A Characterization of Interventional Distributions in Semi-Markovian Causal Models

Transcription

1 A Characterization of Interventional Distribtions in Semi-Markovian Casal Models Jin Tian and Changsng Kang Department of Compter Science Iowa State University Ames, IA {jtian, Jdea Pearl Cognitive Systems Laboratory Compter Science Department University of California, Los Angeles, CA Abstract We offer a complete characterization of the set of distribtions that cold be indced by local interventions on variables governed by a casal Bayesian network of nknown strctre, in which some of the variables remain nmeasred. We show that sch distribtions are constrained by a simply formlated set of ineqalities, from which bonds can be derived on casal effects that are not directly measred in randomized experiments. Introdction The se of graphical models for encoding distribtional and casal information is now fairly standard (Pearl 1988; Spirtes, Glymor, & Scheines 1993; Heckerman & Shachter 1995; Laritzen 2000; Pearl 2000; Dawid 2002). The most common sch representation involves a casal Bayesian network (BN), namely, a directed acyclic graph (DAG) G which, in addition to the sal conditional independence interpretation, is also given a casal interpretation. This additional featre permits one to infer the effects of interventions or actions, called casal effects, sch as those encontered in policy analysis, treatment management, or planning. Specifically, if an external intervention fixes any set T of variables to some constants t, the DAG permits s to infer the reslting post-intervention distribtion, denoted by P t (v), 1 from the pre-intervention distribtion P (v). A complete characterization of the set of interventional distribtions indced by a casal BN of a known strctre has been given in (Pearl 2000, pp.23-4) when all variables are observed. If we do not possess the strctre of the nderlying casal BN, can we still reason abot casal effects? One approach is to identify a set of properties or axioms that characterize casal relations in general, and se those properties as symbolic inferential rles. Assming deterministic fnctional relationships between variables, complete axiomatizations of casal relations sing conterfactals are given in (Galles & Pearl 1998; Halpern 2000). The reslting axioms, however, cannot be directly applied to probabilistic domains in Copyright c 2006, American Association for Artificial Intelligence ( All rights reserved. 1 (Pearl 1995; 2000) sed the notation P (v set(t)), P (v do(t)), or P (v ˆt) for the post-intervention distribtion, while (Laritzen 2000) sed P (v t). their deterministic setting, prior to deriving their probabilistic implications. Additionally, statisticians and philosophers have expressed sspicion of deterministic models as a basis for casal analysis (Dawid 2002), partly becase sch models stand contrary to statistical tradition and partly becase they do not apply to qantm mechanical systems. The casal models treated in this paper are prely stochastic. We seek a characterization for the set of interventional distribtions, P t (v), that cold be indced by some casal BN of nknown strctre. The motivation is two-fold. Assme that we have obtained a collection of experimental distribtions by maniplating varios sets of variables and observing others. We may ask several qestions: (1) Is this collection compatible with the predictions of some nderlying casal BN? That is, can this collection indeed be generated by some casal BN? (2) If we assme that the collection was generated by some nderlying casal BN (even if we do not know its strctre), what can we predict abot new interventions that were not tried experimentally? (that is, abot interventional distribtions that are not in the given collection.) These qestions can be answered by an axiomatization of interventional distribtions generated by casal BNs. When all variables are observed, a complete characterization of the set of interventional distribtions indcible by some casal BN is given in (Tian & Pearl 2002). In this paper, we will seek a characterization of interventional distribtions indcible by Semi-Markovian BNs, a class of Bayesian networks in which some of the variables are nobserved. We identify for properties that are both necessary and sfficient for the existence of a semi-markovian BN capable of generating any given set of interventional distribtions. Casal Bayesian Networks and Interventions A casal Bayesian network, also known as a Markovian model, consists of two mathematical objects: (i) a DAG G, called a casal graph, over a set V = {V 1,..., V n } of vertices, and (ii) a probability distribtion P (v), over the set V of discrete variables that correspond to the vertices in G. 2 The interpretation of sch a graph has two 2 We only consider discrete random variables in this paper.

