Original: Causal Identiﬁcation under Markov equivalence: Calculus, Algorithm, and Completeness
Revised: Causal Identification under Markov equivalence: Calculus, Algorithm, and Completeness

Original: Anonymous Author(s)

Original: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: One common task in many data sciences applications is to answer questions aboutthe effect of new interventions, like: ‘what would happen to Y if we make X equalto x while observing covariates Z
Revised: One common task in many data sciences applications is to answer questions about the effect of new interventions, like: ‘what would happen to Y if we make X equal to x while observing covariates Z

Original: = z ?’.
Revised: = z ?’.

Original: Formally, this is known as conditionaleffect identiﬁcation , where the goal is to determine whether a post-interventionaldistribution is computable from the combination of an observational distributionand assumptions about the underlying domain represented by a causal diagram.
Revised: Formally, this is known as conditional effect identification , where the goal is to determine whether a post-interventional distribution is computable from the combination of an observational distribution and assumptions about the underlying domain represented by a causal diagram.

Original: A plethora of methods was developed for solving this problem, including thecelebrated do-calculus [Pearl, 1995].
Revised: A plethora of methods was developed for solving this problem, including the celebrated do-calculus [Pearl, 1995].

Original: In practice, these results are not alwaysapplicable since they require a fully speciﬁed causal diagram as input, which isusually not available.
Revised: In practice, these results are not always applicable since they require a fully specified causal diagram as input, which is usually not available.

Original: In this paper, we assume as the input of the task a lessinformative structure known as a partial ancestral graph (PAG), which represents a
Revised: In this paper, we assume as the input of the task a less informative structure known as a partial ancestral graph (PAG), which represents a

Original: Markov equivalence class of causal diagrams, learnable from observational data.
Revised: Markov equivalence class of causal diagrams, learnable from observational data.

Original: We make the following contributions under this relaxed setting.
Revised: We make the following contributions under this relaxed setting.

Original: First, we introducea new causal calculus, which subsumes the current state-of-the-art, PAG-calculus.
Revised: First, we introduce a new causal calculus, which subsumes the current state-of-the-art, PAG-calculus.

Original: Second, we develop an algorithm for conditional effect identiﬁcation given a PAGand prove it to be both sound and complete.
Revised: Second, we develop an algorithm for conditional effect identification given a PAG and prove it to be both sound and complete.

Original: In words, failure of the algorithm toidentify a certain effect implies that this effect is not identiﬁable by any method.
Revised: In words, failure of the algorithm to identify a certain effect implies that this effect is not identifiable by any method.

Original: Third, we prove the proposed calculus to be complete for the same task.
Revised: Third, we prove the proposed calculus to be complete for the same task.

Original:  Introduction
Revised:  Introduction

Original: Despite the recent advances in AI and machine learning, the current generation of intelligent systemsstill lacks the pivotal ability to represent, learn, and reason with cause and effect relationships.
Revised: Despite the recent advances in AI and machine learning, the current generation of intelligent systems still lacks the pivotal ability to represent, learn and reason with cause and effect relationships.

Original: Thediscipline of causal inference aims to ‘algorithmitize’ causal reasoning capabilities towards producinghuman-like machine intelligence and rational decision-making [Pearl and Mackenzie, 2018, Pearl,2019, Bareinboim and Pearl, 2016].
Revised: The discipline of causal inference aims to ‘algorithmitize’ causal reasoning capabilities towards producing human-like machine intelligence and rational decision-making [Pearl and Mackenzie, 2018, Pearl, 2019, Bareinboim and Pearl, 2016].

Original: One fundamental type of inference in this setting is concernedwith the effect of new interventions, e.g., ‘what would happen to outcome Y if X were set to 
Revised: One fundamental type of inference in this setting is concerned with the effect of new interventions, e.g., ‘what would happen to outcome Y if X were set to x ?’

Original: More generally, we may be interested in Y ’s distribution in a sub-population picked out by the valueof some covariates Z
Revised: More generally, we may be interested in Y ’s distribution in a sub-population picked out by the value of some covariates Z

Original: = z ’.
Revised: = z ’.

Original: For example, a legislator might be interested in the impact that increasingthe minimum wage ( X = x ) has on proﬁts ( Y ) in small businesses ( Z = z ), which is written incausal language as the interventional distribution P ( y | do ( x ) , z ) , or P x ( y | z ) .
Revised: For example, a legislator might be interested in the impact that increasing the minimum wage ( X = x ) has on profits ( Y ) in small businesses ( Z = z ), which is written in causal language as the interventional distribution P ( y | do ( x ) , z ) , or P x ( y | z ) .

Original: One method capable ofanswering such questions is through controlled experimentation [Fisher, 1951].
Revised: One method capable of answering such questions is through controlled experimentation [Fisher, 1951].

Original: In many practical settings found throughout the empirical sciences, AI, and machine learning, itis not always possible to perform a controlled experiment due to ethical, ﬁnancial, and technicalconsiderations.
Revised: In many practical settings found throughout the empirical sciences, AI, and machine learning, it is not always possible to perform a controlled experiment due to ethical, financial, and technical considerations.

Original: This motivates the study of a problem known as causal effect identiﬁcation [Pearl,[Figure]
Revised: This motivates the study of a problem known as causal effect identification [Pearl,

Original: Z
Revised: W

Original: [Figure]
Revised: X Y Z (a)

Original: W
Revised: W

Original: X
Revised: X

Original: Y
Revised: Y

Revised: Z (b)

Original:  Z  v  (c)
Revised: W X Y Z ◦ v ◦ (c)

Original: 2000, Ch. 3].
Revised: 2000, Ch. 3].

Original: The idea is to use the observational distribution P ( V ) along with assumptions aboutthe underlying domain, articulated in the form of a causal diagram D , to infer the interventionaldistribution P x ( y | z ) when possible.
Revised: The idea is to use the observational distribution P ( V ) along with assumptions about the underlying domain, articulated in the form of a causal diagram D , to infer the interventional distribution P x ( y | z ) when possible.

Original: For instance, Fig.
Revised: For instance, Fig.

Original: 1a represents a causal diagram in which nodescorrespond to measured variables, directed edges represent direct causal relations, and bidirecteddashed edges encode spurious associations due to unmeasured confounders.
Revised: 1a represents a causal diagram in which nodes correspond to measured variables, directed edges represent direct causal relations, and bidirecteddashed edges encode spurious associations due to unmeasured confounders.

Original: A plethora of methodshave been developed to address the identiﬁcation task including the celebrated causal calculusproposed by Pearl [1995] as well as complete algorithms [Tian, 2004, Shpitser and Pearl, 2006,
Revised: A plethora of methods have been developed to address the identification task including the celebrated causal calculus proposed by Pearl [1995] as well as complete algorithms [Tian, 2004, Shpitser and Pearl, 2006,

Original: Huang and Valtorta, 2006].
Revised: Huang and Valtorta, 2006].

Original: For instance, given the causal diagram in Fig.
Revised: For instance, given the causal diagram in Fig.

Original: 1a and the query P x
Revised: 1a and the query P x

Original: ( y | z ) ,the calculus sanctions the identity P x ( y | z )
Revised: ( y | z ) , the calculus sanctions the identity P x ( y | z )

Original: =
Revised: =

Original: P ( y | z, x ) .
Revised: P ( y | z, x ) .

Original: In words, the interventional distribution onthe l.h.s.
Revised: In words, the interventional distribution on the l.h.s.

Original: equates to the observational distribution on the right, which is available as input.
Revised: equates to the observational distribution on the right, which is available as input.

Original: Despitethe power of these results, requiring the diagram as the input of the task is an Achilles heel for thosemethods, since background knowledge is usually not sufﬁcient to pin down the single, true diagram.
Revised: Despite the power of these results, requiring the diagram as the input of the task is an Achilles heel for those methods, since background knowledge is usually not sufficient to pin down the single, true diagram.

Original: To circumvent these challenges, a growing literature develops data-driven methods that attempt tolearn the causal diagram from data ﬁrst, and then perform identiﬁcation from there.
Revised: To circumvent these challenges, a growing literature develops data-driven methods that attempt to learn the causal diagram from data first, and then perform identification from there.

Original: In practice,however, only an equivalence class (EC) of diagrams can be inferred from observational data withoutmaking substantial assumptions about the causal mechanisms [Verma, 1993, Spirtes et al., 2001,
Revised: In practice, however, only an equivalence class (EC) of diagrams can be inferred from observational data without making substantial assumptions about the causal mechanisms [Verma, 1993, Spirtes et al., 2001,

Original: Pearl, 2000].
Revised: Pearl, 2000].

Original: A prominent representation of this class is known as partial ancestral graphs (PAGs)[Zhang, 2008b].
Revised: A prominent representation of this class is known as partial ancestral graphs (PAGs) [Zhang, 2008b].

Original: Fig.
Revised: Fig.

Original: 1c illustrates the PAG learned from observational data consistent with bothcausal diagrams in Figs.
Revised: 1c illustrates the PAG learned from observational data consistent with both causal diagrams in Figs.

Original: 1a and 1b since they are in the same Markov equivalence class.
Revised: 1a and 1b since they are in the same Markov equivalence class.

Original: The directededges in a PAG encode ancestral relations, not necessarily direct, and the circle marks stand forstructural uncertainty.
Revised: The directed edges in a PAG encode ancestral relations, not necessarily direct, and the circle marks stand for structural uncertainty.

Original: Directed edges labeled with v signify the absence of unmeasured confounders.
Revised: Directed edges labeled with v signify the absence of unmeasured confounders.

Original: Causal effect identiﬁcation in a PAG is usually more challenging than from a single diagram due tothe structural uncertainties and the infeasibility of enumerating each member of the EC in most cases.
Revised: Causal effect identification in a PAG is usually more challenging than from a single diagram due to the structural uncertainties and the infeasibility of enumerating each member of the EC in most cases.

Original: The do-calculus was extended for PAGs to account for the inherent structure uncertainties withoutthe need for enumeration [Zhang, 2007].
Revised: The do-calculus was extended for PAGs to account for the inherent structure uncertainties without the need for enumeration [Zhang, 2007].

Original: Still, the calculus falls short of capturing all identiﬁableeffects as we will see in Sec. 3.
Revised: Still, the calculus falls short of capturing all identifiable effects as we will see in Sec. 3.

Original: On the other hand, it is computationally hard to decide whetherthere exists (and, if so, to ﬁnd) a sequence of derivations in the generalized calculus to identify aneffect of interest.
Revised: On the other hand, it is computationally hard to decide whether there exists (and, if so, to find) a sequence of derivations in the generalized calculus to identify an effect of interest.

