Identifying Dynamic Sequential Plans

Transcription

1 Identfyng Dynamc Sequental Plans Jn Tan Department of Computer Scence Iowa State Unversty Ames, IA Abstract We address the problem of dentfyng dynamc sequental plans n the framework of causal Bayesan networks, and show that the problem s reduced to dentfyng causal effects, for whch there are complete dentfcaton algorthms avalable n the lterature. 1 Introducton Ths paper deals wth the problem of evaluatng the effects of sequental plans from a combnaton of nonexpermental data and qualtatve causal assumptons. The causal assumptons wll be represented n the form of an acyclc causal dagram [Sprtes et al., 1993, Heckerman and Shachter, 1995, Laurtzen, 2000, Pearl, 2000] n whch arrows represent the potental exstence of drect causal relatonshps between the correspondng varables. The causal dagram may contan unmeasured varables, and our task s to decde whether we can estmate the effects of a sequence of actons from the observed data. We motvate the study by consderng a medcal treatment problem dscussed n [Pearl and Robns, 1995, Dawd and Ddelez, 2005]. There are a sequence of medcal treatments (X 1,..., X k ) appled to a patent over tme. We have observatons Z before and between the treatments. Doctors may prescrbe a treatment based on prevous treatments and observatons. There s a outcome varable Y (say survval) of specal nterest and we want to estmate the effects of the sequental treatments on Y. In general there may be unobserved confounders that have nfluence on the observed varables. There are dfferent possble strateges for choosng the treatment acton (X ). The smplest acton nvolves fxng the value of X to a partcular value x, called atomc nterventon and denoted by do(x ) n [Pearl, 2000], e.g. fxng the dosage of a treatment rrespectve of any observatons on the patent. A uncondtonal plan conssts of a sequence of pre-defned atomc actons. The problem of dentfyng uncondtonal plans s to compute the dstrbuton of Y under atomc nterventons on a set X of acton varables, denoted by P x (y) = P (y do(x)), a quantty known as the causal effects of X on Y. Suffcent graphcal crtera for the dentfablty of uncondtonal plans are derved n [Pearl and Robns, 1995]. Recently the general problem of dentfyng causal effects P x (y) n a causal dagram contanng unobserved varables has been solved and complete algorthms for dentfcaton are gven n [Tan and Pearl, 2003, Shptser and Pearl, 2006b, Huang and Valtorta, 2006]. In general we may want to use dynamc treatment strateges n whch the values of acton varables (X ) are determned based on the prevously observed varables and prevously taken actons. Suffcent graphcal crtera for the dentfablty of dynamc plans are derved n [Dawd and Ddelez, 2005]. However the dentfablty problem s far from beng solved. In ths paper, we show that the problem of dentfyng dynamc sequental plans can be reduced to the well-studed problem of dentfyng causal effects and therefore essentally solved the sequental plan dentfcaton problem. Although Pearl (2000, Secton 4.2) has suggested that dynamc condtonal plans may be dentfed by dentfyng condtonal causal effects of the form P x (y c), for whch complete dentfcaton algorthms have been developed [Tan, 2004, Shptser and Pearl, 2006a], n ths paper, we wll show that ths gves a suffcent condton for dentfyng dynamc sequental plans but t s not necessary. The rest of the paper s organzed as follows. In Secton 2, we revew the work n [Dawd and Ddelez, 2005] and defne useful notaton. In Secton 3, we formulate the sequental plan problem n the framework of causal Bayesan networks. We