2 components, probabilistic and casal. 3 The probabilistic interpretation views G as representing conditional independence restrictions on P : Each variable is independent of all its non-descendants given its direct parents in the graph. These restrictions imply that the joint probability fnction P (v) = P (v 1,..., v n ) factorizes according to the prodct P (v) = i P (v i pa i ) (1) where pa i are (vales of) the parents of variable V i in G. The casal interpretation views the arrows in G as representing casal inflences between the corresponding variables. In this interpretation, the factorization of (1) still holds, bt the factors are frther assmed to represent atonomos data-generation processes, that is, each conditional probability P (v i pa i ) represents a stochastic process by which the vales of V i are assigned 4 in response to the vales pa i (previosly chosen for V i s parents), and the stochastic variation of this assignment is assmed independent of the variations in all other assignments in the model. Moreover, each assignment process remains invariant to possible changes in the assignment processes that govern other variables in the system. This modlarity assmption enables s to predict the effects of interventions, whenever interventions are described as specific modifications of some factors in the prodct of (1). The simplest sch intervention, called atomic, involves fixing a set T of variables to some constants T = t, which yields the post-intervention distribtion { {i V P t (v) = i T } P (v i pa i ) v consistent with t. 0 v inconsistent with t. (2) Eq. (2) represents a trncated factorization of (1), with factors corresponding to the maniplated variables removed. This trncation follows immediately from (1) since, assming modlarity, the post-intervention probabilities P (v i pa i ) corresponding to variables in T are either 1 or 0, while those corresponding to nmaniplated variables remain naltered. If T stands for a set of treatment variables and Y for an otcome variable in V \ T, then Eq. (2) permits s to calclate the probability P t (y) that event Y = y wold occr if treatment condition T = t were enforced niformly over the poplation. This qantity, often called the casal effect of T on Y, is what we normally assess in a controlled experiment with T randomized, in which the distribtion of Y is estimated for each level t of T. When some variables in a Markovian model are nobserved, the probability distribtion over the observed variables may no longer be decomposed as in Eq. (1). Let 3 A more refined interpretation, called fnctional, is also common (Pearl 2000), which, in addition to interventions, spports conterfactal readings. The fnctional interpretation assmes strictly deterministic, fnctional relationships between variables in the model, some of which may be nobserved. 4 In contrast with fnctional models, here the probability of each V i, not its precise vale, is determined by the other variables in the model. V = {V 1,..., V n } and U = {U 1,..., U n } stand for the sets of observed and nobserved variables respectively. If no U variable is a descendant of any V variable, then the corresponding model is called a semi-markovian model. In a semi-markovian model, the observed probability distribtion, P (v), becomes a mixtre of prodcts: P (v) = P (v i pa i, i )P () (3) i where P A i and U i stand for the sets of the observed and nobserved parents of V i, and the smmation ranges over all the U variables. The post-intervention distribtion, likewise, will be given as a mixtre of trncated prodcts P t(v) 8 X Y < P (v i pa i, i )P () v consistent with t. = {i V : i T } 0 v inconsistent with t. (4) Characterizing Interventional Distribtions Let P denote the set of all interventional distribtions P = {P t (v) T V, t Dm(T ), v Dm(V )} (5) where Dm(T ) represents the domain of T. The set of interventional distribtions indced by a given casal BN mst satisfy some properties. For example the following property P pai (v i ) = P (v i pa i ), for all i, (6) mst hold in all Markovian models, bt may not hold in semi-markovian models. A complete characterization of the set of interventional distribtions indced by a given Markovian model is given in (Pearl 2000, pp.23-4). Now assme that we are given a collection of interventional distribtions, bt the nderlying casal BN, if sch exists, is nknown. We ask whether the collection is compatible with the predictions of some nderlying casal BN. As an example, assme that V consists of two binary variables X and Y with the domain of X being {x 0, x 1 } and the domain of Y being {y 0, y 1 }. Then P consists of the following distribtions P = {P (x, y), P x0 (x, y), P x1 (x, y), P y0 (x, y), P y1 (x, y), P x0,y 0 (x, y), P x0,y 1 (x, y), P x1,y 0 (x, y), P x1,y 1 (x, y)}, where each P t (x, y) is an arbitrary probability distribtion over X, Y with an index t. For this set of distribtions to be indced by some nderlying casal BN sch that each P t (x, y) corresponds to the distribtion of X, Y nder the intervention do(t = t) to the casal BN, they have to satisfy some norms of coherence. For example, it mst be tre that P x0 (x 0 ) = 1. For another example, if the casal graph is X Y then P y0 (x 0 ) = P (x 0 ), and if the casal graph is X Y then P x0 (y 0 ) = P (y 0 ), therefore, it mst be tre that either P y0 (x 0 ) = P (x 0 ) or P x0 (y 0 ) = P (y 0 ), which reflects the constraints that we are considering acyclic models.

