Modeling Causal Reinforcement And Undermining With Noisy-AND Trees

Transcription

1 Modling Causal Rinformnt And Undrmining With Noisy-AND Trs Y. Xiang and N. Jia Univrsity of Gulph, Canada Abstrat. Causal modling, suh as noisy-or, rdus probability paramtrs to b aquird in onstruting a Baysian ntwork. Multipl auss an rinfor ah othr in produing th fft or an undrmin th impat of ah othr. Most xisting ausal modls do not onsidr thir intrations from th prsptiv of rinformnt or undrmining. W show that non of thm an rprsnt both intrations. W prsnt th first xpliit ausal modl that an nod both rinformnt and undrmining and w show how to us suh a modl to support ffiint probability liitation. Introdution A Baysian ntwork (BN) [7] nods onisly th probabilisti knowldg about a larg problm domain. Howvr, whn a variabl has a larg numbr of parnt variabls in th BN, aquisition of th orrsponding onditional probability tabl (CPT) is xponntial on th numbr of parnts. Parl pionrd ida of noisy-or modl [7]. Hnrion [5] addd to noisy- OR modl th laky probability. Diz [] and Srinivas [9] xtndd noisy-or from binary to multi-valud variabls. Hkrman and Brs [4] analyzd a olltion of ausal indpndn rlations that allows ffiint aquisition of onditional probability tabls in BNs. Lmmr and Gossink [6] proposd rntly th rursiv noisy-or modl. Whn multipl auss ar prsnt, thy an rinfor ah othr in produing th fft or thy an undrmin th impat of ah othr. Unlik [6], prvious work do not onsidr ausal intrations among variabls from th prsptiv of rinformnt or undrmining, and modl paramtrs ar limitd to probabilitis of singl aus vnts. All prviously proposd ausal modls, inluding noisy- OR, rursiv noisy-or, noisy-max, noisy-and and noisy-addition, ar limitd to rprsnt ithr rinformnt or undrmining but not both. In this work, w prsnt an noisy-and tr modl that rprsnts arbitrary ausal intrations among a st of auss, som of thm ar rinforing and othrs ar undrmining. Rinformnt and undrmining ar nodd xpliitly to support probability liitation and probabilitis for multi-aus vnts an b inorporatd as modl paramtrs if so dsird. In Stion 2, w introdu th trminology and dfin formally rinformnt and undrmining. Stion 3 prsnts how rinformnt and undrmining an b

2 modld uniformly using noisy-and gats. Stion 4 proposs th noisy-and tr modl and how to us it to obtain ausal probability is dsribd in Stion 5. W prsnt, in Stion 6, how to us noisy-and trs to modl ausal intration whn dfault indpndn assumptions do not hold. W dmonstrat liitation of CPTs with noisy-and trs in Stion 7. Stion 8 ompars rlatd ausal modls with noisy-and trs. 2 Bakground W aim to assss a onditional probability distribution of a variabl x onditiond on a st of variabls Y basd on thir ausal rlation. Th auss that w onsidr ar unrtain auss. Following Lmmr and Gossink [6], an unrtain aus is a aus that an produ an fft but dos not always do so. W dnot a st of binary aus variabls as X = {,..., n } and thir fft variabl (binary) as. For ah i, w dnot i = tru by i and i = fals by i0. Similarly, w dnot = tru by and = fals by 0. W rfr to th vnt that a aus i auss an fft to our as a ausal vnt. W dnot this ausal vnt by { i } or simply i, and w dnot its ngation that i dos not aus as i. Not that ausal vnt i is not just th onurrn of i and. With th abov notation, i is an unrtain aus of if and only if 0 <P( i ) <. W dnot ausal vnt that a st of auss X = {,..., n } auss as {,..., n }, or simply,..., n or x. Whn th aus st is indxd, suh as W i = {,..., n }, th ausal vnt may b dnotd w i. W allow broad intrprtations of ausal vnt by a st of auss, as will b sn in latr stions. For instan, w ar not limitd to th intrprtation in [6]: th fft is ausd by at last on of th auss. Parl [7] rgards a aus as a rtain aus, whos ourrn always auss th fft. H nods th ausal unrtainty using an unrtain inhibitor. Th onjuntion of a rtain aus and an inhibitor in his formulation is quivalnt to an unrtain aus. Whn modling a domain with a BN, th st of all auss of an fft variabl is its parnts. W dnot th st of all auss of by C. To aptur auss that w do not wish to rprsnt xpliitly, w inlud a laky aus variabl in C (as on of through n ). Probability of ausal vnt an b usd to assss CPT P ( C). For xampl, if C = {, 2, 3, 4 }, thn P (, 20, 3, 4 )=P (, 3, 4 ). Not that only aus variabls of valu tru ar inludd in th right-hand sid of th ausal probability. Whn multipl auss ar prsnt, thy may rinfor ah othr in produing th fft. That is, thir ombind influn is gratr than that from only som of thm. Altrnativly, multipl auss may undrmin ah othr in produing th fft. Blow, w dfin rinformnt and undrmining formally. Dfinition Lt R = {W,W 2,...} b a partition of a st X of auss, R b a propr subst of R, and Y b th union of lmnts in R. Sts of auss in R 2

