnt DAGs wll srv s t ss or smpl n sy to unrstn mto or ntyn ompll s. Sn ll t DAGs n t sm quvln lss n t sm jp n rprsnt t sm ontonl npnny normton, t s tus ly srl to v unqu rprsntton to rprsnt t wol lss o quvlnt DAGs. Svrl rpl rprsnttons v n propos. Vrm (Vrm & Prl 1990) propos t noton o rumntry pttrn n omplt pttrn to rtrz t quvln lss usn prtlly rt rp. Anrson (Anrsson, Mn, & Prlmn 1997) sust t noton o ssntl rp, w s spl n rp (Frynr 1990), to rprsnt Mrkov quvlnt DAGs. Mor rntly, Stuny (Stuny 1998) us t noton o lrst n rp (rnt tn t ssntl rp) to rtrz t quvln lss o Bysn ntworks. Sn ll t quvlnt DAGs lso ns t sm jp, w lv tt ts sm jp soul t rprsntton o t wol quvln lss n t soul lso possss t lty to srn rnt DAGs n t sm quvln lss. In t ollown, w prsnt novl lr rprsntton to rprsnt t wol quvln lss s on t noton o Bysn torzton. Dnton 3 Consr DAG n ovr st U = { 1,..., n } o vrls wt ts Bysn torzton s ollows: p(u) = p( 1 ) p( 2 p( 2 ))... p( n p( n )), (2) = p( 1) 1 =, j p( 2,p( 2 )) p(p( 2 ))... p( n,p( n )),(3) p(p( n )) p(,p( )), (4) p(p( j )) FLAIRS

3 wr {, p( )} {p( j )} or ny 1, j n. E p( p( )) n quton (2) s ll tor. Wll t quton n (3) t rton torzton o t DAG. T xprsson n quton (4) s otn y nln ny ppll numrtor n nomntor n quton (3) n s ll t ntrns torzton o t DAG. T ntrns torzton s n prov to t nvrnt proprty o n quvln lss o DAGs. Torm 2 (Won & Wu 2002) Two DAGs r quvlnt n only ty v t sm ntrns torzton. Torm 2 mpls tt ll t DAGs n t sm quvln lss v t sm ntrns torzton. It mmtly ollows: Corollry 1 T ntrns torzton o vn DAG D s unqu n t rtrzs n srs lrlly t wol quvln lss tt t vn DAG D lons to. Exmpl 3 Consr t our DAGs n ovr U = {,,,,,,,, } s sown n F 1. T ntrns torzton or t DAG n () s s ollows n quton (6): p(u) = p() p() p( ) p( ) p( ) p( ) p( ) p( ) p( ) (5) p() p() p() p() p() p() = p() p() p() p() p() p() p() p(). (6) Smlrly, t ntrns torzton or t DAGs n F 1 (), () n (v) n otn n t sm son. It n sly vr tt t our rnt DAGs o v t sm ntrns torzton sown n quton (6). 4. Compll Es Intton In ts ston, w wll prsnt vry smpl mto to nty ompll s n vn DAG s on ts ntrns torzton. T n our propos mto s vry ntutv. As wll monstrt sortly, t ntrns torzton o vn DAG rprsnts t wol quvlnt lss n t ontns ll t normton n to rstor quvlnt DAG (mor prsly, to rstor ts Bysn torzton.) Ts osrvton vs rs to t propos mto or ntyn ompll s. Exmpl 4 Contnu on Exmpl 3 n onsr t ntrns torzton o t Bysn ntwork sown n Fur 1 () rpt s ollows: p() p() p() p() p() p() p(u) = p() p() p() p() p() p() p() p(). (7) W now monstrt ow w n rstor quvlnt DAG (ts Bysn torzton) n t quvln lss rtrz y t ov ntrns torzton. In orr to trnsorm t ntrns torzton n quton (7) nto Bysn torzton, w n to sor ll t nomntors n quton (7) s ollows. p(u) = or p() p() p() p() p() p() p() p() p() p() p() p() p() p(), (8) = p() 1 p() p() p() p() 1 p() p() p() p() p() p() p() p() p(), (9) = p() p( ) p() p( ) p( ) p( ) p( ) p( ), (10) = p() p( ) p() p( ) p( ) p( ) p( ) p( ) p( ), (11) p(u) = p() 1 p() p() 1 p() p() p() p() p() p() p() p() p() p() p(), (12) = p() p() p( ) p( ) p( ) p( ) p( ) p( ) p( ), (13) or