3 Assme that each P t (v) in P is a (indexed) probability distribtion over V. We wold like to know what properties the set of distribtions in P mst satisfy sch that P is compatible with some nderlying casal BN in the sense that each P t (v) corresponds to the post-intervention distribtion of V nder the intervention do(t = t) to the casal BN. (Tian & Pearl 2002) has shown that the following three properties: effectiveness, Markov, and recrsiveness, are both necessary and sfficient for a P set to be indced from a Markovian casal model. Property 1 (Effectiveness) For any set of variables T, P P t (t) = 1. (7) Property 2 (Markov) For any two disjoint sets of variables S 1 and S 2, P v\(s1 s 2)(s 1, s 2 ) = P v\s1 (s 1 )P v\s2 (s 2 ). (8) Definition 1 For two single variables X and Y, define X affects Y, denoted by X Y, as W V, w, x, y, sch that P x,w (y) P w (y). That is, X affects Y if, nder some setting w, intervening on X changes the distribtion of Y. Property 3 (Recrsiveness) For any set of variables {X 0,..., X k } V, (X 0 X 1 )... (X k 1 X k ) (X k X 0 ). (9) These three properties impose constraints on the interventional space P sch that this vast space can be encoded sccinctly, in the form of a single Markovian model. In this paper, we seek a characterization of P set indced from semi-markovian casal models. The effectiveness and recrsiveness properties still hold in semi-markovian models bt the Markov property does not. First some discssions abot the effectiveness and recrsiveness properties. Effectiveness states that, if we force a set of variables T to have the vale t, then the probability of T taking that vale t is one. We give some corollaries of effectiveness that are very sefl dring ftre discssions. For any set of variables S disjoint with T, an immediate corollary of effectiveness reads: P t,s (t) = 1, (10) which follows from P t,s (t) P t,s (t, s) = 1. (11) Eqivalently, if T 1 T, then { 1 if t1 is consistent with t. P t (t 1 ) = 0 if t 1 is inconsistent with t. We frther have that, for T 1 T and S disjoint of T, { Pt (s) if t P t (s, t 1 ) = 1 is consistent with t. 0 if t 1 is inconsistent with t. (12) (13) Recrsiveness is a stochastic version of the (deterministic) recrsiveness axiom given in (Halpern 2000). It comes from restricting the casal models nder stdy to those having acyclic casal graphs. For example, for k = 1 we have X Y (Y X), saying that for any two variables X and Y, either X does not affect Y or Y does not affect X. (Halpern 2000) pointed ot that, recrsiveness can be viewed as a collection of axioms, one for each k, and that the case of k = 1 alone is not enogh to characterize a recrsive model. Recrsiveness defines an order over the set of variables. Define a relation as: X Y if X Y. The transitive closre of,, is a partial order over the set of variables V from the recrsiveness property. Then the following property holds in semi-markovian models. (Note that since a Markovian model is a special type of semi-markovian model, all properties that hold in semi-markovian models also hold in Markovian models.) Property 4 (Directionality) There exists a total order, <, consistent with, sch that P vi,w(s) = P w (s) if X S, X < V i, (14) for any set of variables W disjoint of S. Intitively, directionality implies that an intervention on any variable V i cannot affect earlier variables. If S contains a single variable X, this property is implied by the recrsiveness property, becase if P vi,w(x) P w (x), then