Original: Causal Inference with Non-IID Data using Linear Graphical Models
Revised: Causal Inference with Non-IID Data using Linear Graphical Models

Original: Abstract
Revised: Abstract

Original: Traditional causal inference techniques assume data are independent and identically distributed (IID) and thus ignores interactions among units.
Revised: Traditional causal inference techniques assume data are independent and identically distributed (IID) and thus ignores interactions among units.

Original: However, a unit’s treatment may affect another unit’s outcome (interference), a unit’s treatment may be correlated with another unit’s outcome, or a unit’s treatment and outcome may be spuriously correlated through another unit.
Revised: However, a unit’s treatment may affect another unit’s outcome (interference), a unit’s treatment may be correlated with another unit’s outcome, or a unit’s treatment and outcome may be spuriously correlated through another unit.

Original: To capture such nuances, we model the data generating process using causal graphs and conduct a systematic analysis of the bias caused by different types of interactions when computing causal effects.
Revised: To capture such nuances, we model the data generating process using causal graphs and conduct a systematic analysis of the bias caused by different types of interactions when computing causal effects.

Original: We derive theorems to detect and quantify the interaction bias, and derive conditions under which it is safe to ignore interactions.
Revised: We derive theorems to detect and quantify the interaction bias, and derive conditions under which it is safe to ignore interactions.

Original: Put differently, we present conditions under which causal effects can be computed with negligible bias by assuming that samples are IID.
Revised: Put differently, we present conditions under which causal effects can be computed with negligible bias by assuming that samples are IID.

Original: Furthermore, we develop a method to eliminate bias in cases where blindly assuming IID is expected to yield a significantly biased estimate.
Revised: Furthermore, we develop a method to eliminate bias in cases where blindly assuming IID is expected to yield a significantly biased estimate.

Original: Finally, we test the coverage and performance of our methods through simulations.
Revised: Finally, we test the coverage and performance of our methods through simulations.

Original:  Introduction
Revised:  Introduction

Original: Motivating Example Suppose we are interested in studying the effectiveness of Covid-19 vaccines.
Revised: Motivating Example Suppose we are interested in studying the effectiveness of Covid-19 vaccines.

Original: Specifically, we are interested in the causal effect of vaccine doses, V , on the severity of sickness S .
Revised: Specifically, we are interested in the causal effect of vaccine doses, V , on the severity of sickness S .

Original: A naive method would be building a causal model on V , S and other related factors and estimating the causal effect of V on S using available data.
Revised: A naive method would be building a causal model on V , S and other related factors and estimating the causal effect of V on S using available data.

Original: However, this method may result in biased estimation primarily because traditional causal inference techniques assume that attributes of all units in the sample are independent and identically distributed (IID)[Rubin, 1978] which does not hold true in the pandemic setting since units are not isolated from each other.
Revised: However, this method may result in biased estimation primarily because traditional causal inference techniques assume that attributes of all units in the sample are independent and identically distributed (IID)[Rubin, 1978] which does not hold true in the pandemic setting since units are not isolated from each other.

Original: We exemplify below few instances of this problem that violate IID (Eyre et al.
Revised: We exemplify below few instances of this problem that violate IID (Eyre et al.

Original: Case 1: The vaccination V of a unit i , ( V i ) , decreases their viral load, L i , which in turn decreases the transmission rate of the virus, and hence decreases the severity of sickness S of another unit j , ( S j ) , who comes into contact with i .
Revised: Case 1: The vaccination V of a unit i , ( V i ) , decreases their viral load, L i , which in turn decreases the transmission rate of the virus, and hence decreases the severity of sickness S of another unit j , ( S j ) , who comes into contact with i .

Original: V i causally affects S j .
Revised: V i causally affects S j .

Original: Case 2: V i is affected by the area
Revised: Case 2: V i is affected by the area

Original: A that i lives in, and a contact j who lives in vaccine deprived areas and areas with a higher incidence of Covid-19 infection is more likely to get sick.
Revised: A that i lives in, and a contact j who lives in vaccine deprived areas and areas with a higher incidence of Covid-19 infection is more likely to get sick.

Original: V i and S j are confounded.
Revised: V i and S j are confounded.

Original: Case 3: S i , determines whether or not i is quarantined and thus affects whether i transmits the disease to another unit
Revised: Case 3: S i , determines whether or not i is quarantined and thus affects whether i transmits the disease to another unit

Original: j . S
Revised: j . S

Original: i causally affects S j .
Revised: i causally affects S j .

Original: Such interactions between units plague both observational and experimental studies.
Revised: Such interactions between units plague both observational and experimental studies.

Original: If the latter is performed in a controlled environment where subjects are isolated from each other, the results would not be valid for the target environment, where subjects affect each other, and vice versa.
Revised: If the latter is performed in a controlled environment where subjects are isolated from each other, the results would not be valid for the target environment, where subjects affect each other, and vice versa.

Original: Modeling Interactions A line of existing work that analyzes interactions between units is interference [Cox, 1958].
Revised: Modeling Interactions A line of existing work that analyzes interactions between units is interference [Cox, 1958].

Original: Interference is the phenomenon in which treatment of unit i ( V i ) causally affects the outcome, S j , of another unit j .
Revised: Interference is the phenomenon in which treatment of unit i ( V i ) causally affects the outcome, S j , of another unit j .

Original: In almost all existing literature this is interpreted as there existing a causal pathway from V i to S j .
Revised: In almost all existing literature this is interpreted as there existing a causal pathway from V i to S j .

Original: Case-1 above is a typical example.
Revised: Case-1 above is a typical example.

Original: Clearly, ignoring unit interactions while computing causal effects would result in a biased estimate.
Revised: Clearly, ignoring unit interactions while computing causal effects would result in a biased estimate.

Original: However, we note that interference is not the only type of interaction between units that can yield biased estimates.
Revised: However, we note that interference is not the only type of interaction between units that can yield biased estimates.

Original: For example, in
Revised: For example, in

Original: Case-2 V i and S j are confounded and V i is not a cause of S j .
Revised: Case-2 V i and S j are confounded and V i is not a cause of S j .

Original: Another example is an instance where unit i ′ s treatment affects their own outcome through an attribute of unit j i.e., V i →
Revised: Another example is an instance where unit i ′ s treatment affects their own outcome through an attribute of unit j i.e., V i →

Original: W j → S i , for some W j / ∈ { S j , V j } .
Revised: W j → S i , for some W j / ∈ { S j , V j } .