Original: In a more systematic manner, a complete algorithm has been developed to identifymarginal effects (i.e., P x ( y ) ) given a PAG [Jaber et al., 2019a].
Revised: In a more systematic manner, a complete algorithm has been developed to identify marginal effects (i.e., P x ( y ) ) given a PAG [Jaber et al., 2019a].

Original: This algorithm can be used to identifyconditional effects whenever the joint distribution P x
Revised: This algorithm can be used to identify conditional effects whenever the joint distribution P x

Original: ( y [ z ) is identiﬁable.
Revised: ( y ∪ z ) is identifiable.

Original: Still, many conditionaleffects are identiﬁable even if the corresponding joint effect is not (Sec. 4.2).
Revised: Still, many conditional effects are identifiable even if the corresponding joint effect is not (Sec. 4.2).

Original: Finally, an algorithm toidentify conditional effects has been proposed in [Jaber et al., 2019b], but not proven to be complete.
Revised: Finally, an algorithm to identify conditional effects has been proposed in [Jaber et al., 2019b], but not proven to be complete.

Original: In this paper, we pursue a data-driven formulation for the task of identiﬁcation of any conditionalcausal effect from a combination of an observational distribution and the corresponding PAG (insteadof a fully speciﬁed causal diagram).
Revised: In this paper, we pursue a data-driven formulation for the task of identification of any conditional causal effect from a combination of an observational distribution and the corresponding PAG (instead of a fully specified causal diagram).

Original: Accordingly, we makes the following contributions:
Revised: Accordingly, we makes the following contributions:

Original: We propose a causal calculus for PAGs that subsumes the stat-of-the-art calculus introducedin [Zhang, 2007].
Revised: We propose a causal calculus for PAGs that subsumes the stat-of-the-art calculus introduced in [Zhang, 2007].

Original: We prove the rules are atomic complete , i.e., a rule is not applicable insome causal diagram in the underlying EC whenever it is not applicable given the PAG.2.
Revised: We prove the rules are atomic complete , i e., a rule is not applicable in some causal diagram in the underlying EC whenever it is not applicable given the PAG. 2.

Original: Building on these results, we develop an algorithm for the identiﬁcation of conditional causaleffects given a PAG.
Revised: Building on these results, we develop an algorithm for the identification of conditional causal effects given a PAG.

Original: We prove the algorithm is complete , i.e., the effect is not identiﬁable insome causal diagram in the equivalence class whenever the algorithm fails.3.
Revised: We prove the algorithm is complete , i.e., the effect is not identifiable in some causal diagram in the equivalence class whenever the algorithm fails.

Original: Finally, we prove the calculus is complete for the task of identifying conditional effects
Revised: Finally, we prove the calculus is complete for the task of identifying conditional effects.

Original:  Preliminaries
Revised:  Preliminaries

Original: In this section, we introduce the basic setup and notations.
Revised: In this section, we introduce the basic setup and notations.

Original: Boldface capital letters denote sets ofvariables, while boldface lowercase letters stand for value assignments to those variables.
Revised: Boldface capital letters denote sets of variables, while boldface lowercase letters stand for value assignments to those variables.

Original: Structural Causal Models.
Revised: Structural Causal Models.

Original: We use Structural Causal Models (SCMs) as our basic semanticalframework [Pearl, 2000].
Revised: We use Structural Causal Models (SCMs) as our basic semantical framework [Pearl, 2000].

Original: Formally, an SCM M is a 4-tuple h U , V , F , P ( U ) i , where U is a set ofexogenous (unmeasured) variables and V is a set of endogenous (measured) variables.
Revised: Formally, an SCM M is a 4-tuple ⟨ U , V , F , P ( U ) ⟩ , where U is a set of exogenous (unmeasured) variables and V is a set of endogenous (measured) variables.

Original: F represents acollection of functions such that each endogenous variable V i 2 V is determined by a function f i 2 F .
Revised: F represents a collection of functions such that each endogenous variable V i ∈ V is determined by a function f i ∈ F .

Original: Finally, P ( U ) encodes the uncertainty over the exogenous variables.
Revised: Finally, P ( U ) encodes the uncertainty over the exogenous variables.

Original: Every SCM is associated withone causal diagram where every variable in V [ U is a node, and arrows are drawn between nodes inaccordance with the functions in F .
Revised: Every SCM is associated with one causal diagram where every variable in V ∪ U is a node, and arrows are drawn between nodes in accordance with the functions in F .

Original: Following standard practice, we omit the exogenous nodes andadd a bidirected dashed arc between two endogenous nodes if they share an exogenous parent.
Revised: Following standard practice, we omit the exogenous nodes and add a bidirected dashed arc between two endogenous nodes if they share an exogenous parent.

Original: Weonly consider recursive systems, thus the corresponding diagram is acyclic.
Revised: We only consider recursive systems, thus the corresponding diagram is acyclic.

Original: The marginal distributioninduced over the endogenous variables P ( V ) is called observational.
Revised: The marginal distribution induced over the endogenous variables P ( V ) is called observational.

Original: The d-separation criterioncaptures the conditional independence relations entailed by a causal diagram in P ( V ) .
Revised: The d-separation criterion captures the conditional independence relations entailed by a causal diagram in P ( V ) .

Original: For C ✓ V ,
Revised: For C ⊆ V ,

Original: Q [ C ] denotes the post-intervention distribution of C under an intervention on V \ C , i.e. P v \ c ( c ) .
Revised: Q [ C ] denotes the post-intervention distribution of C under an intervention on V \ C , i.e. P v \ c ( c ) .

Original: Ancestral Graphs.
Revised: Ancestral Graphs.

Original: We now introduce a graphical representation of equivalence classes of causal diagrams.
Revised: We now introduce a graphical representation of equivalence classes of causal diagrams.

Original: A MAG represents a set of causal diagrams with the same set of observed variables that entailthe same conditional independence and ancestral relations among the observed variables [Richardsonand Spirtes, 2002].
Revised: A MAG represents a set of causal diagrams with the same set of observed variables that entail the same conditional independence and ancestral relations among the observed variables [Richardson and Spirtes, 2002].

Original: M-separation extends d-separation to MAGs such that d-separation in a causaldiagram corresponds to m-separation in its unique MAG over the observed variables, and vice versa.
Revised: M-separation extends d-separation to MAGs such that d-separation in a causal diagram corresponds to m-separation in its unique MAG over the observed variables, and vice versa.

Original: Deﬁnition 1 (m-separation) .
Revised: Definition 1 (m-separation) .

Original: A path p between X and Y is active (or m-connecting) relative to Z( X, Y 62 Z ) if every non-collider on p is not in Z , and every collider on p is an ancestor of some
Revised: A path p between X and Y is active (or m-connecting) relative to Z ( X, Y ̸∈ Z ) if every non-collider on p is not in Z , and every collider on p is an ancestor of some

Original: Z 2 Z .
Revised: Z ∈ Z .

Original: X and Y are m-separated by Z if there is no active path between X and Y relative to Z .
Revised: X and Y are m-separated by Z if there is no active path between X and Y relative to Z .

Original: Different MAGs entail the same independence model and hence are Markov equivalent.
Revised: Different MAGs entail the same independence model and hence are Markov equivalent.

Original: A PAGrepresents an equivalence class of MAGs [ M ] , which shares the same adjacencies as every MAG in[ M ] and displays all and only the invariant edge marks.
Revised: A PAG represents an equivalence class of MAGs [ M ] , which shares the same adjacencies as every MAG in [ M ] and displays all and only the invariant edge marks.

Original: A circle indicates an edge mark that is notinvariant.
Revised: A circle indicates an edge mark that is not invariant.

Original: A PAG is learnable from the independence model over the observed variables, and the FCIalgorithm is a standard method to learn such an object [Zhang, 2008b].
Revised: A PAG is learnable from the independence model over the observed variables, and the FCI algorithm is a standard method to learn such an object [Zhang, 2008b].

Original: In this work, an oracle forconditional independences is assumed to be available, which leads to the true PAG.
Revised: In this work, an oracle for conditional independences is assumed to be available, which leads to the true PAG.

Original: Graphical Notions.
Revised: Graphical Notions.

Original: Given a PAG, a path between X and Y is potentially directed (causal) from Xto Y if there is no arrowhead on the path pointing towards X .
Revised: Given a PAG, a path between X and Y is potentially directed (causal) from X to Y if there is no arrowhead on the path pointing towards X .

Original: Y is called a possible descendant of
Revised: Y is called a possible descendant of

Original: X and X a possible ancestor of Y if there is a potentially directed path from X to Y .
Revised: X and X a possible ancestor of Y if there is a potentially directed path from X to Y .

Original: For a set ofnodes X , let An ( X ) ( De ( X ) ) denote the union of X and the set of possible ancestors (descendants)of X .
Revised: For a set of nodes X , let An ( X ) ( De ( X ) ) denote the union of X and the set of possible ancestors (descendants) of X .

Original: Given two sets of nodes X and Y , a path between them is called proper if one of the endpointsis in X and the other is in Y , and no other node on the path is in X or Y .
Revised: Given two sets of nodes X and Y , a path between them is called proper if one of the endpoints is in X and the other is in Y , and no other node on the path is in X or Y .

Original: Let h A, B, C i be anyconsecutive triple along a path p .
Revised: Let ⟨ A, B, C ⟩ be any consecutive triple along a path p .

Original: B is a collider on p if both edges are into B .
Revised: B is a collider on p if both edges are into B .

Original: B is a (deﬁnite)non-collider on p if one of the edges is out of B , or both edges have circle marks at B and there isno edge between A and C .
Revised: B is a (definite) non-collider on p if one of the edges is out of B , or both edges have circle marks at B and there is no edge between A and C .

Original: A path is deﬁnite status if every non-endpoint node along it is either acollider or a non-collider.
Revised: A path is definite status if every non-endpoint node along it is either a collider or a non-collider.

Original: If the edge marks on a path between X and Y are all circles, we call thepath a circle path .
Revised: If the edge marks on a path between X and Y are all circles, we call the path a circle path .

Original: We refer to the closure of nodes connected with circle paths as a bucket .
Revised: We refer to the closure of nodes connected with circle paths as a bucket .

Original: A directed edge X ! Y in a PAG is visible if there exists no causal diagram in the correspondingequivalence class where the relation between X and Y is confounded.
Revised: A directed edge X → Y in a PAG is visible if there exists no causal diagram in the corresponding equivalence class where the relation between X and Y is confounded.