V i X, and therefore V i X, which contradicts the fact that X < V i is consistent with. In Markovian models, the directionality property can be derived from the recrsiveness and Markov properties. Property 5 (Inclsion-Exclsion Ineqalities) For any sbset S 1 V, ( 1) S2 P v\(s1 s 2)(v) 0, v Dm(V ), S 2 V \S 1 where S 2 represents the nmber of variables in S 2. (15) The inclsion-exclsion ineqalities specify 2 V nmber of ineqalities (inclding the trivial one P (v) 0), each hold for all possible instantiations of V. For example, if V = {X, Y, Z}, then the inclsion-exclsion ineqalities specify the following: for all x Dm(X), y Dm(Y ), z Dm(Z), 1 P yz (x) P xz (y) P xy (z) + P z (xy) + P y (xz) + P x (yz) P (xyz) 0 (16) P yz (x) P z (xy) P y (xz) + P (xyz) 0 (17) P xz (y) P z (xy) P x (yz) + P (xyz) 0 (18) P xy (z) P y (xz) P x (yz) + P (xyz) 0 (19) P z (xy) P (xyz) 0 (20) P y (xz) P (xyz) 0 (21) P x (yz) P (xyz) 0 (22) If we assme that a casal order V 1 < V 2... < V n is given sch that Eq. (14) is satisfied, then some of the ineqalities in Eq. (15) can be derived from others. More exactly, we only need the following set of ineqalities. Property 6 (Inclsion-Exclsion Ineqalities with Order) Let V = V \ {V n }. For any sbset S 1 V, X ( 1) S2 P v \(s 1 s 2 )(s 1, s 2, v n) 0, v Dm(V ), S 2 V \S 1 (23)

4 Eq. (23) specifies 2 V 1 nmber of ineqalities. The other half of the ineqalities in Eq. (15) can be derived from Eq. (23) and the following eqation P vns(v \ s) = P s (v \ s) (24) which follows from Eq. (14). Smming the left hand side of Eq. (23) over all the instantiation of V n except v n, we have S 2 V \S 1 ( 1) S2 [P v \(s 1 s 2)(s 1, s 2 ) = P v \(s 1 s 2)(s 1, s 2, v n)] S 2 V \S 1 ( 1) S2 [P v \(s 1 s 2),v n (s 1, s 2 ) = P v \(s 1 s 2)(s 1, s 2, v n)] (from Eq. (24)) S 2 V \S 1 ( 1) S2 P v\(s1 s 2)(s 1, s 2 ). (25) Therefore, we obtain that, for any sbset S 1 V, S 2 V \S 1 ( 1) S2 P v\(s1 s 2)(v) 0, (26) which are the other half of the ineqalities in Eq. (15) besides those in Eq. (23). As an example, assming that V = {X, Y, Z}, and we are given a casal order X < Y < Z, then Directionality and inclsion-exclsion ineqalities specify the following P y (x) = P (x) (27) P z (x, y) = P (x, y) (28) P zx (y) = P x (y) (29) P zy (x) = P y (x) (30) P xy (z) P y (xz) P x (yz) + P (xyz) 0 (31) P y (xz) P (xyz) 0 (32) P x (yz) P (xyz) 0 (33) Theorem 1 (Sondness) Effectiveness, recrsiveness, directionality, and inclsion-exclsion ineqalities hold in all semi-markovian models. See the Appendix A for the proof of sondness. Theorem 2 (Completeness) If a P set satisfies effectiveness, recrsiveness, directionality, and inclsion-exclsion ineqalities, then there exists a semi-markovian model that can generate this P set. See the Appendix B for the proof sketch of completeness. The fll proof is given in (Tian, Kang, & Pearl 2006). Conclsion We have shown that the experimental implications of an nderlying semi-markovian casal model with nknown strctre are flly characterized by for properties. The key element in or characterization is the set of inclsion-exclsion ineqalities Eq. (15). One practical application of this characterization is that any empirical violation of the ineqalities in Eq. (15) wold permit s to conclde that the nderlying model is not semi-markovian; this means that feedback P loops may operate in data generating process, or that the interventions in the experiments are not condcted properly. (e.g., the intervention may not be