Original: In both these cases units interact with each other in a way that might bias the estimation of causal effects although they may not typically be classified as interference.
Revised: In both these cases units interact with each other in a way that might bias the estimation of causal effects although they may not typically be classified as interference.

Original: In spite of the prevalence of such interactions in applications related to health care, infectious diseases, social networks and ad placements, they have not been systematically studied.
Revised: In spite of the prevalence of such interactions in applications related to health care, infectious diseases, social networks and ad placements, they have not been systematically studied.

Original: It is this deficiency that this paper attempts to overcome.
Revised: It is this deficiency that this paper attempts to overcome.

Original: Questions addressed The scenario exemplified above raises several questions regarding the computation of causal effects given non-IID data.
Revised: Questions addressed The scenario exemplified above raises several questions regarding the computation of causal effects given non-IID data.

Original: How can we model different types of interactions among units in the population?
Revised: How can we model different types of interactions among units in the population?

Original: Under what conditions can we safely ignore unit interactions with the guarantee that assuming IID (and applying existing estimation techniques) will result in negligible bias?
Revised: Under what conditions can we safely ignore unit interactions with the guarantee that assuming IID (and applying existing estimation techniques) will result in negligible bias?

Original: If assuming IID would yield a biased estimate, then how can we get rid of this bias?
Revised: If assuming IID would yield a biased estimate, then how can we get rid of this bias?

Original: Our contributions In this work we study causal inference in the presence of interactions among samples using linear models due to the convenience they offer with regard to path analysis.
Revised: Our contributions In this work we study causal inference in the presence of interactions among samples using linear models due to the convenience they offer with regard to path analysis.

Original: We develop interaction models that portray different types of interactions among units and conduct a systematic analysis of the bias caused by different types of interactions.
Revised: We develop interaction models that portray different types of interactions among units and conduct a systematic analysis of the bias caused by different types of interactions.

Original: We derive theorems to detect and quantify the interaction bias.
Revised: We derive theorems to detect and quantify the interaction bias.

Original: We derive conditions under which it is safe to ignore unit interactions when computing the average causal effects.
Revised: We derive conditions under which it is safe to ignore unit interactions when computing the average causal effects.

Original: Furthermore, we develop a method to compute an unbiased estimate of causal effect in cases where blindly assuming IID is expected to yield a significant bias.
Revised: Furthermore, we develop a method to compute an unbiased estimate of causal effect in cases where blindly assuming IID is expected to yield a significant bias.

Original: Finally, we corroborate our findings through simulation studies.
Revised: Finally, we corroborate our findings through simulation studies.

Original: Summary of results in words Blindly assuming that data are IID when in fact they are not, can potentially bias the outcome of a research study.
Revised: Summary of results in words Blindly assuming that data are IID when in fact they are not, can potentially bias the outcome of a research study.

Original: Such bias can occur for the query: causal effect of treatment on outcome, when there is an open (not necessarily directed) path from the treatment of unit i to the outcome of unit j and/or to the outcome of unit i itself such that an intermediate node on the path belongs to unit j .
Revised: Such bias can occur for the query: causal effect of treatment on outcome, when there is an open (not necessarily directed) path from the treatment of unit i to the outcome of unit j and/or to the outcome of unit i itself such that an intermediate node on the path belongs to unit j .

Original: The formula in Theorem 1 quantifies the bias.
Revised: The formula in Theorem 1 quantifies the bias.

Original: Furthermore, only the two types of interaction structures previously mentioned can induce bias.
Revised: Furthermore, only the two types of interaction structures previously mentioned can induce bias.

Original: In the presence of such bias inducing structures it is still possible to compute an unbiased estimate by selecting a subset of samples B such that no biasing paths exist in the interaction graph corresponding to samples in B (Theorem 2).
Revised: In the presence of such bias inducing structures it is still possible to compute an unbiased estimate by selecting a subset of samples B such that no biasing paths exist in the interaction graph corresponding to samples in B (Theorem 2).

Original: More importantly, such a debiasing procedure does not require the selection of IID samples and may contain interactions among them.
Revised: More importantly, such a debiasing procedure does not require the selection of IID samples and may contain interactions among them.

Original: Such a debiasing procedure can also be done in polynomial time (Algorithm 1).
Revised: Such a debiasing procedure can also be done in polynomial time (Algorithm 1).

Original: Empirical analysis: We randomly generate interaction models and show that the bias can be huge if IID is wrongly assumed on non-IID data.
Revised: Empirical analysis: We randomly generate interaction models and show that the bias can be huge if IID is wrongly assumed on non-IID data.

Original: The debiasing method in this paper yields an unbiased estimate.
Revised: The debiasing method in this paper yields an unbiased estimate.

Original: We further show that, as the number of bias-free samples increases, and as the strengths of bias structures decrease, the overall interaction bias decreases.
Revised: We further show that, as the number of bias-free samples increases, and as the strengths of bias structures decrease, the overall interaction bias decreases.

Original:  Preliminaries
Revised:  Preliminaries

Original: Independent and Identically Distributed (IID)
Revised: Independent and Identically Distributed (IID)

Original: If X 1 , . .
Revised: If X 1 , . .

Original: , X n are independent and each has the same marginal distribution with CDF F , we say that X 1 , . . .
Revised: , X n are independent and each has the same marginal distribution with CDF F , we say that X 1 , . . .

Original: X n are IID (independent and identically distributed)
Revised: X n are IID (independent and identically distributed)

Original: [Wasserman, 2013].
Revised: [Wasserman, 2013].

Original: For the sake of simplicity, we use X is IID to refer to all the units of
Revised: For the sake of simplicity, we use X is IID to refer to all the units of

Original: X , X 1 , . . .
Revised: X , X 1 , . . .

Original: , X n , being IID.
Revised: , X n , being IID.

Original: A dataset is IID if all variables in it are IID.
Revised: A dataset is IID if all variables in it are IID.

Original: Linear Causal Models A traditional linear causal model is also known as a linear structural causal model (SCM)
Revised: Linear Causal Models A traditional linear causal model is also known as a linear structural causal model (SCM)

Original: [Brito, 2004, Pearl, 2009, Chen and Pearl, 2014].
Revised: [Brito, 2004, Pearl, 2009, Chen and Pearl, 2014].

Original: The edge coefficients on the causal
Revised: The edge coefficients on the causal

Original: DAG represent direct effects.
Revised: DAG represent direct effects.

Original: An open path is collider-free, i.e., there are no head-to-head arrows on this path.
Revised: An open path is collider-free, i.e., there are no head-to-head arrows on this path.

