Original: ANOCE: A NALYSIS OF C AUSAL E FFECTS WITH M ULTIPLE M EDIATORS VIA C ONSTRAINED S TRUCTURAL L EARNING
Revised: ANOCE: A NALYSIS OF C AUSAL E FFECTS WITH M ULTIPLE M EDIATORS VIA C ONSTRAINED S TRUCTURAL L EARNING

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: In the era of causal revolution, identifying the causal effect of an exposure on the outcome of interest is an important problem in many areas, such as epidemics, medicine, genetics, and economics.
Revised: In the era of causal revolution, identifying the causal effect of an exposure on the outcome of interest is an important problem in many areas, such as epidemics, medicine, genetics, and economics.

Original: Under a general causal graph, the exposure may have a direct effect on the outcome and also an indirect effect regulated by a set of mediators.
Revised: Under a general causal graph, the exposure may have a direct effect on the outcome and also an indirect effect regulated by a set of mediators.

Original: An analysis of causal effects that interprets the causal mechanism contributed through mediators is hence challenging but on demand.
Revised: An analysis of causal effects that interprets the causal mechanism contributed through mediators is hence challenging but on demand.

Original: To the best of our knowledge, there are no feasible algorithms that give an exact decomposition of the indirect effect on the level of individual mediators, due to common interaction among mediators in the complex graph.
Revised: To the best of our knowledge, there are no feasible algorithms that give an exact decomposition of the indirect effect on the level of individual mediators, due to common interaction among mediators in the complex graph.

Original: In this paper, we establish a new statistical framework to comprehensively characterize causal effects with multiple mediators, namely, ANalysis Of Causal Effects (ANOCE), with a newly introduced deﬁnition of the mediator effect, under the linear structure equation model.
Revised: In this paper, we establish a new statistical framework to comprehensively characterize causal effects with multiple mediators, namely, ANalysis Of Causal Effects (ANOCE), with a newly introduced deﬁnition of the mediator effect, under the linear structure equation model.

Original: We further propose a constrained causal structure learning method by incorporating a novel identiﬁcation constraint that speciﬁes the temporal causal relationship of variables.
Revised: We further propose a constrained causal structure learning method by incorporating a novel identiﬁcation constraint that speciﬁes the temporal causal relationship of variables.

Original: The proposed algorithm is applied to investigate the causal effects of 2020 Hubei lockdowns on reducing the spread of the coronavirus in Chinese major cities out of
Revised: The proposed algorithm is applied to investigate the causal effects of 2020 Hubei lockdowns on reducing the spread of the coronavirus in Chinese major cities out of

Original: Hubei.
Revised: Hubei.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: In the era of causal revolution, identifying the causal effect of an exposure on the outcome of interest is an important problem in many areas, such as epidemics (Hernán, 2004), medicine (Hernán et al., 2000), education (Card, 1999), and economics (Panizza & Presbitero, 2014).
Revised: In the era of causal revolution, identifying the causal effect of an exposure on the outcome of interest is an important problem in many areas, such as epidemics (Hernán, 2004), medicine (Hernán et al., 2000), education (Card, 1999), and economics (Panizza & Presbitero, 2014).

Original: Under a general causal graph, the exposure may have a direct effect on the outcome and also an indirect effect regulated by a set of mediators (or intermediate variables).
Revised: Under a general causal graph, the exposure may have a direct effect on the outcome and also an indirect effect regulated by a set of mediators (or intermediate variables).

Original: For instance, during the outbreak of Coronavirus disease 2019 (COVID-19), the Chinese government has taken extreme measures to stop the virus spreading such as locking Wuhan down on Jan 23rd, 2020, followed by 12 other cities in Hubei, known as the “2020 Hubei lockdowns”.
Revised: For instance, during the outbreak of Coronavirus disease 2019 (COVID-19), the Chinese government has taken extreme measures to stop the virus spreading such as locking Wuhan down on Jan 23rd, 2020, followed by 12 other cities in Hubei, known as the “2020 Hubei lockdowns”.

Original: This approach (viewed as the exposure), directly blocked infected people leaving from Hubei; and also stimulated various quarantine measures taken by cities outside of Hubei (as the mediators), which further decreased the migration countrywide in China, and thus indirectly control the spread of COVID-19.
Revised: This approach (viewed as the exposure), directly blocked infected people leaving from Hubei; and also stimulated various quarantine measures taken by cities outside of Hubei (as the mediators), which further decreased the migration countrywide in China, and thus indirectly control the spread of COVID-19.

Original: Quantifying the causal effects of 2020 Hubei lockdowns on reducing the COVID-19 spread regulated by different cities outside Hubei is challenging but of great interest for the current COVID-19 crisis.
Revised: Quantifying the causal effects of 2020 Hubei lockdowns on reducing the COVID-19 spread regulated by different cities outside Hubei is challenging but of great interest for the current COVID-19 crisis.

Original: An analysis of causal effects that interprets the causal mechanism contributed via individual mediators is thus very important.
Revised: An analysis of causal effects that interprets the causal mechanism contributed via individual mediators is thus very important.

Original: Many recent efforts have been made on studying causal effects that are regulated by mediators.
Revised: Many recent efforts have been made on studying causal effects that are regulated by mediators.

Original: Chakrabortty et al.
Revised: Chakrabortty et al.

Original: speciﬁed the individual mediation effect in a sparse high-dimensional causal graphical model.
Revised: speciﬁed the individual mediation effect in a sparse high-dimensional causal graphical model.

Original: However, the sum of marginal individual mediation effect is not equal to the effect of all mediators considered jointly (i.e. the indirect effect) due to the common interaction among mediators (VanderWeele & Vansteelandt, 2014).
Revised: However, the sum of marginal individual mediation effect is not equal to the effect of all mediators considered jointly (i.e. the indirect effect) due to the common interaction among mediators (VanderWeele & Vansteelandt, 2014).

Original: Here, ‘interaction’ means that there exists at least one mediator that is regulated by other mediator(s) (see Figure 1b for illustration), in contrast to the simple ‘parallel’ case (shown in Figure 1a).
Revised: Here, ‘interaction’ means that there exists at least one mediator that is regulated by other mediator(s) (see Figure 1b for illustration), in contrast to the simple ‘parallel’ case (shown in Figure 1a).

Original: Vansteelandt & Daniel (2017) considered an exact decomposition of the indirect effect with a two-mediator setting based on the conditional densities of mediators, while there was no feasible algorithm provided to solve their proposed expressions yet.
Revised: Vansteelandt & Daniel (2017) considered an exact decomposition of the indirect effect with a two-mediator setting based on the conditional densities of mediators, while there was no feasible algorithm provided to solve their proposed expressions yet.

Original: Therefore, a new framework with a computational friendly algorithm that gives an exact decomposition of the indirect effect on the level of individual mediators is desired under the complex causal network.
Revised: Therefore, a new framework with a computational friendly algorithm that gives an exact decomposition of the indirect effect on the level of individual mediators is desired under the complex causal network.

Original: M
Revised: M

Original: Y A
Revised: Y A

Original: M
Revised: M

Original: M p
Revised: M p

Original: M p- (a) A causal graph with parallel
Revised: M p- (a

Original: [Figure]
Revised: [Figure]

Original: M
Revised: M

Original: Y A
Revised: Y A

Original: M
Revised: M

Original: M p
Revised: M p

Original: M p- (b) A causal graph with interacted mediators.
Revised: M p- ( [Figure]

Original: To estimate the underlying causal network, structure learning algorithms of the directed acyclic graph (DAG) are widely used.
Revised: To estimate the underlying causal network, structure learning algorithms of the directed acyclic graph (DAG) are widely used.

Original: Popular methods such as the PC algorithm (Spirtes et al., 2000) that uses conditional independence tests to examine the existence of edges between each pair of variables, require strong assumptions and thus have no guarantee in the ﬁnite sample regime.
Revised: Popular methods such as the PC algorithm (Spirtes et al., 2000) that uses conditional independence tests to examine the existence of edges between each pair of variables, require strong assumptions and thus have no guarantee in the ﬁnite sample regime.

Original: Recently,
Revised: Recently,

Original: Zheng et al.
Revised: Zheng et al.

Original: (2018) opened up another class of causal discovery methods by directly formulating a pure optimization problem over real metrics with a novel characteristic of the acyclicity.
Revised: (2018) opened up another class of causal discovery methods by directly formulating a pure optimization problem over real metrics with a novel characteristic of the acyclicity.

Original: Yu et al.
Revised: Yu et al.

Original: (2019) further extended Zheng et al.
Revised: (2019) further extended Zheng et al.

Original: (2018)’s work with a deep generative model, and showed better performance on the structure learning with weaker assumptions on the noise.
Revised: (2018)’s work with a deep generative model, and showed better performance on the structure learning with weaker assumptions on the noise.

Original: See more follow-up works in Lachapelle et al.
Revised: See more follow-up works in Lachapelle et al.

Original: and Zhu & Chen (2019).
Revised: and Zhu & Chen (2019).

Original: However, the current cutting-edge methods neglect the temporal causal relationship among variables, and thus cannot appropriately represent the causal network with pre-speciﬁed exposure and outcome.
Revised: However, the current cutting-edge methods neglect the temporal causal relationship among variables, and thus cannot appropriately represent the causal network with pre-speciﬁed exposure and outcome.

Original: In this paper, we consider establishing a new statistical framework to comprehensively characterize causal effects with multiple mediators, namely, ANalysis Of Causal Effects (ANOCE), under the linear structure equation model (LSEM).
Revised: In this paper, we consider establishing a new statistical framework to comprehensively characterize causal effects with multiple mediators, namely, ANalysis Of Causal Effects (ANOCE), under the linear structure equation model (LSEM).

Original: Speciﬁcally, we propose two causal effects on the level of individual mediators, the natural direct effect and the natural indirect effect for a mediator, denoted as
Revised: Speciﬁcally, we propose two causal effects on the level of individual mediators, the natural direct effect and the natural indirect effect for a mediator, denoted as

Original: DM and IM , respectively.
Revised: DM and IM , respectively.

Original: Our proposed DM can be interpreted as the direct effect of a particular mediator on the outcome that is not regulated by other mediators, while the IM is the indirect effect of the mediator controlled by its descendant mediators.
Revised: Our proposed DM can be interpreted as the direct effect of a particular mediator on the outcome that is not regulated by other mediators, while the IM is the indirect effect of the mediator controlled by its descendant mediators.

Original: We prove that the DM is valid in the sense that it exactly decomposes the indirect effect of the exposure on the outcome, followed by an
Revised: We prove that the DM is valid in the sense that it exactly decomposes the indirect effect of the exposure on the outcome, followed by an

Original: ANOCE table to explain different sources of causal effects.
Revised: ANOCE table to explain different sources of causal effects.

Original: To bridge the cutting-edge graphical learning approaches with the temporal causal relationship of variables, we extend the variational auto-encoder (VAE) framework in Yu et al.
Revised: To bridge the cutting-edge graphical learning approaches with the temporal causal relationship of variables, we extend the variational auto-encoder (VAE) framework in Yu et al.

Original: with a novel identiﬁcation constraint that speciﬁes the topological order of the exposure and the outcome.
Revised: with a novel identiﬁcation constraint that speciﬁes the topological order of the exposure and the outcome.

Original: The proposed constrained VAE algorithm is then used to estimate causal effects deﬁned in our ANOCE table, named as ‘ANOCE-CVAE’.
Revised: The proposed constrained VAE algorithm is then used to estimate causal effects deﬁned in our ANOCE table, named as ‘ANOCE-CVAE’.

Original: Our contributions can be summarized in the following three aspects:
Revised: Our contributions can be summarized in the following three aspects:

Original: Conceptually, we deﬁne different sources of causal effects through mediators with a newly introduced deﬁnition of direct and indirect mediator effects, and give an exact decomposition of the indirect effect on the level of individual mediators, under the linear structure equation model.
Revised: Conceptually, we deﬁne different sources of causal effects through mediators with a newly introduced deﬁnition of direct and indirect mediator effects, and give an exact decomposition of the indirect effect on the level of individual mediators, under the linear structure equation model.

Original: Methodologically, we incorporate the background knowledge (the temporal causal relationship among variables) when using an optimization approach to the causal discovery.
Revised: Methodologically, we incorporate the background knowledge (the temporal causal relationship among variables) when using an optimization approach to the causal discovery.

Original: Such prior knowledge can be generalized for any measured variable and on the possible set of their parents.
Revised: Such prior knowledge can be generalized for any measured variable and on the possible set of their parents.

Original: Our proposed constrained structural learning can be easily extended to other score-based algorithms.
Revised: Our proposed constrained structural learning can be easily extended to other score-based algorithms.

Original: Practically, extensive simulations are conducted to demonstrate the empirical validity of the proposed algorithm and its competitive performance among existing causal discovery algorithms.
Revised: Practically, extensive simulations are conducted to demonstrate the empirical validity of the proposed algorithm and its competitive performance among existing causal discovery algorithms.

Original: Our method is applied to investigate the causal effects of 2020 Hubei lockdowns on reducing the COVID-19 spread in China, by quantifying the individual effect for each city.
Revised: Our method is applied to investigate the causal effects of 2020 Hubei lockdowns on reducing the COVID-19 spread in China, by quantifying the individual effect for each city.

Original:  B ACKGROUND AND R ELATED W ORK 2. G RAPH T ERMINOLOGY
Revised:  B ACKGROUND AND R ELATED W ORK 2. G RAPH T ERMINOLOGY

Original: Consider a graph G =
Revised: Consider a graph G =

Original: ( X, E ) with a node set X and an edge set E .
Revised: ( X, E ) with a node set X and an edge set E .

Original: There is at most one edge between any pair of nodes.
Revised: There is at most one edge between any pair of nodes.

Original: If there is an edge between X i and X j , then X i and X j are adjacent.
Revised: If there is an edge between X i and X j , then X i and X j are adjacent.

Original: A node X i is said to be a parent of X j if there is a directed edge from X i to X j .
Revised: A node X i is said to be a parent of X j if there is a directed edge from X i to X j .

Original: Let the set of all parents of node X j in G as PA X j ( G ) .
Revised: Let the set of all parents of node X j in G as PA X j ( G ) .

Original: A path from X i to X j in G is a sequence of distinct vertices, π ≡ { a 0 , a 1 , · · · , a L } ⊂ V such that a 0 =
Revised: A path from X i to X j in G is a sequence of distinct vertices, π ≡ { a 0 , a 1 , · · · , a L } ⊂ V such that a 0 =

Original: X i , and a L = X j .
Revised: X i , and a L = X j .

Original: A directed path from X i to X j is a path between X i and X j where all edges are directed toward X j .
Revised: A directed path from X i to X j is a path between X i and X j where all edges are directed toward X j .

Original: A directed cycle is formed by the directed path from X i to X j together with the directed edge X j to X i .
Revised: A directed cycle is formed by the directed path from X i to X j together with the directed edge X j to X i .

Original: A directed graph that does not contain directed cycles is called a directed acyclic graph (DAG).
Revised: A directed graph that does not contain directed cycles is called a directed acyclic graph (DAG).

Original: A directed graph is acyclic if and only if it has a topological ordering.
Revised: A directed graph is acyclic if and only if it has a topological ordering.

Original: Suppose a DAG G =
Revised: Suppose a DAG G =

Original: ( X, E ) that characterizes the causal relationship among | X | = d nodes, where X =
Revised: ( X, E ) that characterizes the causal relationship among | X | = d nodes, where X =

Original: [ X 1 , X 2 , · · · , X d ] (cid:62) represents a random vector and an edge X i
Revised: [ X 1 , X 2 , · · · , X d ] (cid:62) represents a random vector and an edge X i

Original: → X j means that X i is a direct cause of X j .
Revised: → X j means that X i is a direct cause of X j .

Original: Let B = { b i,j } 1 ≤ i ≤ d, 1 ≤ j ≤ d be a d × d matrix, where b i,j is the weight of the edge X i → X j ∈ E , and b i,j = 0 otherwise.
Revised: Let B = { b i,j } 1 ≤ i ≤ d, 1 ≤ j ≤ d be a d × d matrix, where b i,j is the weight of the edge X i → X j ∈ E , and b i,j = 0 otherwise.

Original: Then, we say that G =
Revised: Then, we say that G =

Original: ( X, B ) is a weighted DAG with the node set X and the weighted adjacency matrix B (the edge set E is nested in B ).
Revised: ( X, B ) is a weighted DAG with the node set X and the weighted adjacency matrix B (the edge set E is nested in B ).

Original: 2. R ELATED W ORK
Revised: 2. R ELATED W ORK

Original: Our work connects to the literature of the causal graphical model.
Revised: Our work connects to the literature of the causal graphical model.

Original: Pearl et al.
Revised: Pearl et al.

Original: (2009) provided a comprehensive review of recent advances in the analysis of causes and counterfactuals using ‘dooperator’ by graphical methods.
Revised: (2009) provided a comprehensive review of recent advances in the analysis of causes and counterfactuals using ‘dooperator’ by graphical methods.

Original: Later, Maathuis et al.
Revised: Later, Maathuis et al.

Original: (2009) started to use an unknown DAG without hidden variables to estimate the causal effects from the high-dimensional observational data.
Revised: (2009) started to use an unknown DAG without hidden variables to estimate the causal effects from the high-dimensional observational data.

Original: Nandy et al.
Revised: Nandy et al.

Original: (2017) extended the work of Maathuis et al.
Revised: (2017) extended the work of Maathuis et al.

Original: (2009) with the linear structure equation model, followed by the individual mediation effect deﬁned in Chakrabortty et al.
Revised: (2009) with the linear structure equation model, followed by the individual mediation effect deﬁned in Chakrabortty et al.

Original: All of these models rely on the PC algorithm to search the Markov equivalence class of the partial DAG, and usually require strong assumptions due to the computational limit.
Revised: All of these models rely on the PC algorithm to search the Markov equivalence class of the partial DAG, and usually require strong assumptions due to the computational limit.

Original: Our ANOCE is established under the same causal structure of Chakrabortty et al.
Revised: Our ANOCE is established under the same causal structure of Chakrabortty et al.

Original: but without sparsity and normality assumptions.
Revised: but without sparsity and normality assumptions.

Original: Wide literature on causal discovery can be summarized in three classes.
Revised: Wide literature on causal discovery can be summarized in three classes.

Original: The ﬁrst type focuses on local conditional independence tests to ﬁnd a causal skeleton and then determine the orientation of edges, such as the well-known PC algorithm (Spirtes et al., 2000; Kalisch & Bühlmann, 2007).
Revised: The ﬁrst type focuses on local conditional independence tests to ﬁnd a causal skeleton and then determine the orientation of edges, such as the well-known PC algorithm (Spirtes et al., 2000; Kalisch & Bühlmann, 2007).

Original: However,these methods have no guarantee in the ﬁnite sample regime because of the hardness of testing  conditional independence when the variables are continuous  (Shah & Peters, 2018).
Revised: However, testing the conditional independence of continuous variables is not easy (Shah & Peters, 2018).

Original: The second class speciﬁes properly functional causal models with additional assumptions on data distribution, including the ICA-LiNGAM
Revised: The second class speciﬁes properly functional causal models with additional assumptions on data distribution, including the ICA-LiNGAM

Original: (Shimizu et al., 2006) and the causal additive model (CAM) (Bühlmann et al., 2014).
Revised: (Shimizu et al., 2006) and the causal additive model (CAM) (Bühlmann et al., 2014).