properly randomized or they may have side effects). Another application permits s to bond the effects of ntried interventions from experiments involving axiliary interventions that are easier or cheaper to implement. For example, if we have performed experiments in which X and Y are randomized separately, yielding the distribtions P y (xz) and P x (yz) respectively, then Eq. (19) bonds the experimental distribtion P xy (z) that wold obtain nder a new experimental design where X and Y are randomized simltaneosly. The reslting bond, given by P xy (z) P y (xz) + P x (yz) P (xyz), makes no assmption on the strctre of the nderlying model, or the temporal order of the variable, or the absence of confonding variables in the domain. The fact that or proof constrcts a complete graph does not mean, of corse, that one cannot attempt to extract a more informative graph from P. For example, the set of directed edges can be redced noting that, in every semi- Markovian model, the parents of each V i are a minimal set S i satisfying P si (v i ) = P v\vi (v i ). In words, once we hold fixed the parents of V i, no additional intervention may inflence the probability of V i. Likewise, the set of bidirected arcs can be redced by removing all arcs between a node V i and a maximal set T i of non-descendants of V i satisfying P v\vi (v i ) = P (v\vi)\t i (v i t i ). (34) Indeed, intervening on variables to which V i is not connected by an arc or observing those variables gives s the same information on V i (once we hold fixed all other variables). The qestion remains however whether the removal of these edges from the complete graph indces additional ineqalities and eqalities that need be checked against P. We leave this qestion for ftre work. Acknowledgments The athors thank the anonymos reviewers for helpfl comments. This research was spported in parts by NSF grant IIS and grants from NSF, ONR, AFOSR, and DoD MURI program. Appendix A: Proof of Sondness Theorem (Sondness) Effectiveness, recrsiveness, directionality, and inclsion-exclsion ineqalities hold in all semi-markovian models. Proof: All for properties follow from Eq. (4). Effectiveness From Eq. (4), we have and since P t (T = t ) = 0 for t t, (35) t Dm(T ) P t (t ) = 1, (36) we obtain the effectiveness property of Eq. (7). Recrsiveness Assme that a total order over V that is consistent with the casal graph is V 1 < < V n, sch that

5 V i is a nondescendant of V j if V i < V j. Consider a variable V j and any set of variables S V which does not contain V j. Let B j = {V i V i < V j, V i V \ S} be the set of variables not in S and ordered before V j, and let A j = {V i V j < V i, V i V \ S} be the set of variables not in S and ordered after V j. Then P s (b j, v j, a j ) = P (v i pa i, i )P (v j pa j, j ) and P vj,s(b j, a j ) = {i V i B j} {i V i A j} P (v i pa i, i )P () (37) {i V i B j} {i V i A j} P (v i pa i, i ) P (v i pa i, i )P () (38) Smming both sides of Eq. (37) over all the instantiations of variables in A j and V j, sch that the variable ordered last is smmed first, we obtain P s (b j ) = P (v i pa i, i )P (). (39) {i V i B j} Similarly, smming both sides of Eq. (38) over all the instantiations of variables in A j, we obtain that P vj,s(b j ) = P s (b j ). (40) Since B j is the set of variables ordered before V j, we have that, for any two variables V i < V j and any set of variables S, P vj,s(v i ) = P s (v i ), (41) which states that if V i is ordered before V j then V j does