Original: Note that if there exists an open path from W i to V j , it implies W i ̸⊥⊥ V j .
Revised: Note that if there exists an open path from W i to V j , it implies W i ̸⊥⊥ V j .

Original: The value ‘ V al ( p ) ’ of an open path p in a linear model is defined as the product of the edge coefficients on p .
Revised: The value ‘ V al ( p ) ’ of an open path p in a linear model is defined as the product of the edge coefficients on p .

Original: The root of an open path p is defined as the variable on p that is the ancestor of all variables on p .
Revised: The root of an open path p is defined as the variable on p that is the ancestor of all variables on p .

Original: Average Causal Effects
Revised: Average Causal Effects

Original: In this work the query we are primarily interested in generalizing to the non-IID case is the Average Causal Effect (ACE), also named as the Average Treatment Effect (ATE)
Revised: In this work the query we are primarily interested in generalizing to the non-IID case is the Average Causal Effect (ACE), also named as the Average Treatment Effect (ATE)

Original: [Rubin, 1977, Holland, 1988].
Revised: [Rubin, 1977, Holland, 1988].

Original: For consistency, we use ACE to refer to both.
Revised: For consistency, we use ACE to refer to both.

Original: Given a causal model M , the average causal effect (ACE) of X = t vs X = c
Revised: Given a causal model M , the average causal effect (ACE) of X = t vs X = c

Original: ( t and c are constants) on Y for k units is defined as ACE XY = 1 k (cid:80) i
Revised: ( t and c are constants) on Y for k units is defined as ACE XY = 1 k (cid:80) i

Original: ( Y iX i =
Revised: ( Y iX i =

Original: t − Y iX
Revised: t − Y iX

Original: i = c ) .
Revised: i = c ) .

Original: ACE is defined under the assumption that Y i depends only on factors of unit i (including X i ) Holland [1988].
Revised: ACE is defined under the assumption that Y i depends only on factors of unit i (including X i ) Holland [1988].

Original: Without loss of generality, we assume t = c + 1 1 .
Revised: Without loss of generality, we assume t = c + 1 1 .

Original: In linear models, ACE of X on Y can be identified as β Y X , the linear regression slope of Y on X , if there is no backdoor (non-directed open paths) between X and Y [Pearl et al., 2016, Pearl, 2017].
Revised: In linear models, ACE of X on Y can be identified as β Y X , the linear regression slope of Y on X , if there is no backdoor (non-directed open paths) between X and Y [Pearl et al., 2016, Pearl, 2017].

Original:  Graphical Modeling of Interactions 3.1 Interaction models
Revised:  Graphical Modeling of Interactions 3.1 Interaction models

Original: In this section, we define a graphical model derived from traditional causal models M = <
Revised: In this section, we define a graphical model derived from traditional causal models M = <

Original: G, S > (Pearl [2009], definition 7.1.1).
Revised: G, S > (Pearl [2009], definition 7.1.1).

Original: G is the causal graph and S is the set of structural equations of variables.
Revised: G is the causal graph and S is the set of structural equations of variables.

Original: We refer to the variables in a traditional causal model as generic variables .
Revised: We refer to the variables in a traditional causal model as generic variables .

Original: X, C, Y in
Revised: X, C, Y in

Original: Figure 1 are generic variables.
Revised: Figure 1 are generic variables.

Original: An explicit variable is similar to a generic variable except that it represents an attribute/event of one specific unit (or sample or individual).
Revised: An explicit variable is similar to a generic variable except that it represents an attribute/event of one specific unit (or sample or individual).

Original: For example, “treatment ( X ) ” is a generic variable, and “the treatment of unit i ( X i ) ” is an explicit variable.
Revised: For example, “treatment ( X ) ” is a generic variable, and “the treatment of unit i ( X i ) ” is an explicit variable.

Original: [Figure]
Revised: [Figure]

Revised: [Figure]

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

Original: U
Revised: U

Original: [Figure]
Revised: [Figure]

Original: Y
Revised: Y

Original: [Equation]
Revised: [Equation]

Original: U
Revised: U

Original: [Equation]
Revised: [Equation]

Original: Definition 1 (Interaction model M ∗ ( G ∗ , S ∗ ) ) .
Revised: Definition 1 (Interaction model M ∗ ( G ∗ , S ∗ ) ) .

Original: An interaction model, M ∗ ( G ∗ , S ∗ ) , is a causal model where G ∗ is the interaction network and S ∗ is the set of structural equations defining the data generating process of the observed explicit variables.
Revised: An interaction model, M ∗ ( G ∗ , S ∗ ) , is a causal model where G ∗ is the interaction network and S ∗ is the set of structural equations defining the data generating process of the observed explicit variables.

Original: An interaction network, G ∗ , is a directed acyclic graph with each node representing an explicit variable and each directed edge A i
Revised: An interaction network, G ∗ , is a directed acyclic graph with each node representing an explicit variable and each directed edge A i

Original: → B j representing A i causes B j .
Revised: → B j representing A i causes B j .

Original: An example of interaction model M ∗ ( G ∗ , S ∗ ) , is the interaction network, G ∗ , portrayed in Figureand the structural equations S ∗ (part of) specified beside it; U V i denotes the unobserved exogenous error of an explicit variable V i .
Revised: An example of interaction model M ∗ ( G ∗ , S ∗ ) , is the interaction network, G ∗ , portrayed in Figureand the structural equations S ∗ (part of) specified beside it; U V i denotes the unobserved exogenous error of an explicit variable V i .

Original: Observe that interaction networks allow edges between explicit variables of the same unit (e.g., X 1 → Y 1 ), as well as two distinct units (e.g., C 1 → Y 2 ).
Revised: Observe that interaction networks allow edges between explicit variables of the same unit (e.g., X 1 → Y 1 ), as well as two distinct units (e.g., C 1 → Y 2 ).

Original: We are now ready to define an isolated interaction model for an interaction model M ∗ .
Revised: We are now ready to define an isolated interaction model for an interaction model M ∗ .

Original: It is the “ideal” model constructed from M ∗ by eliminating all interactions between units.
Revised: It is the “ideal” model constructed from M ∗ by eliminating all interactions between units.

Original: Definition 2 (Isolated interaction model IM ∗ ( IG ∗ , IS ∗ ) ) .
Revised: Definition 2 (Isolated interaction model IM ∗ ( IG ∗ , IS ∗ ) ) .

