Actively Identifying Causal Effects with Latent Variables Given Only Response Variable Observable

1 Introduction . Identifying causal effects is one prominent task throughout empirical sciences . In many real problems , we are generally desired to study the causal effect on a specific target only , i.e . response variable , with no need to identify the thorough causal effects involving all variables . For example , a business person wants to maximize profit . The person has many intervention methods such as lowering price , increasing advertisement investment , improving quality , and so on . A valid approach to guide the decisions is to estimate the causal effect of each variable on the response variable . We call it target effect identification in this paper . After identifying all of these causal effects , the person could see which intervention could lead to the most desired response ( profit ) . To achieve target effect identification , an ideal method is to intervene on each decision variable with different attainable values and collect the interventional data of the response variable . However , when there are many decision variables and many attainable values for each variable , it will take enormous number of interventions , which is economically unfeasible . Considering that there is usually a large amount of observational data in reality , we hope to exploit mainly the observational data to achieve it . The main obstacle here is that the causal graph of all involved variables is not available , which makes causal effect identification intractable . Hence , we learn the structure at first . As shown by Verma and Pearl [ 1 ] , only a Markov equivalence class could be learned with observational data without further assumptions , where there are possibly unidentified causal relations resulting in unidentified causal effects . To reveal these effects , we further learn the causal relations in the Markov equivalence class by introducing a small amount of active hard interventions , i.e. , we force some variable to a known value [ 2 , 3 , 4 ] . In practice , generally one can only observe part of variables under interventions rather than all of those , e.g. , a business person can observe his profit , but does not know other companies ’ profit . To make the method applicable in a broader scenario , we consider an extreme situation that only the response variable is observable , which could be trivially extended to any settings with 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . part of variables observed . In addition , there are usually latent confounders in practical problems . For instance , economic level influences both price and advertising cost , but it is hard to evaluate accurately and thus a latent confounder . In summary , in this paper we learn the causal knowledge where there are latent confounders and only the response variable is observable under interventions . This problem is very challenging , because the fact that only the response variable is observable disables the approaches designed for situations where all variables are observable [ 2 , 5 , 6 ] , while the existence of latent confounders disables the typical solutions [ 7 ] . To get a brief understanding of the challenges taken by latent confounders in techniques , consider a graph learned with observational data by traditional causal discovery methods [ 8 ] : If there is no latent confounder , the definite adjacent edges of intervened variable X lead to a definite interventional distribution of the response variable Y by back-door criterion [ 9 ] , which makes it possible to learn the edges of X by the interventional data of Y ; when latent confounders exist , however , the mapping no longer holds . Besides , a partial mixed ancestral graph ( PMAG ) is commonly used to describe the relation between variables in this condition . The back-door criterion may fail in this case since there possibly exist latent confounders influencing the two variables of some directed edges . Hence , we also need to identify whether such latent confounders exist behind the directed edges , in addition to just learning the causal edges . To overcome the difficulties above , we provide a graphical characterization that allows us to efficiently find all possible causal effects of the intervened variable on the response variable in a partial mixed ancestral graph . Guided by the characterization , we learn a local structure and identify the presence of latent confounders behind some directed edges by each intervention , which leads to identifying some causal effects on the response variable . After a few active interventions on different variables , our method ACIC , short for ACtive target effect Identification with latent Confounding , could achieve target effect identification . Due to space limit , we present related work and all the proofs in Appendix . 