Score-based Causal Discovery from Heterogeneous Data

1 INTRODUCTION . Discovering causal relations among variables is a fundamental problem in various fields such as economics , biology , drug testing , and commercial decision making . Because conducting randomized controlled trials is usually expensive or even infeasible , discovering causal relations from observational data , i.e . causal discovery ( Pearl , 2000 ; Spirtes et al. , 2000 ) , has received much attention over the past few decades . Early causal discovery algorithms can be roughly categorized into two types : constraint-based ones ( e.g . PC ( Spirtes et al. , 2000 ) ) and score-based ones ( e.g . GES ( Chickering , 2002 ) ) . In general , these methods can not uniquely identify the causal graph but are guaranteed to output a Markov equivalence class . Since the seminal work by Shimizu et al . ( 2006 ) , several methods have been developed , achieving identifiability of the whole causal structure by making use of constrained Functional Causal Models ( FCMs ) , including the linear non-Gaussian model ( Shimizu et al. , 2006 ) , the nonlinear additive noise model ( Hoyer et al. , 2009 ) , and the post-nonlinear model ( Zhang & Hyvärinen , 2009 ) . Recently , Zheng et al . ( 2018 ) proposed a score-based method that formulates the causal discovery problem as continuous optimization with a structural constraint that ensures acyclicity . Based on the continuous structural constraint , several researchers further proposed to model the causal relations by neural networks ( NNs ) ( Lachapelle et al. , 2019 ; Yu et al. , 2019 ; Zheng et al. , 2019 ) . Another recent work Zhu & Chen ( 2019 ) used reinforcement learning ( RL ) for causal discovery , where the RL agent searches over the graph space and outputs a graph that fits the data best . The above approaches are designed for data from a single domain with a fixed causal model , with the limitation that many of the edge directions can not be determined without strong functional constraints . In addition , the sample size of data from one domain is usually not large enough to guarantee small statistical estimation errors . One way to improve statistical reliability is to combine datasets from multiple domains , such as P-value meta-analyses ( Lee , 2015 ; Marot et al. , 2009 ) . The idea of combining multiple-domain data is commonly seen in learning with mixture of Bayesion networks ( Thiesson et al. , 1998 ) . While mixture of Bayesion networks are usually used for density estimation , the purpose of causal analysis from multiple-domain data is completely different , it aims at discovering the underlying causal graphs for all domains . Regarding causal analysis from multiple-domain data , a challenge is the data heterogeneity problem : the data distribution may vary across domains . For example , in fMRI hippocampus signal analysis , the connection strength among different brain regions may change across different subjects ( domains ) . Due to the distribution shift , directly pooling the data from multiple domains may lead to spurious edges . To tackle the issue , different ways have been investigated , including using sliding windows ( Calhoun et al. , 2014 ) , online change point detection ( Adams & MacKay , 2007 ) , online undirected graph learning ( Talih & Hengartner , 2005 ) , locally stationary structure tracker ( Kummerfeld & Danks , 2013 ) , and regime aware learning ( Bendtsen , 2016 ) . However , these methods may suffer from high estimation variance due to sample scarcity , large type II errors , and a large number of statistical tests . Huang et al . ( 2015 ) recovers causal relations with changing modules by making use of certain types of smoothness of the change , while it does not explicitly locate the changing causal modules . Other similar methods , including Xing et al . ( 2010 ) ; Song et al . ( 2009 ) , can be reduced to online parameter learning because the causal directions are given . By utilizing the invariance property ( Hoover , 1990 ; Tian & Pearl , 2001 ; Peters et al. , 2016 ) and the more general independent change mechanism ( Pearl , 2000 ) , recently , Ghassami et al . ( 2018 ) developed two methods : identical boundaries ( IB ) and minimal changes ( MC ) , for causal discovery from multi-domain data . However , the proposed methods 1 ) assume causal sufficiency ( i.e. , all common causes of variables are measured ) , which is usually not held in real circumstances , 2 ) are designed for linear systems only , 3 ) and are not capable of identifying causal directions from more than ten domains . Huang et al . ( 2019 ) proposed a more general approach called CD-NOD for both linear and nonlinear heterogeneous data , by extending the PC algorithm to tackle the heterogeneity issue . However , inheriting the drawbacks of constraint-based methods , CD-NOD involves a multiple testing problem and is time-consuming due to large number of independence tests . To overcome the limitations of existing works , we propose a Multiple-Domain Score Search ( MDSS ) method for causal discovery from heterogeneous data , which enjoys the following properties . ( 1 ) To avoid spurious edges when combing multi-domain data , MDSS searches over the space of augmented graphs , which includes an additional domain index as a surrogate variable to characterize the distribution