Original: IM ∗ ( IG ∗ , IS ∗ ) is the Isolated interaction model of an interaction model M ∗ ( G ∗ , S ∗ ) if IM satisfies the following conditions:
Revised: IM ∗ ( IG ∗ , IS ∗ ) is the Isolated interaction model of an interaction model M ∗ ( G ∗ , S ∗ ) if IM satisfies the following conditions:

Original: IG ∗ =
Revised: IG ∗ =

Original: G ′ where G ′ is the graph obtained by removing from G ∗ all edges A i
Revised: G ′ where G ′ is the graph obtained by removing from G ∗ all edges A i

Original: → B j , i ̸
Revised: → B j , i ̸

Original: = j , 2.
Revised: = j , 2.

Original: IS ∗ =
Revised: IS ∗ =

Original: S ′ where S ′ is the set of equations obtained by removing from each equation
Revised: S ′ where S ′ is the set of equations obtained by removing from each equation

Original: X i =
Revised: X i =

Original: f
Revised: f

Original: ( Pa ( X i ))
Revised: ( Pa ( X i ))

Original: 2 in S ∗ all terms containing any Y j , ∀
Revised: 2 in S ∗ all terms containing any Y j , ∀

Original: j ̸
Revised: j ̸

Original: = i .
Revised: = i .

Original: For example, the interaction model M ∗ ( G ∗ , S ∗ ) has Figure 2 as G ∗ , and Equations (1-3) as part of
Revised: For example, the interaction model M ∗ ( G ∗ , S ∗ ) has Figure 2 as G ∗ , and Equations (1-3) as part of

Original: S ∗ .
Revised: S ∗ .

Original: The isolated model for M ∗ is denoted IM ∗ ( IG ∗ , IS ∗ ) .
Revised: The isolated model for M ∗ is denoted IM ∗ ( IG ∗ , IS ∗ ) .

Original: IG ∗ is given in Figure 3 below.
Revised: IG ∗ is given in Figure 3 below.

Original: And
Revised: And

Original: IS ∗ for Equations (1-3) are given by Equations (4-6).
Revised: IS ∗ for Equations (1-3) are given by Equations (4-6).

Original: 3.2 Symmetry Assumptions
Revised: 3.2 Symmetry Assumptions

Original: In real-world applications, we will have at our disposal limited (usually just one) observations corresponding to a unit which in turn will make it hard to draw useful conclusions if the model is completely arbitrary.
Revised: In real-world applications, we will have at our disposal limited (usually just one) observations corresponding to a unit which in turn will make it hard to draw useful conclusions if the model is completely arbitrary.

Original: In traditional causal inference techniques this is not a problem since they assume IID, which is assuming for each variable, the distribution is the same and independent for
Revised: In traditional causal inference techniques this is not a problem since they assume IID, which is assuming for each variable, the distribution is the same and independent for

Original: [Figure]
Revised: [Figure]

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original: [Equation]
Revised: [Equation]

Original: all units.
Revised: all units.

Original: While we do not make strong assumptions such as IID, we need to make certain weaker symmetry restrictions (definitions 3, 4), in order to quantify bias and identify ACE .
Revised: While we do not make strong assumptions such as IID, we need to make certain weaker symmetry restrictions (definitions 3, 4), in order to quantify bias and identify ACE .

Original: We only require some of the variables are IID instead of all.
Revised: We only require some of the variables are IID instead of all.

Original: Definition 3 (Balanced interaction model M ∗ ( G ∗ , S ∗ ) ) .
Revised: Definition 3 (Balanced interaction model M ∗ ( G ∗ , S ∗ ) ) .

Original: Let M ∗ ( G ∗ , S ∗ ) be an interaction model with isolated model IM ∗ .
Revised: Let M ∗ ( G ∗ , S ∗ ) be an interaction model with isolated model IM ∗ .

Original: M ∗ is a balanced interaction model if IM ∗ has the same unit-model ( IM ∗ i ( IG ∗ i , IS ∗ i ) ) for every unit i .
Revised: M ∗ is a balanced interaction model if IM ∗ has the same unit-model ( IM ∗ i ( IG ∗ i , IS ∗ i ) ) for every unit i .

Original: Let G ∗ be the graph in Figure 2 and S ∗ be the set of equations (1-3) corresponding to M ∗ ( G ∗ , S ∗ ) .
Revised: Let G ∗ be the graph in Figure 2 and S ∗ be the set of equations (1-3) corresponding to M ∗ ( G ∗ , S ∗ ) .

Original: IG ∗ in Figure 3 is the graph and IS ∗ are the equations (4), (5) and (6) that correspond to IM ∗ , which is the isolated model of M ∗ .
Revised: IG ∗ in Figure 3 is the graph and IS ∗ are the equations (4), (5) and (6) that correspond to IM ∗ , which is the isolated model of M ∗ .

Original: The unit-graph for unit 1 is different from unit 2 .
Revised: The unit-graph for unit 1 is different from unit 2 .

Original: Also, the structural equations for Y 1 and Y 2 of the isolated interaction model (Equations (5) and (6)) are different.
Revised: Also, the structural equations for Y 1 and Y 2 of the isolated interaction model (Equations (5) and (6)) are different.

Original: Hence,
Revised: Hence,

Original: M ∗ is not a balanced interaction model.
Revised: M ∗ is not a balanced interaction model.

Original: [Figure]
Revised: [Figure]

Original: For another example, the interaction model M ∗ is balanced where G ∗ is the graph in Figure 4(a), and
Revised: For another example, the interaction model M ∗ is balanced where G ∗ is the graph in Figure 4(a), and

Original: S ∗ is the set of equations given in Figure 4(c).
Revised: S ∗ is the set of equations given in Figure 4(c).

Original: Remark 1.
Revised: Remark 1.

Original: Note that a balanced interaction model M ∗ does not imply that data generated by it are
Revised: Note that a balanced interaction model M ∗ does not imply that data generated by it are

Original: IID.
Revised: IID.

Original: Being balanced only requires all units share the same causal relationships within each unit itself, but permits interactions and effects from other units.
Revised: Being balanced only requires all units share the same causal relationships within each unit itself, but permits interactions and effects from other units.

Original: For example, the parents of explicit variables Y i and Y j , i ̸
Revised: For example, the parents of explicit variables Y i and Y j , i ̸

Original: =
Revised: =

Original: j can be different in G ∗ i.e., Y i can be caused by a set of variables S k corresponding to unit, k , and Y j can be caused by a distinct set of variables T k .
Revised: j can be different in G ∗ i.e., Y i can be caused by a set of variables S k corresponding to unit, k , and Y j can be caused by a distinct set of variables T k .