Original: The last class, the score-based method, includes the greedy equivalence search (GES) (Chickering, 2002) and the fast GES (fGES)  that use for example Bayesian scores in searching a space of causal models.
Revised: The last class, the score-based method, includes the greedy equivalence search (GES) (Chickering, 2002) and the fast GES (fGES) (Ramsey et al., 2017) that use for example Bayesian scores in searching a space of causal models.

Original: Recently, Zheng et al.
Revised: Recently, Zheng et al.

Original: (2018) opened up another track of score-based methods by constructing an optimization with an acyclicity constraint under the LSEM, i.e. the NOTEARS.
Revised: (2018) opened up another track of score-based methods by constructing an optimization with an acyclicity constraint under the LSEM, i.e. the NOTEARS.

Original: A follow-up work using a VAE parameterized by a graph neural network that generalizes LSEM was proposed in Yu et al.
Revised: A follow-up work using a VAE parameterized by a graph neural network that generalizes LSEM was proposed in Yu et al.

Original: with a more computational friendly constraint, namely DAG-GNN.
Revised: with a more computational friendly constraint, namely DAG-GNN.

Original: Also see Zhu & Chen (2019) and Lachapelle et al.
Revised: Also see Zhu & Chen (2019) and Lachapelle et al.

Original: for other cutting-edge structural learning methods.
Revised: for other cutting-edge structural learning methods.

Original: Our ANOCE-CVAE algorithm extends (though not limits to)
Revised: The improvement of our ANOCE-CVAE over the state-of-the-arts is as follows.

Revised: We consider a new constrained structural learning, by incorporating the background knowledge (the temporal causal relationship among variables) into the score-based algorithms.

Revised: We formulated such prior information as the identiﬁcation constraint and add it as the penalty term in the objective function for the causal discovery.

Original:  the DAG-GNN for an illustration, since it can handle non-linear cases and does not require the normality assumption of the noise.
Revised: In this paper, we typically extend the DAG-GNN for an illustration. Note that the proposed constraint is not limited to the DAG-GNN and can be easily extended to other score-based algorithms.

Original:  A NALYSIS OF C AUSAL E FFECTS 3. S TATISTICAL F RAMEWORK AND A SSUMPTIONS
Revised:  A NALYSIS OF C AUSAL E FFECTS 3. S TATISTICAL F RAMEWORK AND A SSUMPTIONS

Original: Let A be the exposure/treatment, M =
Revised: Let A be the exposure/treatment, M =

Original: [ M 1 , M 2 , · · · , M p ] (cid:62) be mediators with dimension p , and Y be the outcome of interest.
Revised: [ M 1 , M 2 , · · · , M p ] (cid:62) be mediators with dimension p , and Y be the outcome of interest.

Original: Suppose there exists a weighted DAG G =
Revised: Suppose there exists a weighted DAG G =

Original: ( X, B ) that characterizes the causal relationship among X
Revised: ( X, B ) that characterizes the causal relationship among X

Original: =
Revised: =

Original: [ A, M (cid:62) , Y ] (cid:62) , where the dimension of X is d = p + 2 .
Revised: [ A, M (cid:62) , Y ] (cid:62) , where the dimension of X is d = p + 2 .

Original: Let
Revised: Let

Original: Y ∗ ( A = a, M = m ) be the potential outcome that would be observed after receiving treatment a and setting mediators as m , and M ∗ ( A = a ) be the potential mediators that would be observed after receiving treatment a .
Revised: Y ∗ ( A = a, M = m ) be the potential outcome that would be observed after receiving treatment a and setting mediators as m , and M ∗ ( A = a ) be the potential mediators that would be observed after receiving treatment a .

Original: As standard in the causal inference (Rosenbaum & Rubin, 1983), we assume that there is no unmeasured confounder: (A1) the effect of the treatment A on the outcome Y is unconfounded, i.e. Y ∗ ( A
Revised: As standard in the causal inference (Rosenbaum & Rubin, 1983), we assume that there is no unmeasured confounder: (A1) the effect of the treatment A on the outcome Y is unconfounded, i.e. Y ∗ ( A

Original: = a, M = m )
Revised: = a, M = m )

Original: ⊥ A, ∀ a, m ; (A2) the effect of the treatment A on the mediators M is unconfounded, i.e. M ∗ ( A = a ) ⊥ A, ∀ a ; (A3) the effect of the mediators M on the outcome Y is unconfounded given the treatment A , i.e. Y ∗ ( A = a, M = m )
Revised: ⊥ A, ∀ a, m ; (A2) the effect of the treatment A on the mediators M is unconfounded, i.e. M ∗ ( A = a ) ⊥ A, ∀ a ; (A3) the effect of the mediators M on the outcome Y is unconfounded given the treatment A , i.e. Y ∗ ( A = a, M = m )

Original: ⊥ M | A, ∀ a, m .
Revised: ⊥ M | A, ∀ a, m .

Original: In addition, as standard in the graphical causal discovery, we also make the Markov condition, the faithfulness condition, causal sufﬁciency assumption, and the linear structural equation model (LSEM) such that X
Revised: In addition, as standard in the graphical causal discovery, we also make the Markov condition, the faithfulness condition, causal sufﬁciency assumption, and the linear structural equation model (LSEM) such that X

Original: =
Revised: =

Original: [ A, M (cid:62) , Y ] (cid:62) characterized by the pair ( G , (cid:15) ) is generated by
Revised: [ A, M (cid:62) , Y ] (cid:62) characterized by the pair ( G , (cid:15) ) is generated by

Original: [Equation]
Revised: [Equation]

Original: where (cid:15) is a random vector of jointly independent error variables.
Revised: where (cid:15) is a random vector of jointly independent error variables.

Original: Denote all directed paths in G that start with the exposure A and end with the outcome Y as set { π AY ( G ) } .
Revised: Denote all directed paths in G that start with the exposure A and end with the outcome Y as set { π AY ( G ) } .

Original: If there exists at least one directed path π ∗ ∈ { π AY ( G ) } such that the length of π ∗ is larger than 2, we say there is an interaction among mediators, as shown in Figure 1b; otherwise, we call mediators are ‘parallel’ as shown in Figure 1a.
Revised: If there exists at least one directed path π ∗ ∈ { π AY ( G ) } such that the length of π ∗ is larger than 2, we say there is an interaction among mediators, as shown in Figure 1b; otherwise, we call mediators are ‘parallel’ as shown in Figure 1a.

Original: In this paper, we consider all possible causal structures with multiple mediators under assumptions (A1-A3).
Revised: In this paper, we consider all possible causal structures with multiple mediators under assumptions (A1-A3).

Original: We next give the total effect ( TE ), the natural direct effect that is not mediated by mediators ( DE ), and the natural indirect effect that is regulated by mediators ( IE ) deﬁned in Pearl et al.
Revised: We next give the total effect ( TE ), the natural direct effect that is not mediated by mediators ( DE ), and the natural indirect effect that is regulated by mediators ( IE ) deﬁned in Pearl et al.

Original: Deﬁnition 3.1 (Pearl et al., 2009)
Revised: Deﬁnition 3.1 (Pearl et al., 2009)

Original: [Equation]
Revised: [Equation]

Original: where do ( A = a ) is a mathematical operator to simulate physical interventions that hold A constant as a while keeping the rest of the model unchanged, which corresponds to remove edges into A and replace A by the constant a in G .
Revised: where do ( A = a ) is a mathematical operator to simulate physical interventions that hold A constant as a while keeping the rest of the model unchanged, which corresponds to remove edges into A and replace A by the constant a in G .

Original: Here, m ( a ) is the value of M if setting do ( A = a ) , and m ( a +1) is the value of M if setting do ( A = a +1) .
Revised: Here, m ( a ) is the value of M if setting do ( A = a ) , and m ( a +1) is the value of M if setting do ( A = a +1) .

Original: Refer to Pearl et al.
Revised: Refer to Pearl et al.

Original: (2009) for more details of ‘do-operator’.
Revised: (2009) for more details of ‘do-operator’.

Original: Note that in the assumed linear model, the slope of the line is the same everywhere; for convenience and simplicity, we use a and a + 1 to present the change of the treatment of 1 in the deﬁnition.
Revised: Note that in the assumed linear model, the slope of the line is the same everywhere; for convenience and simplicity, we use a and a + 1 to present the change of the treatment of 1 in the deﬁnition.

Original: 3. N ATURAL D IRECT AND I NDIRECT E FFECT FOR I NDIVIDUAL M EDIATORS
Revised: 3. N ATURAL D IRECT AND I NDIRECT E FFECT FOR I NDIVIDUAL M EDIATORS

Original: We ﬁrst give the deﬁnition of the natural direct effect for an individual mediator ( DM ).
Revised: We ﬁrst give the deﬁnition of the natural direct effect for an individual mediator ( DM ).

Original: Deﬁnition 3.2 Natural direct effect for M i :
Revised: Deﬁnition 3.2 Natural direct effect for M i :

Original: [Equation]
Revised: [Equation]

Original: where m ( a ) i is the value of M i when setting do ( A = a ) , Ω i
Revised: where m ( a ) i is the value of M i when setting do ( A = a ) , Ω i

Original: =
Revised: =

Original: M \ M i is the set of mediators except
Revised: M \ M i is the set of mediators except

Original: M i , and o ( a ) i is the value of Ω i when setting do ( A = a ) .
Revised: M i , and o ( a ) i is the value of Ω i when setting do ( A = a ) .

Original: Remark 3.1 From Deﬁnition 3.2, the natural direct effect for M i is the product of the total effect of the treatment A on the mediator M i and the direct effect of the mediator M i on the outcome Y .
Revised: Remark 3.1 From Deﬁnition 3.2, the natural direct effect for M i is the product of the total effect of the treatment A on the mediator M i and the direct effect of the mediator M i on the outcome Y .

Original: The second multiplier is in line with the classical meaning of ‘natural’ in the causal inference literature (Pearl et al., 2009).
Revised: The second multiplier is in line with the classical meaning of ‘natural’ in the causal inference literature (Pearl et al., 2009).

Original: Thus, the DM can be interpreted as the causal effect through a particular mediator from the treatment on the outcome that is not regulated by its descendent mediators.
Revised: Thus, the DM can be interpreted as the causal effect through a particular mediator from the treatment on the outcome that is not regulated by its descendent mediators.

Original: The natural indirect effect for an individual mediator ( IM ) can be deﬁned similarly.
Revised: The natural indirect effect for an individual mediator ( IM ) can be deﬁned similarly.

Original: Deﬁnition 3.3 Natural indirect effect for M i :
Revised: Deﬁnition 3.3 Natural indirect effect for M i :

Original: [Equation]
Revised: [Equation]

Original: Remark 3.2 The second multiplier in the IM i captures the indirect effect of a particular mediator on the outcome regulated by its descendent mediators.
Revised: Remark 3.2 The second multiplier in the IM i captures the indirect effect of a particular mediator on the outcome regulated by its descendent mediators.

Original: We show the individual mediation effect  in
Revised: We show the individual mediation effect ( η ) in

Original: Chakrabortty et al.
Revised: Chakrabortty et al.

Original: (2018) can be decomposed into the DM and the IM in Section F in the appendix.
Revised: (2018) can be decomposed into the DM and the IM in Section F in the appendix when the LSEM assumption holds, i.e., η i = DM

Revised: i + IM i , for i -th mediator.

Original: Next, we give explicit expressions of deﬁned causal effects under the LSEM.
Revised: Next, we give explicit expressions of deﬁned causal effects under the LSEM.

Original: Speciﬁcally, we can write the linear structural model 1 under assumptions (A1-A3) as
Revised: Speciﬁcally, we can write the linear structural model 1 under assumptions (A1-A3) as

Original: [Equation]
Revised: [Equation]

Original: where γ is a scalar, α , β , and 0 p × 1 are p × 1 vectors, B M is a p × p matrix, and (cid:15) ≡ [
Revised: where γ is a scalar, α , β , and 0 p × 1 are p × 1 vectors, B M is a p × p matrix, and (cid:15) ≡ [

Original: (cid:15) A , (cid:15) (cid:62) M , (cid:15) Y ] (cid:62) .
Revised: (cid:15) A , (cid:15) (cid:62) M , (cid:15) Y ] (cid:62) .

Original: Here, γ presents the weight of the edge A → Y , the i -th element of α corresponds to the weight of the edge
Revised: Here, γ presents the weight of the edge A → Y , the i -th element of α corresponds to the weight of the edge

Original: A → M i , and the i -th element of β is the weight of the edge M i
Revised: A → M i , and the i -th element of β is the weight of the edge M i

Original: → Y .
Revised: → Y .

Original: Note that by assumptions (A1-A3), we have the exposure A has no parents and the outcome Y has no descendants, so equivalently, the ﬁrst row and the last column of B (cid:62) are all zeros (i.e., the ﬁrst column and the last row of B are all zeros). {+Notice the exposure can be presented by its own noise, i.e., A = (cid:
+}
Revised: Note that by assumptions (A1-A3), we have the exposure A has no parents and the outcome Y has no descendants, so equivalently, the ﬁrst row and the last column of B (cid:62) are all zeros (i.e., the ﬁrst column and the last row of B are all zeros). {+Notice the exposure can be presented by its own noise, i.e., A = (cid:
+}

Revised: ) A , since A has no parents, so any exposure (with arbitrary noise distribution) will satisfy the LSEM assumption.

Original: Next, we obtain expressions of causal effects under the LSEM in the following theorem.
Revised: Next, we obtain expressions of causal effects under the LSEM in the following theorem.

Original: The proof can be found in Section G.1 of the appendix.
Revised: The proof can be found in Section G.1 of the appendix.

Original: Theorem 3.1 Under assumptions (A1-A3) and Model 1, we have:
Revised: Theorem 3.1 Under assumptions (A1-A3) and Model 1, we have:

Original: the natural direct effect is DE = γ ; 2).
Revised: the natural direct effect is DE = γ ; 2).

Original: the natural indirect effect is IE = β (cid:62) ( I p − B (cid:62) M ) − 1 α , where I p is a p × p identity matrix; 3).
Revised: the natural indirect effect is IE = β (cid:62) ( I p − B (cid:62) M ) − 1 α , where I p is a p × p identity matrix; 3).

Original: the total effect of A on Y is TE = γ + β (cid:62) ( I p − B (cid:62) M ) − 1 α ; 4).
Revised: the total effect of A on Y is TE = γ + β (cid:62) ( I p − B (cid:62) M ) − 1 α ; 4).

Original: the natural direct effect of M i on Y is DM i
Revised: the natural direct effect of M i on Y is DM i

Original: = β
Revised: = β

Original: i { ( I p − B (cid:62) M ) − 1 α } i , where β i is the i -th element of β corresponding to the weight of M i
Revised: i { ( I p − B (cid:62) M ) − 1 α } i , where β i is the i -th element of β corresponding to the weight of M i

Original: → Y , and { ( I p − B (cid:62) M ) − 1 α } i is the i -th element of ( I p − B (cid:62) M ) − 1 α as the total effect of A on M i ,
Revised: → Y , and { ( I p − B (cid:62) M ) − 1 α } i is the i -th element of ( I p − B (cid:62) M ) − 1 α as the total effect of A on M i ,

Original: i
Revised: i

Original: e. E { M i | do ( A = a + 1) } − E { M i | do ( A = a ) } .
Revised: e. E { M i | do ( A = a + 1) } − E { M i | do ( A = a ) } .

Original: Remark 3.3 One may refer to section A in the appendix for the invertibility of I p − B (cid:62) M .
Revised: Remark 3.3 One may refer to section A in the appendix for the invertibility of I p − B (cid:62) M .

Original: Also, a toy example is provided in section E to illustrate how to manually compute the causal effects deﬁned above.
Revised: Also, a toy example is provided in section E to illustrate how to manually compute the causal effects deﬁned above.

Original: Note that there is no explicit expression of the IM due to the complex interaction among mediators, while we provide its theoretical form in Section G.2 with its numerical form in Section B.
Revised: Note that there is no explicit expression of the IM due to the complex interaction among mediators, while we provide its theoretical form in Section G.2 with its numerical form in Section B.

Original: Based on the result 2) and 4) in Theorem 3.1, the IE can be presented as an additive form of DM s, as shown in Theorem 3.2.
Revised: Based on the result 2) and 4) in Theorem 3.1, the IE can be presented as an additive form of DM s, as shown in Theorem 3.2.

Original: Thus, the proposed natural direct effect of individual mediators is valid in the sense that it exactly decomposes the indirect effect of the exposure on the outcome.
Revised: Thus, the proposed natural direct effect of individual mediators is valid in the sense that it exactly decomposes the indirect effect of the exposure on the outcome.

Original: Theorem 3.2 Under assumptions (A1-A3) and Model 1, the IE can be decomposed through DM s:
Revised: Theorem 3.2 Under assumptions (A1-A3) and Model 1, the IE can be decomposed through DM s:

Original: [Equation]
Revised: [Equation]

Original: 3. A NALYSIS OF C AUSAL E FFECTS T ABLE
Revised: 3. A NALYSIS OF C AUSAL E FFECTS T ABLE

Original: Based on the result TE = DE + IE in Pearl et al.
Revised: Based on the result TE = DE + IE in Pearl et al.

Original: (2009) and Theorem 3.2, we summarize the deﬁned causal effects and their relationship in Table 1 for the analysis of causal effects (ANOCE).
Revised: (2009) and Theorem 3.2, we summarize the deﬁned causal effects and their relationship in Table 1 for the analysis of causal effects (ANOCE).

Original: Firstly, the causal effect of A on Y has two sources, the direct effect from A and the indirect effect via p mediators M ( M 1 , · · · , M p ).
Revised: Firstly, the causal effect of A on Y has two sources, the direct effect from A and the indirect effect via p mediators M ( M 1 , · · · , M p ).

Original: Next, the direct source has the degree of freedom ( d.f. ) as 1, while the indirect source has d.f. as p from p mediators.
Revised: Next, the direct source has the degree of freedom ( d.f. ) as 1, while the indirect source has d.f. as p from p mediators.

Original: Note the true d.f. of the indirect effect may be smaller than p , since A may not be regulated by all mediators.
Revised: Note the true d.f. of the indirect effect may be smaller than p , since A may not be regulated by all mediators.

Original: Then, the causal effect for the direct source is the DE and for the indirect source is the IE , where the IE can be further decomposed into p DM s and each component corresponds to the natural direct effect for a speciﬁc mediator.
Revised: Then, the causal effect for the direct source is the DE and for the indirect source is the IE , where the IE can be further decomposed into p DM s and each component corresponds to the natural direct effect for a speciﬁc mediator.

Original: The last row in the table shows that the DE and the IE compose the total effect TE with d.f. as p + 1 .
Revised: The last row in the table shows that the DE and the IE compose the total effect TE with d.f. as p + 1 .

Original: [Table]
Revised: [Table]

Revised: p Total 1 + p

Original:   C ONSTRAINED S TRUCTURAL L EARNING FOR ANOCE
Revised: TE  C ONSTRAINED S TRUCTURAL L EARNING FOR ANOCE

Original: We next estimate the weighted adjacency matrix B with our causal framework under the LSEM to calculate causal effects.
Revised: We next estimate the weighted adjacency matrix B with our causal framework under the LSEM to calculate causal effects.