Original: Which directed edges arevisible is easily decidable by a graphical condition [Zhang, 2008a], so we mark visible edges by v .
Revised: Which directed edges are visible is easily decidable by a graphical condition [Zhang, 2008a], so we mark visible edges by v .

Original: Manipulations in PAGs.
Revised: Manipulations in PAGs.

Original: Let P denote a PAG over V and X ✓ V .
Revised: Let P denote a PAG over V and X ⊆ V .

Original: P X denotes the inducedsubgraph of P over X .
Revised: P X denotes the induced subgraph of P over X .

Original: The X -lower-manipulation of P deletes all those edges that are visible in Pand are out of variables in X , replaces all those edges that are out of variables in X but are invisible in
Revised: The X -lower-manipulation of P deletes all those edges that are visible in P and are out of variables in X , replaces all those edges that are out of variables in X but are invisible in

Original: P with bi-directed edges, and otherwise keeps P as it is.
Revised: P with bi-directed edges, and otherwise keeps P as it is.

Original: The resulting graph is denoted as P X .
Revised: The resulting graph is denoted as P X .

Original: The
Revised: The

Original: X -upper-manipulation of P deletes all those edges in P that are into variables in X , and otherwisekeeps P as it is.
Revised: X -upper-manipulation of P deletes all those edges in P that are into variables in X , and otherwise keeps P as it is.

Original: The resulting graph is denoted as P X . 
Revised: The resulting graph is denoted as P X . X A X

Revised: W

Original: [Figure]
Revised: Z Y ◦ ◦ ◦ ◦ ◦ ◦ v v (a) PAG P . X A

Revised: X

Revised: W

Revised: Z

Revised: Y ◦ ◦ ◦ ◦ v v (b) P W .

Revised: X

Revised: A

Revised: X

Original:  The causal calculus introduced in [Pearl, 1995] is a seminal work that has been instrumental forunderstanding and eventually solving the task of effect identiﬁcation from causal diagrams.
Revised: Z Y ◦ ◦ v (c) P V \{ W } .  Causal Calculus for PAGs The causal calculus introduced in [Pearl, 1995] is a seminal work that has been instrumental for understanding and eventually solving the task of effect identification from causal diagrams.

Original: Zhang[2007] generalized this result to the context of ancestral graphs, where a PAG is taken as the inputof the task, instead of the speciﬁc causal diagram.
Revised: Zhang [2007] generalized this result to the context of ancestral graphs, where a PAG is taken as the input of the task, instead of the specific causal diagram.

Original: In Sec. 3.1, we discuss Zhang’s rules and tryto understand the reasons they are insufﬁcient to solve the identiﬁcation problem in full generality.
Revised: In Sec. 3.1, we discuss Zhang’s rules and try to understand the reasons they are insufficient to solve the identification problem in full generality.

Original: Further, in Sec.3.2, we introduce another generalization of the original calculus and prove that it iscomplete for atomic identiﬁcation.
Revised: Further, in Sec.3.2, we introduce another generalization of the original calculus and prove that it is complete for atomic identification.

Original: This result will be further strengthened in subsequent sections.
Revised: This result will be further strengthened in subsequent sections.

Original: 3.1 Zhang’s Calculus
Revised: 3.1 Zhang’s Calculus

Original: An obvious extension of the m-separation criterion shown in Def. 1 to PAGs blocks all possiblym-connecting paths , as deﬁned next.
Revised: An obvious extension of the m-separation criterion shown in Def. 1 to PAGs blocks all possibly m-connecting paths , as defined next.

Original: Deﬁnition 2 (Possibly m-connecting path) .
Revised: Definition 2 (Possibly m-connecting path) .

Original: In a PAG, a path p between X and Y is a possibly mconnecting path relative to a (possibly empty) set of nodes
Revised: In a PAG, a path p between X and Y is a possibly mconnecting path relative to a (possibly empty) set of nodes

Original: Z
Revised: Z

Original: ( X, Y / 2 Z ) if every deﬁnite non-collideron p is not a member of Z , and every collider on p is a possible ancestor of some member of Z .
Revised: ( X, Y / ∈ Z ) if every definite non-collider on p is not a member of Z , and every collider on p is a possible ancestor of some member of Z .

Original: Xand Y are ˆ m -separated by Z if there is no possibly m-connecting path between them relative to Z .
Revised: X and Y are ˆ m -separated by Z if there is no possibly m-connecting path between them relative to Z .

Original: Using this notion of separation, Zhang [2007] proposed a calculus given a PAG as shown next.
Revised: Using this notion of separation, Zhang [2007] proposed a calculus given a PAG as shown next.

Original: Proposition 1 (Zhang’s Calculus) .
Revised: Proposition 1 (Zhang’s Calculus) .

Original: Let P be the PAG over V , and X , Y , W , Z be disjoint subsetsof V .
Revised: Let P be the PAG over V , and X , Y , W , Z be disjoint subsets of V .

Original: The following rules are valid, in the sense that if the antecedent of the rule holds, then theconsequent holds in every MAG and consequently every causal diagram represented by P.
Revised: The following rules are valid, in the sense that if the antecedent of the rule holds, then the consequent holds in every MAG and consequently every causal diagram represented by P

Original: P ( y | do ( w ) , x , z ) =
Revised: P ( y | do ( w ) , x , z ) =

Original: P ( y | do ( w ) , z ) , if X and Y are ˆ m -separated by W [ Z in P W . 
Revised: P ( y | do ( w ) , z ) , if X and Y are ˆ m -separated by W ∪ Z in P W . 2.

Original: P ( y | do ( w ) , do ( x ) , z ) =
Revised: P ( y | do ( w ) , do ( x ) , z ) =

Original: P ( y | do ( w ) , x , z ) , if X and Y are ˆ m -separated by W [ Z in P W , X .3.
Revised: P ( y | do ( w ) , x , z ) , if X and Y are ˆ m -separated by W ∪ Z in P W , X . 3.

Original: P ( y | do ( w ) , do ( x ) , z ) =
Revised: P ( y | do ( w ) , do ( x ) , z ) =

Original: P ( y | do ( w ) , z ) , if X and Y are ˆ m -separated by W [ Z in P W , X
Revised: P ( y | do ( w ) , z ) , if X and Y are ˆ m -separated by W ∪ Z in P W , X

Original: ( Z ) .where
Revised: ( Z ) .

Original:  X ( Z ) : =
Revised: where X ( Z ) :=

Original: X \ PossAn ( Z ) P W .
Revised: X \ PossAn ( Z ) P W .

Original: In words, rule 1 generalizes m-separation to interventional settings.
Revised: In words, rule 1 generalizes m-separation to interventional settings.

Original: Further, rule 2 licenses alternatinga subset X between intervention and conditioning.
Revised: Further, rule 2 licenses alternating a subset X between intervention and conditioning.

Original: Finally, rule 3 allows the adding/removal of anintervention do ( X = x ) .
Revised: Finally, rule 3 allows the adding/removal of an intervention do ( X = x ) .

Original: The next two examples illustrate the shortcomings of this result, where theﬁrst reveals the drawback of using Def. 2 to establish graphical separation and the second inspectsevaluating X ( Z ) in rule 3 (where the notion of possible ancestors are evoked).
Revised: The next two examples illustrate the shortcomings of this result, where the first reveals the drawback of using Def. 2 to establish graphical separation and the second inspects evaluating X ( Z ) in rule 3 (where the notion of possible ancestors are evoked).

Original: Example 1.
Revised: Example 1.

Original: Consider the PAG P shown in Fig.
Revised: Consider the PAG P shown in Fig.

Original: Since X and Y are not adjacent in P , it is easy to show that X and Y are separable given { Z 1 , Z 2 } in every causal diagram in the equivalence class.
Revised: Since X and Y are not adjacent in P , it is easy to show that X and Y are separable given { Z 1 , Z 2 } in every causal diagram in the equivalence class.

Original: If rule 3 of Pearl’s calculus is used in each diagram, then P x ( y | z 1 , z 2 ) =
Revised: If rule 3 of Pearl’s calculus is used in each diagram, then P x ( y | z 1 , z 2 ) =

Original: P ( y | z 1 , z 2 ) .
Revised: P ( y | z 1 , z 2 ) .

Original: Further, applying rule 2 of do-calculus in each diagram, it’s also the case that P x ( y | z 1 , z 2 ) =
Revised: Further, applying rule 2 of do-calculus in each diagram, it’s also the case that P x ( y | z 1 , z 2 ) =

Original: P ( y | z 1 , z 2 , x ) .
Revised: P ( y | z 1 , z 2 , x ) .

Original: However, due to the possibly m-connecting path h X, Z 1 , Z 2 , Y i , rules 3 and 2 in Prop.are not applicable to P .
Revised: However, due to the possibly m-connecting path ⟨ X, Z 1 , Z 2 , Y ⟩ , rules 3 and 2 in Prop.are not applicable to P .

Original: In other words, even though Pearl’s calculus rules 2 and 3 are applicable to each diagram in the equivalence class, the same results cannot be established by Zhang’s calculus.
Revised: In other words, even though Pearl’s calculus rules 2 and 3 are applicable to each diagram in the equivalence class, the same results cannot be established by Zhang’s calculus.

Original: [Equation]
Revised: [Equation]

Original:  
Revised: ◦ ◦

Original: Example 2.
Revised: Example 2.

Original: Consider the PAG in Fig. 3a, and the evaluation on whether the equality P w,x 1 ,x 2 ( y | z ) =
Revised: Consider the PAG in Fig. 3a, and the evaluation on whether the equality P w,x 1 ,x 2 ( y | z ) =

Original: P w ( y | z ) holds.
Revised: P w ( y | z ) holds.

Original: In order to apply rule 3 of Prop. 1, we need to evaluate whether { X 1 , X 2 } is separatedfrom { Y } given { W, Z } in the manipulated graph in Fig.
Revised: In order to apply rule 3 of Prop. 1, we need to evaluate whether { X 1 , X 2 } is separated from { Y } given { W, Z } in the manipulated graph in Fig.

Original: 3b, which is not true in this case.
Revised: 3b, which is not true in this case.

Original: However,the rule can be improved to be applicable in this case, as we will show later on (Sec. 3.2).
Revised: However, the rule can be improved to be applicable in this case, as we will show later on (Sec. 3.2).

Original: The criticalstep will be the evaluation of the set X ( Z ) from P W .
Revised: The critical step will be the evaluation of the set X ( Z ) from P W .