p(u) = p() 1 p() p() p() p() p() p() p() 1 p() p() p() p() p() p(), (14) = p() p( ) p( ) p( ) p() p( ) p( ) p( ) p( ), (15) In qutons (9), (12), n (14), w v sor nomntor n t ntrns torzton y n pproprt numrtor, ts sorptons rsult n qutons (11), (13), n (15), rsptvly. W tus v nlly otn tr rnt Bysn torztons, w xtly orrspon to t tr quvlnt DAGs sown n F 1 (), () n (v), rsptvly. Tror, w v sussully rstor ll t DAGs tt r quvlnt to t on sown n F 1 (). Tr r w osrvtons tt n m wt rspt to t ov monstrton n Exmpl Drnt DAGs n t sm quvln lss r otn, pnn on ow nomntor n t ntrns torzton s sor. It s ovous tt rnt sorpton wll rsult n rnt Bysn torztons, n, prou rnt ut quvlnt DAGs. 2. It s not tt urn t ours o sorn nomntors, som nomntor s or to sor y prtulr x numrtor, no otr os. In otr wors, som nomntor s no lxlty so tt t s to sor y x numrtor. 518 FLAIRS 2003

4 Bor w mov on, lt s srutnz t nomntor sorpton w v m n Exmpl 4. Not tt or nomntor p(x) to sor y numrtor p(y ), tmust t s tt X Y. Unr ts rstrton, t possl sorpton o t nomntors,,,,, n n summrz y t ollown xprssons, = {, }, (16) = {}, (17) = {,, }, (18) = {,,, }, (19) = {, }, (20) = {, }, (21) wr X = {Y 1,..., Y n } mns tt t nomntor p(x) n possly sor y p(y 1 ),..., p(y n ). W urtr ll t st {Y 1,..., Y n }, not AS(X), t sorpton st or X. It s not tt Y AS(X) X Y. W wll us AS(X) to not ts rnlty. It s lso not tt t st D = {X p(x ) s nomntor n t ntrns torzton} s multst, n w wll us X to not t numr o ourrn o X n t multst D. Rll tt t DAG o Bysn ntwork n rwn s on ts Bysn torzton, mor prsly, s on ts tor p( p( )), yrtn n rom no n p( ) to t no. Follown ts ln o rsonn, t ollows tt nomntor, sy p(x), s no o ut s or to sor y x numrtors, sy p(y ), tn t s xpt tt t s y x, wr y Y X, x X, wll ppr n vry rsultn Bysn torztons. Tror, ty wll ompll s y nton. For nstn, n xprsson (17), t only numrtor tt s ppll to sor p() s p(), n, t tor p( ) wll otn y ts sorpton n t s n wll ppr n ll possl rsultn Bysn torztons, w mpls tt ty r ompll s s n vr y F 1. Spl ttnton soul p to t two ntl nomntors p() n p() n xprssons (20) n (21), rsptvly. Altou ts two nomntors r synttlly ntl, ot o tm v to sor n orr to otn t Bysn torzton. T ppll numrtors or ot o tm s t sorpton st {, }, w ontns xtly two lmnts, t sm numr o t ourrn o p() s nomntors. Ts nts tt on o t p() must sor y p(), n t otr p() must sor y p(), nootr os. Ts sorptons mply tt t tors p( ) n p( ) wll otn n t s,,, n wll ppr n ll possl rsultn Bysn torztons, w mpls tt ty r ompll s s n vr y F 1 s wll. Sn t numrtors p(), p(), n p() v n snt to sor t nomntors p(), p(), n p(), rsptvly, ts ns t sorpton st or rom {,,, } sown n xprsson (19) to t nw rn snlton st {}, w mpls tt t nomntor p() wll v to sor y t numrtor p() to otn p( ). Tror, t s lso ompll s s n vr y F 1. T ov nlyss rsult n t ollown rn xprssons or t sorptons o nomntors, ontrstn wt tos n t xprssons (16-21), = {, }, (22) = {}, (23) = {, }, (24) = {}, (25) = {, }, (26) = {, }, (27) rom w t ompll s n oun out rt wy nomntor n only sor y x nomntor. Bs on t ov sussons, w tus propos t ollown prour to nty ompll s n vn DAG. T orrtnss o t prour wll vn sortly. PROCEDURE Fn-Compll-Es Input: vn DAG D. Output: Compll s n D ollt n t st E. { 1: Otn t ntrns torzton o t vn D, n lt E =. 