not affect V i, based on or definition of X affects Y. Therefore, we have that if V j affects V i then V j is ordered before V i, or V j V i V j < V i. (42) Recrsive property (9) then follows from (42) becase the relation < is a total order. Directionality Let < be a total order consistent with the casal graph. In the above proof for recrsiveness, we have shown that Eq. (40) and (42) hold. (42) means that the total order < is consistent with. And Eq. (14) follows immediately from Eq. (40). Inclsion-Exclsion Ineqalities We se the following eqation k (1 a i ) i=1 = 1 i a i + i,j a i a j + ( 1) k a 1 a k. (43) Take a j = P (v j pa j, j ), we have that X Y P (v i pa i, i ) {i V i S 1 } Y {j V j V \S 1 } = X since for all V i V (1 P (v j pa j, j ))P () S 2 V \S 1 ( 1) S2 P v\(s1 s 2 )(s 1, s 2) 0 (44) 0 P (v i pa i, i ) 1. (45) Appendix B: Proof Sketch of Completeness From directionality property, there exists a total order on V, V 1 < V 2 < < V n, sch that P vi,w(s) = P w (s) if X S, X < V i, (46) We will constrct a casal model consistent with this order. Let the domain of each variable V j be Dm(V j ) = {v 1 j,..., v dj j } where d j is the nmber of vales V j can take. We will constrct a fnctional model in the form of v j = f j (v 1,..., v j 1, r j ), j = 1,..., n. (47) For discrete variables, the nmber of possible fnctions is finite. We will se the response variable representation (Balke & Pearl 1994b; 1994a) (called mapping variable in (Heckerman & Shachter 1995)). Let the domain of r j be Dm(r j ) = {1, 2,..., Dm(r j ) } where Dm(r j ) = d d1 dj 1 j. We will constrct a model of the form P (v) = P (v j v 1,..., v j 1, r j )P (r 1,..., r n ) r 1,...,r n j (48) For a fnctional model, each of the probabilities P (v j v 1,..., v j 1, r j ) wold be either 0 or 1, and only non-zero vales contribte to the smmation in Eq. (48). For a fixed vale of v 1,..., v j, let D j (v j v 1,..., v j 1 ) Dm(r j ) be the set of vales of r j sch that P (v j v 1,..., v j 1, r j ) = 1. Then P (v) = P (r 1,..., r n ), (49) r 1 D 1 r n D n and for any T V P v\t (v) = r i Dm(r i ) {i V i T } r i D i {i V i T } P (r 1,..., r n ). (50) If we can constrct a distribtion P (r 1,..., r n ) sch that Eq. (50) holds for any T V and v Dm(V ), then we have a semi-markovian model that can indce the P set.

6 Given a distribtion P (r 1,..., r n ), we will define an event A ij i1,...,ij 1 j as the event that r j is in D j (v ij j vi1 1,..., vij 1 ), and we will think of the event A ij i1,...,ij 1 j as a set in the space Dm(r 1 ) Dm(r n ). Then Eqs. (49) and (50) become P (v i1 1,..., vin n ) = P (A i1 1 Ai2 i1 2 An in i1,...,in 1 ), (51) and P v\t (v) = P ( {k V k T } A i k i 1,...,i k 1 k ). (52) For a fixed i 1,..., i n 1, the set of events An jn i1,...,in 1, j n = 1,..., d n are mtally exclsive and exhastive. For a fixed i 1,..., i n 1, letting A k be a shorthand notation for A i k i 1,...,i k 1 k and letting A k represent the event not A k, the set of events \ \ \ A k A k A j n n, I {1,..., n 1}, j n = 1,..., d n k I k I (53) are mtally exclsive and exhastive, and ths form a partition of the space Dm(r 1 ) Dm(r n ). The probabilities of these events can be compted from Eq. (52) sing the inclsion-exclsion principle, and we obtain P ( A k A k A j n n ) k I = k I S 2 V \S 1 ( 1) S2 P v \(s 1 s 2)(s 1, s 2, v n ), (54) where V = V \ {V n }, and S 1 = {V i i I}. From the Inclsion-Exclsion Ineqalities with Order given in Eq. (23), we have a valid assignment of probabilities to each of the mtally exclsive and exhastive events in (53). It is not hard to see that the eqations (52) for V n V \ T lead to constraints in the form of, for each S V, P vns(v \ s) = P s (v \ s). (55) These constraints are satisfied by the P set since Eq.