Original: 3.2 A New Calculus
Revised: 3.2 A New Calculus

Original: Building on the analysis of the calculus proposed in [Zhang, 2007], we introduce next a set of rulescentered around blocking deﬁnite m-connecting paths , as deﬁned next.[Figure]
Revised: Building on the analysis of the calculus proposed in [Zhang, 2007], we introduce next a set of rules centered around blocking definite m-connecting paths , as defined next.

Original: Deﬁnition 3 (Deﬁnite m-connecting path) .
Revised: Definition 3 (Definite m-connecting path) .

Original: In a PAG, a path p between X and Y is a deﬁnitem-connecting path relative to a (possibly empty) set Z ( X, Y 62 Z ) if p is deﬁnite status, every deﬁnitenon-collider on p is not a member of Z , and every collider on p is an ancestor of some member of Z .
Revised: In a PAG, a path p between X and Y is a definite m-connecting path relative to a (possibly empty) set Z ( X, Y ̸∈ Z ) if p is definite status, every definite non-collider on p is not a member of Z , and every collider on p is an ancestor of some member of Z .

Original: X and Y are m-separated by Z if there is no deﬁnite m-connecting path between them relative to Z .
Revised: X and Y are m-separated by Z if there is no definite m-connecting path between them relative to Z .

Original: It is easy to see that every deﬁnite m-connecting path is a possibly m-connecting path, accordingto Def. 2; however, the converse is not true.
Revised: It is easy to see that every definite m-connecting path is a possibly m-connecting path, according to Def. 2; however, the converse is not true.

Original: For example, given the PAG in Figure 2, we have twodeﬁnite status paths between X and Y .
Revised: For example, given the PAG in Figure 2, we have two definite status paths between X and Y .

Original: The ﬁrst is X  Z 1  Y and the second is X  Z 2  Ywhere Z 1 and Z 2 are deﬁnite non-colliders.
Revised: The first is X ◦−◦ Z 1 ◦−◦ Y and the second is X ◦−◦ Z 2 ◦−◦ Y where Z 1 and Z 2 are definite non-colliders.

Original: Given set Z = { Z 1 , Z 2 } , Z blocks all deﬁnite statuspaths between X and Y .
Revised: Given set Z = { Z 1 , Z 2 } , Z blocks all definite status paths between X and Y .

Original: Alternatively, the path X  Z 1  Z 2  Y is not deﬁnite status since
Revised: Alternatively, the path X ◦−◦ Z 1 ◦−◦ Z 2 ◦−◦ Y is not definite status since

Original: Z 1 , Z 2 are not colliders or non-colliders on this path.
Revised: Z 1 , Z 2 are not colliders or non-colliders on this path.

Original: Hence, the path is a possibly m-connectingpath relative to Z by Def. 

Revised: Hence, the path is a possibly m-connecting path relative to Z by Def.

Original: but not a deﬁnite m-connecting path by Def. 3.
Revised: but not a definite m-connecting path by Def. 3.

Original: We are now ready to use this new deﬁnition and formulate a more powerful calculus.
Revised: We are now ready to use this new definition and formulate a more powerful calculus.

Original: Theorem 1.
Revised: Theorem 1.

Original: Let P be the PAG over V , and X , Y , W , Z be disjoint subsets of V .
Revised: Let P be the PAG over V , and X , Y , W , Z be disjoint subsets of V .

Original: The followingrules are valid, in the sense that if the antecedent of the rule holds, then the consequent holds in every
Revised: The following rules are valid, in the sense that if the antecedent of the rule holds, then the consequent holds in every

Original: MAG and consequently every causal diagram represented by P .[-
-]
Revised: MAG and consequently every causal diagram represented by P .[-
-]

Original: P ( y | do ( w ) , x , z ) =
Revised: P ( y | do ( w ) , x , z ) =

Original: P ( y | do ( w ) , z ) , if X and Y are m -separated by W [ Z in P W .2.
Revised: P ( y | do ( w ) , z ) , if X and Y are m -separated by W ∪ Z in P W . 2.

Original: P ( y | do ( w ) , do ( x ) , z ) =
Revised: P ( y | do ( w ) , do ( x ) , z ) =

Original: P ( y | do ( w ) , x , z ) , if X and Y are m -separated by W [ Z in P W , X .3.
Revised: P ( y | do ( w ) , x , z ) , if X and Y are m -separated by W ∪ Z in P W , X . 3.

Original: P ( y | do ( w ) , do ( x ) , z ) =
Revised: P ( y | do ( w ) , do ( x ) , z ) =

Original: P ( y | do ( w ) , z ) , if X and Y are m -separated by W [ Z in P W , X ( Z ) .
Revised: P ( y | do ( w ) , z ) , if X and Y are m -separated by W ∪ Z in P W , X ( Z ) .

Original: where X ( Z ) :=
Revised: where X ( Z ) :=

Original: X \ PossAn ( Z ) P V \ W .
Revised: X \ PossAn ( Z ) P V \ W .

Original: A few observations are important at this point.
Revised: A few observations are important at this point.

Original: Despite the visual similarity to Prop. 1, there are twopivotal differences between these calculi.
Revised: Despite the visual similarity to Prop. 1, there are two pivotal differences between these calculi.

Original: First, Thm. 1 only requires blocking the deﬁnite statuspaths, hence the use of ‘m-separation’ in Thm. 1 instead of ‘ ˆ m -separation’.
Revised: First, Thm. 1 only requires blocking the definite status paths, hence the use of ‘m-separation’ in Thm. 1 instead of ‘ ˆ m -separation’.

Original: Consider the PAG P in
Revised: Consider the PAG P in

Original: Fig.
Revised: Fig.

Original: We want to evaluate whether P x ( y | z 1 , z 2 ) =
Revised: We want to evaluate whether P x ( y | z 1 , z 2 ) =

Original: P ( y | x, z 1 , z 2 ) by applying Rule 2 in Theorem1.
Revised: P ( y | x, z 1 , z 2 ) by applying Rule 2 in Theorem 1.

Original: Since all the edges in the PAG are circle edges, then P X = P .
Revised: Since all the edges in the PAG are circle edges, then P X = P .

Original: As discussed earlier, the set
Revised: As discussed earlier, the set

Original: Z = { Z 1 , Z 2 } blocks all the deﬁnite status path between X and Y .
Revised: Z = { Z 1 , Z 2 } blocks all the definite status path between X and Y .

Original: Hence, X and Y are m-separatedby Z and P x ( y | z 1 , z 2 )
Revised: Hence, X and Y are m-separated by Z and P x ( y | z 1 , z 2 )

Original: =
Revised: =

Original: P ( y | x, z 1 , z 2 ) holds true by Rule 2.
Revised: P ( y | x, z 1 , z 2 ) holds true by Rule 2.

Original: Second, Thm. 1 deﬁnes X ( Z ) as the subset of X that is not in the possible ancestors of Z in P V \ W ,as opposed to P W in Prop. 1.
Revised: Second, Thm. 1 defines X ( Z ) as the subset of X that is not in the possible ancestors of Z in P V \ W , as opposed to P W in Prop. 1.

Original: We revisit the query in Ex. 2 to clarify this subtle but signiﬁcantdifference.
Revised: We revisit the query in Ex. 2 to clarify this subtle but significant difference.

Original: Given the PAG in Fig.
Revised: Given the PAG in Fig.

Original: 3a, we want to evaluate whether P w,x 1 ,x 2 ( y | z ) =
Revised: 3a, we want to evaluate whether P w,x 1 ,x 2 ( y | z ) =

Original: P w ( y | z ) byapplying Rule 3 from Thm.
Revised: P w ( y | z ) by applying Rule 3 from Thm.

Original: 1 instead of Prop. 1.
Revised: 1 instead of Prop. 1.

Original: Fig.
Revised: Fig.

Original: 3c shows P V \{ W } where
Revised: 3c shows P V \{ W } where

Original: X =
Revised: X =

Original: { X 1 , X 2 } arenot possible ancestors of Z .
Revised: { X 1 , X 2 } are not possible ancestors of Z .

Original: Therefore, X ( Z ) =
Revised: Therefore, X ( Z ) =

Original: X , the edges into X 1 are cut in P W, X ( Z ) , and Xand Y are m-separated therein.
Revised: X , the edges into X 1 are cut in P W, X ( Z ) , and X and Y are m-separated therein.

Original: Third, the proof of the Theorem 1 is provided in the appendix, but it follows from the relationshipbetween m-connecting path in a manipulated MAG to a deﬁnite m-connecting path in the corresponding manipulated PAG.
Revised: Third, the proof of the Theorem 1 is provided in the appendix, but it follows from the relationship between m-connecting path in a manipulated MAG to a definite m-connecting path in the corresponding manipulated PAG.

Original: It was conjectured in [Zhang, 2008a, Footnote 15] that, for X , Y ⇢ V , if thereis an m-connecting path in M Y , X , then there is a deﬁnite m-connecting path in P Y , X .
Revised: It was conjectured in [Zhang, 2008a, Footnote 15] that, for X , Y ⊂ V , if there is an m-connecting path in M Y , X , then there is a definite m-connecting path in P Y , X .

Original: In this work,we prove that conjecture to be true for the special class of manipulations required in the rules of the calculus.
Revised: In this work, we prove that conjecture to be true for the special class of manipulations required in the rules of the calculus.

Original: Finally, the next proposition establishes the necessity of the antecedents in Thm.
Revised: Finally, the next proposition establishes the necessity of the antecedents in Thm.

Original: 1 in orderto apply the corresponding rule given every diagram in the equivalence class.
Revised: 1 in order to apply the corresponding rule given every diagram in the equivalence class.

Original: Theorem 2 (Atomic Completeness) .
Revised: Theorem 2 (Atomic Completeness) .

Original: The calculus in Theorem 1 is atomically complete; meaning,whenever a rule is not applicable given a PAG, then the corresponding rule in Pearl’s calculus is notapplicable given some causal diagram in the Markov equivalence class.
Revised: The calculus in Theorem 1 is atomically complete; meaning, whenever a rule is not applicable given a PAG, then the corresponding rule in Pearl’s calculus is not applicable given some causal diagram in the Markov equivalence class.

Original: For instance, considering PAG P in Fig.
Revised: For instance, considering PAG P in Fig.

Original: 1c, we note that ( Z ⇢⇢ ?? Y | X ) P X,Z , which means rule 2 is notapplicable.
Revised: 1c, we note that ( Z (cid:26)(cid:26) ⊥⊥ Y | X ) P X,Z , which means rule 2 is not applicable.