Original: However, for M ∗ to be balanced it is required that for all distinct units i and j , all Y i have the same relationship with i ’s explicit variables as Y j with j ’s variables.
Revised: However, for M ∗ to be balanced it is required that for all distinct units i and j , all Y i have the same relationship with i ’s explicit variables as Y j with j ’s variables.

Original: We further note that if M ∗∗ is balanced then all the unit-models IM ∗ i ( IG ∗ i , IS ∗ i ) in definition 3 are identical (with no edges between IG ∗ i and IG ∗ j ), and can be succinctly represented by a (single) causal model M ( G, S ) where G and S can be constructed from any IG ∗ i and IS ∗ i by replacing explicit variables with generic variables.
Revised: We further note that if M ∗∗ is balanced then all the unit-models IM ∗ i ( IG ∗ i , IS ∗ i ) in definition 3 are identical (with no edges between IG ∗ i and IG ∗ j ), and can be succinctly represented by a (single) causal model M ( G, S ) where G and S can be constructed from any IG ∗ i and IS ∗ i by replacing explicit variables with generic variables.

Original: In addition to the assumption that the isolated components being the same, it would be helpful if we also have symmetrical assumptions on the underlying distributions of specific sets of variables.
Revised: In addition to the assumption that the isolated components being the same, it would be helpful if we also have symmetrical assumptions on the underlying distributions of specific sets of variables.

Original: For example, it is reasonable to assume all units’ treatments have the same distribution, i.e., for any treatment X
Revised: For example, it is reasonable to assume all units’ treatments have the same distribution, i.e., for any treatment X

Original: = x , all units have an equal chance of getting the treatment X
Revised: = x , all units have an equal chance of getting the treatment X

Original: = x .
Revised: = x .

Original: Definition 4 (Ancestral same-distribution condition (ASDC)) .
Revised: Definition 4 (Ancestral same-distribution condition (ASDC)) .

Original: In the interaction network G ∗ a balanced interaction model, generic variable W to satisfies the ancestral same-distribution condition (ASDC) if for all unit i , 1)
Revised: In the interaction network G ∗ a balanced interaction model, generic variable W to satisfies the ancestral same-distribution condition (ASDC) if for all unit i , 1)

Original: Pa ( W i ) satisfies ASDC, and 2)
Revised: Pa ( W i ) satisfies ASDC, and 2)

Original: Pa ( W i ) ⊆
Revised: Pa ( W i ) ⊆

Original: V ( i ) , and 3) for any different unit j ̸
Revised: V ( i ) , and 3) for any different unit j ̸

Original: = i , Pa ( W i ) and Pa ( W j ) have the same set of generic variables, and their exogenous errors
Revised: = i , Pa ( W i ) and Pa ( W j ) have the same set of generic variables, and their exogenous errors

Original: U W i
Revised: U W i

Original: and U W j have the same distribution.
Revised: and U W j have the same distribution.

Original: (When i=j, the condition is automatically satisfied.)
Revised: (When i=j, the condition is automatically satisfied.)

Original: For example, in Figures 4(a) and 4(b), X satisfies ASDC in the former (assuming the condition on exogenous errors is satisfied) but not in the latter, since in the latter Pa ( X 1 ) ̸ =
Revised: For example, in Figures 4(a) and 4(b), X satisfies ASDC in the former (assuming the condition on exogenous errors is satisfied) but not in the latter, since in the latter Pa ( X 1 ) ̸ =

Original: Pa ( X 2 ) .
Revised: Pa ( X 2 ) .

Original: ASDC implies IID as stated in the following lemma.
Revised: ASDC implies IID as stated in the following lemma.

Original: Lemma 1.
Revised: Lemma 1.

Original: If W satisfies ASDC, then any two explicit variables W i and W j are IID (Independent and Identically Distributed.)
Revised: If W satisfies ASDC, then any two explicit variables W i and W j are IID (Independent and Identically Distributed.)

Original: Remark 2.
Revised: Remark 2.

Original: The descendants of an ASDC variable need not be IID.
Revised: The descendants of an ASDC variable need not be IID.

Original: For example, in Figure 4(a),
Revised: For example, in Figure 4(a),

Original: X satisfies ASDC, and Y i and Y j are descendants of X i and X j .
Revised: X satisfies ASDC, and Y i and Y j are descendants of X i and X j .

Original: Y i and Y j have different sets of parents, making their distributions different, so Y is non-IID.
Revised: Y i and Y j have different sets of parents, making their distributions different, so Y is non-IID.

Original: 3.3 Quantity of Interest: True Average Causal Effect (TACE)
Revised: 3.3 Quantity of Interest: True Average Causal Effect (TACE)

Original: We generalize traditional ACE to the non-IID setting.
Revised: We generalize traditional ACE to the non-IID setting.

Original: Examine the interactions depicted in Figure 5(c).
Revised: Examine the interactions depicted in Figure 5(c).

Original: Unit i ’s
Revised: Unit i ’s

Original: treatment X i affects their outcome through unit j ’s outcome Y j .
Revised: treatment X i affects their outcome through unit j ’s outcome Y j .

Original: X i
Revised: X i

Original: → Y j → Y i is a “spurious” causal path.
Revised: → Y j → Y i is a “spurious” causal path.

Original: We are interested in computing the ACE of a unit’s treatment on their outcome, excluding the effects transmitted via spurious paths from their neighbors/contacts.
Revised: We are interested in computing the ACE of a unit’s treatment on their outcome, excluding the effects transmitted via spurious paths from their neighbors/contacts.

Original: In an experimental setting, interactions can be eliminated by isolating all subjects.
Revised: In an experimental setting, interactions can be eliminated by isolating all subjects.

Original: In an observational setting, where we are given the non-IID data, we are interested in computing the average causal effect of treatment on outcome as if all units were isolated .
Revised: In an observational setting, where we are given the non-IID data, we are interested in computing the average causal effect of treatment on outcome as if all units were isolated .

Original: We present the formal definition below.
Revised: We present the formal definition below.

Original: Definition 5 (True Average Causal Effect ( TACE XY )) .
Revised: Definition 5 (True Average Causal Effect ( TACE XY )) .

Original: Let M ∗ be an interaction model.
Revised: Let M ∗ be an interaction model.