3 ar said to rinfor ah othr, if for vry subst R R, it holds that P ( y ) P ( x ). Othrwis, sts of auss in R ar said to undrmin ah othr. Whn ah R i is a singlton, rinformnt orrsponds to positiv ausality in [6] and undrmining orrsponds to inhibition. Hn, rinformnt and undrmining ar mor gnral. Thy allow modling of rinformnt of sts of auss whn auss in som st ar undrmining. Similarly, thy allow modling of undrmining of sts of auss whn auss in som st ar rinforing. This will bom mor lar in Stion 4. 3 Noisy-AND Gats for Rinformnt and Undrmining W propos to modl rinformnt as wll as undrmining uniformly basd on AND gat, whih w rfr to as noisy-and gat. It builds on prvious work with noisy-or [7] and noisy-and [2], but taks a diffrnt prsptiv from rinforing and undrmining intrations among unrtain auss. W assum that, by dfault, sts of rinforing auss R = {W,..., W m }, whr W i and W j ar disjoint for all i and j, satisfy failur onjuntion: ( w,..., w m )=( w )... ( w m ). () That is, sts of rinforing auss fail to produ fft if ah st of auss has faild to produ th fft. W also assum that, by dfault, sts of rinforing auss satisfy failur indpndn: P ( w,..., w m )=P ( w )... P ( w m ). (2) That is, failur vnts w,..., w m ar indpndnt of ah othr.... n... n (a),..., n (b),..., n Fig.. Noisy-AND gat. W modl th dfault rinforing intration graphially with th noisy-and gat in Figur (a), whr ah W i = { i } is a singlton, m = n, failur onjuntion is xprssd by th AND gat, and failur indpndn is xprssd by lak of dirt onntion btwn individual failur vnts. Th following Lmma onfirms thir rinformnt. Du to spa limit, w omit proofs for all formal rsults. 3

4 Lmma Lt R = {W,W 2,...} b a partition of a st X of unrtain auss of fft and sts in R satisfy Eqns () and (2). Thn, intration among sts of auss in R is rinforing. Whn ah W i is a singlton, Eqn (2) an b altrnativly writtn as P (,..., n )= n ( P ( i )), (3) whih is th noisy-or modl [7]. Thrfor, Lmma also formalizs rlation btwn noisy-or and rinformnt. W rfr to th noisy-and gat in Figur (a) as th dfault modl for rinformnt. Th dfault modl rprsnts only on possibl rinformnt among sts of auss. W prsnt rprsntation for diffrnt rinformnts in Stion 6. Nxt, w onsidr undrmining. W assum that, by dfault, sts of undrmining auss satisfy suss onjuntion: i= x =( w )... ( w m ). (4) That is, whn sts of undrmining auss sud in ausing th fft in undrmining way, ah st of auss must hav bn fftiv. W mphasiz that th suss ours in an undrmining way. If any st of auss has ourrd but has faild to b fftiv, it would not undrmin th othr sts of auss. W also assum that, by dfault, sts of undrmining auss sud indpndntly, i.., P ( x )=P ( w )... P ( w m ). (5) Th following lmma onfirms thir undrmining intration, whos proof is straightforward. Lmma 2 Lt R = {W,W 2,...} b a partition of a st X of unrtain auss of fft and sts in R satisfy Eqns (4) and (5). Thn, intration among sts of auss in R is undrmining. Again, th dfault modl rprsnts only on possibl undrmining intration among sts of auss. W dsrib rprsntation of othr undrmining intrations in Stion 6. 4 Noisy-AND Trs Considr two sts X and Y of auss that rinfor ah othr. It is possibl that auss within X undrmin ah othr, and so do auss within Y. In gnral, suh intrplay of ausal intrations of diffrnt naturs an form a hirarhy. In this stion, w prsnt a graphial rprsntation to modl suh a hirarhy. It is basd on noisy-and gats and has a tr topology. W trm it noisy-and tr. W assum that a domain xprt is omfortabl to assss rinforing and 4