2 show how to reduce the problem of dentfyng dynamc sequental plans nto a problem of dentfyng causal effects n Secton 4, and dscuss n Secton 5 the problem versus that of dentfyng uncondtonal plans and condtonal causal effects. Secton 6 concludes the paper. 2 Prevous Work and Notaton Dawd and Ddelez (2005) formulated the problem of dentfyng dynamc sequental plans n the framework of regme ndcators and nfluence dagrams. An nfluence dagram (ID) s a DAG over a set V = {V 1,..., V n } of varables that also ncludes regme ndcators as specal nodes of ther own called decson nodes [Dawd, 2002]. We assume that all varables are dscrete. The DAG s assumed to represent condtonal ndependence assertons that each varable s ndependent of all ts non-descendants gven ts drect parents n the graph. 1 These assertons mply that the jont probablty functon P (v) = P (v 1,..., v n ) factorzes accordng to the product [Pearl, 1988] P (v) = P (v pa ) (1) where pa are (values of) the parents of varable V n the graph. 2 The queston of causal nference s consdered as a problem of nference across dfferent regmes, n whch we may ntervene n certan varables n certan ways and observe the behavor of other varables. Regme ndcators are used to represent dfferent types of nterventons. Here we wll roughly follow the notaton used n [Ddelez et al., 2006]. The regme ndcator for an nterventon n a varable V s denoted by σ V and can take values n a set of strateges. Under strategy σ V, the condtonal probablty P (v pa ) s changed to P (v pa ; σ V ). We wll consder the followng types of nterventons. Idle regme σ V = : No nterventon takes place, therefore P (v pa ; σ V = ) = P (v pa ). The dle regme s also called the observatonal regme under whch we wll assume that observatonal data has been collected. Therefore P (v pa ) can be estmated from data f V and P a are observed. 1 We use famly relatonshps such as parents, chldren, and ancestors to descrbe the obvous graphcal relatonshps. 2 We use uppercase letters to represent varables or sets of varables, and use correspondng lowercase letters to represent ther values (nstantatons). Atomc nterventon σ V = do(v ): The strategy of settng V to a fxed value v, denoted by do(v = v ) or smply do(v ) n Pearl (2000), such that P (v pa ; σ V = do(v )) = δ(v, v ), where δ(v, v ) s one f v = v and zero otherwse. Condtonal nterventon σ V = do(g(c)): In general, V may be made to respond n a specfed way to some set C of prevously observed varables, denoted by do(v = g(c)) n Pearl (2000), such that P (v pa ; σ V = do(g(c))) = δ(v, g(c)), where g(.) s a pre-specfed determnstc functon and the varables n C can not be descendants of V. Random nterventon σ V = d C : More generally, we may let V take on a random value accordng to some dstrbuton possbly dependng on some set C of prevously observed varables such that P (v pa ; σ V = d C ) = P (v c), where P (v c) s a pre-specfed probablty dstrbuton and the varables n C can not be descendants of V. In a sequental decson problem, we may ntervene, at least n prncple, n a set of varables X = {X } V, called control varables or acton varables, and are nterested n the response of a varable Y, called response varable or outcome varable. Let Z be the rest of observed varables whch are often called covarates. The varables are assumed to be ordered n a sequence (L 1, X 1,..., L K, X K, Y ) where L Z are the set of observed covarates after X 1 and before X. We denote L = (L 1,..., L ) and X = (X 1,..., X ). Gven an nterventon strategy σ X = {σ X }, under a condton called smple stablty whch says that the observed covarates L and the outcome Y are ndependent of how acton varables are generated once all earler observables ( L 1, X 1 for L ; X, Z for Y ) are gven, Dawd and Ddelez (2005) show that the postnterventon dstrbuton of Y s dentfed as P (y; σ X ) P (y x, z) P (l l 1, x 1 ) x,z P (x x 1, l ; σ X ), (2) where P (x x 1, l ; σ X ) are determned by the chosen regme and the other quanttes can be estmated from