Original: True average causal effect of X on Y , denoted as TACE XY , is defined as the ACE of X on Y in the isolated interaction model IM ∗ corresponding to M ∗ .
Revised: True average causal effect of X on Y , denoted as TACE XY , is defined as the ACE of X on Y in the isolated interaction model IM ∗ corresponding to M ∗ .

Original: TACE is the non-IID version of ACE and is the same as ACE in a traditional causal model where all samples are isolated.
Revised: TACE is the non-IID version of ACE and is the same as ACE in a traditional causal model where all samples are isolated.

Original: Again, without loss of generality, we assume the difference between the treatment value and the outcome value is 1 , i.e. treatment is X
Revised: Again, without loss of generality, we assume the difference between the treatment value and the outcome value is 1 , i.e. treatment is X

Original: = c
Revised: = c

Original: + 

Revised: +

Original: and the outcome is X = c .
Revised: and the outcome is X = c .

Original:  Defining, Quantifying, Detecting and Removing Interaction Bias for TACE
Revised:  Defining, Quantifying, Detecting and Removing Interaction Bias for TACE

Original: Almost all machine learning algorithms including those that employ causal techniques assume that data are IID (Schölkopf [2022], section 3).
Revised: Almost all machine learning algorithms including those that employ causal techniques assume that data are IID (Schölkopf [2022], section 3).

Original: In other words, the theoretical and performance guarantees of these algorithms are based on data being IID.
Revised: In other words, the theoretical and performance guarantees of these algorithms are based on data being IID.

Original: As such it would be useful to determine conditions under which an algorithm meant for IID data can be applied on non-IID data with the certainty that the resulting bias would be negligible.
Revised: As such it would be useful to determine conditions under which an algorithm meant for IID data can be applied on non-IID data with the certainty that the resulting bias would be negligible.

Original: We formally define interaction bias below.
Revised: We formally define interaction bias below.

Original: Definition 6 (Interaction bias) .
Revised: Definition 6 (Interaction bias) .

Original: Let balanced model M ∗ be the true model that generated the (available) non-IID dataset D .
Revised: Let balanced model M ∗ be the true model that generated the (available) non-IID dataset D .

Original: Let Q denote the query of interest and let Q ∗ be its true value.
Revised: Let Q denote the query of interest and let Q ∗ be its true value.

Original: Let A denote an algorithm that outputs an unbiased estimate of Q given data that are IID and the causal graph that generated the IID data.
Revised: Let A denote an algorithm that outputs an unbiased estimate of Q given data that are IID and the causal graph that generated the IID data.

Original: Let G † denote an approximate causal graph constructed under the assumption that D is IID such that no assumption in G † is refuted by D .
Revised: Let G † denote an approximate causal graph constructed under the assumption that D is IID such that no assumption in G † is refuted by D .

Original: Let ˆ Q be the estimate computed by A using G † and D as input.
Revised: Let ˆ Q be the estimate computed by A using G † and D as input.

Original: Interaction bias is given by || Q ∗ − ˆ Q || .
Revised: Interaction bias is given by || Q ∗ − ˆ Q || .

Original: 4.1 Quantifying Bias
Revised: 4.1 Quantifying Bias

Original: We define the two main types of problematic graphical structures in a linear interaction network that introduces bias in the estimation of TACE .
Revised: We define the two main types of problematic graphical structures in a linear interaction network that introduces bias in the estimation of TACE .

Original: Definition 7 (Deflecting bias structure) .
Revised: Definition 7 (Deflecting bias structure) .

Original: A deflecting bias structure for TACE XY in an interaction network G ∗ is an open path between X j and Y i for i ̸
Revised: A deflecting bias structure for TACE XY in an interaction network G ∗ is an open path between X j and Y i for i ̸

Original: = j .
Revised: = j .

Original: Deflecting bias structures are open paths from one unit to another unit.
Revised: Deflecting bias structures are open paths from one unit to another unit.

Original: For example, Figures 5(a) and 5(b) contain deflecting bias structures.
Revised: For example, Figures 5(a) and 5(b) contain deflecting bias structures.

Original: The interaction network in Figure 5(a) has a directed open path between X j and Y i , and the interaction network in Figure 5(b) has a confounded open path between X j and Y i .
Revised: The interaction network in Figure 5(a) has a directed open path between X j and Y i , and the interaction network in Figure 5(b) has a confounded open path between X j and Y i .

Original: Definition 8 (Reflecting bias structure) .
Revised: Definition 8 (Reflecting bias structure) .

Original: A reflecting bias structure for TACE XY in an interaction network G ∗ is an open path between X i and Y i through some explicit variable W j with i ̸
Revised: A reflecting bias structure for TACE XY in an interaction network G ∗ is an open path between X i and Y i through some explicit variable W j with i ̸

Original: = j .
Revised: = j .

Original: Reflecting bias structures are open paths that go from a unit through another unit and back to the same unit.
Revised: Reflecting bias structures are open paths that go from a unit through another unit and back to the same unit.

Original: For example, Figures 5(c) and 5(d) contain a reflecting bias structure.
Revised: For example, Figures 5(c) and 5(d) contain a reflecting bias structure.

Original: In each of them, there is an open path from X i to Y i through Y j .
Revised: In each of them, there is an open path from X i to Y i through Y j .

Original: In some cases, there can be a deflecting bias structure embedded in a reflecting bias structure, as in Figures 5(c) and 5(d).
Revised: In some cases, there can be a deflecting bias structure embedded in a reflecting bias structure, as in Figures 5(c) and 5(d).

Original: However, this is not necessary.
Revised: However, this is not necessary.

Original: Figure 5(e) contains only a reflecting bias structure ( X i → C j → Y i ) but no deflecting bias structure.
Revised: Figure 5(e) contains only a reflecting bias structure ( X i → C j → Y i ) but no deflecting bias structure.

Original: [Figure]
Revised: [Figure]

Original: Theorem 1.
Revised: Theorem 1.

Original: Let M ∗ ( G ∗ , S ∗ ) be a balanced interaction model in which treatment variable X i and outcome variable Y i are not confounded by any variable in V i , ∀ i .
Revised: Let M ∗ ( G ∗ , S ∗ ) be a balanced interaction model in which treatment variable X i and outcome variable Y i are not confounded by any variable in V i , ∀ i .

Original: Let D be the available data generated by M ∗ and let G † be the approximate graph constructed using D .
Revised: Let D be the available data generated by M ∗ and let G † be the approximate graph constructed using D .

Original: Let TACE XY be identifiable in G † and be given by β Y X , the regression coefficient of Y on X .
Revised: Let TACE XY be identifiable in G † and be given by β Y X , the regression coefficient of Y on X .