5 undrmining intrations among auss aording to som partial ordr and is abl to artiulat th hirarhy. For xampl, onsidr a patint in th pross to rovr from a disas D. Taking mdiin M hlps rovry and so dos rgular xris. Patint s normal dit ontains minrals that failitat rovry but taking with mdiin M rdus fftivnss of both. Th auss and fft involvd ar dfind as follows: : Rovry from disas D within a partiular tim priod. : Taking mdiin M. 2 : Rgular xris. 3 : Patint taks his/hr normal dit. For th purpos of prognosis, on nds to assss P (, 2, 3 ). To as th task, a physiian may onsidr first undrmining intration btwn and 3. (S)h thn onsidrs rinforing intration btwn sts {, 3 } and { 2 }. Thus, th physiian has artiulatd an ordr for stpwis assssmnt. In addition, th physiian also asssss P ( )=0.85,P( 2 )=0.8,P( 3 )=0.7. If this is all th information that th physiian an provid, th ausal intration an b modld as th noisy-and tr in Figur 2 (a). 3 2,3 2,2 3 (a), 2,3 (b), 2,3 Fig. 2. (a) Noisy-AND tr modl of disas xampl. (b) Altrnativ modl. From th uppr AND gat and Eqn (5), w driv P (, 3 )=0.595, an fft of undrmining. Th output of th uppr AND gat is ngatd (shown by th whit oval) bfor ntring th lowr AND gat and th orrsponding vnt has probability P (, 3 )= From th lowr AND gat and Eqn (2), w driv P (, 2, 3 )=P (, 3 )P ( 2 )=0.08, and P (, 2, 3 )=0.99. Th following dfins a noisy-and tr in gnral. Dfinition 2 Lt b an fft and X = {,..., n } b a st of unrtain auss that is known to hav ourrd. An noisy-and tr for modling ausal intration among lmnts of X is a dirtd tr whr th following holds: 5

6 . Thr ar two typs of nods on th tr. An vnt nod is shown as a blak oval and a gat nod is shown as an AND gat. Eah vnt nod has an in-dgr and an out-dgr. Eah gat has an in-dgr 2 and an out-dgr. 2. Evry link onnts an vnt nod with a gat nod. Thr ar two typ of links: forward links and ngation links. Eah link is dirtd from its tail nod to its had nod onsistntly along th input-to-output stram of gats. A forward link is shown as a lin and is impliitly dirtd. A ngation link is shown as a lin with a whit oval at th had and is xpliitly dirtd. 3. All trminal nods ar vnt nods and ah is labld by a ausal vnt in th form y or y. Exatly on trminal nod, alld th laf, is onntd to th output of a gat and has y = x. Eah othr trminal nod is onntd to th input of a gat and is a root. For ah root, y is a propr subst of x, it holds i y i = x with i indxing roots, and for vry two roots with y j and y k, it holds y j y k =. 4. Multipl inputs of a gat g must b in on of th following ass: (a) Eah is ithr onntd by a forward link to a nod labld with y, or by a ngation link to a nod labld with y. Output of g is onntd by a forward link to a nod labld with i y i. (b) Eah is ithr onntd by a forward link to a nod labld with y, or by a ngation link to a nod labld with y. Output of g is onntd by a forward link to a nod labld with i y i. Dgr rstrition in Condition nsurs that an vnt rprsnts th output of no mor than on gat and is onntd to th input of no mor than on gat. Condition 4 nsurs that inputs to ah gat ithr all orrsponds to ausal vnts in th form of y, or all orrsponds to ausal vnts in th form of y. Smantially, 4 (a) orrsponds to undrmining sts of auss and 4 (b) orrsponds to rinforing sts. 5 Noisy-AND Tr Evaluation A noisy-and tr an b usd to valuat P ( x ) givn P ( y) or P ( y) for ah root nod. Th omputation an b prformd rursivly by domposing th noisy-and tr into subtrs. Th following lmma shows that suh domposition is valid. Lmma 3 Lt T b a noisy-and tr, th laf of T b v, and th gat onntd to v b g. Ltv and g b dltd from T, as wll as th links inoming to g. In th rmaining graph, ah omponnt is ithr an isolatd vnt nod or a noisy- AND tr. A noisy-and tr an b valuatd aording to th following algorithm. Algorithm GtCausalEvntProb(T) Input: A noisy-and tr T. 6