Original: Clearly, the diagram in Fig.
Revised: Clearly, the diagram in Fig.

Original: 1a is in the equivalence class of P and the corresponding ruleof Pearl’s calculus is not applicable due to the latent confounder between Z and Y .
Revised: 1a is in the equivalence class of P and the corresponding rule of Pearl’s calculus is not applicable due to the latent confounder between Z and Y .

Original:  Effect Identiﬁcation: A Complete Algorithm
Revised:  Effect Identification: A Complete Algorithm

Original: It is challenging to use the calculus rules in Thm.
Revised: It is challenging to use the calculus rules in Thm.

Original: 1 to identify causal effects since it is computationallyhard to decide whether there exists (and, if so, to ﬁnd) a sequence of derivations in the generalizedcalculus to identify an effect of interest.
Revised: 1 to identify causal effects since it is computationally hard to decide whether there exists (and, if so, to find) a sequence of derivations in the generalized calculus to identify an effect of interest.

Original: The goal of this section is to formulate an algorithm toidentify conditional causal effects.
Revised: The goal of this section is to formulate an algorithm to identify conditional causal effects.

Original: The next deﬁnition formalizes the notion of identiﬁability from a
Revised: The next definition formalizes the notion of identifiability from a

Original: PAG, generalizing the causal-diagram-speciﬁc notion introduced in [Pearl, 2000, Tian, 2004].
Revised: PAG, generalizing the causal-diagram-specific notion introduced in [Pearl, 2000, Tian, 2004].

Original: Deﬁnition 4 (Causal-Effect Identiﬁability) .
Revised: Definition 4 (Causal-Effect Identifiability) .

Original: Let X , Y , Y be disjoint sets of endogenous variables, V .
Revised: Let X , Y , Z be disjoint sets of endogenous variables, V .

Original: The causal effect of X on Y conditioned on Z is said to be identiﬁable from a PAG P if the quantity
Revised: The causal effect of X on Y conditioned on Z is said to be identifiable from a PAG P if the quantity

Original: P x ( y | z ) can be computed uniquely from the observational distribution P ( V ) given every causaldiagram D in the Markov equivalence class represented by P .
Revised: P x ( y | z ) can be computed uniquely from the observational distribution P ( V ) given every causal diagram D in the Markov equivalence class represented by P .

Original: The remainder of the section is organized as follows.
Revised: The remainder of the section is organized as follows.

Original: Sec.
Revised: Sec.

Original: 4.1 introduces a version of the IDPalgorithm [Jaber et al., 2019a] to identify marginal causal effects.
Revised: 4.1 introduces a version of the IDP algorithm [Jaber et al., 2019a] to identify marginal causal effects.

Original: The attractiveness of this versionis that it yields simpler expressions whenever the effect is identiﬁable while preserving the sameexpressive power, i.e., completeness for marginal identiﬁcation.
Revised: The attractiveness of this version is that it yields simpler expressions whenever the effect is identifiable while preserving the same expressive power, i.e., completeness for marginal identification.

Original: Sec.
Revised: Sec.

Original: 4.2 utilizes the new algorithmalong with the calculus in Thm.
Revised: 4.2 utilizes the new algorithm along with the calculus in Thm.

Original: 1 to formulate a complete algorithm for conditional identiﬁcation.
Revised: 1 to formulate a complete algorithm for conditional identification.

Original: 4.1 Marginal Effect Identiﬁcation
Revised: 4.1 Marginal Effect Identification

Original: We introduce the notion of pc-component next, which generalizes the notion of c-component that isinstrumental to solve identiﬁcaiton problems in a causal diagram [Tian and Pearl, 2002].
Revised: We introduce the notion of pc-component next, which generalizes the notion of c-component that is instrumental to solve identification problems in a causal diagram [Tian and Pearl, 2002].

Original: Deﬁnition 5 (PC-Component) .
Revised: Definition 5 (PC-Component) .

Original: In a PAG, or any induced subgraph thereof, two nodes are in the samepossible c-component (pc-component) if there is a path between them such that (1) all non-endpointnodes along the path are colliders, and (2) none of the edges is visible.
Revised: In a PAG, or any induced subgraph thereof, two nodes are in the same possible c-component (pc-component) if there is a path between them such that (1) all non-endpoint nodes along the path are colliders, and (2) none of the edges is visible.

Original: Following Def. 5, e.g., W and Z in Fig.
Revised: Following Def. 5, e.g., W and Z in Fig.

Original: 1c are in the same pc-component due to W ! X  Z .
Revised: 1c are in the same pc-component due to W ◦→ X ←◦ Z .

Original: Bycontrast, X, Y are not in the same pc-component since the direct edge between them is visible and Zalong h X, Z, Y i is not a collider.
Revised: By contrast, X, Y are not in the same pc-component since the direct edge between them is visible and Z along ⟨ X, Z, Y ⟩ is not a collider.

Original: Building on pc-components, we deﬁne the key notion of regions.
Revised: Building on pc-components, we define the key notion of regions.

Original: Deﬁnition 6 (Region R CA ) .
Revised: Definition 6 (Region R CA ) .

Original: Given PAG P over V , and A ✓ C ✓ V .
Revised: Given PAG P over V , and A ⊆ C ⊆ V .

Original: The region of A with respectto C , denoted R CA , is the union of the buckets that contain nodes in the pc-component of A in P C .
Revised: The region of A with respect to C , denoted R CA , is the union of the buckets that contain nodes in the pc-component of A in P C .

Original: A region expands a pc-component and will prove to be useful in the identiﬁcation algorithm.
Revised: A region expands a pc-component and will prove to be useful in the identification algorithm.

Original: Forexample, the pc-component of X in Fig. 2 is { X, Z 1 , Z 2 } and the region R V X
Revised: For example, the pc-component of X in Fig. 2 is { X, Z 1 , Z 2 } and the region R V X

Original: =
Revised: =

Original: { X, Z 1 , Z 2 , Y } .
Revised: { X, Z 1 , Z 2 , Y } .

Original: Building further on these deﬁnitions and the new calculus, we derive a new identiﬁcation criterion.
Revised: Building further on these definitions and the new calculus, we derive a new identification criterion.

Original: Proposition 2.
Revised: Proposition 2.

Original: Let P denote a PAG over V , T be a union of a subset of the buckets in P , and X ⇢ Tbe a bucket.
Revised: Let P denote a PAG over V , T be a union of a subset of the buckets in P , and X ⊂ T be a bucket.

Original: Given P v \ t (i.e., an observational expression for Q [ T ] ), Q [ T \ X ] is identiﬁable by thefollowing expression if, in P T , C X \ PossDe ( X ) ✓ X , where C X is the pc-component of X .
Revised: Given P v \ t (i.e., an observational expression for Q [ T ] ), Q [ T \ X ] is identifiable by the following expression if, in P T , C X ∩ PossDe ( X ) ⊆ X , where C X is the pc-component of X .

Original: [Equation]
Revised: [Equation]

Original: Note the interventions are over buckets which may or may not be single nodes
Revised: Note the interventions are over buckets which may or may not be single nodes.

Original: Since there is littleto no causal  a bucket marginal  subsets Algorithm Input: PAG P and two disjoint sets X , Y ⇢ V Output: Expression for P x ( y ) or FAIL 1: Let D = PossAn
Revised: Since there is little to no causal information inside a bucket, marginal effects of interventions over subsets of buckets are not identifiable.

Original: ( Y ) P V \ X 2: return

Original: P d \ y I

Original: DENTIFY ( D , V , P ) 3: function

Original: I DENTIFY ( C , T , Q = Q [ T ] ) 4: if C = ; then return5: if C =

Original: T then return Q /*

Original: In P T , let B denote a bucket, and let C B denote the pc-component of B */ 6:

Original: if 9 B ⇢ T \ C such that C B \ PossDe ( B ) P T ✓ B then 7:

Original: Compute Q [ T \ B ] from Q ; .

Original: via Prop.8: return I DENTIFY ( C , T \ B , Q [ T \ B ] ) 9: else if 9 B ⇢ C such that R B 6 =

Original: C then .

Original: R B is equivalent to R CB 10: return I DENTIFY ( R B , T ,Q ) ⇥ I DENTIFY ( R C \R B , T ,Q )

Original: I DENTIFY ( R B \ R C \R B , T ,Q ) 11: else throw FAIL are not identiﬁable.

Original: Also, the input distribution is possibly interventional which licenses recursiveapplications of the criterion.
Revised: Also, the input distribution is possibly interventional which licenses recursive applications of the criterion.

Original: The next example illustrates the power of the new criterion.
Revised: The next example illustrates the power of the new criterion.

Original: Example 3.
Revised: Example 3.

Original: Consider PAG P in Fig.
Revised: Consider PAG P in Fig.

Original: 3a and the query P x 1 ,x 2 ,w ( y, z, a ) .
Revised: 3a and the query P x 1 ,x 2 ,w ( y, z, a ) .

Original: Starting with the observational distribution P ( V ) as input, let
Revised: Starting with the observational distribution P ( V ) as input, let

Original: T =
Revised: T =

Original: V and X =
Revised: V and X =

Original: { X 1 , W } .
Revised: { X 1 , W } .

Original: We have C X
Revised: We have C X

Original: = {
Revised: = {

Original: X 1 , W, A, X 2 } ,
Revised: X 1 , W, A, X 2 } ,

Revised: Input: PAG P and two disjoint sets X , Y ⊂ V

Revised: Output: Expression for P x ( y ) or FAIL 1: Let D = PossAn

Revised: ( Y ) P V \ X 2: return (cid:80)

Revised: d \ y I DENTIFY

Original: PossDe ( X )  = 
Revised: ( D , V , P ) 3: function I DENTIFY ( C , T , Q = Q [ T ] ) 4: if C = ∅ then return5: if C = T then return Q /*

Revised: In P T , let B denote a bucket, and let C B denote the pc-component of B */ 6: if ∃ B ⊂ T \ C such that C B ∩ PossDe ( B ) P T ⊆ B then 7:

Revised: Compute Q [ T \ B ] from Q ; ▷ via Prop.8: return I DENTIFY ( C , T \ B , Q [ T \ B ] ) 9: else if ∃ B ⊂ C such that R B ̸

Revised: = C then ▷ R B is equivalent to R CB 10: return I DENTIFY ( R B , T ,Q ) ×

Revised: I DENTIFY

Original:  { X 1 , W, Z, Y } , and C X \ PossDe ( X ) =
Revised: ( R C \R B , T ,Q ) I DENTIFY ( R B ∩R C \R B , T ,Q ) 11: else throw FAIL PossDe ( X ) = { X 1 , W, Z, Y } , and C X ∩ PossDe ( X ) =

Original: X .
Revised: X .