Original: Let α denote the true value of TACE X,Y in M ∗ .
Revised: Let α denote the true value of TACE X,Y in M ∗ .

Original: If X satisfies ASDC then the interaction bias is given by,
Revised: If X satisfies ASDC then the interaction bias is given by,

Original: [Equation]
Revised: [Equation]

Original: [ ji ] V al ( p ) σ 2 Rp σ 2 X (cid:12)(cid:12)(cid:12) , where P [ iji ] is the set of reflecting bias structures between X i and Y i through any explicit variable
Revised: [ ji ] V al ( p ) σ 2 Rp σ 2 X (cid:12)(cid:12)(cid:12) , where P [ iji ] is the set of reflecting bias structures between X i and Y i through any explicit variable

Original: W j of unit j with i ̸
Revised: W j of unit j with i ̸

Original: = j , P [ ji ] is the set of deflecting bias structures between X j and Y i with i ̸
Revised: = j , P [ ji ] is the set of deflecting bias structures between X j and Y i with i ̸

Original: = j , and R p is the root of path p .
Revised: = j , and R p is the root of path p .

Original: It follows from Theorem 1 that in a balanced interaction model in which no X i and Y i are confounded by any variable in V i , the reflecting and deflecting structures are the only two structures that will bias the identification of TACE .
Revised: It follows from Theorem 1 that in a balanced interaction model in which no X i and Y i are confounded by any variable in V i , the reflecting and deflecting structures are the only two structures that will bias the identification of TACE .

Original: Note that although definition of interaction bias (Definition 6) on
Revised: Note that although definition of interaction bias (Definition 6) on

Original: TACE is for any unbiased estimator for ACE , we focus only on the ordinary least squares estimator in this paper.
Revised: TACE is for any unbiased estimator for ACE , we focus only on the ordinary least squares estimator in this paper.

Original: This is because among the class of unbiased linear estimators, the OLS estimator has the minimum variance
Revised: This is because among the class of unbiased linear estimators, the OLS estimator has the minimum variance

Original: [Johnson et al., 2014].
Revised: [Johnson et al., 2014].

Original: We exemplify theorem 1.
Revised: We exemplify theorem 1.

Original: [Figure]
Revised: [Figure]

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original: [Equation]
Revised: [Equation]

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

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

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

Original: Example 1.
Revised: Example 1.

Original: Figure 6 shows an example of an interaction model with 4 units where X 1 , . . .
Revised: Figure 6 shows an example of an interaction model with 4 units where X 1 , . . .

Original: , X 5 are the treatments, and Y 1 , . . .
Revised: , X 5 are the treatments, and Y 1 , . . .

Original: , Y 5 the outcomes.
Revised: , Y 5 the outcomes.

Original: The numbers on the edges are the edge coefficients.
Revised: The numbers on the edges are the edge coefficients.

Original: C satisfies ASDC, and C i for i
Revised: C satisfies ASDC, and C i for i

Original: = 1 , 2 , 3 , 5 are omitted from the graph for simplicity.
Revised: = 1 , 2 , 3 , 5 are omitted from the graph for simplicity.

Original: Suppose we want to estimate the ACE of X on Y as if the units were isolated: Input: the interaction network G ∗ as shown in Figure 6 (no parameter i.e., S ∗ is not an input), Output: the TACE XY (should equal to 2 ).
Revised: Suppose we want to estimate the ACE of X on Y as if the units were isolated: Input: the interaction network G ∗ as shown in Figure 6 (no parameter i.e., S ∗ is not an input), Output: the TACE XY (should equal to 2 ).

Original: If we estimate ACE XY ignoring the connections between units, our estimator will be ˆ β Y X , with Y =
Revised: If we estimate ACE XY ignoring the connections between units, our estimator will be ˆ β Y X , with Y =

Original: { Y 1 , . . .
Revised: { Y 1 , . . .

Original: , Y 5 } and X =
Revised: , Y 5 } and X =

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

Original: , X 5 } .
Revised: , X 5 } .

Original: This is because ignoring the connections, the graph becomes X i
Revised: This is because ignoring the connections, the graph becomes X i

Original: → Y i separated for i = 1 , . . .
Revised: → Y i separated for i = 1 , . . .

Original: , 5 , so is essentially X
Revised: , 5 , so is essentially X

Original: → Y Pearl [2009].
Revised: → Y Pearl [2009].

Original: However, by Theorem 1,
Revised: However, by Theorem 1,

Original: [Equation]
Revised: [Equation]

Original: Hence, the result is biased, and does not give us what we want.
Revised: Hence, the result is biased, and does not give us what we want.

Original: We show later in Theorem 2 how to compute an unbiased estimate of TACE .
Revised: We show later in Theorem 2 how to compute an unbiased estimate of TACE .

Original: 4.2 Detecting Bias
Revised: 4.2 Detecting Bias

Original: In this section, we provide a graphical criterion resulting from Theorem 1, to detect interaction bias.
Revised: In this section, we provide a graphical criterion resulting from Theorem 1, to detect interaction bias.

Original: Corollary 1.
Revised: Corollary 1.

Original: Let M ∗∗ ( G ∗∗ , S ) be a balanced interaction model in which X satisfies ASDC and
Revised: Let M ∗∗ ( G ∗∗ , S ) be a balanced interaction model in which X satisfies ASDC and

Original: TACE is identified as β Y X =
Revised: TACE is identified as β Y X =

Original: α in the approximate graph, then interaction bias exists iff G ∗∗ contains a reflecting or deflecting bias structure.
Revised: α in the approximate graph, then interaction bias exists iff G ∗∗ contains a reflecting or deflecting bias structure.

Original: For example, Figure 7 contains both reflecting and deflecting bias structures.
Revised: For example, Figure 7 contains both reflecting and deflecting bias structures.

Original: Figure 8 does not contain any bias structure.
Revised: Figure 8 does not contain any bias structure.

Original: So Figure 7 has interaction bias and Figure 8 does not.
Revised: So Figure 7 has interaction bias and Figure 8 does not.

Original: Note that the interactions in Figure 8 do not qualify as bias structures by Definitions 7 and 8.
Revised: Note that the interactions in Figure 8 do not qualify as bias structures by Definitions 7 and 8.

Original: 4.3 Removing Bias
Revised: 4.3 Removing Bias

Original: Theorem 2 presents a technique for computing an unbiased estimate of TACE in cases where theorem 1 predicts significant bias.
Revised: Theorem 2 presents a technique for computing an unbiased estimate of TACE in cases where theorem 1 predicts significant bias.