7 dnot laf of T by v and gat onntd to v by g; for ah nod w dirtly onntd to input of g, do if probability P (w) for vnt at w is not spifid, dnot sub-and-tr with w as th laf by T w ; P (w) =GtCausalEvntP rob(t w ); if (w, g) is a forward link, P (w) =P (w); ls P (w) = P (w); P (v) = w P (w); rturn P (v); Th following thorm stablishs soundnss of GtCausalEvntProb. W dfin th dpth of a noisy-and tr to b th maximum numbr of gat nods ontaind in a path from a root to th laf. Thorm Lt T b a noisy-and tr whr probability for ah root nod is spifid in th rang (0, ) and P (v) b rturnd by GtCausalEvntP rob(t ). Thn P (v) is a probability in th rang (0, ) and it ombins givn probabilitis aording to rinformnt or undrmining spifid by th topology of T. Not that th topology of T is a ruial pi of knowldg. For th abov xampl, suppos th physiian artiulats a diffrnt ordr, whih is shown in Figur 2 (b). Th physiian fls that rinforing intration btwn and 2 should b onsidrd first. Th undrmining intration btwn sts {, 2 } and { 3 } should thn b onsidrd. Applying GtCausalEvntProb, w obtain P (, 2 )=0.03 and P (, 2, 3 )= Rlaxing Dfault Assumptions A noisy-and tr assums, by dfault, failur indpndn for rinforing sts of auss and suss indpndn for undrmining sts of auss. For givn sts of auss, th xprt may disagr with suh assumptions. This may manifst in trms of disagrmnt of th xprt with output vnt probability of a noisy-and gat. Whn this ours, noisy-and tr rprsntation allows asy modifiation by dlting th orrsponding AND gat from th tr. In partiular, lt g b th gat in qustion and its output b onntd to nod v. If th xprt disagrs with th vnt probability omputd for nod v, th ntir subtr with v as th laf an b disardd by dlting th link (g, v). Nod v rmains in th rsultant nw noisy-and tr as a root nod. Th xprt an thn spify a propr vnt probability for v. For instan, with th noisy-and tr in Figur 2 (a), suppos that th xprt disagrs with P (, 3 )= Instad, (s)h fls that 0.4 is mor appropriat. Not that this assignmnt is onsistnt with th undrmining intration btwn and 3, but th dgr of undrmining is diffrnt from what th dfault assumption ditats. W an thn rmov root nods labld by and 3 as wll as th gat that thy ar onntd to. As th rsult, nod, 3 boms a root nod and P (, 3 )=0.4 an 7