Original: Hence, the criterion in Prop. 2 is applicable and we have P x ( y, z, a, x 2 ) =
Revised: Hence, the criterion in Prop. 2 is applicable and we have P x ( y, z, a, x 2 ) =

Original: [Equation]
Revised: P ( v ) P ( x 1 ,w | a,x 2 ) =

Original:  y, z | a, w ) after simpliﬁcation.
Revised: P ( a, x 2 ) × P ( y, z | a, w ) after simplification.

Original: Next, we consider intervening on X 2 given P x 1 ,w ( y, z, a, x 2 ) .
Revised: Next, we consider intervening on X 2 given P x 1 ,w ( y, z, a, x 2 ) .

Original: Notice X 2 is disconnected from theother nodes in P V \{ X 1 ,W } and it trivially satisﬁes the criterion in Prop. 2.
Revised: Notice X 2 is disconnected from the other nodes in P V \{ X 1 ,W } and it trivially satisfies the criterion in Prop. 2.

Original: Therefore, we get theexpression P x 1 ,x 2 ,w ( y, z, a ) =
Revised: Therefore, we get the expression P x 1 , [Equation] [Equation]

Original: P x 1 ,w ( y,z,a,x 2 ) P x 1 ,w ( x 2 | y,z,a ) = 
Revised: [Equation] [Equation] = [Equation]

Original:  P ( a ) ⇥ P ( y, z | w, a ) after simpliﬁcation.
Revised: ,z,a ) = P ( a ) × P ( y, z | w, a ) after simplification.

Original: A more general criterion was introduced in [Jaber et al., 2018a, Thm. 1] based on the possiblechildren of the intervention bucket X instead of the possible descendants.
Revised: A more general criterion was introduced in [Jaber et al., 2018, Thm. 1] based on the possible children of the intervention bucket X instead of the possible descendants.

Original: However, the correspondingexpression is convoluted and usually large, which could be intractable even if the effect is identiﬁable.
Revised: However, the corresponding expression is convoluted and usually large, which could be intractable even if the effect is identifiable.

Original: Alg. 1 shows the proposed version of IDP , which builds on the new criterion (Prop. 2).
Revised: Alg. 1 shows the proposed version of IDP , which builds on the new criterion (Prop. 2).

Original: Speciﬁcally,the key difference between this algorithm and the one proposed in [Jaber et al., 2019a] is in Lines 6-7,where the criterion in Prop. 2 is used as opposed to that in [Jaber et al., 2018a, Thm. 1].
Revised: Specifically, the key difference between this algorithm and the one proposed in [Jaber et al., 2019a] is in Lines 6-7, where the criterion in Prop. 2 is used as opposed to that in [Jaber et al., 2018, Thm. 1].

Original: Interestinglyenough, the new criterion is “just right,” namely, it is also sufﬁcient to obtain a complete algorithmfor marginal effect identiﬁcation, as shown in the next result.
Revised: Interestingly enough, the new criterion is “just right,” namely, it is also sufficient to obtain a complete algorithm for marginal effect identification, as shown in the next result.

Original: Theorem 3 (completeness) .
Revised: Theorem 3 (completeness) .

Original: Alg. 1 is complete for identifying marginal effects
Revised: Alg. 1 is complete for identifying marginal effects

Original: P x ( y ) .
Revised: P x ( y ) .

Original: Moreover, thecalculus in Thm. 1, together with standard probability manipulations are complete for the same task.
Revised: Moreover, the calculus in Thm. 1, together with standard probability manipulations are complete for the same task.

Original: 4.2 Conditional Effect Identiﬁcation
Revised: 4.2 Conditional Effect Identification

Original: We start by making a couple of observations, and then build on those observations to formulatean algorithm to identify conditional causal effects.
Revised: We start by making a couple of observations, and then build on those observations to formulate an algorithm to identify conditional causal effects.

Original: The proposed algorithm leverages the calculusin Thm. 1 and the IDP algorithm in Alg.1.
Revised: The proposed algorithm leverages the calculus in Thm. 1 and the IDP algorithm in Alg.

Original: Obs.
Revised: Obs.

Original: 1 notes that a conditional effect P x ( y | z ) can berewritten as
Revised: 1 notes that a conditional effect P x ( y | z ) can be rewritten as

Original: P x ( y , z )
Revised: [Equation] y ′

Original: Py P x ( y  , z ) , and hence it is identiﬁable if P x ( y , z ) is identiﬁable by Alg. 1.
Revised: P x ( y ′ , z ) , and hence it is identifiable if P x ( y , z ) is identifiable by Alg. 1.

Original: Observation 1 (Marginal Effect) .
Revised: Observation 1 (Marginal Effect) .

Original: Consider PAG P 1 in Fig.
Revised: Consider PAG P 1 in Fig.

Original: 4a where the goal is to identify thecausal effect P b ( a, c | d ) .
Revised: 4a where the goal is to identify the causal effect P b ( a, c | d ) .

Original: We notice that the effect P b ( a, c, d ) is identiﬁable using the IDP algorithm.
Revised: We notice that the effect P b ( a, c, d ) is identifiable using the IDP algorithm.

Original: Let E := P ( a, d ) ⇥ P ( c | b, d ) denote the expression for the marginal effect P b ( a, c, d ) which can beobtained from IDP .
Revised: Let E := P ( a, d ) × P ( c | b, d ) denote the expression for the marginal effect P b ( a, c, d ) which can be obtained from IDP .

Original: Consequently, the target effect can be computed using the expression E/ 
Revised: Consequently, the target effect can be computed using the expression E/ (cid:80) a ′ ,c ′ E .

Original:  P a,c E .
Revised: Whenever the marginal effect P x ( y , z ) is not identifiable using Alg. 1, Observations 2 and 3 propose techniques to identify the conditional effect using the calculus in Thm. 1, namely rule 2. Obs. 2 uses rule 2 of Thm. 1, when applicable, to move variables from the conditioning to the intervention set. The marginal effect of the resulting conditional query turns out to be identifiable, and consequently does the conditional effect. We note that the work in [Shpitser and Pearl, 2006] uses the same trick to formulate an algorithm for conditional effect identification given a causal diagram.

Original: A
Revised: A

Original: B
Revised: B

Original: C
Revised: C

Original: D  v  (a) PAG P 1 .
Revised: D ◦ v ◦ (a) PAG P 1 .

Original: X
Revised: X

Original: Z
Revised: Z

Original: T
Revised: T

Original: W
Revised: W

Original: Y       v
Revised: Y ◦ ◦ ◦ ◦ ◦ ◦ v

Original: [Figure]
Revised: [Figure]

Original: X
Revised: X

Original: Z
Revised: Z

Original: T
Revised: T

Original: W
Revised: W

Original: Y
Revised: Y

Original: [Figure]
Revised: [Figure]

Original: Algorithm
Revised: Algorithm

Original: Input: PAG P and three disjoint sets X , Y , Z ⇢ V
Revised: Input: PAG P and three disjoint sets X , Y , Z ⊂ V

Original: Output: Expression for P x ( y | z ) or FAIL 1: D 
Revised: Output: Expression for P x ( y | z ) or FAIL 1: D ← PossAn

Original: PossAn ( Y [ Z ) P V \ X /* Let B 1 , . . .
Revised: ( Y ∪ Z ) P V \ X /* Let B 1 , . . .

Original: B m denote the buckets in P */ 2: while 9 B i s.t.
Revised: B m denote the buckets in P */ 2: while ∃ B i s.t.

Original: B i \ D 6 = ; ^
Revised: B i ∩ D ̸ = ∅ ∧ B

Original: B i 6✓ D do 3:
Revised: i ̸⊆ D do 3:

Original: X  B i \ X 4: if ( X 0 ?? Y | ( X \ X 0 ) 
Revised: X ′ ← B i ∩ X 4: if ( X ′ ⊥⊥ Y | ( X \ X ′ ) ∪ Z ) P X \ X ′ , X

Original: [ Z ) P X \ X 0 , X 0 then 5: x  x \ x 0 ; z  z [ x. Apply rule 2 of Thm.6:
Revised: ′ then 5: x ← x \ x ′ ; z ← z ∪ x ′ ▷ Apply rule 2 of Thm.6:

Original: D  PossAn ( Y [ Z ) P V \ X 7: else throw FAIL /* Let Z 1 , . . .
Revised: D ← PossAn ( Y ∪ Z ) P V \ X 7: else throw FAIL /* Let Z 1 , . . .

Original: Z m partition Z such that Z
Revised: Z m partition Z such that Z

Original: i := Z \ B i */ 8: while 9 Z i s.t.
Revised: i := Z ∩ B i */ 8: while ∃ Z i s.t.

Original: ( Z i ?? Y | X [ ( Z \ Z i ))
Revised: ( Z i ⊥⊥ Y | X ∪ ( Z \ Z i ))

Original: P X , Z i do 9: x  x [ z i ; z  z \ z i . Apply rule 2 of Thm.10: E  IDP ( P , x , y [ z ) 11: return E/ P y E
Revised: P X , Z i do 9: x ← x ∪ z i ; z ← z \ z i ▷ Apply rule 2 of Thm.10: E ← IDP ( P , x , y ∪ z ) 11: return E/ (cid:80) y ′

Original: Whenever the marginal effect P x ( y , z ) is not identiﬁable using Alg. 1, Observations 2 and 3 proposetechniques to identify the conditional effect using the calculus in Thm. 1, namely rule 2.

Original: Obs.

Original: 2 usesrule 2 of Thm. 1, when applicable, to move variables from the conditioning to the intervention set. The marginal effect of the resulting conditional query turns out to be identiﬁable, and consequentlydoes the conditional effect. We note that the work in [Shpitser and Pearl, 2006] uses the same trick toformulate an algorithm for conditional effect identiﬁcation given a causal diagram.
Revised: E

Original: Observation 2 (Flip Observations to Interventions) .
Revised: Observation 2 (Flip Observations to Interventions) .

Original: Consider PAG P 1 in Fig.
Revised: Consider PAG P 1 in Fig.

Original: 4a and the causalquery P a ( c | b, d ) .
Revised: 4a and the causal query P a ( c | b, d ) .