Original: To better capture the sampling distribution faithful to the DAG, we consider a deep generative model that generalizes the LSEM instead of using a regression that heavily relies on assumptions of noise (see more discussion in Section 2.2).
Revised: To better capture the sampling distribution faithful to the DAG, we consider a deep generative model that generalizes the LSEM instead of using a regression that heavily relies on assumptions of noise (see more discussion in Section 2.2).

Original: Speciﬁcally, the LSEM 1 can be rewritten as ( I p +2 − B (cid:62) )
Revised: Speciﬁcally, the LSEM 1 can be rewritten as ( I p +2 − B (cid:62) )

Original: X = (cid:15) , where I p +2 is a ( p +2) × ( p +2) identity matrix.
Revised: X = (cid:15) , where I p +2 is a ( p +2) × ( p +2) identity matrix.

Original: Inversely, we have
Revised: Inversely, we have

Original: X =
Revised: X =

Original: ( I p +2 − B (cid:62) ) − 1 (cid:15) .
Revised: ( I p +2 − B (cid:62) ) − 1 (cid:15) .

Original: Following the VAE architecture in Yu et al.
Revised: Following the VAE architecture in Yu et al.

Original: (2019), we treat the random error (cid:15) as the independent latent variables to generate X , by two multilayer perceptrons as the encoder and the decoder, with weights denoted as θ .
Revised: (2019), we treat the random error (cid:15) as the independent latent variables to generate X , by two multilayer perceptrons as the encoder and the decoder, with weights denoted as θ .

Original: We adopt their acyclicity constraint on B as,
Revised: We adopt their acyclicity constraint on B as,

Original: [Equation]
Revised: [Equation]

Original: where tr ( · ) is the trace of a matrix, t is a hyperparameter that depends on an estimation of the largest eigenvalue of B , and • denotes for the element-wise square.
Revised: where tr ( · ) is the trace of a matrix, t is a hyperparameter that depends on an estimation of the largest eigenvalue of B , and • denotes for the element-wise square.

Original: Next, to incorporate the background knowledge of the temporal causal relationship among variables, we propose an identiﬁcation constraint that indicates the topological order of the exposure and the outcome.
Revised: Next, to incorporate the background knowledge of the temporal causal relationship among variables, we propose an identiﬁcation constraint that indicates the topological order of the exposure and the outcome.

Original: As mentioned in Equation 3, under assumptions (A1-A3), the exposure A has no parents, i.e. PA A ( G ) = ∅ , and the outcome Y has no descendants, i.e. Y (cid:54)∈ PA X ( G ) .
Revised: As mentioned in Equation 3, under assumptions (A1-A3), the exposure A has no parents, i.e. PA A ( G ) = ∅ , and the outcome Y has no descendants, i.e. Y (cid:54)∈ PA X ( G ) .

Original: Or equivalently, we have the ﬁrst column and the last row of B should equal to zero.
Revised: Or equivalently, we have the ﬁrst column and the last row of B should equal to zero.

Original: Therefore, the matrix B must satisfy
Revised: Therefore, the matrix B must satisfy

Original: [Equation]
Revised: [Equation]

Original: where b i,j is the element of the matrix B in i -th row and j -th column.
Revised: where b i,j is the element of the matrix B in i -th row and j -th column.

Original: The above constraint forces the topological order of the exposure as 1 while the outcome as p + 2 , under which the DAG is searched within a restricted regime.
Revised: The above constraint forces the topological order of the exposure as 1 while the outcome as p + 2 , under which the DAG is searched within a restricted regime.

Original: The prior knowledge in 4 can be generalized for any measured variable and on the possible set of their parents, by connecting the topological order to the weighted matrix B .
Revised: The prior knowledge in 5 can be generalized for any measured variable and on the possible set of their parents, by connecting the topological order to the weighted matrix B .

Original: Following Yu et al.
Revised: Following Yu et al.

Original: (2019), the objective function is the evidence lower bound with two constraints:
Revised: (2019), the objective function is the evidence lower bound with two constraints:

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: where D KL ( ·||· ) is the Kullback-Leibler divergence, p ( (cid:15) ) is the prior distribution of (cid:15) , q ( (cid:15) | X i ) is the reconstructed empirical posterior distribution of (cid:15) , and p ( X i | (cid:15) ) is the likelihood function.
Revised: where D KL ( ·||· ) is the Kullback-Leibler divergence, p ( (cid:15) ) is the prior distribution of (cid:15) , q ( (cid:15) | X i ) is the reconstructed empirical posterior distribution of (cid:15) , and p ( X i | (cid:15) ) is the likelihood function.

Original: Then, we have the loss function based on the augmented Lagrangian as
Revised: Then, we have the loss function based on the augmented Lagrangian as

Original: [Equation]
Revised: [Equation]

Original: where λ 1 and λ 2 are Lagrange multipliers, and c and d are penalty terms.
Revised: where λ 1 and λ 2 are Lagrange multipliers, and c and d are penalty terms.

Original: To minimize the loss inand satisfy both h 1 ( B ) = 0 and h 2 ( B ) = 0 , we simultaneously update λ 1 and λ 2 and increase c and d to inﬁnity, by modifying the basic technique in Yu et al.
Revised: To minimize the loss inand satisfy both h 1 ( B ) = 0 and h 2 ( B ) = 0 , we simultaneously update λ 1 and λ 2 and increase c and d to inﬁnity, by modifying the basic technique in Yu et al.

Original: Here, the minimization can be solved using a blackbox stochastic optimization such as ‘Adam’ in Kingma & Ba (2014).
Revised: Here, the minimization can be solved using a blackbox stochastic optimization such as ‘Adam’ in Kingma & Ba (2014).

Original: Denote the estimated matrix as (cid:98) B from the above constrained structural learning.
Revised: Denote the estimated matrix as (cid:98) B from the above constrained structural learning.

Original: Under Theorem 3.1, we can estimate causal effects in the ANOCE table based on the learned (cid:98) B .
Revised: Under Theorem 3.1, we can estimate causal effects in the ANOCE table based on the learned (cid:98) B .

Original: We name the above algorithm as
Revised: We name the above algorithm as

Original: ANOCE-CVAE, with a detail pseudocode provided in Section B.
Revised: ANOCE-CVAE, with a detail pseudocode provided in Section B.

Original: Remark 4.1 We incorporate the temporal causal relationship among variables when using an optimization approach to the causal discovery.
Revised: Remark 4.1 We incorporate the temporal causal relationship among variables when using an optimization approach to the causal discovery.

Original: Such constrained structural learning is not limited to the VAE framework and can be extended to any score-based algorithms.
Revised: Such constrained structural learning is not limited to the VAE framework and can be extended to any score-based algorithms.

Original: For instance, one can add constraint 4 into the objective function in Zheng et al.
Revised: For instance, one can add constraint 5 into the objective function in Zheng et al.

Original: (2018) or the reward in Zhu & Chen (2019).
Revised: (2018) or the reward in Zhu & Chen (2019).

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: We conduct extensive simulation studies to investigate the proposed method on learning causal effects with multiple mediators, followed by a comparison to the popular structural learning algorithms.
Revised: We conduct extensive simulation studies to investigate the proposed method on learning causal effects with multiple mediators, followed by a comparison to the popular structural learning algorithms.

Original: 5. S IMULATION S TUDIES
Revised: 5. S IMULATION S TUDIES

Original: Scenarios are generated as follows.
Revised: Scenarios are generated as follows.

Original: In Scenario 1 to 3, we ﬁx the dimension ofthe mediators M as p = 10 while increasing the complexity of theunderlying true graph to examine the sensitivity of our algorithm to sparsity.
Revised: In Scenario 1 to 3, we ﬁx the dimension of M as p = 10 while increasing the complexity of the true graph to examine the sensitivity of our algorithm to sparsity.

Original: Speciﬁcally, Scenario 1 represents the simplest causal graph with only one edge ( A → Y ) shown in Figure 2a; and Scenario 2 has a fully connectedcausal graph with independent mediators (corresponding to the parallel case, i.e. B M = 

Revised: Speciﬁcally, Scenario 1 is the simplest causal graph with only one edge ( A → Y ) shown in Figure 2a; and Scenario 2 has a fully connected graph with independent mediators (corresponding to the parallel case, i.e. B M =

Original: p × p ) illustrated in Figure 2d.
Revised: p × p ) illustrated in Figure 2d.

Original: In Scenario 3, we consider interacted mediators such that B M (cid:54) = 

Revised: In Scenario 3, we consider interacted mediators such that B M (cid:54) =

Original: p × p , as demonstrated in Figure 2g. For Scenario 4, we allow p = 30 with interacted mediators to examine the stability of our method under the high-dimensional setting. 
Revised: p × p , as demonstrated in Figure 2g. For Scenario 4, we allow p = 30 with interacted mediators to examine the stability of our method under the high-dimensional setting. Here, the true

Revised: DAGs in Scenarios 3 and 4 are generated from the Erd ˝ os-Re ´ nyi (ER) model with an expected degree as 2 .

Revised: Note that we consider fully identiﬁable models in the experiments so that it is meaningful to evaluate causal effects from the estimated graph.

Original: Here, { A, M, Y } is generated from Model 1 with normal noises in Scenario 1-4.
Revised: The synthetic datasets { A, M, Y } are generated from Model 1 with Gaussian errors in Scenario 1-4.

Original: We also set A ∈ {− 1 , 1 } in Scenario 4 to show that our algorithm is capable to handle both discrete and continuous exposure, denoted as Scenario 4 ∗ .
Revised: We also set A ∈ {− 1 , 1 } in Scenario 4 to show that our algorithm is capable to handle both discrete and continuous exposure, denoted as Scenario 4 ∗ .

Original: The sample size n is chosen from { 50 , 500 } to be consistent with the scale of our real data analysis.
Revised: The sample size n is chosen from { 50 , 500 } to be consistent with the scale of our real data.

Original: More details of the simulation setting and the implementationare reported in Section C in the appendix to save space.
Revised: See more details of the data generation in Section C.1 and the implementation in Section C.2 in the appendix. [Figure]

Original: The averaged estimated matrix (cid:98) B (cid:62) by the ANOCE-CVAE under different settings is illustrated in
Revised: under the ANOCE-CVAE: (a-c) correspond to Scenario 1; (d-f) correspond to Scenario 2; (g-i) correspond to Scenario 3; (j) is the shared true DAG for Scenario 4 and 4 ∗ ; (k-l) are results under Scenario 4; (m-n) are results under Scenario 4 ∗ . (a) True. [Figure] [Figure]

Original: Figure 2 over 100 replications.
Revised: [Figure] [Figure] [Figure] [Figure]

Original: The numerical results are summarized in Table 2 (for Scenario 1 to 3)
Revised: [Figure] [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Revised: The averaged estimated matrix (cid:98) B (cid:62) over 100 replications under the proposed ANOCE-CVAE is illustrated in Figure 2.

Revised: The numerical results are summarized in Table 2 (for Scenario 1 to 3) and

Original: and Table 3 (for Scenario 4 and 4 ∗ ) in the appendix, including the bias of the estimated TE , DE , 
Revised: Table 3 (for Scenario 4 and 4 ∗ ) in the appendix, including the bias of the estimated TE , DE , IE ,

Original: IE , DM and IM for each mediator with their standard error.
Revised: DM and IM for each mediator with their standard error.

Original: It can be observed that our proposed method could correctly identify most of the edges in the causal graph when n = 500 in almost all cases.
Revised: It can be observed that our proposed method could correctly identify most of the edges in the causal graph when n = 500 in almost all cases.

Original: Based on Table 2 and 3, the estimated causal effects are close to the true values as the sample size increases, indicating the good performance of our proposed method on identifying the causal effects regardless of the sparsity, the distribution of the exposure, and the dimension of mediators.
Revised: Based on Table 2 and 3, the estimated causal effects are close to the true values as the sample size increases, indicating the good performance of our proposed method on identifying the causal effects regardless of the sparsity, the distribution of the exposure, and the dimension of mediators.

Original: 5. C OMPARISON
Revised: 5. C OMPARISON

Original: We  compare our approach against the PC (Spirtes et al., 2000), the ICA-LiNGAM
Revised: We next compare our approach against the PC (Spirtes et al., 2000), the ICA-LiNGAM

Original: (Shimizu et al., 2006), the NOTEARS (Zheng et al., 2018), and the DAG-GNN (Yu et al., 2019), using a moderate setting (Scenario 3).
Revised: (Shimizu et al., 2006), the NOTEARS (Zheng et al., 2018), and the DAG-GNN (Yu et al., 2019).

Original: Default hyper-parameters of these implementations are used unless otherwise stated.
Revised: Random graphs are generated from both the ER and the Scale-Free (SF) networks with the expected degree as 1, 2, and 4, denoted as Cases ER1, ER2, ER4, SF1, SF2, and SF4, respectively. To be consistent with Section 5.1, we refer Scenario 3 (generated by the ER with the degree as 2) as ER2, and set d = 12 (i.e. p = 10 ) with Gaussian errors for other ﬁve cases, under n = 500 . Details of the data generation and the implementation of each method are reported in Section C.1 and C.2.

Original: Here,to be consistent with the NOTEARS and the DAG-GNN, wealso use a graph threshold as 0.3  to prune the noise edges.
Revised: Here, we use a graph threshold as 0.3 (commonly used in other methods) and 0.4 to prune the noise edges for a fair comparison.

Original: The averaged estimated matrix (cid:98) B (cid:62) over 100 replications under different methods is shown in Figure 3 with n = 500 .
Revised: The averaged estimated matrix (cid:98) B (cid:62) over 100 replications under different methods is shown in Figure 3 with a graph threshold as 0.3 for Scenario 3 (i.e., Case ER2) as an illustration. See other cases in Figures 5 to 15 in the appendix.

Original: All the numerical results  are reported in Table 4 of the appendix, including the false discovery rate, the true positive rate, and the structural Hamming distance.
Revised: All the numerical results of six cases are reported in Tables 4 and 5 in the appendix, including the false discovery rate, the true positive rate, and the structural Hamming distance.

Original: Figure 3 shows our algorithm performs the best among ﬁve methods, followed by the  DAG-GNN.
Revised: It is shown our algorithm performs the best among ﬁve methods in most cases, followed by the NOTEARS and the DAG-GNN.

Original: Theresults of the comparison study not only support the choice of the extension on the score-based method (by comparing the results of the DAG-GNN and the NOTEARS with other methods), but also validate the necessity of our method on the causal discovery with the background knowledge (by comparing the results of our method with the DAG-GNN).
Revised: The comparison studies not only support the choice of the extension on the score-based algorithm (by comparing the results of the NOTEARS and the DAG-GNN with other methods), but also validate the improvement of our method over the DAG-GNN by introducing the background knowledge in the causal discovery.

Original:  R EAL D ATA A NALYSIS : COVID-19 O UTBREAK
Revised:  R EAL D ATA A NALYSIS : COVID-19 O UTBREAK

Original: From early Jan 2020 to late Feb 2020 (the Spring Festival period), COVID-19 spread to every province-level division of China, exacerbated by the Chinese new year migration and human to human transmission.
Revised: From early Jan 2020 to late Feb 2020 (the Spring Festival period), COVID-19 spread to every province-level division of China, exacerbated by the Chinese new year migration and human to human transmission.

Original: The Chinese Government locked Hubei down on Jan 24th, which directly blocked infected people leaving from Hubei, and also indirectly control the spread of COVID-19.
Revised: The Chinese Government locked Hubei down on Jan 24th, which directly blocked infected people leaving from Hubei, and also indirectly control the spread of COVID-19.

Original: Here, for simplicity, we attribute the causal effects of the measures taken by cities outside of Hubei to the original and main action of interest, i.e. Hubei lockdowns.

Original: Thus, the COVID-19 example satisﬁes the considered causality framework for studying the causal mediator effects.

Original: [Figure]
Revised: [Figure]

Original: S

Original: h

Original: e n z

Original: h e n G u a n g z

Original: h
Revised: n

Original: o u B e iji n g C h e n gd u S h a n g h a i D o n gg u a n S u
Revised: [Figure]

Original: z h 
Revised: h a

Original: o u
Revised: [Figure]

Original: X i a n H a n g z
Revised: gd e N a n y a n g Y

Original: h o u
Revised: i c hun

Original: Z h e ng z h
Revised: X i n y a n

Original: o u C h o n gq i n g C h a n g s h a N a n ji n g K un m i n g T i a n ji n H e f e i N a nn i n g W e n z h
Revised: [Figure]

Original: o u N a n c h a n g Z h o uk o
Revised: A n q i n g J i u

Original: u F u y a n g S h a n gq i u Y u e y a n g Z hu m a d i a n C h a n gd e N a n y a n g Y
Revised: [Figure]

Original: i c hun X i n y a ng A n q i n g J i u ji a n g DM
Revised: n

Original: IM 1.

Original: [Figure]
Revised: [Figure]

Revised: Here, for simplicity, we attribute the causal effects of the measures taken by cities outside of Hubei to the original and main action of interest, i.e. Hubei lockdowns.

Revised: Thus, the COVID-19 example satisﬁes the considered causality framework for studying the causal mediator effects.

Original: We collect the data from the National Health Commission (NHC) of China and Baidu Qianxi for analysis.
Revised: We collect the data from the National Health Commission (NHC) of China and Baidu Qianxi for analysis.

Original: Speciﬁcally, let the exposure A
Revised: Speciﬁcally, let the exposure A

Original: as if Hubei is on lockdown, 0 for unlocked (before and on
Revised: as if Hubei is on lockdown, 0 for unlocked (before and on

Original: Jan 23rd), and 1 for locked (on and after Jan 24th).
Revised: Jan 23rd), and 1 for locked (on and after Jan 24th).

Original: We select 30 candidate cities outside Hubei that contain most potential infected people, as mediators M .
Revised: We select 30 candidate cities outside Hubei that contain most potential infected people, as mediators M .

Original: The daily migration scale index (MSI) of each city is used as the value of each mediator, which is the migration magnitude of large groups of people from one geographical area to another (Chen et al., 2020) and is comparable among cities.
Revised: The daily migration scale index (MSI) of each city is used as the value of each mediator, which is the migration magnitude of large groups of people from one geographical area to another (Chen et al., 2020) and is comparable among cities.

Original: Lastly, we use the daily increase rate of conﬁrmed cases out of Hubei to characterize the severity of the virus spreading with a one-week delay (due to the diagnose and incubation period of COVID- (Lauer et al., 2020)): Y t =Conﬁrmed cases out of Hubei t +8 −
Revised: Lastly, we use the daily increase rate of conﬁrmed cases out of Hubei to characterize the severity of the virus spreading with a one-week delay (due to the diagnose and incubation period of COVID- (Lauer et al., 2020)): Y t =

Original: Conﬁrmed cases out ofHubei t
Revised: Conﬁrmed cases out of

Original: [-+
-]
Revised: [-+
-][Equation]

Original:  Conﬁrmed cases out ofHubei t
Revised: 8 − Conﬁrmed cases out of

Original: + .
Revised: [Equation]

Original:  Here, the time t starts from Jan 12th to Feb 20th, 2020, since Jan 19th, 2020 is the earliest date with an available number of conﬁrmed cases out of Hubei (to compute Y t =1 on Jan 12th), and after Feb 20th, 2020, the pandemic was under control outside Hubei with the evidence of the work resumption in China.
Revised: [Equation] cases out of [Equation] Here, the time t starts from Jan 12th to Feb 20th, 2020, since Jan 19th, 2020 is the earliest date with an available number of conﬁrmed cases out of Hubei (to compute Y t =1 on Jan 12th), and after Feb 20th, 2020, the pandemic was under control outside Hubei with the evidence of the work resumption in China.

