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: Based on the ancestral graph framework, the authors tackle the problem of identifying an interventional distribution $f (Y | do (X))$ by only observing the outcome $Y$ after each possible intervention. This work extended previous work by Wang et al. assuming no latent variables on the graph.