(46) holds. For each fixed vale i 1,..., i n 1, we have a probability assignment to the set of mtally exclsive and exhastive events k I A k k I A k A j n n given by Eq.(54). Are these assignments consistent for different vales of i 1,..., i n 1? In other words, does there exist a distribtion P (r 1,..., r n ) that satisfies the assignments in Eq.(54) for i 1 = 1,..., d 1,..., i n 1 = 1,..., d n 1? If the answer is yes, then there exists a distribtion P (r 1,..., r n ) sch that Eq. (52) holds for any T V and v Dm(V ), and therefore there exists a semi-markovian model that can indce the P set. For a fixed vale i 1,..., i n 1, we consider another (finer) partition of the space Dm(r 1 ) Dm(r n ), denoted by K i1,...,i n 1, A j1 1 Aj2 i1 2 A jn i1,...,in 1 n, j 1 = 1,..., d 1,..., j n = 1,..., d n. (56) We se P (K i1,...,i n 1 ) to denote a probability assignment that assigns a probability vale to each set in K i1,...,i n 1. We can show the following Lemma 1 Given the probability assignments in Eq. (54), there exist probability assignments P (K i1,...,i n 1 ) for i 1 = 1,..., d 1,..., i n 1 = 1,..., d n 1, sch that, for two different partitions K i1,...,i n 1 and K i 1,...,i n 1, if i 1 = i 1,..., i k = i k, then P (K i 1,...,i n 1 ) and P (K i 1,...,i n 1 ) indce the same probabilities P (A j1 1 A j2 i1 2 A j k+1 i 1,...,i k k+1 ). Then we can show that Lemma 2 There exists a distribtion P (r 1,..., r n ) sch that all the probability assignments P (K i1,...,i n 1 ) in Lemma 1 are satisfied for i 1 = 1,..., d 1,..., i n 1 = 1,..., d n 1. The completeness Theorem 2 follows from Lemma 2. References Balke, A., and Pearl, J. 1994a. Conterfactal probabilities: Comptational methods, bonds, and applications. In Uncertainty in Artificial Intelligence Balke, A., and Pearl, J. 1994b. Probabilistic evalation of conterfactal qeries. In Proceedings of the National Conference on Artificial Intelligence Dawid, A Inflence diagrams for casal modelling and inference. International Statistical Review 70(2). Galles, D., and Pearl, J An axiomatic characterization of casal conterfactals. Fondations of Science 3(1): Halpern, J. Y Axiomatizing casal reasoning. Jornal of Artificial Intelligence Research 12: Heckerman, D., and Shachter, R Decision-theoretic fondations for casal reasoning. Jornal of Artificial Intelligence Research 3: Laritzen, S Graphical models for casal inference. In Complex Stochastic Systems. London/Boca Raton: Chapman and Hall/CRC Press. chapter 2, Pearl, J Probabilistic Reasoning in Intelligent Systems. San Mateo, CA: Morgan Kafmann. Pearl, J Casal diagrams for empirical research. Biometrika 82: Pearl, J Casality: Models, Reasoning, and Inference. NY: Cambridge University Press. Spirtes, P.; Glymor, C.; and Scheines, R Casation, Prediction, and Search. New York: Springer-Verlag. Tian, J., and Pearl, J A new characterization of the experimental implications of casal Bayesian networks. In Proceedings of the National Conference on Artificial Intelligence (AAAI). Tian, J.; Kang, C.; and Pearl, J A characterization of interventional distribtions in semimarkovian casal models. Technical report, Department of Compter Science, Iowa State University. jtian/papers/aaai06-tech.pdf.