Original: The ﬁnal dataset yields a total of 38 records.
Revised: The ﬁnal dataset yields a total of 38 records.

Original: More details of data collection can be found in Section D of the appendix.
Revised: More details of data collection can be found in Section D of the appendix.

Original: The proposed algorithm is applied to the COVID-19 data with 100 replications by setting different random seeds in the neural network.
Revised: The proposed algorithm is applied to the COVID-19 data with 100 replications by setting different random seeds in the neural network.

Original: The estimated weighted adjacency matrix is shown in Figure 4a, with the detailed ANOCE table reported in Table 5 in the appendix.
Revised: The estimated weighted adjacency matrix is shown in Figure 4a, with the detailed ANOCE table reported in Table 6 in the appendix.

Original: The total effect of
Revised: The total effect of

Original: Hubei lockdowns on the daily increase rate of conﬁrmed cases outside Hubei as -0.497, where the direct effect is -0.078 and the indirect effect is -0.419.
Revised: Hubei lockdowns on the daily increase rate of conﬁrmed cases outside Hubei as -0.497, where the direct effect is -0.078 and the indirect effect is -0.419.

Original: In other words, by locking Hubei down, China successfully reduced 49.7% of the daily new cases outside Hubei; 84% of which is the indirect effect contributed via the reduced migration of cities (the mediators) out of Hubei, and the rest 16% owes to the direct effect of Hubei lockdowns since infected people were constrained in Hubei after the lockdown.
Revised: In other words, by locking Hubei down, China successfully reduced 49.7% of the daily new cases outside Hubei; 84% of which is the indirect effect contributed via the reduced migration of cities (the mediators) out of Hubei, and the rest 16% owes to the direct effect of Hubei lockdowns since infected people were constrained in Hubei after the lockdown.

Original: Thus, the lockdown is effective in reducing the COVID-19 spread in China.
Revised: Thus, the lockdown is effective in reducing the COVID-19 spread in China.

Original: The total indirect effect of the lockdown can be further broken down by cities’ direct effects ( DM s, corresponds to the intensity of transmission within a particular city).
Revised: The total indirect effect of the lockdown can be further broken down by cities’ direct effects ( DM s, corresponds to the intensity of transmission within a particular city).

Original: We compare cities’ DM s with their associated indirect effects ( IM s, describes the secondary migration from a particular city to other places) in Figure 4b, where a positive effect means spreading the virus while negative means control.
Revised: We compare cities’ DM s with their associated indirect effects ( IM s, describes the secondary migration from a particular city to other places) in Figure 4b, where a positive effect means spreading the virus while negative means control.

Original: Note that the selected 30 cities are ordered by their cumulative MSI during the data period.
Revised: Note that the selected 30 cities are ordered by their cumulative MSI during the data period.

Original: From Figure 4b, the majority of cities have a negative DM (colored in blue), which implies the infection within cities outside Hubei have been effectively controlled under the lockdown.
Revised: From Figure 4b, the majority of cities have a negative DM (colored in blue), which implies the infection within cities outside Hubei have been effectively controlled under the lockdown.

Original: There are more cities with a positive IM s (red), which is in line with the intuition that the secondary migration among cities may exacerbate the pandemic.
Revised: There are more cities with a positive IM s (red), which is in line with the intuition that the secondary migration among cities may exacerbate the pandemic.

Original: The positive effects (red) are more likely located at the ﬁrst 20 nodes, which corresponds to the cities with large MSI, while the last 10 cities with relatively small MSI are almost all blue.
Revised: The positive effects (red) are more likely located at the ﬁrst 20 nodes, which corresponds to the cities with large MSI, while the last 10 cities with relatively small MSI are almost all blue.

Original: This accords with the migration peak among big cities during the Spring Festival period that aggravated the spread of the virus.
Revised: This accords with the migration peak among big cities during the Spring Festival period that aggravated the spread of the virus.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: We conclude our paper with the following discussions.
Revised: We conclude our paper with the following discussions.

Original: First, our proposed method can be extended to a non-linear model via a transformation on the outcome, and a generalized deﬁnition of the DM from a graphical perspective is given in Section F.2 of the appendix without the LSEM assumption.
Revised: First, the proposed DM can be extended beyond the LSEM assumption. A generalized deﬁnition of the DM from a graphical perspective is given in Section F.2 of the appendix without the LSEM.

Original: Second, due to possibly unmeasured confounders in our real data, such as cities’ features and periodic effect, we may consider extending our model with a new topological order that contains confounders for a wider utility, such as forcing the topological order of k confounders as 1 to k followed by the exposure as 1 + k .
Revised: Second, due to possibly unmeasured confounders in our real data, such as cities’ features and periodic effect, we may consider extending our model with a new topological order that contains confounders for a wider utility, such as forcing the topological order of k confounders as 1 to k followed by the exposure as 1 + k .

Original: Third, our proposed identiﬁcation constraint can be generalized to other background knowledge and can be applied to any score-based structural learning methods.
Revised: Third, our proposed identiﬁcation constraint can be generalized to other background knowledge.

Original: A A DDITIONAL G RAPH T ERMINOLOGY
Revised: A A DDITIONAL G RAPH T ERMINOLOGY

Original: Given the node set, the weighted DAG can be uniquely determined by its weighted adjacency matrix, i.e., there is a one-to-one transformation between G and B .
Revised: Given the node set, the weighted DAG can be uniquely determined by its weighted adjacency matrix, i.e., there is a one-to-one transformation between G and B .

Original: Suppose the graph nodes X in G are sorted in its topological order (corresponding to elementary transformation of the matrix), then the matrix B is strictly upper triangular with the diagonal elements as 0.
Revised: Suppose the graph nodes X in G are sorted in its topological order (corresponding to elementary transformation of the matrix), then the matrix B is strictly upper triangular with the diagonal elements as 0.

Original: Therefore, for an identity matrix
Revised: Therefore, for an identity matrix

Original: I with the same dimension as B , I − B (cid:62) is invertible since all its diagonal elements are 1 (positive).
Revised: I with the same dimension as B , I − B (cid:62) is invertible since all its diagonal elements are 1 (positive).

Original: B A LGORITHM : ANOCE-CVAE
Revised: B A LGORITHM : ANOCE-CVAE

Original: The ﬁrst part of the ANOCE-CVAE algorithm is on learning causal DAG from the observational data in the constrained space, by minimizing the loss function in Equation 6 using blackbox stochastic optimization solvers.
Revised: The ﬁrst part of the ANOCE-CVAE algorithm is on learning causal DAG from the observational data in the constrained space, by minimizing the loss function in Equation 7 using blackbox stochastic optimization solvers.

Original: Here, to minimize the loss in 6 and satisfy both h 1 ( B ) = 0 and h 2 ( B ) = 0 , we simultaneously update λ 1 and λ 2 and increase c and d to inﬁnity, by modifying the basic technique in Yu et al.
Revised: Here, to minimize the loss in 7 and satisfy both h 1 ( B ) = 0 and h 2 ( B ) = 0 , we simultaneously update λ 1 and λ 2 and increase c and d to inﬁnity, by modifying the basic technique in Yu et al.

Original: (2019), corresponding to Part One.
Revised: (2019), corresponding to Part One.

Original: II.A.b and Part One.
Revised: II.A.b and Part One.

Original: II.B in Algorithm 1.
Revised: II.B in Algorithm 1.

Original: The second part is to estimate causal effects in the ANOCE table from the learned causal structure, based on the results from Theorem 3.1.
Revised: The second part is to estimate causal effects in the ANOCE table from the learned causal structure, based on the results from Theorem 3.1.

Original: Here, we numerically calculate the natural indirect effect for mediator IM based on Corollary F.1 in step IV of the second part.
Revised: Here, we numerically calculate the natural indirect effect for mediator IM based on Corollary F.1 in step IV of the second part.

Original: Global
Revised: Global

Original: Dataset X =
Revised: Dataset X =

Original: { A, M, Y } , sample size n , dimension of mediators p , max iteration K , number of epoch H , original learning rate r 0 , tolerance of constraint to zero δ , parameter update bound U , tuning parameters ρ and ω , and penalty terms c and d ;
Revised: { A, M, Y } , sample size n , dimension of mediators p , max iteration K , number of epoch H , original learning rate r 0 , tolerance of constraint to zero δ , parameter update bound U , tuning parameters ρ and ω , and penalty terms c and d ;

Original: Local mean and standard variance
Revised: Local mean and standard variance

Original: of
Revised: of

Original: µ (cid:15) and σ
Revised: µ (cid:15) and σ

Original: , mean and standard variance of X µ X σ X weights in multilayer perceptrons of encoder and decoder θ = { W (1)
Revised: , mean and standard variance of X µ X σ X weights in multilayer perceptrons of encoder and decoder θ = { W (1)

Original: W (2) and IE , and natural direct and indirect effect for mediator DM and IM
Revised: W (2) and IE , and natural direct and indirect effect for mediator DM and IM

Original: Elseif
Revised: Elseif

Original: B. h
Revised: B. h

Original: h
Revised: h

Original: C. If h new<
Revised: C. If h new<

Original: and
Revised: and

Original: new
Revised: new

Original: III.
Revised: III.

Original: Output (cid:98) B ← B ;
Revised: Output (cid:98) B ← B ;

Original: Algorithm
Revised: Algorithm

Original: Part Two: Estimate causal effects in ANOCE based on matrix (cid:98) B ;
Revised: Part Two: Estimate causal effects in ANOCE based on matrix (cid:98) B ;

Original: I.
Revised: I.

Original: According to Equation
Revised: According to Equation

Original: : A. Get (cid:98) γ as the direct effect DE ; B. Get (cid:98) α as the effect of A on M , (cid:98) β , and the inside matrix (cid:98) B M ; II.
Revised: : A. Get (cid:98) γ as the direct effect DE ; B. Get (cid:98) α as the effect of A on M , (cid:98) β , and the inside matrix (cid:98) B M ; II.

Original: Get (cid:98) ζ ≡ ( I p − B (cid:62) M ) − 1 (cid:98) α that represents the causal effect of A on M ; III.
Revised: Get (cid:98) ζ ≡ ( I p − B (cid:62) M ) − 1 (cid:98) α that represents the causal effect of A on M ; III.

Original: Get (cid:98) β
Revised: Get (cid:98) β

Original: (cid:62) (cid:98) ζ that represents the total natural indirect effect IE ;
Revised: (cid:62) (cid:98) ζ that represents the total natural indirect effect IE ;

Original: For each mediator
Revised: For each mediator

Original: i , i

Original: p
Revised: p

Original: Deﬁne the natural direct effect
Revised: Deﬁne the natural direct effect

Original: M i

Original: DM [
Revised: DM [

Original: (cid:98) α
Revised: (cid:98) α

Original: ] (cid:98)
Revised: ] (cid:98)

Original: ; IV.
Revised: ; IV.

Original: Get the natural indirect effect for mediator:
Revised: Get the natural indirect effect for mediator:

Original: For each mediator M i , i = 1 , · · · , p : A. Delete M i from the matrix (cid:98) B and get (cid:99) B (cid:48) i ; B. Repeat step II.
Revised: For each mediator M i , i = 1 , · · · , p : A. Delete M i from the matrix (cid:98) B and get (cid:99) B (cid:48) i ; B. Repeat step II.

Original: with reduced matrix (cid:99) B (cid:48)
Revised: with reduced matrix (cid:99) B (cid:48)

Original: i and get (cid:98) β (cid:48) and (cid:98) ζ (cid:48) ; C. Calculate the effect difference as the total mediation effect (cid:98) β (cid:62) (cid:98) ζ − (cid:98) β
Revised: i and get (cid:98) β (cid:48) and (cid:98) ζ (cid:48) ; C. Calculate the effect difference as the total mediation effect (cid:98) β (cid:62) (cid:98) ζ − (cid:98) β

Original: (cid:48)(cid:62) (cid:98) ζ (cid:48)
Revised: (cid:48)(cid:62) (cid:98) ζ (cid:48)

Original: D. Deﬁne the natural indirect effect for M i as IM [ i ] = { (cid:98) β (cid:62) (cid:98) ζ − (cid:98) β
Revised: D. Deﬁne the natural indirect effect for M i as IM [ i ] = { (cid:98) β (cid:62) (cid:98) ζ − (cid:98) β

Original: (cid:48)(cid:62) (cid:98) ζ (cid:48) } − DM [ i ] ; V. Deﬁne the total effect TE =
Revised: (cid:48)(cid:62) (cid:98) ζ (cid:48) } − DM [ i ] ; V. Deﬁne the total effect TE =

Original: + (cid:98) β (cid:62) (cid:98)ζ .
Revised: + (cid:98) β (cid:62) (cid:98)

Original: C
Revised: C

Original: A
Revised: A

Original: DDITIONAL S IMULATION S TUDIES
Revised: DDITIONAL S IMULATION S TUDIES

Original: In this section, we give more details on simulation studies to investigate the ﬁnite sample performance of the proposed method for learning causal effects with multiple mediators, in comparison to the popular causal discovery methods, including the PC, the ICA-LiNGAM, the NOTEARS, and the
Revised: In this section, we give more details on simulation studies to investigate the ﬁnite sample performance of the proposed method for learning causal effects with multiple mediators, in comparison to the popular causal discovery methods, including the PC, the ICA-LiNGAM, the NOTEARS, and the

Original: DAG-GNN.
Revised: DAG-GNN.

Original: The computing infrastructure used is a virtual machine in the compute engine of Google
Revised: The computing infrastructure used is a virtual machine in the compute engine of Google

Original: Cloud Platform with 8 processor cores and 32GB memory.
Revised: Cloud Platform with 8 processor cores and 32GB memory.

Original: The average runtime for each result is around 1 to 2 hours.
Revised: The average runtime for each result is around 1 to 2 hours.

Original: C. D ATA G ENERATION
Revised: C. D ATA G ENERATION

Original: We ﬁrst generate a random DAG by the Erd˝os-Re´nyi model with the  expected node degree as 2, and then remove all in-edges (from precedent nodes) of the ﬁrst node as A; and remove all out-edges (from descendent nodes) of the last node as Y ; then the remained nodes are the mediators M .
Revised: We ﬁrst generate a random DAG from the Erd ˝ os-Re ´ nyi (ER) or the Scale-Free (SF) network (Barabási & Albert, 1999) with an expected node degree. Then, we remove all in-edges (from precedent nodes) of the ﬁrst node as A and remove all out-edges (from descendent nodes) of the last node as Y , and thus, the remaining nodes are the mediators M .

Original: Edges in DAGs for all scenarios are randomly assigned with weights ( w ∈ {− 1 , 1 } with equal probability) to obtain the weighted adjacency matrix B .
Revised: Edges in DAGs for all scenarios are randomly assigned with weights ( w ∈ {− 1 , 1 } with equal probability) to obtain the weighted adjacency matrix B .

Original: The experiment data sets are generated fromModel 3, where the error variables in (cid:15) ≡
Revised: Speciﬁcally, the true DAGs in Scenarios 3 and 4 are generated from the Erd˝os-Re´nyi (ER) model with an expected degree as 2, where we set number of nodes d = 12 (i.e. p = 10 ) in Scenarioand d = 32 (i.e. p = 30 ) in Scenario 4. Note that we consider fully identiﬁable models in Section 5.1 so that it is meaningful to evaluate causal effects from the estimated graph. In Section 5.2, we repeat the above generation procedure with d = 12 to generate the true graph from both the ER and the Scale-Free (SF) networks with the expected degree as 1, 2, and 4, denoted as Cases ER1, ER2, ER4, SF1, SF2, and SF4, respectively.

Revised: Here, to be consistent with Section 5.1, we refer Scenario(generated by the ER with the degree as 2) as Case ER2.

Revised: The synthetic datasets { A, M, Y } are generated from Model 1, where the error variables in (cid:15) ≡

Original: [ (cid:15) A , (cid:15) (cid:62) M , (cid:15) Y ] (cid:62) independently follow a normal distribution with mean 0 and noise 0.5.
Revised: [ (cid:15) A , (cid:15) (cid:62) M , (cid:15) Y ] (cid:62) independently follow a normal distribution with mean 0 and noise 0.5 except for the binary exposure in Scenario 4 ∗ .

Original: Here, we add a baseline of 1.0 on the outcome Y .
Revised: Here, we add a baseline of 1.0 on the outcome Y .

Original: Note that thegaussian exposure in
Revised: Note that the

Original:  Scenario 4 and the binary exposure  Scenario 4 ∗ have the same mean and noise and thus their results are comparable.
Revised: Gaussian exposure in Scenario 4 and the binary exposure in Scenario 4 ∗ have the same mean and noise and thus their results are comparable.

Original: C. I MPLEMENTATION OF ANOCE-CVAE
Revised: C. I MPLEMENTATION D ETAILS

Original: For each setting, we apply the proposed method with 100 replications.
Revised: We detail the implementation for the proposed ANOCE-CVAE and comparison partners as follows:

Original: Here, the ANOCE-CVAE is implemented based on PyTorch (Paszke et al., 2017), using Adam (Kingma & Ba, 2014) to minimize the loss function in Equation 6.
Revised: • ANOCE-CVAE: The ANOCE-CVAE is implemented based on PyTorch (Paszke et al., 2017), using Adam (Kingma & Ba, 2014) to minimize the loss function in Equation 7.

Original: We set the batch size as 25 for n = 50 and 100 for n = 500 with hidden nodes as p 2 , the initial learning rate as 0 .
Revised: We set the batch size as 25 for n = 50 and 100 for n = 500 with hidden nodes as p 2 , the initial learning rate as 0 .

Original: 003 with an update rule as r ← r/ { log ( c ) + log ( d ) + 0 .
Revised: 003 with an update rule as r ← r/ { log ( c ) + log ( d ) + 0 .

Original: 01 } where c and d are penalty terms for two constraints, and the parameter update bound as U = 10 20 , for all settings.
Revised: 01 } where c and d are penalty terms for two constraints, and the parameter update bound as U = 10 20 , for all settings.

Original: Following the recommendation of Yu et al.
Revised: Following the recommendation of Yu et al.

Original: , we ﬁnd that tuning parameters ρ = 0 .
Revised: (2019), we ﬁnd that their tuned parameters ρ = 0 .

Original: 25 and ω = 10 also work well in our settings, and we adopt the Huber-norm regularization of B for a better convergence.
Revised: 25 and ω = 10 also work well in our settings, and we adopt the Huber-norm regularization of B for a better convergence.

Original: Here, the variational posterior and the likelihood are parameterized as Gaussian with unit noise to approximate the underlying true model.
Revised: Here, the variational posterior and the likelihood are parameterized as Gaussian with unit noise to approximate the underlying true model.

Original: [Table] [Table] [Table] [Table]
Revised: The code is publicly available at an anonymous repository at https://github.com/anoce-cvae/ANOCE-CVAE .

Original: [Table]
Revised: • PC (Spirtes et al., 2000):

Original: [Table] [Table] [Table] [Table]
Revised: We set the Fisher-z test for the PC algorithm with the p-value as 0.01 for all settings.

Original: [Table] [Table] [Table]
Revised: The implementation is available through the py-causal package at https://github.com/bd2kccd/py-causal , written in highly optimized Java codes.