shift . ( 2 ) The changing causal modules can be immediately identified from the recovered augmented graph . ( 3 ) Benefiting from causal invariance and the independent change mechanism , MDSS uses a novel Multiple-Domain Score ( MDS ) to help identify more causal directions beyond those in the Markov equivalence class from distribution-shifted data . ( 4 ) MDSS can be readily incorporated into off-the-shelf search strategies and is time-efficient and applicable to both linear and nonlinear data . ( 5 ) Theoretically , we show that MDSS is guaranteed to find the correct graph skeleton asymptotically , and further identify more causal directions than other traditional score-based and constraint-based algorithms . Empirical studies on both synthetic and real data prove the efficacy of our method . 2 METHODOLOGY . In this section , we start from a brief introduction to causal discovery and distribution shifts ( Section 2.1 ) , and then in Section 2.2 and 2.3 , we introduce our proposed Multiple-Domain Score Search ( MDSS ) . In Section 2.2 , MDSS starts with a predefined graph search algorithm to learn the skeleton of the causal graph , with the linear Bayesian information criterion ( BIC ) score or nonlinear generalized score ( GS ( Huang et al. , 2018 ) ) on the augmented causal system . Then in Section 2.3 , MDSS further identifies causal directions with Multiple-Domain Score ( MDS ) based on the identified skeleton of the graph from Section 2.2 . Both theoretically and empirically , we show that MDSS can identify more directions compared to algorithms that are designed for i.i.d . or stationary data . 2.1 BACKGROUND IN CAUSAL DISCOVERY AND DISTRIBUTION SHIFTS . The basic causal discovery problem can be formulated as follows : Suppose there are d observable random variables , i.e . V = ( V1 , ... , Vd ) . Each random variable satisfies the following generating process : Vi = fi ( PAi , ) , where fi is a function to model the causal relation between Vi and its parents PAi , and i is a noise variable with non-zero variance . All the noise variables are independent of each other . The task of causal discovery is to recover the causal adjacency matrix B given the observed data matrix X ∈ RT×d , where Bij = 1 indicates that Vi is a parent of Vj , and T is the sample size . We denote the underlying causal graph over V as G0 . For each Vi , we call P ( Vi|PAi ) its causal module . For a single domain , the joint probability can be factorized as P ( V ) = ∏d i=1 P ( Vi|PAi ) . Suppose there are n domains with distribution shifts ( i.e . P ( V ) changes across domains ) , which implies that some causal modules change across domains . The changes may be caused by the variation of functional models , causal strength , or noise variance . Furthermore , we have the following assumptions . Assumption 1 . The changes of causal modules can be represented as functions of domain index C , denoted by g ( C ) , Assumption 2 . There is no confounder in each single dataset , but we allow the changes of different causal modules being dependent . Remark : If changes in several causal modules are dependent , it can be regarded as special `` confounders '' that simultaneously affect these causal modules . As a consequence of such confounders , previous causal discovery algorithms designed for i.i.d . or stationary data may output erroneous edges . See section 3.1 for an illustration . Thus , causal discovery from multiple-domain data with distribution shifts ( i.e . , heterogeneous data ) can be much more difficult than that from single-domain data . 2.2 SKELETON ESTIMATION ON AUGMENTED GRAPHS . With Assumptions 1 and 2 , it is natural to consider g ( C ) as an extra variable in order to remove any potential influence caused by these special confounders . We assume that there are L such confounders ( g1 ( C ) , ... , gL ( C ) ) . The causal relation between each observable variable Vi and its parents PAi can be formalized with Vi = fi ( PAi , gi ( C ) , θi ( C ) , i ) , ( 1 ) where gi ( C ) ⊆ { gl ( C ) } Ll=1 is the set of confounders that influence Vi , θi ( C ) are the effective parameters in Vi ’ s causal module that are also assumed to be functions of C and are mutually independent for all variables . Let G0 be the underlying causal graph over V. We denote the graph resulting from adding arrows gi ( C ) → Vi and θi ( C ) → Vi on G0 for each Vi in V as Gaug over V ∪ { gl ( C ) } Ll=1 ∪ { θi ( C ) } d i=1 . We call Gaug an augmented graph ( see Figure 1 ( d ) as an example ) , which satisfies the following assumption . Assumption 3 . The joint distribution over V ∪ { gl ( C ) } Ll=1 ∪ { θi ( C ) } d i=1 is Markov and faithful to Gaug . To remove the potential influence from confounders and recover causal relations from multiple domains , one way is to perform causal discovery algorithms on the augmented graph . While { gl ( C ) } Ll=1 and { θi ( C ) } d i=1 are not directly observed , we take C as a surrogate variable ( Huang et al. , 2019 ) for them because C is always available as a domain index . Given Assumptions 1 , 2 and 3 , one can apply any score-based method over V∪ { C } to recover the causal relations among variables V as if { gl ( C ) } Ll=1∪ { θi ( C ) } d i=1 were known . For simplicity , we denote the graph over V ∪ { C } as augmented graph as well . Since C is the domain index , P ( C ) follows a discrete uniform distribution . Correspondingly , the generating process of non-stationary data can be considered as follows : First we generate random values from P ( C ) , and then we generate data points over V according to the SEM in Equation 1 . Finally , generated data points are sorted in ascending order according to the values of C ( i.e. , data points having the same value of C are regarded as belonging to the same domain ) . In other words , we observe the distribution P ( V|C ) , where P ( V|C ) may change across different values of C , resulting in non-stationary data . Note that if we do not include C into the system explicitly , samples of V are not i.i.d . However , after explicitly including the domain index C into the system , P ( V , C ) is fixed , and thus the pooled data are i.i.d . samples from distribution P ( V , C ) . Before stating our main result , we first give the definitions of globally consistent scoring criterion and locally consistent scoring criterion , which will be used in the paper . Definition 1 ( Globally Consistent Scoring Criterion ) . Let D be a dataset consisting of T i.i.d . samples from some distribution P ( · ) . LetH and G be any DAGs . A scoring criterion S is globally consistent if the following two properties hold as T →∞ : 1 . IfH contains P and G does not contain P , then S ( H , D ) > S ( G , D ) 1 . 2 . IfH and G both contain P , and G contains fewer parameters thanH , then S ( H , D ) < S ( G , D ) . Definition 2 ( Locally Consistent Scoring Criterion ) . Let D be a dataset consisting of T i.i.d . samples from some distribution P ( · ) . Let G be any DAG , and let G′ be the DAG that results from adding the edge Vi → Vj on G. A scoring criterion S ( G , D ) is locally consistent if the following two properties hold as T →∞ : 1 . If Vj 6⊥Vi|PAGj , then S ( G′ , D ) > S ( G , D ) . 2 . If Vj ⊥Vi|PAGj , then S ( G′ , D ) < S ( G , D ) . 1Here , larger score means the corresponding graph is closer to the equivalent class of the true DAG , while the MDS defined in Section 2.3 should be regarded as a type of `` loss function '' which needs to be minimized . It has been shown that the BIC score and the GS score are both globally and locally consistent ( Chickering , 2002 ; Huang et al. , 2018 ) . The procedure for skeleton estimation on augmented graphs is described in Algorithm 1 . The predefined graph search algorithms will be discussed in Section 2.4 . Apart from the recovered skeleton over V , the changing modules can be detected as well in Step 4 of Algorithm 1 . It is important to note that we allow causal relations to be either linear or nonlinear . If they are nonlinear , we apply GS as a score function . When they are linear , although we can also use GS , we use linear BIC instead because it is less likely to be overfitting for linear data and is computationally more efficient . Algorithm 1 Skeleton Search on Augmented Graph Input : n datasets , each has T observations , d variables and index C. Output : skeleton S of Gaug ’ s subgraph G1 over V , and variables VC ∈ V that are connected with C. 1 : Pool all datasets with an extra surrogate variable C to form a data matrix X ∈ RnT× ( d+1 ) . 2 : Use the predefined graph search algorithm with BIC or GS plus acyclicity constraints to recover the augmented graph . Eliminate any direction Vi → C in the graph with the prior that any variable Vi does not affect domain index . This step leads to the recovered augmented graph Gaug . 3 : Discard the index variable in Gaug to obtain the induced subgraph G1 . Discard the directions in G1 and output the skeleton S of G1 . 4 : Detect changing causal modules by inspecting Gaug recovered in Step 2 , and output VC . The validity of searching on augmented graph is guaranteed by Theorem 1 . Theorem 1 . Let D be the pooling of all datasets , DC be the augmented dataset with the domain index as an extra random variable . Let G0 be the underlying causal graph for the distribution of D over V , GC be the underlying causal graph for the distribution of DC over V ∪ C. If we denote G′C as the graph after the following modifications on GC : 1. adding any edges , 2. deleting any edges or 3. reversing any edges that changes the conditional dependence relation of GC , then we have S ( GC , DC ) > S ( G′C , DC ) , where S is any globally consistent scoring criterion . Proof of the theorem is given in Appendix A.1 . Intuitively , this theorem means we will obtain an augmented graph that is in the same Markov equivalence class as the true augmented graph if we maximize the score .

Summary: This paper proposes strategies for learning the structure of multiple sets of data observed over a common set of variables which may exhibit distribution shift. The authors address this problem by augmenting the dataset with an indicator variable which indicates membership to  dataset. After augmenting the dataset standard algorithms for structure learning are applied, with the additional restriction that the indicator variable may only be an ancestor. The authors provide theory that shows the procedure consistently estimates the local structures. The authors then show how the additional information obtained from the structure learned with the context variable can be used to disambiguate directions. Experimental results show the efficacy of the proposed approach.