3 observatonal data. Eq. (2) s known as the G-formula, and has been obtaned n [Robns, 1986, Robns, 1987] n the framework of potental response models. When there are unobserved confounders, the smple stablty may not hold. Dawd and Ddelez (2005) makes extended stablty assumpton whch essentally s (smple) stablty wth respect to the extended doman that ncludes unobserved U varables gnorng the dstncton between Z and U. The G-formula (2) no longer holds unless we nclude unobserved U varables, but then the condtonal probabltes nvolvng U varables can no longer be estmated from the data. Suffcent graphcal crtera for dentfyng P (y; σ X ) are derved. The crtera were obtaned by dentfyng graphcal condtons under whch the smple stablty can be reganed such that the G-formula can be used, and by extendng the work n [Pearl and Robns, 1995] to dynamc plans. 3 Problem Formulaton In ths paper, we wll formulate the sequental plan problem n the framework of causal Bayesan networks. A causal Bayesan network (CBN) conssts of a DAG G over a set V = {V 1,..., V n } of varables, called a causal dagram. The nterpretaton of such a graph has two components, probablstc and causal. The probablstc nterpretaton vews G as representng condtonal ndependence assertons such that the jont probablty functon P (v) = P (v 1,..., v n ) factorzes accordng to Eq. (1). The causal nterpretaton vews the drected edges n G as representng causal nfluences between the correspondng varables. In ths nterpretaton, the factorzaton of (1) stll holds, but the factors are further assumed to represent autonomous data-generaton processes, that s, each condtonal probablty P (v pa ) represents a stochastc process by whch the values of V are chosen n response to the values pa (prevously chosen for V s parents), and the stochastc varaton of ths assgnment s assumed ndependent of the varatons n all other assgnments. Moreover, each assgnment process remans nvarant to possble changes n the assgnment processes that govern other varables n the system. Ths modularty assumpton enables us to predct the effects of nterventons, whenever nterventons are descrbed as specfc modfcatons of some factors n the product of (1). We typcally assume that every varable V can potentally be manpulated by external nterventon. So we mght thnk of a CBN as an ID such that each node s (mplctly) ponted to by a correspondng regme/nterventon ndcator. In a sequental decson problem, we may ntervene n a set of acton varables X = {X } V, and are nterested n the response of a set of outcome varables Y. Assume that all the varables V are observed and let the rest of covarate varables be Z = {Z } = V \ (X Y ). Gven an nterventon strategy σ X = {σ X }, by modularty assumpton, we can predct the effects of σ X as P (v; σ X ) = P (y pa y ) P (z pa z ) P (x pa x ; σ X ), (3) where, by modularty assumpton, those condtonal probabltes correspondng to unmanpulated varables reman unaltered. We note that Dawd and Ddelez s (2005) smple stablty assumpton leads to Eq. (3) n the framework of CBNs. We see that, gven a CBN, whenever all varables n V are observed, the consequence of an nterventon strategy on the outcome varables Y s computed as P (y; σ X ) P (y pa y ) P (z pa z ) x,z P (x pa x ; σ X ), (4) where P (x pa x ; σ X ) are determned by the chosen regme and the other quanttes can be estmated from observatonal data. We note that the G-formula (2) can be reduced to Eq. (4) by usng the condtonal ndependence relatonshps mpled by the CBN that each varable s ndependent of all ts non-descendants gven ts parents. In general we may be concerned wth confoundng effects due to unobserved nfluental varables. In the