Original: [Table] [Table] [Table]
Revised: Also see examples here https://github.com/bd2kccd/py-causal/blob/development/ example/py-causal%20-%20PC-ALL%20in%20Action.ipynb . • ICA-LiNGAM

Original: [Table]
Revised: (Shimizu et al., 2006):

Original: [Table] [Table] [Table]
Revised: The ICA-LiNGAM assumes linear non-Gaussian additive model to recover the weighted adjacency matrix.

Original: [Table] [Table]
Revised: We implement the ICA-LiNGAM with default hyper-parameters through the lingam package for all settings.

Original: [Table] [Table] [Table]
Revised: See their repository at https: //github.com/

Original: [Table] [Table] [Table] [Table]
Revised: cdt15/lingam .

Original: [Table]
Revised: • NOTEARS

Original: [Table]
Revised: (Zheng et al., 2018):

Original: [Table] [Table] [Table]
Revised: The NOTEARS estimates the weighted adjacency matrix by formulating the optimization with an acyclicity constraint.

Original: [Table] [Table] [Table] [Table]
Revised: The implementation is available at their repository at https://github.com/xunzheng/notears .

Original: [Table] [Table] [Table]
Revised: We set the loss function as the least square error with the L 1 regularization. We ﬁnd the NOTEARS is sensitive to the choice of the L 1 regularization in our settings.

Original: [Table] [Table] [Table] [Table]
Revised: For a fair comparison, we set the L 1 penalty parameter as 0.

Original: [Table] [Table] [Table] [Table]
Revised: (instead of the default 0.1) for all settings, which achieves an overall good performance in most cases.

Original: IM
Revised: Note the author modiﬁed their acyclicity constraint in their codes to be the one used in Yu et al.

Original: [Table] [Table]
Revised: (i.e. Equation 4).

Original: [Table] [Table] [Table]
Revised: We also use the same acyclicity constraint for NOTEARS, DAGGNN, and our method for a fair comparison.

Original: [Table]
Revised: Other hyper-parameters are set as default in their repository.

Original: [Table] [Table] [Table] IM
Revised: • DAG-GNN (Yu et al., 2019):

Original: [Table]
Revised: The DAG-GNN incorporates the variational auto-encoder into causal discovery with a modiﬁed smooth characterization on acyclicity in the evidence lower bound as the loss function.

Original: [Table]
Revised: Codes are available at their repository at https://github.com/ fishmoon1234/DAG-GNN based on PyTorch (Paszke et al., 2017).

Original: [Table] [Table]
Revised: We set the same hyperparameters used in our ANOCE-CVAE for a fair comparison.

Original: [Table] [Table]
Revised: Speciﬁcally, we use Adam (Kingma & Ba, 2014) to minimize the loss function, and set the batch size as 25 for n = 50 and 100 for n = 500 with hidden nodes as p 2 .

Original: IM
Revised: The initial learning rate is set as 0 .

Original: [Table] [Table] [Table] [Table]
Revised: 003 with an update rule as r ← r/ { log ( c 0 ) + 0 .

Original: [Table]
Revised: 01 } where c 0 is penalty term for the acyclicity constraint. The rest settings are the same as the default in their codes.

Original: [Table] [Table]
Revised: In the comparison studies (see Section 5.2 and C.4), we use a uniform graph threshold as 0.

Original: [Table] [Table] [Table] [Table]
Revised: (commonly used in current literature) for all algorithms to prune the noise edges for a fair comparison.

Original: IM [Table] [Table] [Table]
Revised: In addition, we also provide the results under the graph threshold as 0.4 for additional comparison.

Original: C. A DDITIONAL R ESULTS OF ANOCE-CVAE
Revised: C. A DDITIONAL R ESULTS OF ANOCE-CVAE

Revised: In this section, we provide additional simulation results for the ANOCE-CVAE.

Original: The results are summarized in Table 2 (for Scenario 1 to 3) and Table 3 (for Scenario 4 and 4 ∗ ), including the bias of the estimated TE , DE , IE , DM and IM for each mediator with their standard error, over 100 replications.
Revised: Following Section 5.1, the numerical results are summarized in Table 2 (for Scenario 1 to 3) and Table 3 (for Scenarioand 4 ∗ ), including the bias of the estimated TE , DE , IE , DM and IM for each mediator with their standard error, over 100 replications.

Original: Note that due to limited space, we save the numerical results of the IM in Table 3.
Revised: Note that due to limited space, we save the numerical results of the IM in Table 3.

Original: From the results in Table 2, it is clear that the estimated TE , DE , IE , DM and IM for each mediator are close to the true values as the sample size increases in Scenario 1 to 3, which indicates the good performance of our proposed method on identifying the causal effects regardless of the sparsity.
Revised: From the results in Table 2, it is clear that the estimated TE , DE , IE , DM and IM for each mediator are close to the true values as the sample size increases in Scenario 1 to 3, which indicates the good performance of our proposed method on identifying the causal effects regardless of the sparsity.

Original: With the expected node degree increasing, one can observe a slightly larger bias and standard error of the estimated causal effects as expected, as shown in Table 2.
Revised: With the expected node degree increasing, one can observe a slightly larger bias and standard error of the estimated causal effects as expected, as shown in Table 2.

Original: Based on Table 3, the results of Scenario 4 and Scenario 4 ∗ are merely identical under different sample sizes, indicating our proposed method can handle either discrete or continuous exposure.
Revised: Based on Table 3, the results of Scenario 4 and Scenario 4 ∗ are merely identical under different sample sizes, indicating our proposed method can handle either discrete or continuous exposure.

Original: In addition, by comparing the results of Scenario 3 and 4 where we ﬁx the expected node degree as 2, one can observe a slightly larger bias of the estimated causal effects but of a similar small scale, as the dimension of mediators p increases, which implies the stability of our method under the high-dimensional setting.
Revised: In addition, by comparing the results of Scenario 3 and 4 where we ﬁx the expected node degree as 2, one can observe a slightly larger bias of the estimated causal effects but of a similar small scale, as the dimension of mediators p increases, which implies the stability of our method under the high-dimensional setting [Table]

Original: [Table]
Revised: [Table]

Original: DM0.02 (0.01) 0.09 (0.01) 0.14 (0.01) 0.15 (0.01) 
Revised: DM0.01 (0.00) 0.00 (0.00) 0.68 (0.03) 0.02 (0.01) 0.44 (0.02) 0.10 (0.01) DM0.01 (0.00) 0.00 (0.00) 0.67 (0.02) 0.02 (0.02) 0.27 (0.04) 0.03 (0.01) DM0.00 (0.00) 0.00 (0.00) 0.62 (0.03) 0.03 (0.01) 0.67 (0.03) 0.16 (0.03) DM0.00 (0.00) 0.00 (0.00) 0.66 (0.03) 0.01 (0.01) 0.59 (0.03) 0.14 (0.03)

Original: DM0.00  (0.01) 0.00 (0.00)  0.01 (0.01)  0.00 (0.00)
Revised: DM0.00 (0.00) 0.00 (0.00) 0.65 (0.02) 0.01 (0.01) 0.00 (0.00) 0.00 (0.00) DM0.01 (0.00) 0.00 (0.00) 0.66 (0.03) 0.01 (0.01) 0.10 (0.02) 0.05 (0.01) IM0.00 (0.00) 0.00 (0.00) 0.08 (0.01) 0.08 (0.01) 0.76 (0.03) 0.16 (0.04) IM0.00 (0.00) 0.00 (0.00) 0.09 (0.01) 0.01 (0.01) 0.00 (0.00) 0.00 (0.00)

Original: DM0.02 (0.00) 0.01 (0.01) 0.06 (0.01) 0.01 (0.01) 
Revised: IM0.00 (0.00) 0.00 (0.00) 0.03 (0.01) 0.03 ( [Table]

Revised: [Table]

Original:  DM0.25 (0.02) 0.11 (0.02) 0.51 (0.04) 0.16 (0.03)
Revised: DM0.02 (0.01) 0.09 (0.01) 0.14 (0.01) 0.15 (0.01) DM0.00 (0.01) 0.00 (0.00) 0.01 (0.01) 0.00 (0.00) DM0.02 (0.00) 0.01 (0.01) 0.06 (0.01) 0.01 (0.01) DM0.25 (0.02) 0.11 (0.02) 0.51 (0.04) 0.16 (0.03)

Original: DM0.02 (0.01) 0.10 (0.01) 0.09 (0.02) 0.16 (0.01)
Revised: DM0.02 (0.01) 0.10 (0.01) 0.09 (0.02) 0.16 (0.01)

Original: DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)
Revised: DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)

Original: DM0.09 (0.01) 0.10 (0.02) 0.18 (0.03) 0.13 (0.02)
Revised: DM0.09 (0.01) 0.10 (0.02) 0.18 (0.03) 0.13 (0.02)

Original: DM0.03 (0.01) 0.07 (0.02) 0.04 (0.03) 0.06 (0.03)
Revised: DM0.03 (0.01) 0.07 (0.02) 0.04 (0.03) 0.06 (0.03)

Original: DM0.01 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)
Revised: DM0.01 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)

Original: DM0.02 (0.00) 0.01 (0.00) 0.00 (0.00) 0.00 (0.00)
Revised: DM0.02 (0.00) 0.01 (0.00) 0.00 (0.00) 0.00 (0.00)

Original: DM0.01 (0.00) 0.00 (0.00) 0.08 (0.01) 0.01 (0.01)
Revised: DM0.01 (0.00) 0.00 (0.00) 0.08 (0.01) 0.01 (0.01)

Original: DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)
Revised: DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)

Original: DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)
Revised: DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)

Original: DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 
Revised: DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) DM0.01 (0.01) 0.02 (0.01) 0.01 (0.01) 0.01 (0.00)

Original: DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)
Revised: DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00) DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)

Original: DM0.01 (0.01) 0.02 (0.01) 0.01 (0.01) 0.01 (0.00)
Revised: [Figure]

Original: DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)
Revised: [Figure] [Figure] [Figure]

Original: DM0.00 (0.00) 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Revised: This section provides more results on comparison studies against the existing methods.

Revised: The data generation and implementation details are provided in Section C.1 and C.2.

Revised: Following Section 5.

Revised: and C.2, we use a graph thresholds as 0.3 (commonly used in current literature) or 0.4 (for additional comparison) in all algorithms to prune the noise edges for a fair comparison.

Revised: The estimated graphs (after pruning) are evaluated by three metrics: the false discovery rate (FDR), the true positive rate (TPR), and the structural Hamming distance (SHD, the smallest number of edge additions, deletions, and reversals to convert the estimated graph into the true DAG).

Original: 0 2 4 6 8 102 48(b
Revised: Here, the SHD takes into account both false positives and negatives and a lower SHD indicates a better estimate of the causal graph. The FDR, TPR, and SHD of the averaged estimated matrix (cid:98) B (cid:62) with their standard deviation over 100 replications are reported in Table 4 for the graph threshold as 0.3 and in Tablefor the graph threshold as 0.4, under different methods for all six cases with sample size n = 500 . [Table] [Table]

Original: [Figure]
Revised: [Figure]

Original: ANOCE 0 2 4 6 8 102 48(c) PC 0 2 4 6 8 102 48(d) ICA 0 2 4 6 8 102 48(e)
Revised: [Figure] [Figure]

Original: [-NOTEARS 
-]
Revised: [-NOTEARS 
-][Figure]

Original: 2 4 6 8 102 48(f) [Figure] 
Revised: [Figure] [Figure] 2 48(b)

Original: [Figure] [Figure] [Figure] [Figure]
Revised: ANOCE 0 2 4 6 8 102 48(c) PC 0 2 4 6 8 102 48(d)

Original: ICA 0 2 4 6 8 102 48(e)
Revised: ICA 0 2 4 6 8 102 48(e)

Original: NOTEARS 

Revised: NOTEARS

Original: 2 4 6 8 102 48(f)
Revised: 2 4 6 8 102 48(f)

Original: DAG
Revised: DAG-GNN

Original: [Figure]
Revised: Besides Figure 3 for Case ER2 with a graph threshold as 0.3 shown in the main text, we also illustrate the averaged estimated matrix (cid:98) B (cid:62) over 100 replications under different methods for Case ER1, ER4, SF1, SF2, and SF4 with a graph threshold as 0.3 in Figure 5, 6, 7, 8, 9, respectively, and for Case ER1, ER2, ER4, SF1, SF2, and SF4 with a graph threshold as 0.4 in Figure 10, 11, 12, 13, 14, 15, respectively, under n = 500 .

Original: [Figure]
Revised: [Figure]

Original: (e) NOTEARS [-
 (f) DAG-GNN C. A DDITIONAL C OMPARISON S TUDIES This section provides more results on comparison studies against-] the existing methods.
Revised: From Table 4 and 5, it is clear that our algorithm performs the best among the ﬁve methods in most cases, followed by the other two score-based methods, i.e. the NOTEARS [-
 (f) DAG-GNN C. A DDITIONAL C OMPARISON S TUDIES This section provides more results on comparison studies against-]and the DAG-GNN.

Original: Here, we also include a binary setting by replacing the gaussian exposure in Scenario 3 with the binary exposure as Scenario 3 ∗ . Default hyper-parameters of these implementations are used unless otherwise stated. The estimated graphs (after pruning) are evaluated by three metrics: the false discovery rate (FDR), the true positive rate (TPR), and the structural Hamming distance (SHD, the smallest number of edge additions, deletions, and reversals to convert the estimated graph into the true DAG).
Revised: While the traditional methods (the PC and the ICA-LiNGAM) perform the worst with large SHD and small

Original: Here, the SHD takes into account both false positives and negatives and a lower SHD indicates a better estimate of the causal graph. We illustrate the averaged estimated matrix (cid:98) B (cid:62) (with a graph threshold as 0.3) over 100 replications under different methods in Figure 5 for Scenario 3 with n = 50 , 6 for Scenario 3 ∗ with n = 50 , and 7 for Scenario 3 ∗ with n = 500 . Note the results of Scenario 3 with n =is reported in Figure 3.
Revised: TPR. This ﬁnding supports the choice of the extension on the score-based method.

Original: The FDR, TPR, and SHD of the averaged estimated matrix (cid:98) B (cid:62) with their standard deviation are reported in Table 4 over 100 replications under different methods.
Revised: Moreover, by comparing our performance with the DAG-GNN, one can observe a substantial gain in terms of the SHD and the TPR in most cases with comparable FDR.

Original: From Table 4, it is clear that our algorithm performs best among ﬁve methods, followed by the DAG-GNN and the NOTEARS.
Revised: This validates the improvement of our method over the DAG-GNN by introducing the background knowledge in the causal discovery.

Original: Under the ANOCE-CVAE, the FDR and SHD for the averaged estimated matrix (cid:98) B (cid:62) are close to 0 and the TPR approaches to 1 in all scenarios, when n = 500 .

Original: Another support is illustrated in Figure 3 and 7,  where the averaged estimated matrix (cid:98) B (cid:62) under the ANOCE-CVAE is approximately the same as the ground true graph B (cid:62)in all scenarios when n = 500 .
Revised: Another supports are illustrated in Figure 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15 for different settings, where the averaged estimated matrix (cid:98) B (cid:62) under the ANOCE-CVAE is approximately the same as the ground true graph B (cid:62) when n = 500 .

Original: However, thetraditional methods (the PC and the ICA-LiNGAM) can hardly recognize the true causal pattern even under n = 500 .
Revised: However, the PC and the ICA-LiNGAM can hardly recognize the true causal pattern.

Original: In addition, it can be observed that under the ANOCE-CVAE, the ﬁrst row and the last column of  the estimated (cid:98) B (cid:62) are all zeros with different sample sizes, which corresponds to our identiﬁcation constraint successfully satisﬁed; while other methods fail toidentify the treatment and the outcome when the sample size is small from Figure 5 and 6. [Table] [Figure]
Revised: In addition, methods have a slightly better performance in terms of FDR and SHD while a slightly worse performance in terms of TPR under the graph threshold as 0.4, in comparison to the results under the graph threshold as 0.3.

Original: D A DDITIONAL R EAL D ATA R ESULTS
Revised: D A DDITIONAL R EAL D ATA R ESULTS

Original: In this section, we provide additional real data analysis on the COVID-19.
Revised: In this section, we provide additional real data analysis on the COVID-19.

Original: D. D ATA C OLLECTION
Revised: D. D ATA C OLLECTION

Original: To better characterize the causality of the virus spreading under the Hubei lockdowns in China, we assume: 1) Hubei was the centre of the COVID-19 outbreak in China (Zhou et al., 2020); 2) the decreased migration outside Hubei was largely stimulated by the lockdown; 3) individual who departed to one destination would not return to the original departure due to travel restrictions in
Revised: To better characterize the causality of the virus spreading under the Hubei lockdowns in China, we assume: 1) Hubei was the centre of the COVID-19 outbreak in China (Zhou et al., 2020); 2) the decreased migration outside Hubei was largely stimulated by the lockdown; 3) individual who departed to one destination would not return to the original departure due to travel restrictions in

Original: China.
Revised: China.

Original: Under the above assumptions, it is reasonable to use a temporal causal relationship to describe the spread of COVID-19 under 2020 Hubei lockdowns as in Figure 8.
Revised: Under the above assumptions, it is reasonable to use a temporal causal relationship to describe the spread of COVID-19 under 2020 Hubei lockdowns as in Figure 16.

Original: Next, we give more details on how to present components in Figure 8 with appropriate variables.
Revised: Next, we give more details on how to present components in Figure 16 with appropriate variables.

Original: First, we set the exposure A as if Hubei is on lockdown, 0 for unlocked (before and on Jan 23rd) and 1 for locked (on and after Jan 24th).
Revised: First, we set the exposure A as if Hubei is on lockdown, 0 for unlocked (before and on Jan 23rd) and 1 for locked (on and after Jan 24th).

Original: To select the candidate cities that contain most potential infected people, we rank cities outside Hubei by their received Wuhan (identiﬁed over 60% cases in China reported by the NHC) migration between Jan 1st, 2020 and Jan 22nd, 2020 (before the lockdown), and choose the top 30 cities (account for 69.17% of total Wuhan migration) as mediators M , to control the noise.
Revised: To select the candidate cities that contain most potential infected people, we rank cities outside Hubei by their received Wuhan (identiﬁed over 60% cases in China reported by the NHC) migration between Jan 1st, 2020 and Jan 22nd, 2020 (before the lockdown), and choose the top 30 cities (account for 69.17% of total Wuhan migration) as mediators M , to control the noise.

Original: We use the daily migration scale index (MSI) of each city as the value of each mediator, deﬁned in Baidu Qianxi to describe the migration magnitude.
Revised: We use the daily migration scale index (MSI) of each city as the value of each mediator, deﬁned in Baidu Qianxi to describe the migration magnitude.

Original: By noticing the following facts: 1) it took usually 2 days to diagnose the COVID-19; 2) the estimated median incubation period is 5 days (Lauer et al., 2020); the outcome of interest Y is deﬁned as the daily increase rate of conﬁrmed cases out of Hubei with a one-week (7=2+5) delay.
Revised: By noticing the following facts: 1) it took usually 2 days to diagnose the COVID-19; 2) the estimated median incubation period is 5 days (Lauer et al., 2020); the outcome of interest Y is deﬁned as the daily increase rate of conﬁrmed cases out of Hubei with a one-week (7=2+5) delay.

Original: We delete extreme data points (two outbreaks at jails), and the ﬁnal dataset yields a total of 38 records.
Revised: We delete extreme data points (two outbreaks at jails), and the ﬁnal dataset yields a total of 38 records.