8 b assignd to it. Applying GtCausalEvntProb to th nw noisy-and tr, w obtain P (, 2, 3 )=0.88. This flxibility of noisy-and tr allows it to b usd in an intrativ way, inrasing its xprssiv powr as a tool for probability liitation: An xprt an start by artiulating a noisy-and tr whr ah root is labld by a singl aus i. Th dfault assumptions on failur and suss indpndn now allow omputation of probability for ah non-root ausal vnt. This an b viwd as th first approximation of th xprt s subjtiv blif. Th xprt an thn xamin ah omputd vnt probability and did if it is onsistnt with his/hr blif. Upon idntifiation of disagrmnt ovr a nod v onntd to th output of a gat g, th xprt an tra bakward to input vnts onntd to g. Th xprt will did whthr (s)h disagrs with th probabilitis of any input vnts. If no suh disagrmnt is idntifid, thn th xprt must b disagring with th dgr of rinformnt or undrmining implid by th assumption on failur or suss indpndn. (S)h an thn assss a probability for th output vnt as w illustratd abov. Not that this assssmnt, with th omputd probability as rfrn, is asir than an assssmnt to b mad from vauum. On th othr hand, if disagrmnt with th probability of an input vnt is idntifid, th prossing ontinus by traing furthr bak towards root nods. It is possibl that as th xprt tras disagrmnts, maks modifiations to vnt probabilitis, and dlts subtrs, a dp noisy-and tr startd with boms vry shallow in th nd. Many root nod labls now onsist of a subst of auss, instad of a singl on at th start. Th topology of th rsultant noisy-and tr boms vry diffrnt. This dos not man that th original noisy-and tr was wrong. It has disappard aftr srving its usful rol in liitation. 7 Eliitation of CPTs With Noisy-AND Trs W dmonstrat how to us noisy-and trs to liit CPTs in BNs with an xampl. Considr an fft (hild) variabl with a st of svn auss (parnts) in a BN:,..., 7. Suppos that a domain xprt idntifis th following thr substs of auss and intration within ah subst: Subst s : and 2 ar undrmining ah othr. Subst s 2 : 2, 3 and 4 ar rinforing ah othr. Subst s 3 : 6 and 7 ar rinforing ah othr. Th xprt asssss that intration btwn substs s and s 2 is also undrmining and, togthr as a group, thy rinfor s 3. Without furthr quantitativ information, ths assssmnts produ th noisy-and tr in Figur 3 (a). Suppos that th following probabilitis for singl-aus vnts ar also providd: P ( )=0.65,P( 2 )=0.35,P( 3 )=0.8, P ( 4 )=0.3,P( 5 )=0.6,P( 6 )=0.75,P( 7 )=

9 2 g, 2 3 g g2,, g 4,...,,..., 2,..., 2 n,..., 3 n... 3 n,..., 2 n,,..., n, 2 n,,,,, g 5, 6 7,,..., 2 n (a),,,,,, (b) Fig. 3. (a) An xampl noisy-and tr. (b) Graphial modl for rursiv noisy-or. To assss P (,..., 7 ), w apply GtCausalEvntProb to obtain P (,..., 7 )=P (,..., 7 )=0.92. To assss P (, 2, 30, 4, 5, 6, 7 ), liminat nod 3 from Figur 3 (a) and modify output labls for g 2, g 3 and g 5. Th valuation givs P (, 2, 30, 4, 5, 6, 7 )=P (, 2, 4, 5, 6, 7 )= W hav usd th sam noisy-and tr to assss both probabilitis abov. This is not nssary. That is, noisy-and trs do not rquir that diffrnt ausal probabilitis to b assssd using th sam tr. If th xprt fls that a partiular ombination of a subst of auss follows a diffrnt pattrn of intration, a distint noisy-and tr an b usd, without produing invalid CPT. Commonly, w xpt that on tr an b usd for assssmnt of all probabilitis in a CPT. If th xprt is happy with th rsult, th omplxity of his/hr assssmnt task is only O(n), whr n is th numbr of auss. Suppos that th xprt blivs that is too high for P (, 2, 30, 4, 5, 6, 7 ) and (s)h attributs to th output from gat g 4 P ( 6, 7 )= 0.3 as too low. Instad, (s)h blivs 0.2 is a bttr assssmnt. In rspons, w rmov th subtr with g 4 as th laf and spify 0.2 as th probability for th nw root vnt nod 6, 7. GtCausalEvntProb now gnrats P (, 2, 30, 4, 5, 6, 7 )= Rlatd Modls Of Causal Intration W ompar noisy-and trs with rlatd modls of ausal intration. As w hav dfind rinformnt and undrmining undr th binary ontxt, th following analysis is rstritd to suh ontxt if appropriat. Som modls of ausal intration ar limitd to rprsnt ithr rinformnt or undrmining but not both. Noisy-MAX modl [] boms noisy-or modl whn variabls ar binary. Thrfor, from Lmma, whn domain is binary, noisy-max rprsnts only rinforing intration. 9