presence of unobserved confounders, the dstrbuton over observed varables can no longer factorze accordng to (1). Lettng V = Y Z X and U = {U 1,..., U n } stand for the sets of observed and unobserved varables, respectvely, the observed probablty dstrbuton, P (v), becomes a mxture of products: P (v) P (v pa v ) P (u pa u ). u { V V } { U U} (5) We stll make modularty assumpton n the CBN wth unobserved varables, and the effects of an nterventon strategy σ X on the outcome varables Y can be expressed as P (y; σ X ) P (y pa y ) P (z pa z ) x,z,u P (x pa x ; σ X ) P (u pa u ). (6)

4 We note that Dawd and Ddelez s (2005) extended stablty assumpton leads to Eq. (6) n the framework of CBNs. In (6), the quanttes P (y pa y ) and P (z pa z ) (and P (u pa u )) may nvolve elements of U and may not be estmable from data. Then the queston of dentfablty arses,.e., whether t s possble to express P (y; σ X ) as a functon of the observed dstrbuton P (v). Defnton 1 [Plan Identfablty] A sequental plan s sad to be dentfable f P (y; σ X ) s unquely computable from the observed dstrbuton P (v). 4 Identfcaton of Sequental Plans Frst we make the followng assumpton about the type of nterventons we wll consder. Assumpton 1 P (x pa x ; σ X ) does not depend on the unobserved varable. That s, for condtonal nterventon σ X = do(g(c)) or random nterventon σ X = d C, we requre C X Z. Ths assumpton corresponds to Condton 6.6 or 7.2 n [Dawd and Ddelez, 2005]. Under Assumpton 1, Eq. (6) becomes P (y; σ X ) P (x pa x ; σ X ) P (y pa y ) x,z u P (z pa z ) P (u pa u ) (7) P (x pa x ; σ X )P x (y, z) (8) x,z Obvously a suffcent condton for P (y; σ X ) beng dentfable s that the causal effect P x (y, z) s dentfable. In partcular, a smple suffcent condton for P x (y, z) beng dentfable s f all the parents of acton (X) varables are observables, whch s Condton 6.3 n [Dawd and Ddelez, 2005]. Proposton 1 If all the parents of acton (X) varables are observables, then P (y; σ X ) s dentfable [Dawd and Ddelez, 2005]. Proof: If all the parents of acton (X) varables are observables, then P (x pa x ) contans no unobserved (U) varables, and Eq. (5) can be wrtten as P (v) = P (x pa x )P x (y, z), (9) from whch we obtan that P x (y, z) s dentfed as p(x, y, z) P x (y, z) = P (x (10) pa x ) = P (v v ), (11) { V Y Z} where we have used the chan rule assumng an order of V varables that s consstent wth the DAG and v denotes the V varables ordered ahead of V. Hence the sequental plan s dentfed as P (y; σ X ) x,z P (x pa x ; σ X ) whch s essentally the G-formula (2). { V Y Z} P (v v ), (12) In general P x (y, z) beng dentfable s not a necessary condton for P (y; σ X ) beng dentfable. Eq. (7) may be smplfed n that a factor P (z pa z ) may be summed out (usng z P (z pa z ) = 1) f Z does not appear n any other factors (graphcally, f Z does not have any chldren). We can derve stronger dentfcaton crteron by summng out as many factors as possble from Eq. (7). Before presentng our result, we frst ntroduce some notaton. Followng [Tan and Pearl, 2003], for any observed set S V of varables, we defne the quantty Q[S] to denote the post-nterventon dstrbuton of S under atomc nterventons to all other varables: Q[S](v) = P v\s (s) u { V S} P (v pa v ) { U U} P (u pa u ). (13) For convenence, we wll often wrte Q[S](v) as Q[S]. Eq. (7) can be wrtten as P (y; σ X ) x,z P (x pa x ; σ X )Q[Y Z]. (14) Let G σx denote the manpulated graph under the nterventon strategy σ X, whch can be constructed from the orgnal causal graph G as follows: For an atomc nterventon σ X = do(x ), cut off all the arrows enterng X ; For a condtonal nterventon σ X = do(g (c )) or a random nterventon σ X = d C, cut off all the arrows enterng X and then add an arrow enterng X from each varable n C.