Original: [Figure] [Figure]
Revised: D. I NITIAL D ATA A NALYSIS

Original: Figure 9a demonstrates the location of selected cities on the Chinese map with the color representing its cumulative conﬁrmed cases by March 1st, 2020.
Revised: Figure 17a demonstrates the location of selected cities on the Chinese map with the color representing its cumulative conﬁrmed cases by March 1st, 2020.

Original: It can be observed that the selected mediators are either located around Hubei province or are big central cities such as Beijing that has a large population and migration scale.
Revised: It can be observed that the selected mediators are either located around Hubei province or are big central cities such as Beijing that has a large population and migration scale.

Original: In addition, we provide the partial heat map of the correlation matrix between cities and the treatment A or the outcome Y as illustrated in Figure 9b.
Revised: In addition, we provide the partial heat map of the correlation matrix between cities and the treatment A or the outcome Y as illustrated in Figure 17b.

Original: Here, one could observe that cities’ MSIs are highly positive correlated with the lockdown while are highly negative correlated with the daily increase rate of conﬁrmed cases. 
Revised: Here, one could observe that cities’ MSIs are highly positive correlated with the lockdown while are highly negative correlated with the daily increase rate of conﬁrmed cases. [Figure] S h

Original: D. A DDITIONAL R ESULTS
Revised: e n z h e n G u a n g z h o u B e iji n g C

Revised: h e n gd u

Revised: S h a n g h a

Revised: i

Revised: D

Revised: o n gg u a n

Revised: S u z h

Revised: o u

Revised: X i a n H a n g z h o u

Revised: Z h e n g z h o u

Revised: C h o n gq i n g

Revised: C h a n g s h

Revised: a

Revised: N a n ji n g K un m i n g T i a n ji n H e f e i N a nn i n g W e n z h o u N a

Revised: [Figure]

Revised: c h a n g

Revised: Z h o u k o u F u y a n g S h a n gq i u Y u

Revised: e y a n g Z hu m a d

Revised: i a n

Revised: C h a n gd e

Revised: N a n y a n g Y

Revised: i c hun

Revised: X

Revised: i n y a n g A n q i n g J

Revised: i u ji a n g

Original:  Table 5 lists all the numerical facts of the selected cities, including the population (million), the cumulative migration scale index (MSI) during the data period (Jan 12th to Feb 20th, 2020), the ratio of received Wuhan migration between Jan 1st to Jan 22nd, 2020 (before the lockdown), the cumulative conﬁrmed cases by March 1st, 2020, and cities’ direct ( DM ) and indirect effect ( IM ).
Revised: A Y 1. [Figure] D. A DDITIONAL R ESULTS Table 6 lists all the numerical facts of the selected cities, including the population (million), the cumulative migration scale index (MSI) during the data period (Jan 12th to Feb 20th, 2020), the ratio of received Wuhan migration between Jan 1st to Jan 22nd, 2020 (before the lockdown), the cumulative conﬁrmed cases by March 1st, 2020, and cities’ direct ( DM ) and indirect effect ( IM ).

Original: Note the selected 30 cities in Table 5 are sorted according to their cumulative MSI, and its order is used as the order of the mediators in this paper.
Revised: Note the selected 30 cities in Table 6 are sorted according to their cumulative MSI, and its order is used as the order of the mediators in this paper.

Original: It can be seen from Table 5 that the population, the cumulative MSI, the ratio of received Wuhan migration, and the conﬁrmed cases are highly correlated among selected cities as expected.
Revised: It can be seen from Table 6 that the population, the cumulative MSI, the ratio of received Wuhan migration, and the conﬁrmed cases are highly correlated among selected cities as expected.

Revised: Note that we list these factors to assist the interpretation of the results, while none of them is used to estimate the DAG of interest.

Revised: Therefore, similar values of factors don’t necessarily imply similar causal effects.

Original: Besides the general causal pattern of cities’ estimated DM s and IM s stated in the main text, we provide more interpretation on the level of the individual city.
Revised: Besides the general causal pattern of cities’ estimated DM s and IM s stated in the main text, we provide more interpretation on the level of the individual city.

Original: Here, we compare three groups of cities that are of similar scale (population and migration) or geographic position, to specify our results.
Revised: Here, we compare three groups of cities that are of similar scale (population and migration) or geographic position, to specify our results.

Original: First, we compare the results between Beijing and Shanghai, where both cities have comparable population scale and are the center of politic or economic in China (the correlation between the MSI of Beijing and the outcome yields the same value as of Shanghai as 0.50).
Revised: First, we compare the results between Beijing and Shanghai, where both cities have comparable population scale and are the center of politic or economic in China (the correlation between the MSI of Beijing and the outcome yields the same value as of Shanghai as 0.50).

Original: It can be seen from Tablethat Beijing and Shanghai have similar positive indirect effects as 0.247 and 0.235, respectively, while
Revised: It can be seen from Tablethat Beijing and Shanghai have similar positive indirect effects as 0.247 and 0.235, respectively, while

Original: Shanghai has a slightly higher direct effect on controlling the virus as -0.069, which is possibly due to the smaller MSI of Shanghai.
Revised: Shanghai has a slightly higher direct effect on controlling the virus as -0.069, which is possibly due to the smaller MSI of Shanghai.

Original: Second, we compare three cities located at Guangdong province (see
Revised: Second, we compare three cities located at Guangdong province (see

Original: Figure 10a and 10b), including Shenzhen, Guangzhou, and Dongguan.
Revised: Figure 18a and 18b), including Shenzhen, Guangzhou, and Dongguan.

Original: All three cities have positive effects, among which Guangzhou yields the highest direct effect as 0.847, followed by Dongguan.
Revised: All three cities have positive effects, among which Guangzhou yields the highest direct effect as 0.847, followed by Dongguan.

Original: The effect size of these cities agrees with their correlation coefﬁcients with the outcome, where the correlation between Guangzhou and Y is 0.61, followed by Dongguan as 0.58 and Shenzhen as 0.50.
Revised: The effect size of these cities agrees with their correlation coefﬁcients with the outcome, where the correlation between Guangzhou and Y is 0.61, followed by Dongguan as 0.58 and Shenzhen as 0.50.

Original: The last comparison is among cities in southeastern China (see Figure 10a and 10b), including
Revised: The last comparison is among cities in southeastern China (see Figure 18a and 18b), including

Original: Suzhou, Hangzhou, and Wenzhou, all of which have a negative direct effect on the virus control and a positive indirect effect for the virus spread.
Revised: Suzhou, Hangzhou, and Wenzhou, all of which have a negative direct effect on the virus control and a positive indirect effect for the virus spread.

Original: Here, Wenzhou achieves the largest absolute value of the negative direct effect as -0.650, which conforms to its strict local shelter in home order after the
Revised: Here, Wenzhou achieves the largest absolute value of the negative direct effect as -0.650, which conforms to its strict local shelter in home order after the

Original: Hubei lockdowns, where the correlation between 
Revised: Hubei lockdowns, where the correlation between Wenzhou and the outcome achieves the highest value as 0.83.

Original: [Table]
Revised: [Table]

Original: Kunming 6.
Revised: Kunming 6.

Original: Tianjin 12.
Revised: Tianjin 12.

Original: Hefei 5.
Revised: Hefei 5.

Original: Nanning 6.
Revised: Nanning 6.

Original: Wenzhou 9.
Revised: Wenzhou 9.

Original: Nanchang 5.
Revised: Nanchang 5.

Original: Zhoukou 8.
Revised: Zhoukou 8.

Original: Fuyang 7.
Revised: Fuyang 7.

Original: Shangqiu 7.
Revised: Shangqiu 7.

Original: Yueyang 5.
Revised: Yueyang 5.

Original: Zhumadian 7.
Revised: Zhumadian 7.

Original: Changde 5.
Revised: Changde 5.

Original: Nanyang 10.
Revised: Nanyang 10.

Original: Yichun 5.
Revised: Yichun 5.

Original: Xinyang 6.
Revised: Xinyang 6.

Original: [Table]
Revised: [Table]

Original: We summarize all different sources of causal effects in Table 5 as the ANOCE table of 2020 Hubei lockdowns on reducing the COVID-19 spread in China.
Revised: We summarize all different sources of causal effects in Table 6 as the ANOCE table of 2020 Hubei lockdowns on reducing the COVID-19 spread in China.

Original: Note that due to cities’ different levels of control measures on the coronavirus outside Hubei as well as other possible confounders, there are some inconsistency between cities’ DMs and their cumulative MSIs.
Revised: Note that due to cities’ different levels of control measures on the coronavirus outside Hubei as well as other possible confounders, there are some inconsistency between cities’ DMs and their cumulative MSIs.

Original: We leave the extension with confounders for further investigation.
Revised: We leave the extension with confounders for further investigation.

Revised: Further interpretation of the real data analysis requires domain experts.

Original: One can refer to Figure 10a and 10b for cities’ DM s and IM s on the Chinese map.
Revised: One can refer to Figure 18a and 18b for cities’ DM s and IM s on the Chinese map.

Original: To check the reasonability of our results, we plot the estimated weighted matrix in Figure 4a, where the ﬁrst node (indexed 0) represents the Hubei lockdowns, the last node (indexed 31) is the daily increase rate out of Hubei, and the middle 30 nodes (indexed 1-30) correspond to 30 selected cities in Figure 4b.
Revised: To check the reasonability of our results, we plot the estimated weighted matrix in Figure 4a, where the ﬁrst node (indexed 0) represents the Hubei lockdowns, the last node (indexed 31) is the daily increase rate out of Hubei, and the middle 30 nodes (indexed 1-30) correspond to 30 selected cities in Figure 4b.

Original: From Figure 4a, we can observe: • 1).
Revised: From Figure 4a, we can observe: • 1).

Original: The color of the ﬁrst column is almost all blue, indicatingthat locking Hubei down can reduce the migration of selected cities; • 2). An approximate red upper
Revised: The color of the ﬁrst column is almost all blue, indicating

Original: [Figure]
Revised: [Figure]

Original: [-A 1 2 5 6   11       20 21      
 Y-]
Revised: [-A 1 2 5 6   11       20 21      
 Y-]that locking Hubei down can reduce the migration of selected cities; • 2).

Original: [Figure] triangular among ﬁrst 20 nodes implies a migration trend with positive effects among central cities with large MSI, i.e. relatively smaller cities tend to have positive effects on other cities with relatively larger MSI; • 3).
Revised: An approximate red upper triangular among ﬁrst 20 nodes implies a migration trend with positive effects among central cities with large MSI, i.e. relatively smaller cities tend to have positive effects on other cities with relatively larger MSI; • 3).

Original: An approximate red lower triangular among last 10 nodes indicates a weaker migration trend with an opposite direction among non-central cities with small MSI; • 4).
Revised: An approximate red lower triangular among last 10 nodes indicates a weaker migration trend with an opposite direction among non-central cities with small MSI; • 4).

Original: There is also an almost all blue rectangle in the right top of the estimated matrix, showing non-central cities tend to have negative effects on central cities.
Revised: There is also an almost all blue rectangle in the right top of the estimated matrix, showing non-central cities tend to have negative effects on central cities.

Original: All the above ﬁndings are accords with the migration trend during the Spring Festival  and intensive mutual communications among central cities in China.
Revised: Overall speaking, all the above ﬁnding accords with the migration trend during the Spring Festival period and intensive mutual communications among central cities in China, though there are also some noisy causal directions opposite with the main trend in each area due to possible confounders and identiﬁability issue in the linear Gaussian model.

Original: In addition, we provide the estimated DAG for the coronavirus data in Figure 10c as the complete spreading network among major cities outside Hubei, where the exposure A , mediators M (cities’ corresponding index can be found in Table 5), and the outcome Y are colored in red, blue and green, respectively.
Revised: In addition, we provide the estimated DAG for the coronavirus data in Figure 18c as the complete spreading network among major cities outside Hubei, where the exposure A , mediators M (cities’ corresponding index can be found in Table 6), and the outcome Y are colored in red, blue and green, respectively.

Original: It can be observed that the in-degree is larger than the out-degree for nodes with small index, while an opposite rule is applied for nodes with large index.
Revised: It can be observed that the in-degree is larger than the out-degree for nodes with small index, while an opposite rule is applied for nodes with large index.

Original: This ﬁnding is consistent with the migration trend identiﬁed in our main text.
Revised: This ﬁnding is consistent with the migration trend identiﬁed in our main text.

Original: Lastly, we give the spreading network among cities that received most Wuhan migration during the data period, including Beijing, Shanghai, Guangzhou,
Revised: Lastly, we give the spreading network among cities that received most Wuhan migration during the data period, including Beijing, Shanghai, Guangzhou,

Original: Shenzhen, Chengdu, Chongqing, Zhengzhou, Changsha, and Xinyang, plus Wuhan, in Figure 10d, to illustrate the partial interaction trend among cities.
Revised: Shenzhen, Chengdu, Chongqing, Zhengzhou, Changsha, and Xinyang, plus Wuhan, in Figure 18d, to illustrate the partial interaction trend among cities.

Revised: Each node refers to a city with the color of the node presenting the percentage of received Wuhan migration, ranging from 2.57% to 5.50%.

Original: E T OY E XAMPLE
Revised: M Y A M M

Original:  Example E.1 Here, we use a toy example of a weighted DAG G toy under the LSEM given in Figure 11 to better demonstrate our deﬁnitions.
Revised: M 4 0. E T OY E XAMPLE Example E.1 Here, we use a toy example of a weighted DAG G toy under the LSEM given in Figure 19 to better demonstrate our deﬁnitions.

Original: In G toy , p = 4 mediators are included, where M 1 ← 0 .
Revised: In G toy , p = 4 mediators are included, where M 1 ← 0 .

Original: 2 A + (cid:15) 1 , M 2 ← 0 .
Revised: 2 A + (cid:15) 1 , M 2 ← 0 .

Original: 2 M 1 + 0 .
Revised: 2 M 1 + 0 .

Original: 5 M 4 + (cid:15) 2 ,
Revised: 5 M 4 + (cid:15) 2 ,

Original: M 3 ← 0 .
Revised: M 3 ← 0 .

Original: 4 A + (cid:15) 3 , M 4 ← 0 .
Revised: 4 A + (cid:15) 3 , M 4 ← 0 .

Original: 5 M 3 + (cid:15) 4 , Y = 0 .
Revised: 5 M 3 + (cid:15) 4 , Y = 0 .

Original: 5 A
Revised: 5 A

Original: + 1 . 1 M 

Revised: + 1 . 1 M

Original: + 0 .
Revised: + 0 .

Original: 7 M 3 .
Revised: 7 M 3 .

Original: There are 4 directed path from
Revised: There are 4 directed path from

Original: A to Y :
Revised: A to Y :

Original: M

Original: Y A

Original: M M M 4 0. 1) ‘ A → Y ’ with length 1; 2) ‘
Revised: 1) ‘ A → Y ’ with length 1; 2) ‘

Original: A → M 3 → Y ’ with length 2; 3) ‘ A → M 1 → M 2 → Y ’ with length 3; 4) ‘ A → M 3 → M 4 → M 2 → Y ’ with length 4.
Revised: A → M 3 → Y ’ with length 2; 3) ‘ A → M 1 → M 2 → Y ’ with length 3; 4) ‘ A → M 3 → M 4 → M 2 → Y ’ with length 4.

Original: Since there exists 3 directed path π ∗ ∈ { π AY ( G toy ) } such that the length of π ∗ is larger than 2, we have the mediators in G toy are interacted.
Revised: Since there exists 3 directed path π ∗ ∈ { π AY ( G toy ) } such that the length of π ∗ is larger than 2, we have the mediators in G toy are interacted.

Original: From the weighted DAG G toy , we have the direct effect of A on Y is γ toy = 0 .
Revised: From the weighted DAG G toy , we have the direct effect of A on Y is γ toy = 0 .

Original: 5 , α toy ≡ [0 . 5 , 0 , 0 . 4 , 0] (cid:62) , β toy ≡ [0 , 1 . 1 , 0 .
Revised: 5 , α toy ≡ [0 . 5 , 0 , 0 . 4 , 0] (cid:62) , β toy ≡ [0 , 1 . 1 , 0 .

Original: 7 , 0] (cid:62) , and
Revised: 7 , 0] (cid:62) , and

Original: B (cid:62) Mtoy =   0 0 .
Revised: B (cid:62) Mtoy =   0 0 .

Original: 0  0 0 0 .
Revised: 0  0 0 0 .

Original:  .
Revised:  .

Original: Note one may recover G toy from the weight matrix B toy as long as the order of the vertices in B toy is given.
Revised: Note one may recover G toy from the weight matrix B toy as long as the order of the vertices in B toy is given.

Original: Then, we have( I p − B (cid:62) Mtoy ) − 1 =      0 . 20 . 
Revised: Then, we have

Original:  .
Revised: [Equation]

Original: Thus, the indirect effect of A on Y is β toy (cid:62) ( I p − B (cid:62) Mtoy ) − 1 α toy =[0 , 1 . 1 , 0 .       0 .    [0 . 5 , 0 , 0 . 4 , 0] (cid:62) =
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: Thus, the indirect effect of A on Y is

Original: [Equation]
Revised: [Equation]

Original: From the Equation E.1 and Theorem 3.1, we have the DM of M 2 and M 3 is 0 .
Revised: From the Equation E.1 and Theorem 3.1, we have the DM of M 2 and M 3 is 0 .

Original: 22 and 0 .
Revised: 22 and 0 .

Original: 28 , respectively, while other DM are 0.
Revised: 28 , respectively, while other DM are 0.

Original: Note that there is no explicit expression of the natural indirect effect through the directed path from A to Y ( IM ) due to the complex interaction among mediators, while we provide its theoretical form in Equation G.6 based on the path method in Wright (1921) and
Revised: Note that there is no explicit expression of the natural indirect effect through the directed path from A to Y ( IM ) due to the complex interaction among mediators, while we provide its theoretical form in Equation G.6 based on the path method in Wright (1921) and

Original: Nandy et al.
Revised: Nandy et al.

Original: Speciﬁcally, for each directed path from A to Y , we have:
Revised: Speciﬁcally, for each directed path from A to Y , we have:

Original: 1) the direct effect through ‘ A → Y ’ ( DE ): 0.5; 2) the effect of path ‘ A → M 3 → Y ’: 0 . 4 × 0 .
Revised: 1) the direct effect through ‘ A → Y ’ ( DE ): 0.5; 2) the effect of path ‘ A → M 3 → Y ’: 0 . 4 × 0 .

Original: 7 = 0 .
Revised: 7 = 0 .

Original: 28 ; 3) the effect of path ‘ A → M 1 → M 2 → Y ’: 0 . 5 × 0 .
Revised: 28 ; 3) the effect of path ‘ A → M 1 → M 2 → Y ’: 0 . 5 × 0 .

Original: 2 × 1 . 1 = 0 .
Revised: 2 × 1 . 1 = 0 .

Original: 11 ; 4) the effect of path ‘ A → M 3 → M 4 → M 2 → Y ’: 0 .
Revised: 11 ; 4) the effect of path ‘ A → M 3 → M 4 → M 2 → Y ’: 0 .

Original: 4 × 0 .
Revised: 4 × 0 .

Original: 5 × 0 .
Revised: 5 × 0 .

Original: 5 × 1 . 1 = 0 .
Revised: 5 × 1 . 1 = 0 .