10 Similarly, noisy-min modl [2] boms noisy-and whn variabls ar binary. Hn, aording to Lmma 2, whn domain is binary, noisy-min rprsnts only undrmining intration. Lmmr and Gossink [6] proposd RNOR to modl rinformnt. To assss fft probability du to a st of auss, RNOR modl an ombin ausal probabilitis du to substs of auss, whr ah subst may not b singlton. Thir ombination at subst lvl has inflund our thinking in formulation of noisy-and trs. Aording to RNOR, for a st of auss X = {,..., n },if P (,..., n ) is not providd by th xprt, it is stimatd as P (,..., n )= n i= P (,..., i,, i+,,..., n ) P (,..., i,, i+2,,..., n ) (6) as long as auss in X ar rinforing. Howvr, if auss in X ar undrmining, th rsult from th quation may not b a valid probability. No graphial rprsntation of RNOR was proposd in [6]. W prsnt a graphial modl whih rvals th indpndn assumption undrlying RNOR. Using failur vnts, w rwrit Eqn (6) blow: P (,..., n )= = = i= n i= P (,..., i,, i+,,..., n ) P (,..., i,, i+2,,..., n ) n P (( i+, ) (,..., i,, i+2,,..., n )) (8) P (,..., i,, i+2,,..., n ) n P ( i+,,..., i,, i+2,,..., n ) (9) i= Figur 3 (b) shows th graphial modl of RNOR basd on Eqn (7) and Eqn (9). A gat rprsnting onditioning has bn introdud and is shown as a triangl with a vrtial bar in th ntr. W rfr to th gat as a COND gat. Th output of a COND gat is th vnt of its lft input vnt onditiond on its right input vnt. Not that i+,,..., i,, i+2,,..., n is a wll dfind vnt. Eah input vnt to a COND gat is assoiatd with a ral potntial. Its output vnt is assignd a potntial dfind by th division of th two input potntials (th on in th lft dividd by that in th right). For th AND gat, its output vnt is assignd a potntial dfind by th produt of potntials of its inputs. Inputs of ah gat ar not onntd in any path othr than through th gat. Eqn (9) and Figur 3 (b) rval that RNOR modl assums that onditional failur vnt dnotd by i+,,..., i,, i+2,,..., n (whr i runs from to n) is indpndnt of ah othr. This is not surprising as RNOR is drivd from rwriting Eqn (3) and it assums failur indpndn among all auss. Howvr, whn RNOR is usd rursivly by rplaing dfault probabilitis on input of som COND gats, th indpndn assumption is invalidatd, (7) 0

11 whil th topology of th graphial modl and th rul of probability ombination (Eqn (6)) rmain and do not rflt suh invalidation. On th othr hand, indpndn assumptions mad in noisy-and trs ar loal to ah gat. Assumption mad rlativ to a gat govrns only th probability ombination at th output of th gat and is indpndnt of th assumptions mad at othr gats. Whn th dfault probability produd by a gat is rplad and th orrsponding subtr rmovd, it dos not invalidat any indpndn assumptions at othr gats in th rmaining noisy-and tr. That is, modifiation of a noisy-and tr dos not invalidat th ohrn of th undrlying indpndn assumptions. Noisy-addition [3] an rprsnt nithr rinformnt nor undrmining. Considr a noisy-addr with two binary auss and 2 whos domains ar {0, }. It has th following DAG modl, whr i and i 2 ar intrmdiat variabls and fft = i + i 2 : i i 2 2 Th modl assums P (i j =0 j = 0) = and 0 <P(i j = j =)< for j =, 2. For simpliity, w assum P (i = =)=P (i 2 = 2 =) and dnot thir valu by q. Not that P ( = =)=P (i = = ). To did whthr this modl an rprsnt rinformnt or undrmining, P ( = =, 2 = ) should b ompard with q. W driv th following: P ( = =, 2 =) = P (i =0,i 2 = =, 2 =)+P (i =,i 2 =0 =, 2 =) = P (i =0 =)P (i 2 = 2 =)+P (i = =)P (i 2 =0 2 =) Dnoting P ( = =, 2 =)byr, w hav r =2q( q). If q<0.5, thn r> q. Ifq>0.5, thn r<q. By Dfinition, if a ausal modl is rinforing, thn no mattr what valu P ( y ) is, th rlation P ( y ) P ( x ) must hold and rvrs of th inquality must hold for undrmining. Bing unabl to maintain th inquality aross th ntir rang of valus for P ( y ) implis that noisy-addition is unabl to rprsnt ithr rinformnt or undrmining. Noisy-AND trs diffr from thos onsidrd by Hkrman and Brs [4] in that th amhanisti modl has ssntially a star topology and othr thr modls (domposabl, multiply domposabl and tmporal) ar ssntially binary trs. Whn th binary tr is instantiatd aording to noisy-or, noisy-and, noisy-max, noisy-min, noisy-addition, it inhrits limitations of ths modls as disussd abov. In ths modls, ah root nod must b a singl aus variabl, whil noisy-and trs allow a root nod to rprsnt a ausal vnt of multipl auss. Parl [8] analyzd ausation using funtional ausal modls. Our work is onsistnt with his funtional approah and in partiular proposs noisy-and trs as a usful boolan funtional modl.