5 Based on Eq. (14), we obtan the followng suffcent crteron for dentfyng P (y; σ X ). Theorem 1 Let Z D be the set of varables n Z that are ancestors of Y n G σx. P (y; σ X ) s dentfable f the causal effects Q[Y Z D ] = P x,z\zd (y, z D ) s dentfable. Proof: Let X D be the set of varables n X that are ancestors of Y n G σx. Then all the non-ancestors of Y can be summed out from Eq. (14) leadng to P (y; σ X ) P (x pa x ; σ X )Q[Y Z D ] x D,z D { X X D} x D,z D { X X D} P (x pa x ; σ X )P x,z\zd (y, z D ). (15) (16) We conjecture that the condton n Theorem 1 s also necessary. It mght appear that Eq. (15) can be further smplfed as follows. Let Z σxd be the set of Z varables that appear n the term { X P (x X D} pa x ; σ X ) (the set of condtonng varables n the strategy σ XD ). Then the term { X P (x X D} pa x ; σ X ) s a functon of X D and Z σxd. Eq. (15) becomes P (y; σ X ) x D,z σxd { X X D} x D,z σxd g(x D, z σxd ) P (x pa x ; σ X ) z D\z σxd Q[Y Z D ] (17) z D\z σxd Q[Y Z D ] (18) From Eq. (18), P (y; σ X ) s dentfable f z D\z σxd Q[Y Z D ] s dentfable, and ntutvely, the condton appears to be necessary too, snce the term g(x D, z σxd ) s specfed externally and no more factors can be summed out (as far as the functon g(.) s not ndependent of any varables n z σxd ). A strct proof of ths necessty s stll under study. On the other hand, due to the fact that none of the factors correspondng to the varables n Z D \ Z σxd can be summed out from Q[Y Z D ], t has been shown that z D\z σxd Q[Y Z D ] can be dentfed only va dentfyng Q[Y Z D ] and t s a f and only f condton (Lemma 11 n [Huang and Valtorta, 2006]). So from the pont of vew of dentfyng P (y; σ X ) the reducton from Eq. (15) to Eq. (17) s not necessary. U Z X 1 X 2 Y Fgure 1: An example causal graph We therefore have reduced the problem of dentfyng dynamc sequental plans P (y; σ X ) nto that of dentfyng causal effects Q[Y Z D ] whle the latter problem has been solved and complete algorthms are gven n [Tan and Pearl, 2003, Shptser and Pearl, 2006b, Huang and Valtorta, 2006]. We demonstrate the applcaton of Theorem 1 and the dentfcaton process wth an example. Consder the problem of dentfyng P (y; σ X1, σ X2 ) n Fgure 1, whch was studed n [Dawd and Ddelez, 2005] and s troublng to the methods presented theren. Theorem 1 calls for dentfyng Q[{Y, Z}] whch can be shown to be dentfable. We are gven the observatonal dstrbuton where P (v) = P (y x 1, x 2, z)p (x 2 )Q[{Z, X 1 }], (19) Q[{Z, X 1 }] u We want to compute P (z x 2, u)p (x 1 u)p (u). (20) P (y; σ X1, σ X2 ) P (x 1 ; σ X1 )P (x 2 ; σ X2 )P (y x 1, x 2, z)q[{z}], x 1,x 2,z (21) whch calls for computng Q[{Z}]. From Eq. (20), t s clear that From Eq. (19), we obtan Q[{Z}] x 1 Q[{Z, X 1 }]. (22) Q[{Z, X 1 }] = P (z, x 1 x 2 ) (23) Therefore Q[{Z}] s dentfed and we fnally obtan P (y; σ X1, σ X2 ) P (x 1 ; σ X1 )P (x 2 ; σ X2 )P (y x 1, x 2, z)p (z x 2 ). x 1,x 2,z (24)