2: Lt D = { 1,..., m } multst, wr p( ) s nomntor n t ntrns torzton otn n stp 1. 3: For D, =1,..., m, omput s sorpton st AS( ); 4: For D s.t. =1, =1,..., m, I AS( ) s snlton st, AS( j )=AS( j ) AS( ), or ll j, 5: For D s.t. > 1, =1,..., m, I AS( ) =, AS( j )=AS( j ) AS( ), or ll j, 6: For D, =1,..., m, I k = AS( ) =, E = E {p(y j ) Y j AS( ),j =1,..., k}, 7: Rturn E. } Compll s, rom t prsptv o t ntrns torzton o vn DAG, s otn y ntyn wtr nomntor p(x) n sor y only on numrtor or n possly sor y multpl numrtors. I nomntor p(x) n only sor y only on numrtor p(y ), tn t tor p(y X X) wll ppr n ll possl rsultn Bysn torztons, tror, t s y x, wr y Y X, x X, wll ompll s. Stp 1-5 n t prour omputs t sorpton st or nomntor. In stp 6, or ny nomntor p(x) n ts sorpton st AS(X), wtry to s wtr t n sor y unqu numrtor or y multpl os o rnt numrtors. Tr r two ss: 1. p(x) s nomntor su tt X = 1, onsr AS(X). FLAIRS

5 I AS(X) > 1, notr wors, p(x) n sor y t lst two rnt numrtors, tn tr s no n to v ny ompll s prou y sorn p(x). I AS(X) =1, notr wors, p(x) wll sor y only on nomntor p(y ), tn t tor p(y X X) otn wll v rs to ompll s y x, wr y Y X, x X. 2. p(x) s nomntor su tt X > 1, onsr AS(X). I AS(X) = X, tn tr xsts on-to-on orrsponn twn mmr o Y AS(X) n ourrn o p(x) s nomntor, n w s vry p(x) wll sor y unqu p(y ). Tror, ompll s wll otn y su sorptons. I AS(X) > X, tn t s mor numrtor p(y ), wr Y AS(X) tn t nomntor p(x), n w s nomntor s not unquly sor y numrtor. Tror, tr s no n to otn t ompll s. T nlyss o t two ss s n xtly ptur y stp 6 n t prour. T ov susson tully provs t ollown torm. Torm 3 Gvn DAG D, t output o t prour Fn-Compll-Es ov s xtly t ompll s n D. T prour Fn-Compll-Es only nvolvs som st oprtons n n mplmnt sly. Stuny, M Bysn ntworks rom t pont o vw o n rps. In Prons o t 14t Conrn on Unrtnty n Artl Intllrn (UAI-98), Morn Kumnn. Vrm, T., n Prl, J Equvln n syntss o usl mols. In Sxt Conrn on Unrtnty n Artl Intlln, GE Corport Rsr n Dvlopmnt. Won, S., n Wu, D An lr rtrzton o quvlnt ysn ntworks. In Numnn;, M. M. B., n Stur, R., s., Intllnt Inormton Prossn, T 17t IFIP Worl Computr Conrss, Strm 8, IIP-2002, Kluwr Am Pulsr. 5. Conluson In ts ppr, w v monstrt tt t noton o ntrns torzton n srv s t lr rtrzton o t quvln lss o DAGs, s on w w v prsnt smpl mto or ntyn ompll s n Bysn ntwork. T nw mto s ntutvly smpl n n sly mplmnt. It rvls t mportn o t ntrns torzton o n quvln lss o Bysn ntworks. T pplton o ntrns torzton n ts ppr susts mor rsr on t torzton o t jp n y Bysn ntwork s wort pursun. Rrns Anrsson, S.; Mn, D.; n Prlmn, M A rtrzton o mrkov quvln lsss or yl rps. T Annls o Sttsts 25(2): Ckrn, D A trnsormtonl rtrzton o quvlnt ysn ntwork struturs. In Elvnt Conrn on Unrtnty n Artl Intlln, Morn Kumnn Pulsrs. Frynr, M T n rp mrkov proprty. Snnvn Journl o Sttsts 11: Prl, J Prolst Rsonn n Intllnt Systms: Ntworks o Plusl Inrn. Sn Frnso, Clorn: Morn Kumnn Pulsrs. 520 FLAIRS 2003