12 9 Conlusions Causal intrations may b rinforing or undrmining. Distintion of thm an failitat ausal modling and CPT liitation in onstrution of Baysian ntworks. W hav shown that xisting ausal modls an modl ithr on typ of intrations (suh as noisy-or, noisy-and, noisy-max, noisy-min and rursiv noisy-or) or non of thm (suh as noisy-addition). W prsnt th first xpliit ausal modl, trmd noisy-and trs, that an nod both rinformnt and undrmining. Furthrmor, xisting ausal modls, xpt rursiv noisy-or, limit modl paramtrs to probabilitis of singl aus vnts, and rursiv noisy-or introdus inonsistnt dpndn assumptions whn probabilitis of multi-aus vnts ar intgratd through rursion. On th othr hand, noisy-and trs intgrat probabilitis of both singl aus vnts and multi-aus vnts ohrntly. Thrfor, noisy-and trs provid a simpl yt powrful nw approah for knowldg liitation in probabilisti graphial modls. Aknowldgmnts Th finanial support from National Sins and Enginring Rsarh Counil (NSERC) of Canada through Disovry Grant is aknowldgd. Rfrns. F.J. Diz. Paramtr adjustmnt in Bays ntworks: Th gnralizd noisy orgat. In D. Hkrman and A. Mamdani, ditors, Pro. 9th Conf. on Unrtainty in Artifiial Intllign, pags Morgan Kaufmann, S.F. Galan and F.J. Diz. Modling dynami ausal intratiosn with Baysian ntworks: tmporal noisy gats. In Pro. 2nd Intr. Workshop on Causal Ntworks, pags 5, D. Hkrman. Causal indpndn for knowldg aquisition and infrn. In D. Hkrman and A. Mamdani, ditors, Pro. 9th Conf. on Unrtainty in Artifiial Intllign, pags Morgan Kaufmann, D. Hkrman and J.S. Brs. Causal indpndn for probabilisti assssmnt and infrn using Baysian ntworks. IEEE Trans. on Systm, Man and Cybrntis, 26(6):826 83, M. Hnrion. Som pratial issus in onstruting blif ntworks. In L.N. Kanal, T.S. Lvitt, and J.F. Lmmr, ditors, Unrtainty in Artifiial Intllign 3, pags Elsvir Sin Publishrs, J.F. Lmmr and D.E. Gossink. Rursiv noisy OR - a rul for stimating omplx probabilisti intrations. IEEE SMC, Part B, 34(6): , J. Parl. Probabilisti Rasoning in Intllignt Systms: Ntworks of Plausibl Infrn. Morgan Kaufmann, J. Parl. Causality: Modls, Rasoning, and Infrn. Cambridg Univrsity Prss, S. Srinivas. A gnralization of noisy-or modl. In D. Hkrman and A. Mamdani, ditors, Pro. 9th Conf. on Unrtainty in Artifiial Intllign, pags Morgan Kaufmann,