6 X 1 Y U 1 X 2 Z (a) G U 2 X 1 Y U 1 X 2 Z (b) G σx Fgure 2: An example causal graph 5 Dscusson 5.1 Uncondtonal plans are easer In general dentfyng dynamc plans s more dffcult than dentfyng uncondtonal plans that nvolve only atomc nterventons. Let an nterventon strategy σ X consst of all atomc nterventons. Then the manpulated graph G σ X s a subgraph of G σx. Let Z D be the set of varables n Z that are ancestors of Y n G σ X. Then Z D s a subset of Z D. We have P (y; σ X) = P x (y) z D U 2 Q[Y Z D]. (25) Therefore dentfyng P x (y) calls for dentfyng Q[Y Z D ] whle P (y; σ X) calls for dentfyng Q[Y Z D ]. Now we have Q[Y Z D] = z D\z D Q[Y Z D ] (26) The factors of the varables n Z D \Z D are summed out from Q[Y Z D ] snce the varables n Z D \ Z D can be ancestors of Y only through X s. In general, whenever Q[Y Z D ] (and therefore P (y; σ X )) s dentfable, then P x (y) s dentfable. However, t s possble that P x (y) s dentfable but Q[Y Z D ] (and therefore P (y; σ X )) s not. We demonstrate ths pont wth an example. Consder the problem of dentfyng P (y; σ X1, σ X2 ) n Fgure 2(a), whch was studed n [Pearl and Robns, 1995]. If P (x 2 x 1, z; σ X2 ) depends on Z, say σ X2 = do(g(x 1, z)), then Z D = {Z}, and by Theorem 1, to dentfy P (y; σ X1, σ X2 ) we need to dentfy Q[{Y, Z}], whch can be shown to be not dentfable (by theorems n [Huang and Valtorta, 2006]). More specfcally, gven the observatonal dstrbuton P (v) = P (x 2 x 1, z)q[{x 1, Z, Y }], (27) we want to dentfy P (y; σ X1, σ X2 ) P (x 1 ; σ X1 )P (x 2 x 1, z; σ X2 )Q[{Y, Z}] (28) x 1,x 2,z From Eq. (28), we see that f P (x 2 x 1, z; σ X2 ) depends on Z, then the dentfablty of P (y; σ X1, σ X2 ) depends on the dentfablty of Q[{Y, Z}]. We therefore conclude that P (y; σ X1, σ X2 ) s not dentfable. On the other hand, f P (x 2 x 1, z; σ X2 ) s ndependent of Z, say P (x 2 x 1, z; σ X2 ) = P (x 2 x 1 ) (or σ X2 = do(x 2 )), then the set Z D of varables n Z that are ancestors of Y n G σx becomes empty (see Fgure 2(b)), and, by Theorem 1, the dentfablty of P (y; σ X1, σ X2 ) depends on the dentfablty of Q[{Y }]. In fact, n ths case, Eq. (28) becomes P (y; σ X1, σ X2 = d X1 ) P (x 1 ; σ X1 )P (x 2 x 1 ) Q[{Y, Z}] x 1,x 2 z P (x 1 ; σ X1 )P (x 2 x 1 )Q[{Y }] (29) x 1,x 2 From Eq. (27) we obtan Q[{X 1, Z, Y }] = P (v)/p (x 2 x 1, z) = P (y x 1, x 2, z)p (x 1, z). (30) It can be shown (or confrmed) that Q[{Y }] z Q[{X 1, Z, Y }] y,z Q[{X 1, Z, Y }] We obtan z P (y x 1, x 2, z)p (z x 1 ). (31) P (y; σ X1, σ X2 = d X1 ) P (x 1 ; σ X1 )P (x 2 x 1 ) P (y x 1, x 2, z)p (z x 1 ). x 1,x 2 z (32) And n partcular, the uncondtonal plan s dentfed as P x1,x 2 (y) = Q[{Y }] z P (y x 1, x 2, z)p (z x 1 ). 5.2 Identfcaton va condtonal causal effects? (33) Pearl (2000) has suggested that dynamc sequental plans nvolvng condtonal and random nterventons