Original: Unlike the case in Obs. 1, the marginal effect P a ( b, c, d ) is not identiﬁable by the
Revised: Unlike the case in Obs. 1, the marginal effect P a ( b, c, d ) is not identifiable by the

Original: IDP algorithm.
Revised: IDP algorithm.

Original: Using rule 2 of Thm.
Revised: Using rule 2 of Thm.

Original: 1, we have ( B ?? C | D ) P A,B
Revised: 1, we have ( B ⊥⊥ C | D ) P A,B

Original: and we move B from conditioningto intervention, i.e., P a ( c | b, d ) =
Revised: and we move B from conditioning to intervention, i.e., P a ( c | b, d ) =

Original: P a,b ( c | d ) .
Revised: P a,b ( c | d ) .

Original: The marginal effect P a,b ( c, d ) is identiﬁable by IDP andwe get the expression E :
Revised: The marginal effect P a,b ( c, d ) is identifiable by IDP and we get the expression E :

Original: = P ( d ) ⇥ P ( c | b, d ) .
Revised: = P ( d ) × P ( c | b, d ) .

Original: Hence, we have P a ( c | b, d ) =
Revised: Hence, we have P a ( c | b, d ) =

Original: P a,b ( c | d ) = E/ P c  E .
Revised: P a,b ( c | d ) = E/ (cid:80) c ′ E .

Original: Finally, Obs. 3 comes as a surprise since it requires ﬂipping interventions to observations, contraryto Obs.
Revised: Finally, Obs. 3 comes as a surprise since it requires flipping interventions to observations, contrary to Obs.

Original: A key graphical structure in the PAG that requires such a treatment is the presence of aproper possibly directed path from X to Y [ Z that starts with a circle edge.
Revised: A key graphical structure in the PAG that requires such a treatment is the presence of a proper possibly directed path from X to Y ∪ Z that starts with a circle edge.

Original: Observation 3 (Flip Interventions to Observations) .
Revised: Observation 3 (Flip Interventions to Observations) .

Original: We revisit the query in Example.
Revised: We revisit the query in Example.

Original: First, themarginal effect P x ( y, z 1 , z 2 ) is not identiﬁable by the IDP algorithm.
Revised: First, the marginal effect P x ( y, z 1 , z 2 ) is not identifiable by the IDP algorithm.

Original: Also, we cannot use rule 2 in
Revised: Also, we cannot use rule 2 in

Original: Thm. 1 to ﬂip Z 1 or Z 2 into interventions since they are both adjacent to Y with a circle edge (  ).
Revised: Thm. 1 to flip Z 1 or Z 2 into interventions since they are both adjacent to Y with a circle edge ( ◦−◦ ).

Original: However, we can use rule 2 to ﬂip X to the conditioning set since there are no deﬁnite m-connectingpaths between X and Y given { Z 1 , Z 2 } in the PAG.
Revised: However, we can use rule 2 to flip X to the conditioning set since there are no definite m-connecting paths between X and Y given { Z 1 , Z 2 } in the PAG.

Original: Hence, we obtain P x
Revised: Hence, we obtain P x

Original: ( y | z 1 , z 2 ) =
Revised: ( y | z 1 , z 2 ) =

Original: P ( y | z 1 , z 2 , x ) .
Revised: P ( y | z 1 , z 2 , x ) .

Original: Alternatively, consider PAG P 2 in Fig.
Revised: Alternatively, consider PAG P 2 in Fig.

Original: 4b with the causal query P x
Revised: 4b with the causal query P x

Original: ( y | z ) .
Revised: ( y | z ) .

Original: We cannot use rule 2 toﬂip X to an observation since h X, W, Y i is active given Z in P 2 X .
Revised: We cannot use rule 2 to flip X to an observation since ⟨ X, W, Y ⟩ is active given Z in P 2 X .

Original: In fact, the causal diagram G in
Revised: In fact, the causal diagram G in

Original: Fig.
Revised: Fig.

Original: 4c is in the equivalence class of P 2 and such that P x ( y | z ) is provably not identiﬁable [Shpitserand Pearl, 2006, Corol. 2].
Revised: 4c is in the equivalence class of P 2 and such that P x ( y | z ) is provably not identifiable [Shpitser and Pearl, 2006, Corol. 2].

Original: Hence, the effect is not identiﬁable given P 2 according to Def. 4.
Revised: Hence, the effect is not identifiable given P 2 according to Def. 4.

Revised: Putting these observations together, we formulate the CIDP algorithm (Alg. 2) for identifying conditional causal effects given a PAG.

Original: A
Revised: The algorithm is divided into three phases. In Phase I (lines 1-7), Obs. 3 is used to check for proper possibly directed paths from X to Y ∪ Z that start with a circle edge. This is checked algorithmically by computing D = PossAn ( Y ∪ Z ) P V \ X , iteratively, and checking if some bucket B i in P intersects with, but is not a subset of, D .

Revised: If such a bucket exists, CIDP flips B i ∩ X from interventions to observations using rule 2, when applicable, else the algorithm throws a fail and the effect is not computable.

Revised: In Phase II (lines 8-9), Obs. 2 is used to flip the subset of observations in each bucket into interventions by applying rule 2 of Thm. 1,

Revised: A

Original: B
Revised: B

Original: C G
Revised: C G

Original: E
Revised: E

Original: F
Revised: F

Original: Y       v  v (a) PAG P .
Revised: Y ◦ ◦◦ ◦ ◦ ◦ ◦ v ◦ v (a) PAG P .

Original: A
Revised: A

Original: B
Revised: B

Original: C G
Revised: C G

Original: E
Revised: E

Original: F
Revised: F

Original: Y      v  v (b) P F,A
Revised: Y ◦ ◦ ◦ ◦ ◦ v ◦ v (b) P F,A

Original: A
Revised: A

Original: B
Revised: B

Original: C G
Revised: C G

Original: E
Revised: E

Original: F
Revised: F

Original: Y       v (c) P F,E

Original: Putting these observations together, we formulate the CIDP algorithm (Alg. 2) for identifyingconditional causal effects given a PAG.

Original: The algorithm is divided into three phases.

Original: In Phase I(lines 1-7), Obs. 3 is used to check for proper possibly directed paths from X to Y [ Z that start witha circle edge. This is checked algorithmically by computing D = PossAn ( Y [ Z ) P V \ X , iteratively,and checking if some bucket B i in P intersects with, but is not a subset of, D .
Revised: Y ◦ ◦ ◦ ◦ ◦ ◦ v (c) P F,E whenever applicable.

Original: If such a bucketexists, CIDP ﬂips B i \ X from interventions to observations using rule 2, when applicable, elsethe algorithm throws a fail and the effect is not computable.

Original: In Phase II (lines 8-9), Obs. 2 is usedto ﬂip the subset of observations in each bucket into interventions by applying rule 2 of Thm.

Original: 1,whenever applicable.

Original: Finally, in Phase III (line 10), the marginal effect P x ( y [ z ) is computed fromthe modiﬁed sets X and Z , using the IDP algorithm in Alg.
Revised: Finally, in Phase III (line 10), the marginal effect P x ( y ∪ z ) is computed from the modified sets X and Z , using the IDP algorithm in Alg.

Original: If the call is successful, an expressionfor the conditional effect is returned at line 11.
Revised: If the call is successful, an expression for the conditional effect is returned at line 11.

Original: The example below illustrates CIDP in action.
Revised: The example below illustrates CIDP in action.

Original: Anempirical evaluation of CIDP is provided inApp. G; the R package will be made available.
Revised: An empirical evaluation of CIDP is provided in the full report [Jaber et al., 2022].

Original: Example 4.
Revised: Example 4.

Original: Consider PAG P in Fig.
Revised: Consider PAG P in Fig.

Original: 5a and the conditional query P x
Revised: 5a and the conditional query P x

Original: ( y | z ) :
Revised: ( y | z ) :

Original: = P a,f ( y | b, e ) .
Revised: = P a,f ( y | b, e ) .

Original: In
Revised: In

Original: Phase I, we have D = PossAn
Revised: Phase I, we have D = PossAn

Original: ( Y [ Z ) P V \ X =
Revised: ( Y ∪ Z ) P V \ X =

Original: { Y, B, E, C, G } , and the bucket { A, B } satisﬁes theconditions at line 2 since A 62 D .
Revised: { Y, B, E, C, G } , and the bucket { A, B } satisfies the conditions at line 2 since A ̸∈ D .

Original: In P F,A (as shown in Fig.
Revised: In P F,A (as shown in Fig.

Original: 5b), X [-
-]
Revised: 5b), X [-
-]′ =

Original: = { A } is m-separated from Ygiven { B, E, F } which satisﬁes the if condition at line 4.
Revised: { A } is m-separated from Y given { B, E, F } which satisfies the if condition at line 4.

Original: Hence, we ﬂip A to the conditioning set Zvia rule 2 of Thm.
Revised: Hence, we flip A to the conditioning set Z via rule 2 of Thm.

Original: 1 to obtain the updated query P x
Revised: 1 to obtain the updated query P x

Original: ( y | z ) =
Revised: ( y | z ) =

Original: P f ( y | a, b, e ) .
Revised: P f ( y | a, b, e ) .

Original: In Phase II (lines 8-9), let Z 1 = { E } and Z 2 =
Revised: In Phase II (lines 8-9), let Z 1 = { E } and Z 2 =

Original: { A, B } .
Revised: { A, B } .

Original: In P F,E (see Fig.
Revised: In P F,E (see Fig.

Original: 5c)
Revised: 5c)

Original: , we have E m-separated from Y given { F } [ Z 2which satisﬁes the if condition at line 8.
Revised: , we have E m-separated from Y given { F } ∪ Z 2 which satisfies the if condition at line 8.

Original: Hence, we ﬂip E to the intervention set using rule 2 of Thm.
Revised: Hence, we flip E to the intervention set using rule 2 of Thm.

Original: 1and we get the updated query P x
Revised: and we get the updated query P x

Original: ( y | z ) =
Revised: ( y | z ) =

Original: P e,f ( y | a, b ) .
Revised: P e,f ( y | a, b ) .

Original: Next, we check if Z  2 is m-separated from Ygiven Z 1 in P Z 1 , Z2 which does not hold due to a bidirected edge between B and E .
Revised: Next, we check if Z = Z 2 is m-separated from Y given X in P X , Z which does not hold due to a bidirected edge between B and Y .

Original: Hence, rule 2is not applicable and Z2 remains in the conditioning set.
Revised: Hence, rule 2 is not applicable and Z remains in the conditioning set.