Original: 11 ; 5) so the indirect effect of A on Y ( IE ) is 0 .
Revised: 11 ; 5) so the indirect effect of A on Y ( IE ) is 0 .

Original: 11 + 0 .
Revised: 11 + 0 .

Original: 11 + 0 .
Revised: 11 + 0 .

Original: 28 = 0 . 5 ; 6) and the total effect of A on Y ( TE ) is 0 .
Revised: 28 = 0 . 5 ; 6) and the total effect of A on Y ( TE ) is 0 .

Original: 5 + 0 .
Revised: 5 + 0 .

Original: 5 = 1 . 0 ; 7) the indirect effect for M 1 ( IM 1 ) corresponds to the effect of path ‘ A → M 1 → M 2 → Y ’, thus is 
Revised: 5 = 1 . 0 ; 7) the indirect effect for M 1 ( IM 1 ) corresponds to the effect of path ‘ A → M 1 → M 2 → Y ’, thus is 0 .

Original: 11 ; 8) the indirect effect for M 2 ( IM 2 ) is zero since there is no path ﬁrst goes through M 2 followed by other mediators; 9) the indirect effect for M 3 ( IM 3 ) corresponds to the effect of path ‘ A → M 3 → M 4 → M 2 → Y ’, thus is 0 .
Revised: 11 ; 8) the indirect effect for M 2 ( IM 2 ) is zero since there is no path ﬁrst goes through M 2 followed by other mediators; 9) the indirect effect for M 3 ( IM 3 ) corresponds to the effect of path ‘ A → M 3 → M 4 → M 2 → Y ’, thus is 0 .

Original: 11 ; 10) the indirect effect for M 4 ( IM 4 ) corresponds to the effect of path ‘ A → M 3 → M 4 → M 2 → Y ’, thus is 0 .
Revised: 11 ; 10) the indirect effect for M 4 ( IM 4 ) corresponds to the effect of path ‘ A → M 3 → M 4 → M 2 → Y ’, thus is 0 .

Original: We can calculate the last edge-speciﬁc effect directly from its deﬁnition.
Revised: We can calculate the last edge-speciﬁc effect directly from its deﬁnition.

Original: Since there is no M 1 → Y and M 4 → Y in G toy , we have LE toy= LE toy= 0 .
Revised: Since there is no M 1 → Y and M 4 → Y in G toy , we have LE toy= LE toy= 0 .

Original: By deleting M 2 → Y in G toy , the total effect reduced by 0 .
Revised: By deleting M 2 → Y in G toy , the total effect reduced by 0 .

Original: 11 + 0 .
Revised: 11 + 0 .

Original: 11 = 0 .
Revised: 11 = 0 .

Original: 22 , so LE toy= 0 . 22 ; similarly, after deleting M 3 → Y , the total effect decreases 0.28, thus LE toy= 0 .
Revised: 22 , so LE toy= 0 . 22 ; similarly, after deleting M 3 → Y , the total effect decreases 0.28, thus LE toy= 0 .

Original: One may notice that the last edge-speciﬁc effects are equal to the DM s, and the additive of the last edge-speciﬁc effects is exact the last step of calculation β toy (cid:62) ( I p − B (cid:62) Mtoy ) − 1 α toy in Equation E.1.
Revised: One may notice that the last edge-speciﬁc effects are equal to the DM s, and the additive of the last edge-speciﬁc effects is exact the last step of calculation β toy (cid:62) ( I p − B (cid:62) Mtoy ) − 1 α toy in Equation E.1.

Original: F C ONNECTION TO L ITERATURE
Revised: F C ONNECTION TO L ITERATURE

Original: We establish the connection between our proposed method to the literature from three different angles.
Revised: We establish the connection between our proposed method to the literature from three different angles.

Original: First, we show that the individual mediation effect deﬁned in Chakrabortty et al.
Revised: First, we show that the individual mediation effect deﬁned in Chakrabortty et al.

Original: (2018) can be decomposed into our deﬁned DM and IM .
Revised: (2018) can be decomposed into our deﬁned DM and IM when the LSEM assumption holds.

Original: Next, we give an equivalent deﬁnition of the DM through a type of special edge (last edge) in the causal graph.
Revised: Next, we give an equivalent deﬁnition of the DM through a type of special edge (last edge) in the causal graph.

Original: Lastly, we prove that the proposed DM is consistent with the interventional effect via a particular mediator deﬁned in Vansteelandt & Daniel (2017) under the LSEM.
Revised: Lastly, we prove that the proposed DM is consistent with the interventional effect via a particular mediator deﬁned in Vansteelandt & Daniel (2017) under the LSEM.

Original: F. F ROM I NDIVIDUAL M EDIATION V IEWPOINT
Revised: F. F ROM I NDIVIDUAL M EDIATION V IEWPOINT

Original: Chakrabortty et al.
Revised: Chakrabortty et al.

Original: (2018) deﬁned the individual mediation effect under the LSEM as follows.
Revised: (2018) deﬁned the individual mediation effect under the LSEM as follows.

Original: Deﬁnition F.1 (Chakrabortty et al., 2018)
Revised: Deﬁnition F.1 (Chakrabortty et al., 2018)

Original: Individual mediation effect for M i :
Revised: Individual mediation effect for M i :

Original: [Equation]
Revised: [Equation]

Revised: F.

Original:  In the following theorem, we show that the summation of η i is strictly larger than the IE if the mediators are not parallel.
Revised: [Equation] In the following theorem, we show that the summation of η i is strictly larger than the IE if the mediators are not parallel.

Original: The proof is given in Section G.2.
Revised: The proof is given in Section G.2.

Original: Theorem F.

Revised: Theorem F.

Original: If there exists at least one directed path π ∗ ∈ { π AY ( G ) } such that the length of π ∗ is larger than 2, and the element in B is nonnegative, then
Revised: If there exists at least one directed path π ∗ ∈ { π AY ( G ) } such that the length of π ∗ is larger than 2, and the element in B is nonnegative, then

Original: [Equation]
Revised: [Equation]

Original: Remark F.1 From the above theorem, it is clear that the mediator effect deﬁned in Chakrabortty et al.
Revised: Remark F.1 From the above theorem, it is clear that the mediator effect deﬁned in Chakrabortty et al.

Original: (2018) is not appropriate for interpreting the decomposition of the indirect effect, when there exists interaction among mediators (a common situation as described in the introduction).
Revised: (2018) is not appropriate for interpreting the decomposition of the indirect effect, when there exists interaction among mediators (a common situation as described in the introduction).

Original: Here, we keep the condition that the element in B is nonnegative, as the multiple count mediation effects in
Revised: Here, we keep the condition that the element in B is nonnegative, as the multiple count mediation effects in

Original: Chakrabortty et al.
Revised: Chakrabortty et al.

Original: (2018) may cancel out in some cases and their summation would equal to the IE by chance.
Revised: (2018) may cancel out in some cases and their summation would equal to the IE by chance.

Original: Inspired by the proof of Theorem F.1, the mediator effect η i can be decomposed into two parts, the natural direct and indirect effect for i -th mediator, as shown in the following corollary.
Revised: Inspired by the proof of Theorem F.1, the mediator effect η i can be decomposed into two parts, the natural direct and indirect effect for i -th mediator, as shown in the following corollary.

Original: Corollary F.1 Under assumptions (A1-A3) and Model 1, we have
Revised: Corollary F.1 Under assumptions (A1-A3) and Model 1, we have

Original: [Equation]
Revised: [Equation]

Original: Remark F.2 Corollary F.1 together with the deﬁnition of η i in Chakrabortty et al.
Revised: Remark F.2 Corollary F.1 together with the deﬁnition of η i in Chakrabortty et al.

Original: (2018) provides a feasible way to numerically calculate the natural indirect effect IM i .
Revised: (2018) provides a feasible way to numerically calculate the natural indirect effect IM i .

Original: Speciﬁcally, by deleting the mediator M i in the causal graph, the reduced treatment effect corresponds to η i , then IM i =
Revised: Speciﬁcally, by deleting the mediator M i in the causal graph, the reduced treatment effect corresponds to η i , then IM i =

Original: η 
Revised: η i −

Revised: DM i , where the explicit expression of the DM i is provided in Theorem 3.1.

Revised: See more implementation details in Section B.

Original: F. F ROM G RAPHICAL P ERSPECTIVE
Revised: F. F ROM G RAPHICAL P ERSPECTIVE

Original: Next, we give the deﬁnition of the edge-speciﬁc effect following Avin et al.
Revised: Next, we give the deﬁnition of the edge-speciﬁc effect following Avin et al.

Original: Suppose a directed edge of interest as X i → X j in a weighted DAG G .
Revised: Suppose a directed edge of interest as X i → X j in a weighted DAG G .

Original: Deﬁne a new weighted DAG G (cid:48)
Revised: Deﬁne a new weighted DAG G (cid:48)

Original: i,j by deleting the directed edge X i → X j in G , i.e. G (cid:48)
Revised: i,j by deleting the directed edge X i → X j in G , i.e. G (cid:48)

Original: i,j ≡ G \ ( X i → X j ) .
Revised: i,j ≡ G \ ( X i → X j ) .

Original: Deﬁnition F.2 (Avin et al., 2005)
Revised: Deﬁnition F.2 (Avin et al., 2005)

Original: Edge-speciﬁc effect:
Revised: Edge-speciﬁc effect:

Original: [Equation]
Revised: [Equation]

Original: where TE G means the total effect in graph G .
Revised: where TE G means the total effect in graph G .

Original: We next give an equivalent deﬁnition of our proposed DM from a graphical perspective.
Revised: We next give an equivalent deﬁnition of our proposed DM from a graphical perspective.

Original: Let the edge in G that starts with i -th mediator and ends with node Y , i.e. M i
Revised: Let the edge in G that starts with i -th mediator and ends with node Y , i.e. M i

Original: → Y , as the i -th last edge.
Revised: → Y , as the i -th last edge.

Original: Denote the graph G deleting the i -th last edge ( M i → Y ) as G (cid:48)
Revised: Denote the graph G deleting the i -th last edge ( M i → Y ) as G (cid:48)

Original: i .
Revised: i .

Original: We deﬁne the i th last edge-speciﬁc effect as
Revised: We deﬁne the i th last edge-speciﬁc effect as

Original: Deﬁnition F.

Revised: Deﬁnition F.

Original: Last edge-speciﬁc effect for M i :
Revised: Last edge-speciﬁc effect for M i :

Original: [Equation]
Revised: [Equation]

Original: if there exists edge
Revised: , otherwise .

Original: [Equation] Y in G ; [Equation] otherwise
Revised: (F

Original: [Equation]
Revised: [Equation]

Original: By Theorem 3.1, we have ( I p − B (cid:62) M ) − 1 α is the causal effect of A on M .
Revised: By Theorem 3.1, we have ( I p − B (cid:62) M ) − 1 α is the causal effect of A on M .

Original: Let ζ ≡ ( I p − B (cid:62) M ) − 1 α , with its i -th element ζ i ≡ { ( I p − B (cid:62) M ) − 1 α } i .
Revised: Let ζ ≡ ( I p − B (cid:62) M ) − 1 α , with its i -th element ζ i ≡ { ( I p − B (cid:62) M ) − 1 α } i .

Original: Next, we show that the i -th last edge-speciﬁc effect can be presented as β i ζ i under the LSEM in the following theorem, where β i is the i -th element of the vector β and corresponds to the weight of the edge M i
Revised: Next, we show that the i -th last edge-speciﬁc effect can be presented as β i ζ i under the LSEM in the following theorem, where β i is the i -th element of the vector β and corresponds to the weight of the edge M i

Original: → Y .
Revised: → Y .

Original: The proof can be found in Section
Revised: The proof can be found in Section

Original: G.3.
Revised: G.3.

Original: Theorem F.

Revised: Theorem F.

Original: Under assumptions (A1-A3) and Model 1, we have
Revised: Under assumptions (A1-A3) and Model 1, we have

Original: [Equation]
Revised: [Equation]

Original: Based on Theorem 3.1, the natural direct effect of M i on Y can be expressed as DM i = β
Revised: Based on Theorem 3.1, the natural direct effect of M i on Y can be expressed as DM i = β

Original: i ζ i .
Revised: i ζ i .

Original: Thus, with the result of Theorem F.2, it is easy to show the following corollary.
Revised: Thus, with the result of Theorem F.2, it is easy to show the following corollary.

Original: Corollary F.

Revised: Corollary F.

Original: Under assumptions (A1-A3) and Model 1, the natural direct effect of M i is equal to the i -th last edge-speciﬁc effect:
Revised: Under assumptions (A1-A3) and Model 1, the natural direct effect of M i is equal to the i -th last edge-speciﬁc effect:

Original: [Equation]
Revised: [Equation]

Original: Remark F.

Revised: Remark F.

Original: Here, both deﬁnitions describe the direct impact of one mediator M i on the outcome.
Revised: Here, both deﬁnitions describe the direct impact of one mediator M i on the outcome.

Original: The natural direct effect of a particular mediator M i can be understood as the inﬂuence when removing the direct edge between M i and Y .
Revised: The natural direct effect of a particular mediator M i can be understood as the inﬂuence when removing the direct edge between M i and Y .

Original: Thus, we have the equivalence between two deﬁnitions.
Revised: Thus, we have the equivalence between two deﬁnitions.

Original: Then, we can decompose the total natural indirect effect into p last edge-speciﬁc effects or p DM s as the following additive form, based on Theorem 3.2 and Corollary F.2.
Revised: Then, we can decompose the total natural indirect effect into p last edge-speciﬁc effects or p DM s as the following additive form, based on Theorem 3.2 and Corollary F.2.

Original: Corollary F.

Revised: Corollary F.

Original: Under assumptions (A1-A3) and Model 1, we have
Revised: Under assumptions (A1-A3) and Model 1, we have

Original: [Equation]
Revised: [Equation]

Original:  In fact, based on the uniqueness of each last edge, the natural indirect effect can be decomposed into p last edge-speciﬁc effect regardless of the LSEM setting through the graphical perspective.
Revised: = LE i . [Equation] In fact, based on the uniqueness of each last edge, the natural indirect effect can be decomposed into p last edge-speciﬁc effect regardless of the LSEM setting through the graphical perspective.

Original: We give the following intuitive conclusion.
Revised: We give the following intuitive conclusion.

Original: The proof can be found in Section G.4.
Revised: The proof can be found in Section G.4.

Original: Theorem F.

Revised: Theorem F.

Original: The IE can be decomposed through LE s as:
Revised: The IE can be decomposed through LE s as:

Original: [Equation]
Revised: [Equation]

Original: Remark F.4 One can view the last edge-speciﬁc effect as the generalized deﬁnition of the natural direct effect for mediator without the LSEM 
Revised: Remark F.4 One can view the last edge-speciﬁc effect as the generalized deﬁnition of the natural direct effect for mediator without the LSEM assumption.

Original: F. F ROM I NTERVENTIONAL E FFECT L EVEL
Revised: F. F ROM I NTERVENTIONAL E FFECT L EVEL

Original: Finally, we show the consistency of our deﬁned DM to the interventional effect via a particular mediator deﬁned in Vansteelandt & Daniel (2017) under the LSEM.
Revised: Finally, we show the consistency of our deﬁned DM to the interventional effect via a particular mediator deﬁned in Vansteelandt & Daniel (2017) under the LSEM.

Original: Deﬁnition F.4 (Vansteelandt & Daniel, 2017)
Revised: Deﬁnition F.4 (Vansteelandt & Daniel, 2017)

Original: Under assumptions (A1-A3), the interventional effect via M i is
Revised: Under assumptions (A1-A3), the interventional effect via M i is

Original: [Equation]
Revised: [Equation]

Original: where M i is the support of M i , o i =
Revised: where M i is the support of M i , o i =

Original: [ m 1 , · · · , m i − 1 , m i +1 , · · · , m p ] , P ( M = m |
Revised: [ m 1 , · · · , m i − 1 , m i +1 , · · · , m p ] , P ( M = m |

Original: A = a ) is the probability of M = m when setting A = a .
Revised: A = a ) is the probability of M = m when setting A = a .

Original: Theorem F.4 Under assumptions (A1-A3) and Model 1, we have
Revised: Theorem F.4 Under assumptions (A1-A3) and Model 1, we have

Original: [Equation]
Revised: [Equation]

Original: Remark F.

Revised: Remark F.

Original: The proof can be found in Section G.5.
Revised: The proof can be found in Section G.5.

Original: Based on Deﬁnition 3.2 and Equation F.10, both the proposed DM and the effect deﬁned in Vansteelandt & Daniel (2017) contain the information of the causal effect of A on the mediator M i , i.e. P
Revised: Based on Deﬁnition 3.2 and Equation F.10, both the proposed DM and the effect deﬁned in Vansteelandt & Daniel (2017) contain the information of the causal effect of A on the mediator M i , i.e. P

Original: ( M i = m i
Revised: ( M i = m i

Original: |
Revised: |

Original: A = a + 1) − P ( M i = m i |
Revised: A = a + 1) − P ( M i = m i |

Original: A
Revised: A

Original: = a ) .
Revised: = a ) .

Original: G T ECHNICAL P ROOFS G. P ROOF OF T HEOREM 3. 
Revised: G T ECHNICAL P ROOFS G. P ROOF OF T HEOREM 3.

Original: Proof G.

Revised: Proof G.

Original: In this proof, we will give the explicit expressions of causal effects deﬁned under the
Revised: In this proof, we will give the explicit expressions of causal effects deﬁned under the

Original: LSEM.
Revised: LSEM.

Original: First, Equation 3 is equivalent to
Revised: First, Equation 3 is equivalent to

Original: [Equation]
Revised: [Equation]

Original: Based on M = α A + B (cid:62) M M
Revised: Based on M = α A + B (cid:62) M M

Original: +
Revised: +

Original: cid:15) M , by moving B (cid:62) M M to the left-hand side, we have
Revised: cid:15) M , by moving B (cid:62) M M to the left-hand side, we have

Original: [Equation]
Revised: [Equation]

Original: Suppose the mediators are sorted in the topological order (a series of elementary transformation of the matrix), then the matrix B (cid:62) M is strictly upper triangular with the diagonal element as 0.
Revised: Suppose the mediators are sorted in the topological order (a series of elementary transformation of the matrix), then the matrix B (cid:62) M is strictly upper triangular with the diagonal element as 0.

Original: Thus, we have I p − B (cid:62) M is invertible, then I p − B (cid:62) M under its original order should be also invertible (any invertible matrix after elementary transformation is still invertible).
Revised: Thus, we have I p − B (cid:62) M is invertible, then I p − B (cid:62) M under its original order should be also invertible (any invertible matrix after elementary transformation is still invertible).

Original: Therefore, we can rewrite M as a purely function of A plus the error term as follows.
Revised: Therefore, we can rewrite M as a purely function of A plus the error term as follows.

Original: [Equation]
Revised: [Equation]

Original: Then we replace mediators in Equation G.1 with Equation G.2 and obtain
Revised: Then we replace mediators in Equation G.1 with Equation G.2 and obtain

Original: [Equation]
Revised: [Equation]

Original: Next, we show how to get the explicit expressions of E { Y | do ( A = a ) } under the LSEM.
Revised: Next, we show how to get the explicit expressions of E { Y | do ( A = a ) } under the LSEM.