7 may be dentfed by dentfyng condtonal causal effects of the form P x (y z). For nterventons on a sngle varable X, we can show [Pearl, 2000, Secton 4.2] that and P (y; σ X = do(g(z))) z P (y; σ X = d Z ) x,z P x (y z) x=g(z) P (z), P x (y z)p (x z)p (z). Therefore t appears that P (y; σ X ) s dentfable f and only f P x (y z) s dentfable. Ths dea was generalzed to dynamc sequental plans. Consder a plan nvolvng a sequence of condtonal nterventons σ X = do(g (C )). Let Z σx = Z ( C ) be the set of condtonng varables n the strategy σ X. Pearl (2006) shows that P (y; σ X ) z σx P xz (y z σx )P xz (z σx ), (34) where x z are the values attaned by X when the condtonng set Z σx takes the values z σx. Pearl then suggests that sequental condtonal plans can be dentfed by dentfyng condtonal causal effects P x (y z) and P x (z). Ths motvated the study of the dentfablty of condtonal causal effects and complete algorthms have been developed n [Tan, 2004, Shptser and Pearl, 2006a]. Next we show that although the dentfcaton of P xz (y z σx ) and P xz (z σx ) s suffcent for dentfyng P (y; σ X ), t s nonetheless not necessary. Rewrte Eq. (34) n the followng P (y; σ X ) δ(x, g (C ))P x (y, z σx ) (35) x,z σx δ(x, g (C )) Q[Y, Z] (36) x,z σx z\z σx Comparng the reducton from Eq. (14) nto (17) wth the reducton from (14) to (36), we obtan that f X D = X then (36) s equvalent to (17), otherwse Eq. (36) may be further reduced n that more factors could be summed out from Q[Y, Z]. We obtan the followng concluson If all the varables n X are ancestors of Y n G σx, then the sequental plan P (y; σ X ) can be dentfed by dentfyng the causal effects P x (y, z σx ), otherwse t s possble that P (y; σ X ) s dentfable even f P x (y, z σx ) s not. We demonstrate ths pont wth an example. Consder the problem of dentfyng P (y; σ X ) where Z 1 U 1 U 2 Z 3 X 1 Z 2 X 2 X 3 Y (a) G Z 1 U 1 U 2 Z 3 X 1 Z 2 X 2 X 3 Y (b) G σx Fgure 3: An example causal graph σ X = {σ X1 = do(g 1 (Z 1 )), σ X2 = do(g 2 (Z 2 )), σ X3 = do(g 3 (Z 3 ))} n Fgure 3(a). The graph G σx s shown n Fgure 3(b). We have Z D = {Z 1, Z 3 }, and Theorem 1 calls for dentfyng Q[{Y, Z 1, Z 3 }] whch can be shown to be dentfable. On the other hand, P x1x 2x 3 (y, z 1, z 2, z 3 ) = Q[{Y, Z 1, Z 2, Z 3 }] s not dentfable. More specfcally, gven the observatonal dstrbuton P (v) =P (y x 1, x 3, z 3 )P (x 3 x 2, z 3 )P (z 1 )Q[{X 2, Z 3 }] Q[{X 1, Z 2 }], (37) we want to dentfy P (y; σ X ) δ(x, g (z ))P (y x 1, x 3, z 3 )P (z 1 ) x,z Q[{Z 3 }]Q[{Z 2 }] (38) x 1,x 3,z 1,z 3 δ(x 1, g 1 (z 1 ))δ(x 3, g 3 (z 3 ))P (y x 1, x 3, z 3 ) P (z 1 )Q[{Z 3 }], (39) where Q[{Z 3 }] can be dentfed as We obtan Q[{Z 3 }] u 2 P (z 3 u 2 )P (u 2 ) = P (z 3 ). (40) P (y; σ X ) z 1,z 3 P (y g 1 (z 1 ), g 3 (z 3 ), z 3 )P (z 1 )P (z 3 ). On the other hand, P x1x 2x 3 (y, z 1, z 2, z 3 ) = P (y x 1, x 3, z 3 )P (z 1 )Q[{Z 3 }]Q[{Z 2 }] (41) = P (y x 1, x 3, z 3 )P (z 1 )P (z 3 )Q[{Z 2 }] (42)

8 s not dentfable snce Q[{Z 2 }] s not dentfable. In fact the condtonal causal effect P x1x 2x 3 (y z 1, z 2, z 3 ) = P (y x 1, x 3, z 3 ) (43) s dentfable but the causal effect P x1x 2x 3 (z 1, z 2, z 3 ) = P (z 1 )P (z 3 )Q[{Z 2 }] (44) s not dentfable. 