Original: Finally, we call IDPin Alg. 1 to computethe marginal effect P e,f
Revised: Finally, we call IDP to compute the marginal effect P e,f

Original: ( y, a, b ) , if possible.
Revised: ( y, a, b ) , if possible.

Original: The effect is identiﬁable with the simpliﬁed expression
Revised: The effect is identifiable with the simplified expression

Original: P ( y | b, e, f ) ⇥ P ( a, b ) .
Revised: P ( y | b, e, f ) × P ( a, b ) .

Original: Hence, P a,f ( y | b, e ) =
Revised: Hence, P a,f ( y | b, e ) =

Original: P e,f ( y | a, b ) =
Revised: P e,f ( y | a, b ) =

Original: P ( y | b,e,f ) ⇥ P ( a,b ) P y  P ( y  | b,e,f ) ⇥ P ( a,b ) =
Revised: P ( y | b,e,f ) × P ( a,b ) (cid:80) y ′ P ( y ′ | b,e,f ) × P ( a,b ) =

Original: P ( y | b, e, f ) .
Revised: P ( y | b, e, f ) .

Original: The soundness of Alg. 2 follows from that of Alg.
Revised: The soundness of Alg. 2 follows from that of Alg.

Original: 1 and Thm.
Revised: 1 and Thm.

Original: Next, we turn to its completeness.
Revised: Next, we turn to its completeness.

Original: According to Def. 4, whenever CIDP fails, we need to establish one of two conditions for completeness.
Revised: According to Def. 4, whenever CIDP fails, we need to establish one of two conditions for completeness.

Original: Either there exist two causal diagrams in the equivalence class with different identiﬁcations, orthe effect is not identiﬁable in some causal diagram according to the criterion in [Shpitser and Pearl,2006, Corol. 2].
Revised: Either there exist two causal diagrams in the equivalence class with different identifications, or the effect is not identifiable in some causal diagram according to the criterion in [Shpitser and Pearl, 2006, Corol. 2].

Original: Thm.
Revised: Thm.

Original: 4 establishes completeness by proving that the latter is always the case.
Revised: 4 establishes completeness by proving that the latter is always the case.

Original: Thisresult along with the completeness of the calculus rules for the identiﬁcation of marginal effects (see
Revised: This result along with the completeness of the calculus rules for the identification of marginal effects (see

Original: Thm. 3) implies that the rules are complete for conditional effects as well.
Revised: Thm. 3) implies that the rules are complete for conditional effects as well.

Original: Theorem 4 (completeness) .
Revised: Theorem 4 (completeness) .

Original: Alg. 2 is complete for identifying conditional effects
Revised: Alg. 2 is complete for identifying conditional effects

Original: P x ( y | z ) .
Revised: P x ( y | z ) .

Original: Also, thecalculus in Thm.
Revised: Also, the calculus in Thm.

Original: 1, together with standard probability manipulations are complete for the same task.
Revised: 1, together with standard probability manipulations are complete for the same task.

Original:  Conclusions
Revised:  Discussion

Revised: In this work, an oracle for conditional independences is assumed to be available, which leads to the true PAG.

Revised: Assuming the presence of an oracle for conditional independence encapsulates the challenge of dealing with finite data and of testing for conditional independence thereof.

Revised: Another challenge lies in the computational complexity of learning the PAG in the first place [Colombo et al., 2012], and estimating the expression when the effect is identifiable [Jung et al., 2021].

Revised: In light of this, it is important to make the distinction between the task of causal effect identification and that of causal effect estimation.

Revised: This set of results is concerned with the first task (causal identification), which asks whether a target conditional effect is uniquely computable from P ( V ) , the observational distribution, and given a

Revised: PAG learnable from P ( V ) .

Revised: The objective of CIDP in Algorithm 2 is to decide whether the effect is identifiable and provide an expression for it when the answer is yes, while being agnostic as to whether P ( V ) can be accurately estimated from the available samples.

Revised: An interesting direction for future work is to investigate the setting where the causal effect is not identifiable given the PAG, according to Def. 4, and study whether the causal effect can be bounded [Balke and Pearl, 1997].

Revised: As for the second task, the estimation of the conditional causal effect using the identification formula provided by Algorithm 2 poses several challenges under finite data.

Revised: The number of samples sufficient to identify a given effect would depend on the size of the expression, among other factors, and naive methods for estimation exacerbate this problem.

Revised: Recent work such as [Jung et al., 2021] proposes a double machine learning estimator for marginal effects that are identifiable given a PAG.

Revised: An interesting direction of work is to generalize this approach to conditional causal effects that are identifiable by CIDP .

Revised:  Conclusions

Original: In this work, we investigate the problem of identifying conditional interventional distributionsgiven a Markov equivalence class of causal diagrams represented by a PAG.
Revised: In this work, we investigate the problem of identifying conditional interventional distributions given a Markov equivalence class of causal diagrams represented by a PAG.

Original: We introduce a newgeneralization of the do-calculus for identiﬁcation of interventional distributions in PAGs (Thm. 1)and show it to be atomically complete (Thm. 2).
Revised: We introduce a new generalization of the do-calculus for identification of interventional distributions in PAGs (Thm. 1) and show it to be atomically complete (Thm. 2).

Original: Building on these results, we develop the CIDPalgorithm (Alg. 2), which is both sound and complete, i.e., it identiﬁes any conditional effects of theform P x ( y | z ) that is identiﬁable (Thm. 4).
Revised: Building on these results, we develop the CIDP algorithm (Alg. 2), which is both sound and complete, i.e., it identifies any conditional effects of the form P x ( y | z ) that is identifiable (Thm. 4).

Original: Finally, we show that the new calculus rules, along withstandard probability manipulations, are complete for the same task.
Revised: Finally, we show that the new calculus rules, along with standard probability manipulations, are complete for the same task.

Original: These results close the problemof effect identiﬁcation under Markov equivalence in that they completely delineate the theoreticalboundaries of what is, in principle, computable from a certain data collection.
Revised: These results close the problem of effect identification under Markov equivalence in that they completely delineate the theoretical boundaries of what is, in principle, computable from a certain data collection.

Original: We expect the newlyintroduced machinery to help data scientists to identify novel effects in real world settings.
Revised: We expect the newly introduced machinery to help data scientists to identify novel effects in real world settings.

Revised: Acknowledgments and Disclosure of Funding

Revised: Bareinboim and Ribeiro’s research was supported in part by the NSF, ONR, DoE, Amazon, JP

Revised: Morgan, and The Alfred P. Sloan Foundation.

Revised: Zhang’s research was supported by the RGC of Hong

Revised: Kong under GRF13602720.

Original: Checklist
Revised: Checklist

Original: The checklist follows the references.
Revised: The checklist follows the references.

Original: Please read the checklist guidelines carefully for information onhow to answer these questions.
Revised: Please read the checklist guidelines carefully for information on how to answer these questions.

Original: For each question, change the default [TODO] to [Yes] , [No] , or[N/A] .
Revised: For each question, change the default [TODO] to [Yes] , [No] , or [N/A] .

Original: You are strongly encouraged to include a justiﬁcation to your answer , either by referencingthe appropriate section of your paper or providing a brief inline description.
Revised: You are strongly encouraged to include a justification to your answer , either by referencing the appropriate section of your paper or providing a brief inline description.

Original: For example:
Revised: For example:

Original: • Did you include the license to the code and datasets?
Revised: • Did you include the license to the code and datasets?

Original: [Yes] See Section ...•
Revised: [Yes] See Section ... •

Original: Did you include the license to the code and datasets?
Revised: Did you include the license to the code and datasets?

Original: [No] The code and the data areproprietary.
Revised: [No] The code and the data are proprietary.

Original: • Did you include the license to the code and datasets?
Revised: • Did you include the license to the code and datasets?

Original: [N/A]
Revised: [N/A]

Original: Please do not modify the questions and only use the provided macros for your answers.
Revised: Please do not modify the questions and only use the provided macros for your answers.

Original: Note that the
Revised: Note that the

Original: Checklist section does not count towards the page limit.
Revised: Checklist section does not count towards the page limit.

Original: In your paper, please delete this instructionsblock and only keep the Checklist section heading above along with the questions/answers below.For
Revised: In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.

Original:  all authors... (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s contributions and scope?
Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?

Original: [Yes] The three contributions claimed in the abstract and reiterated at the end of the introduction are reﬂected in Section 3 for contribution 1,and in Section 4 for contributions 2 and 3.
Revised: [Yes] The three contributions claimed in the abstract and reiterated at the end of the introduction are reflected in Section 3 for contribution 1, and in Section 4 for contributions 2 and 3.

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [Yes] See Section 2, lines 103–104.
Revised: [Yes] See Section 5.

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [N/A] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
Revised: [N/A] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?

Original: [Yes] 
Revised: [Yes] 2.

Original: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?
Revised: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes] The assumptionin lines 103-104 leads to the true PAG which is what the theoretical results results arebased on.
Revised: [Yes] The assumption in lines 103-104 leads to the true PAG which is what the theoretical results are based on.

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes] The proofs can befound in the appendix.
Revised: [Yes] The proofs can be found in the full report [Jaber et al., 2022].

Original: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?
Revised: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Original: [No] The code will be made freely available with the camera-ready version of the paper in case of acceptance.
Revised: [No] The code will be made freely available with the camera-ready version of the paper in case of acceptance.

Original: (b) Did you specify all the training details (e g., data splits, hyperparameters, how theywere chosen)?
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [Yes] (c) Did you report error bars (e g., with respect to the random seed after running experi- ments multiple times)?
Revised: [Yes] (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [Yes] (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
Revised: [Yes] (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?

Original: [Yes] 
Revised: [Yes] 4.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?[N/A]
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?

Original: [Table]

Original:  (b) Did you mention the license of the assets?[N/A]
Revised: [N/A] (b) Did you mention the license of the assets?

Original:  (c) Did you include any new assets either in the supplemental material or as a URL?[N/A]
Revised: [N/A] (c) Did you include any new assets either in the supplemental material or as a URL?

Original:  (d) Did you discuss whether and how consent was obtained from people whose data you’re[Table] using/curating?
Revised: [N/A] (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?

Original: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identiﬁable information or offensive content?
Revised: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?

Original: [N/A] 
Revised: [N/A] 5.

Original: If you used crowdsourcing or conducted research with human subjects...
Revised: If you used crowdsourcing or conducted research with human subjects...

Original: (a) Did you include the full text of instructions given to participants and screenshots, if applicable?
Revised: (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
Revised: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Original: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?
Revised: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?

Original: [N/A]
Revised: [N/A]