2 Preliminary . We guide readers to Appendix A.1 for the notions including causal graph ( G = ( V , E ) ) , mixed graph , mark , arrowhead , tail , circle ( ◦ ) , partial mixed graph , parent , child , spouse , possible directed path , ( possible ) ancestor , PossAn ( X , G ) , An ( X , G ) , ( possible ) descendant , PossDe ( X , G ) , De ( X , G ) , adjacent , Adj ( X , G ) , almost directed cycle , collider path , minimal path , ancestral graph , m-separation ( m-connecting or active ) , maximum ancestral graph ( MAG , M ) , discriminating path , Markov equivalent , Markov equivalence class ( MEC ) , partial ancestral graph ( PAG ) , visible , MX , PX , causal effect ( P ( Y |do ( X ) ) ) , and the common assumptions including positivity and no selection bias . ∗ is a wildcard that denotes any of the marks . For a partial mixed graph , we say it is a partial mixed ancestral graph ( PMAG ) if there is no directed or almost directed cycle , and denote it by P . Note both MAG and PAG are special cases of PMAG , thus P could also denote PAG . The relation between PMAG and PAG is similar to that between partial directed acyclic graph ( PDAG ) and completed partial directed acyclic graph ( CPDAG ) when there is no latent confounder [ 10 ] . MAG M is consistent to PMAG P if it belongs to the MEC represented by P ( detailed in Appendix A.1 ) . For a directed edge Vi → Vj in a graph , we say the edge is pure if the two variables are not influenced by common latent confounders , denoted by Vi p−→ Vj . We use a purity matrix to characterize the purity of each edge in a PMAG G , in which 1 is in ( i , j ) entry only if the directed edge Vi → Vj in G is pure , and 0 otherwise . Note that 0 does not imply that the edge is impure . It is also possible that we are unaware of whether it is pure or the edge is not directed . In the literature , a graphical characterization to imply that there are no latent confounders behind a directed edge in an MAG or PAG is proposed by Zhang [ 11 ] , where they call it by visibility . Visibility is sufficient but not necessary for purity , which is detailed in Appendix A.2 . Thus , given a PMAG , we initialize the purity matrix by setting ( i , j ) entry to 1 if Vi → Vj is visible and 0 otherwise . For an MAGM , MX˜denotes the graph by removing the directed edges out of X inM that are labeled to be pure in purity matrix . For a PMAG P , see Appendix A.4 for the definition of PX˜ . Since the graph is continuously learned by interventions in this paper , the graph in the process is a PMAG . Hence we present Prop . 1 to guide causal effect identification in a PMAG with the consideration of purity matrix rather than only in a PAG or MAG . It is based on the generalized back-door criterion ( GBC ) and the graphical condition for the causal effect identifiability by GBC in MAG or PAG proposed by Maathuis et al . [ 12 ] . Before that , we introduce D-SEP ( X , Y , G ) in Def . 1 . Definition 1 ( D-SEP ( X , Y , G ) [ 12 ] ) . Let X and Y be two distinct vertices in a mixed graph G. We say that V ∈ D-SEP ( X , Y , G ) if V 6= X , and there is a collider path between X and V in G , such that every vertex on this path ( including V ) is an ancestor of X or Y in G. Proposition 1 . Let G be a PMAG and W be a purity matrix of G. Suppose X ∈ An ( Y , G ) and Y are two distinct vertices in G. There exists a generalized back-door set relative to ( X , Y ) and ( G , W ) if and only if D-SEP ( X , Y , GX˜ ) ∩ PossDe ( X , G ) = ∅ . Moreover , if the set exists , D-SEP ( X , Y , GX˜ ) is such a set . Denote D-SEP ( X , Y , GX˜ ) by D , the causal effect is P ( Y |do ( X = x ) ) = ∫ D P ( D ) P ( Y |D , X = x ) dD . ( 1 ) There are two possible unidentifiable cases for P ( Y |do ( X ) ) by GBC in Prop . 1 . One is that there are many possible causal effects due to the missing of exact structure information , but we do not know which is correct . In this case , we could address it by learning the structure with interventional data . The other unidentifiable case is that the causal effect is unidentifiable by GBC even if we know the MAG and the purity of each edge . We return “ Fail ” for such X because GBC is not sufficient for identifying the causal effect in this case , and we say GBC fails ( to identify P ( Y |do ( X ) ) ) . 3 The Proposed Approach . In this paper , we assume faithfulness , positivity , and no selection bias . Denote the decision variables by X1 , · · · , Xp and the response variable by Y . Given the observational data of these variables , our goal is to achieve target effect identification by generalized back-door criterion , i.e. , we aim to identify P ( Y |do ( Xi ) ) by GBC for each variable Xi , i = 1 , 2 , · · · , p if GBC does not fail . With the observational data , we can learn a PAG P that the true MAG is consistent to by FCI algorithm [ 8 ] . And we initialize a purity matrix by setting ( i , j ) entry to 1 if the directed edge between the two variables is visible in P and 0 otherwise . The causal effects of some variables on Y are possibly unidentifiable in P by Prop . 1 . To identify these effects , we further introduce active interventions and observe Y under those . Beginning from P , in each round our method selects one variable X from Xi , i = 1 , · · · , p to intervene and exploits the interventional data of Y to learn the structure , including learning an updated PMAG by revealing circles and updating the purity matrix . The process repeats until identifying all effects P ( Y |do ( Xi ) ) , i = 1 , 2 , · · · , p ( identifying failing of GBC is also included ) . Since it is recursive , we just present the method to learn the structure by interventional data in one round . The criterion to select the intervention variable in each round is given at the end .

Summary: This paper develops a method for identifying causal effects on a response variable from observational and interventional data, with the restriction that under interventions only the response variable can be measured. The method does not assume away latent confounders. Theoretical justifications and some empirical evaluations of the method are presented .