6 Concluson We present a method for dentfyng dynamc sequental plans. A closed-form expresson for the probablty of the outcome varables under a dynamc plan can be obtaned n terms of the observed dstrbuton, by usng the algorthms for dentfyng causal effects avalable n the lterature. Acknowledgments Ths research was partly supported by NSF grant IIS References [Dawd and Ddelez, 2005] A.P. Dawd and V. Ddelez. Identfyng the consequences of dynamc treatment strateges. Techncal report, Department of Statstcal Scence, Unversty College London, UK, [Dawd, 2002] A.P. Dawd. Influence dagrams for causal modellng and nference. Internatonal Statstcal Revew, 70(2), [Ddelez et al., 2006] V. Ddelez, A.P. Dawd, and S. Genelett. Drect and ndrect effects of sequental treatments. In Proceedngs of the 22nd Annual Conference on Uncertanty n Artfcal Intellgence, pages , [Heckerman and Shachter, 1995] D. Heckerman and R. Shachter. Decson-theoretc foundatons for causal reasonng. Journal of Artfcal Intellgence Research, 3: , [Huang and Valtorta, 2006] Y. Huang and M. Valtorta. Identfablty n causal bayesan networks: A sound and complete algorthm. In Proceedngs of the Twenty-Frst Natonal Conference on Artfcal Intellgence, pages , Menlo Park, CA, July AAAI Press. [Laurtzen, 2000] S. Laurtzen. Graphcal models for causal nference. In O.E. Barndorff-Nelsen, D. Cox, and C. Kluppelberg, edtors, Complex Stochastc Systems, chapter 2, pages Chapman and Hall/CRC Press, London/Boca Raton, [Pearl and Robns, 1995] J. Pearl and J.M. Robns. Probablstc evaluaton of sequental plans from causal models wth hdden varables. In P. Besnard and S. Hanks, edtors, Uncertanty n Artfcal Intellgence 11, pages Morgan Kaufmann, San Francsco, [Pearl, 1988] J. Pearl. Probablstc Reasonng n Intellgent Systems. Morgan Kaufmann, San Mateo, CA, [Pearl, 2000] J. Pearl. Causalty: Models, Reasonng, and Inference. Cambrdge Unversty Press, NY, [Pearl, 2006] J. Pearl. Comment on dentfyng condtonal plans [Robns, 1986] J.M. Robns. A new approach to causal nference n mortalty studes wth a sustaned exposure perod applcatons to control of the healthy workers survvor effect. Mathematcal Modelng, 7: , [Robns, 1987] J.M. Robns. A graphcal approach to the dentfcaton and estmaton of causal parameters n mortalty studes wth sustaned exposure perods. Journal of Chronc Dseases, 40(Suppl 2):139S 161S, [Shptser and Pearl, 2006a] I. Shptser and J. Pearl. Identfcaton of condtonal nterventonal dstrbutons. In R. Dechter and T.S. Rchardson, edtors, Proceedngs of the Twenty-Second Conference on Uncertanty n Artfcal Intellgence, pages AUAI Press, July [Shptser and Pearl, 2006b] I. Shptser and J. Pearl. Identfcaton of jont nterventonal dstrbutons n recursve sem-markovan causal models. In Proceedngs of the Twenty-Frst Natonal Conference on Artfcal Intellgence, pages , Menlo Park, CA, July AAAI Press. [Sprtes et al., 1993] P. Sprtes, C. Glymour, and R. Schenes. Causaton, Predcton, and Search. Sprnger-Verlag, New York, [Tan and Pearl, 2003] J. Tan and J. Pearl. On the dentfcaton of causal effects. Techncal Report R-290-L, Department of Computer Scence, Unversty of Calforna, Los Angeles, jtan/r290-l.pdf. [Tan, 2004] J. Tan. Identfyng condtonal causal effects. In Proceedngs of the Conference on Uncertanty n Artfcal Intellgence (UAI), 2004.