Original: Following the results in Rosenbaum & Rubin (1983), under the assumption (A2), we have P { M | do ( A = a ) } =
Revised: Following the results in Rosenbaum & Rubin (1983), under the assumption (A2), we have P { M | do ( A = a ) } =

Original: P ( M |
Revised: P ( M |

Original: A
Revised: A

Original: = a ) , and thus,
Revised: = a ) , and thus,

Original: E { M | do ( A = a ) } =
Revised: E { M | do ( A = a ) } =

Original: E ( M |
Revised: E ( M |

Original: A = a ) .
Revised: A = a ) .

Original: Similarly, we can get E { Y | do ( A = a ) } =
Revised: Similarly, we can get E { Y | do ( A = a ) } =

Original: E ( Y |
Revised: E ( Y |

Original: A = a ) under the assumption (A1), and
Revised: A = a ) under the assumption (A1), and

Original: E { Y | do ( A = a, M = m ) }
Revised: E { Y | do ( A = a, M = m ) }

Original: =
Revised: =

Original: E ( Y |
Revised: E ( Y |

Original: A = a, M = m ) under the assumption (A3).
Revised: A = a, M = m ) under the assumption (A3).

Original: Based on above results and Equation G.3, we have
Revised: Based on above results and Equation G.3, we have

Original: [Equation]
Revised: [Equation]

Original:  where the ﬁrst ‘=’ is held under the assumption (A1), the second and forth ‘=’ are given by Equation
Revised: (G.4) where the ﬁrst ‘=’ is held under the assumption (A1), the second and forth ‘=’ are given by Equation

Original: G.3 that Y
Revised: G.3 that Y

Original: =
Revised: =

Original: γA + β (cid:62) M + (cid:15) Y and M =
Revised: γA + β (cid:62) M + (cid:15) Y and M =

Original: ( I p [Equation] B 
Revised: ( I p − B (cid:62) M ) − 1 α A

Original: [Equation]
Revised: + ( I p − B (cid:62) M ) − 1 (cid:15) M .

Original: Following the same calculation procedure of E { Y | do ( A = a ) } , we next give the natural direct effect under assumptions (A1-A3) and Model 1 as
Revised: Following the same calculation procedure of E { Y | do ( A = a ) } , we next give the natural direct effect under assumptions (A1-A3) and Model 1 as

Original: [Equation]
Revised: [Equation]

Original: where the ﬁrst ‘=’ is given by the deﬁnition of the DE .
Revised: where the ﬁrst ‘=’ is given by the deﬁnition of the DE .

Original: Similarly, the natural indirect effect is
Revised: Similarly, the natural indirect effect is

Original: [Equation]

Original: Thus

Original: [Equation]
Revised: [Equation]

Original:  the total effect of A on Y is
Revised: Thus, the total effect of A on Y is

Original: [Equation]
Revised: [Equation]

Original: Finally, we give the expression for the natural direct effect of M i on Y under the LSEM.
Revised: Finally, we give the expression for the natural direct effect of M i on Y under the LSEM.

Original: Based on the assumption (A2) and Equation G.2, we have
Revised: Based on the assumption (A2) and Equation G.2, we have

Original: [Equation]
Revised: [Equation]

Original: where { ( I p − B (cid:62) M ) − 1 α } i is the i -th element of the vector ( I p − B (cid:62) M ) − 1 α .
Revised: where { ( I p − B (cid:62) M ) − 1 α } i is the i -th element of the vector ( I p − B (cid:62) M ) − 1 α .

Original: Then, based on Y = γA + β (cid:62) M + (cid:15) Y and the assumption (A3) [Equation] we have,[Equation]
Revised: Then, based on Y = γA + β (cid:62) M + (cid:15) Y and the assumption (A3), we have,

Original: | A

Original: [Equation]

Original: =

Original: [Equation]

Original:    m

Original: [Equation]
Revised: [Equation]

Original: m

Original: [Equation]

Original: m

Original: [Equation]

Original:    −

Original: [Equation]

Original:    m

Original: [Equation]

Original: m (

Original: [Equation]

Original: m ( a

Original: [Equation]

Original:    =

Original: [Equation]

Original: , where 1 i is a p × 1 vector with the i -th element as 1 while others qual to 0, and β i is the i -th element of the vector β .
Revised: where 1 i is a p × 1 vector with the i -th element as 1 while others qual to 0, and β i is the i -th element of the vector β .

Original: Thus, we have
Revised: Thus, we have

Original: [Equation]
Revised: [Equation]

Original: G. P ROOF OF T HEOREM F. 
Revised: G. P ROOF OF T HEOREM F.

Original: Proof G.2 1.
Revised: Proof G.2 1.

Original: If there is no directed path π ∗ ∈ { π AY ( G ) } such that the length of π ∗ is larger than 2, i.e. the length of π ∗ ∈ { π AY ( G ) } is either 1 or 2.
Revised: If there is no directed path π ∗ ∈ { π AY ( G ) } such that the length of π ∗ is larger than 2, i.e. the length of π ∗ ∈ { π AY ( G ) } is either 1 or 2.

Original: Here, the path with length 1 corresponds to
Revised: Here, the path with length 1 corresponds to

Original: A → Y , and paths with length 2 are A → M i → Y with possibly i
Revised: A → Y , and paths with length 2 are A → M i → Y with possibly i

Original: = 1 , · · · , p .
Revised: = 1 , · · · , p .

Original: Thus, there is no interaction among mediators.
Revised: Thus, there is no interaction among mediators.

Original: By the deﬁnition of the LSEM, we have B M = 

Revised: By the deﬁnition of the LSEM, we have B M =

Original: p × p , where 0 p × p is a p × p zero matrix.
Revised: p × p , where 0 p × p is a p × p zero matrix.

Original: Following the path method (the causal effect of X i on X j along a directed path from X i → X j in G can be calculated by multiplying all edge weights along the path) illustrated in Wright (1921) and Nandy et al.
Revised: Following the path method (the causal effect of X i on X j along a directed path from X i → X j in G can be calculated by multiplying all edge weights along the path) illustrated in Wright (1921) and Nandy et al.

Original: (2017), we could obtain (cid:80) η i = (cid:80) i β i
Revised: (2017), we could obtain (cid:80) η i = (cid:80) i β i

Original: α i = IE .
Revised: α i = IE .

Original: (See a toy example provided in section E to illustrate how to use the path method to manually compute the causal effects.)
Revised: (See a toy example provided in section E to illustrate how to use the path method to manually compute the causal effects.)

Original: If there exists at least one directed path π ∗ ∈ { π AY ( G ) } such that the length of π ∗ is larger than 2, and the element in B is nonnegative, we have B M (cid:54) = 

Revised: If there exists at least one directed path π ∗ ∈ { π AY ( G ) } such that the length of π ∗ is larger than 2, and the element in B is nonnegative, we have B M (cid:54) =

Original: p × p .
Revised: p × p .

Original: Without loss of generality, suppose there exists M i ∈ M with a set of directed path that starts with A , contains M i , then goes through other mediators, and ends with Y , denoted each path in such set as π i,
Revised: Without loss of generality, suppose there exists M i ∈ M with a set of directed path that starts with A , contains M i , then goes through other mediators, and ends with Y , denoted each path in such set as π i,

Original: j = { A → · · · → M i · · · → · · · →
Revised: j = { A → · · · → M i · · · → · · · →

Original: Y } for j = 1 , · · · , n i , where n i is the size of such path set for M i , and the weights of edges in π i,j is positive.
Revised: Y } for j = 1 , · · · , n i , where n i is the size of such path set for M i , and the weights of edges in π i,j is positive.

Original: Note the set { π i,j } excludes the paths end with M i → Y .
Revised: Note the set { π i,j } excludes the paths end with M i → Y .

Original: Let e π i,j denote the causal effect of A on Y through directed path π i,j .
Revised: Let e π i,j denote the causal effect of A on Y through directed path π i,j .

Original: Based on the path method in
Revised: Based on the path method in

Original: Wright (1921) and Nandy et al.
Revised: Wright (1921) and Nandy et al.

Original: (2017) with the deﬁnition of IM i , we have its theoretical form as 
Revised: (2017) with the deﬁnition of IM i , we have its theoretical form as [Equation]

Original: [Equation]
Revised: [Equation]

Original: By Equation G.5 and the deﬁnition of η i , we have its ﬁrst multiplier as
Revised: By Equation G.5 and the deﬁnition of η i , we have its ﬁrst multiplier as

Original: [Equation]
Revised: [Equation]

Original: which is also the ﬁrst multiplier in both DM i and IM i .
Revised: which is also the ﬁrst multiplier in both DM i and IM i .

Original: And the second multiplier of η i can be expressed as
Revised: And the second multiplier of η i can be expressed as

Original: [Equation]
Revised: [Equation]

Original: where m ( a ) i is the value of M i when setting do ( A = a ) .
Revised: where m ( a ) i is the value of M i when setting do ( A = a ) .

Original: Here, the ﬁrst ‘=’ is valid since m i can be arbitrary number, and the second and third ‘=’ are based on the equivalent interventions.
Revised: Here, the ﬁrst ‘=’ is valid since m i can be arbitrary number, and the second and third ‘=’ are based on the equivalent interventions.

Original: Based on the technique of plus and minus the same term
Revised: Based on the technique of plus and minus the same term, we decompose the second multiplier of η i into two parts as follows

Original: [Equation]
Revised: [Equation]

Original: we decompose the second multiplier of η i into two parts as follows
Revised: the second multiplier of DM i [Equation] the second multiplier of

Original: [Equation]
Revised: [Equation]

Original: where Ω i =
Revised: where Ω i =

Original: M \ M i is the sets of mediators except M i , and o ( a ) i is the value of Ω i when setting do ( A = a ) .
Revised: M \ M i is the sets of mediators except M i , and o ( a ) i is the value of Ω i when setting do ( A = a ) .

Original: Here, the ﬁrst term in the above equation corresponds to the second multiplier of DM i , while the second term is the second multiplier of IM i .
Revised: Here, the ﬁrst term in the above equation corresponds to the second multiplier of DM i , while the second term is the second multiplier of IM i .

Original: Thus, the summation of η i is
Revised: Thus, the summation of η i is

Original: [Equation]
Revised: [Equation]

Original: where the ﬁrst ‘=’ is from Deﬁnition F.1, the second ‘=’ is given by Equation G.7 and Deﬁnition 3.
Revised: where the ﬁrst ‘=’ is from Deﬁnition F.1, the second ‘=’ is given by Equation G.7 and Deﬁnition 3.

Original: and 3.3, and the last ‘=’ comes from Theorem 3.2 and the theoretical form of IM in Equation G.6.
Revised: and 3.3, and the last ‘=’ comes from Theorem 3.2 and the theoretical form of IM in Equation G.6.

Original: Here, we have e π i,j > 0 since the weights of edges in π i,j is positive based on the path method in
Revised: Here, we have e π i,j > 0 since the weights of edges in π i,j is positive based on the path method in

Original: Wright (1921) and Nandy et al.
Revised: Wright (1921) and Nandy et al.

Original: Then, (cid:80) i (cid:80) n i j =1 e π i,j is also strictly larger than 0.
Revised: Then, (cid:80) i (cid:80) n i j =1 e π i,j is also strictly larger than 0.

Original: Therefore, we have
Revised: Therefore, we have

Original: [Equation]
Revised: [Equation]

Original:  P ROOF OF T HEOREM F. 
Revised: G. P ROOF OF T HEOREM F.

Original: Proof G.3 1.
Revised: Proof G.3 1.

Original: If there doesn’t exist edge M i
Revised: If there doesn’t exist edge M i

Original: → Y in G , then by deﬁnition we have β i = 0 .
Revised: → Y in G , then by deﬁnition we have β i = 0 .

Original: Thus,
Revised: Thus,

Original: LE
Revised: LE

Original: [Equation]
Revised: [Equation]

Original: If there exists edge M i → Y in G .
Revised: If there exists edge M i → Y in G .

Original: Suppose there is a directed path set with size m i associated to the edge M i
Revised: Suppose there is a directed path set with size m i associated to the edge M i

Original: → Y , where each directed path ˜
Revised: → Y , where each directed path ˜

Original: π i,j starts with node A and ends with M i
Revised: π i,j starts with node A and ends with M i

Original: → Y , denoted as ˜ π i,j = { A → · · · → · · · → M i → Y } for j = 1 , · · · , m i .
Revised: → Y , denoted as ˜ π i,j = { A → · · · → · · · → M i → Y } for j = 1 , · · · , m i .

Original: Let e ˜
Revised: Let e ˜

Original: π
Revised: π

Original: i,j denote the causal effect of A on Y through directed path ˜
Revised: i,j denote the causal effect of A on Y through directed path ˜

Original: π i,j , e ( A,M i ) ˜
Revised: π i,j , e ( A,M i ) ˜

Original: π i,j be the causal effect of
Revised: π i,j be the causal effect of

Original: A on M i through directed path ˜
Revised: A on M i through directed path ˜

Original: π i,j , and e ( M i ,Y ) is the causal effect of M i on Y through directed edge M i → Y .
Revised: π i,j , and e ( M i ,Y ) is the causal effect of M i on Y through directed edge M i → Y .

Original: Following the path method in Wright (1921) and Nandy et al.
Revised: Following the path method in Wright (1921) and Nandy et al.

Original: (2017), we have e ˜ π
Revised: (2017), we have e ˜ π

Original: i,j = e ( A,M i ) ˜ π i,j e ( M i ,Y ) .
Revised: [Equation]

Original: Then the i -th last edge-speciﬁc effect is equal to the summation of the effect through each path ˜
Revised: Then the i -th last edge-speciﬁc effect is equal to the summation of the effect through each path ˜

Original: π i,j , i.e.,
Revised: π i,j , i.e.,

Original: LE

Original: [Equation]
Revised: [Equation]

Original: Here, by the similar argument based on the path method, we have e ( M i ,Y )
Revised: Here, by the similar argument based on the path method, we have e ( M i ,Y )

Original: =
Revised: =

Original: β i and (cid:80) n i j =1 e ( A,M i ) ˜
Revised: β i and (cid:80) n i j =1 e ( A,M i ) ˜

Original: π i,j as the total causal effect of A on M i .
Revised: π i,j as the total causal effect of A on M i .

Original: Recall that ζ i ≡ { ( I p − B (cid:62) M ) − 1 α } i is the causal effect of A on M i .
Revised: Recall that ζ i ≡ { ( I p − B (cid:62) M ) − 1 α } i is the causal effect of A on M i .

Original: Therefore, the i -th LE is the product of the causal effect of A on M i and the causal effect of M i on Y , i.e., 
Revised: Therefore, the i -th LE is the product of the causal effect of A on M i and the causal effect of M i on Y , i.e., [Equation] i [Equation]

Revised: i

Original: [Equation]
Revised: [Equation]

Original: G. P ROOF OF T HEOREM F. 
Revised: G. P ROOF OF T HEOREM F.

Original: Proof G.4 Given a general DAG G with nodes { A, M, Y } , let the union of all directed paths that contain the i -th last edge as τ i =
Revised: Proof G.4 Given a general DAG G with nodes { A, M, Y } , let the union of all directed paths that contain the i -th last edge as τ i =

Original: π : A → · · · → M i → Y } , i = 1 , · · · p .
Revised: π : A → · · · → M i → Y } , i = 1 , · · · p .

Original: Here, we have τ i =
Revised: Here, we have τ i =

Original: ˜ π
Revised: ˜ π

Original: i,j } 1 ≤ j ≤ m j established in Section G.3.
Revised: i,j } 1 ≤ j ≤ m j established in Section G.3.

Original: It is clear that the union set of τ i in G is equal to the set of all directed paths start with A and end with node Y (except A → Y ) in G as (cid:91)i τ i = [Equation]
Revised: It is clear that the union set of τ i in G is equal to the set of all directed paths start with A and end with node Y (except A → Y ) in G as (cid:91)

Original: AY
Revised: i

Original: [Equation]
Revised: [Equation]

Original: G ) } \ { A → Y } .

Original: Also, based on the uniqueness of each last edge, τ i is pairwise disjoint, i.e.
Revised: Also, based on the uniqueness of each last edge, τ i is pairwise disjoint, i.e.

Original: [Equation]
Revised: [Equation]

Original: Since the IE is deﬁned as the total causal effect of A on Y that goes through mediators, we have the
Revised: Since the IE is deﬁned as the total causal effect of A on Y that goes through mediators, we have the

Original: IE equal to the causal effect that goes through the set { π AY ( G ) } \ { A → Y } , i.e. the IE equal to the causal effect that goes through set (cid:83) i τ i .
Revised: IE equal to the causal effect that goes through the set { π AY ( G ) } \ { A → Y } , i.e. the IE equal to the causal effect that goes through set (cid:83) i τ i .

Original: Based on the mutual disjoint property of τ i , we have the
Revised: Based on the mutual disjoint property of τ i , we have the

Original: IE is exactly the summation of the causal effect through τ i .
Revised: IE is exactly the summation of the causal effect through τ i .

Original: Lastly, from the deﬁnition of LE i , we have
Revised: Lastly, from the deﬁnition of LE i , we have

Original: [Equation]
Revised: [Equation]

Original: G. P ROOF OF T HEOREM F. 
Revised: G. P ROOF OF T HEOREM F.

Original: Proof G.5 The proof of the consistency of our deﬁned DM to the interventional effect ξ i can be completed based on Equation 3 under assumptions (A1-A3) and Model 1.
Revised: Proof G.5 The proof of the consistency of our deﬁned DM to the interventional effect ξ i can be completed based on Equation 3 under assumptions (A1-A3) and Model 1.

Original: Recall the deﬁnition in Equation F.10, we have
Revised: Recall the deﬁnition in Equation F.10, we have

Original: [Equation]
Revised: [Equation]

Original: = a ) P ( M i = m i | A = a ) (cid:111) .
Revised: [Equation]

Original: Given A = a , the value of M i is m ( a ) i and Ω i takes o ( a ) i ; while when setting A = a + 1 , the value of M i is m ( i ) a +1 .
Revised: Given A = a , the value of M i is m ( a ) i and Ω i takes o ( a ) i ; while when setting A = a + 1 , the value of M i is m ( i ) a +1 .

Original: Therefore, we have P ( M i
Revised: Therefore, we have P ( M i

Original: = m i
Revised: = m i

Original: | A = a ) = 1 if m i
Revised: | A = a ) = 1 if m i

Original: = m ( a ) i otherwise is 0, and
Revised: = m ( a ) i otherwise is 0, and

Original: P (Ω i
Revised: P (Ω i

Original: =
Revised: =

Original: o i | A = a ) = 

Revised: o i | A = a ) =

Original: if o i
Revised: if o i

Original: = o ( a ) i otherwise is 0.
Revised: = o ( a ) i otherwise is 0.

Original: Under assumptions (A1-A3), we haveξ
Revised: Under assumptions (A1-A3), we have

Original: [Equation]
Revised: [Equation]

Original: Then, based on the LSEM that Y =
Revised: Then, based on the LSEM that Y =

Original: γA + β (cid:62) M + (cid:15) Y , we can further obtain that[Equation]  [Equation]
Revised: γA + β (cid:62) M + (cid:15) Y , we can further obtain that

Original:  

Original: [Equation]
Revised: [Equation]

Original: 

Original: [Equation]

Original: From Equation G.2, we haveξ
Revised: From Equation G.2, we have

Original: [Equation]
Revised: [Equation]

Original: Thus, under assumptions (A1-A3) and Model 1, we have
Revised: Thus, under assumptions (A1-A3) and Model 1, we have

Original: [Equation]
Revised: [Equation]