Original: It proceeds by applying linear regression on a set of samples B that satisfy the condition that no bias inducing structures exist between any two distinct units i and j .
Revised: It proceeds by applying linear regression on a set of samples B that satisfy the condition that no bias inducing structures exist between any two distinct units i and j .

Original: In particular, a subset of samples/units B is termed as a bias-free subset for TACE XY if no reflecting bias structures exist for any i ∈ S and no deflecting bias structures exist in G ∗ S where G ∗ S is the latent projection of G ∗ on B (Definition 2.6.1, Pearl [2009]).
Revised: In particular, a subset of samples/units B is termed as a bias-free subset for TACE XY if no reflecting bias structures exist for any i ∈ S and no deflecting bias structures exist in G ∗ S where G ∗ S is the latent projection of G ∗ on B (Definition 2.6.1, Pearl [2009]).

Original: For example in figure 6, B comprises of units 2 & 5 and G ∗ S is X 2 → Y 2 X 5 → Y 5 .
Revised: For example in figure 6, B comprises of units 2 & 5 and G ∗ S is X 2 → Y 2 X 5 → Y 5 .

Original: However, B is not unique for a given interaction network.
Revised: However, B is not unique for a given interaction network.

Original: Another candidate for B is units 2 & 4 and the associated G ∗ S is X 2 → Y 2 X 4 → Y 4 .
Revised: Another candidate for B is units 2 & 4 and the associated G ∗ S is X 2 → Y 2 X 4 → Y 4 .

Original: An algorithm for constructing B is presented in Algorithm 1, with an example and discussion in the appendix.
Revised: An algorithm for constructing B is presented in Algorithm 1, with an example and discussion in the appendix.

Original: This algorithm starts by randomly initializing B with a sample.
Revised: This algorithm starts by randomly initializing B with a sample.

Original: Then it goes through the rest of the samples and adds a sample to B if its inclusion does not create bias structures in the resultant graph,
Revised: Then it goes through the rest of the samples and adds a sample to B if its inclusion does not create bias structures in the resultant graph,

Original: G ∗ S .
Revised: G ∗ S .

Original: Algorithm 1 Select a bias-free subset B from an interaction network G ∗ and return the largest subset from t iterations
Revised: Algorithm 1 Select a bias-free subset B from an interaction network G ∗ and return the largest subset from t iterations

Original: Input
Revised: Input

Original: Output
Revised: Output

Original: 1: function F IND S UB ( G ∗ , t ) 2:
Revised: 1: function F IND S UB ( G ∗ , t ) 2:

Original: B = ∅ 3: for i = 1 , . .
Revised: B = ∅ 3: for i = 1 , . .

Original: . , t do 4:
Revised: . , t do 4:

Original: Units = randomly sorted list 1 , . .
Revised: Units = randomly sorted list 1 , . .

Original: . , n 5:
Revised: . , n 5:

Original: B = { Units [1] } (The indices for Units start from 1) 6: for i = 2 , . .
Revised: B = { Units [1] } (The indices for Units start from 1) 6: for i = 2 , . .

Original: . , n do 7: if Units [ i ] has no reflecting bias structure in G ∗ then 8: if Units [ i ] has no deflecting bias structure in G ∗ with an element in B then 9:
Revised: . , n do 7: if Units [ i ] has no reflecting bias structure in G ∗ then 8: if Units [ i ] has no deflecting bias structure in G ∗ with an element in B then 9:

Original: B = B ∪ { Unit [ i ] } 10:
Revised: B = B ∪ { Unit [ i ] } 10:

Original: B = B ∪ { B } 11: return Largest B in B
Revised: B = B ∪ { B } 11: return Largest B in B

Original: Theorem 2.
Revised: Theorem 2.

Original: Let G ∗ be an interaction network.
Revised: Let G ∗ be an interaction network.

Original: Given the conditions in Theorem 1 and ‘ B ’ a bias-free subset for G ∗ , TACE XY =
Revised: Given the conditions in Theorem 1 and ‘ B ’ a bias-free subset for G ∗ , TACE XY =

Original: E [ ˆ β Y X ] where the regression coefficient is calculated using only samples in set B .
Revised: E [ ˆ β Y X ] where the regression coefficient is calculated using only samples in set B .

Original: Note that bias-free subset of samples B used in Theorem 2 is not always IID .
Revised: Note that bias-free subset of samples B used in Theorem 2 is not always IID .

Original: While we insist that no reflecting or deflecting bias structures exist in G ∗ S , we do not restrict other forms of interactions among these samples.
Revised: While we insist that no reflecting or deflecting bias structures exist in G ∗ S , we do not restrict other forms of interactions among these samples.

Original: For example, in Figure 8, Units {1, 2, 3} constitute a bias-free subset.
Revised: For example, in Figure 8, Units {1, 2, 3} constitute a bias-free subset.

Original: In this case, Y is not IID ( Y 1 and Y 2 are dependent, Y 2 and Y 3 are dependent) and hence the bias-free subset is non-IID.
Revised: In this case, Y is not IID ( Y 1 and Y 2 are dependent, Y 2 and Y 3 are dependent) and hence the bias-free subset is non-IID.

Original: Also note that to compute an unbiased estimate using Theorem 2, we have at our disposal a smaller set of samples; so the variance of estimation will be larger.
Revised: Also note that to compute an unbiased estimate using Theorem 2, we have at our disposal a smaller set of samples; so the variance of estimation will be larger.

Original: There is a trade off between ignoring interaction (large bias, small variance), and using theorem 2 (no bias, large variance).
Revised: There is a trade off between ignoring interaction (large bias, small variance), and using theorem 2 (no bias, large variance).

Original: It remains future work to quantify the variance of the estimator in Theorem 2 for different interaction models, but in
Revised: It remains future work to quantify the variance of the estimator in Theorem 2 for different interaction models, but in

Original: Section 5, we provide simulation results and case analysis study to empirically show its performance.
Revised: Section 5, we provide simulation results and case analysis study to empirically show its performance.

Original: Applicability of theorems 1 & 2 to real world problems: A natural question that arises at this juncture is whether we need an entire interaction network to apply these results to real world problems.
Revised: Applicability of theorems 1 & 2 to real world problems: A natural question that arises at this juncture is whether we need an entire interaction network to apply these results to real world problems.

Original: Theorem 1 quantifies bias and in doing so reveals to us if and how various factors such as sample size and strength of connections (value of path coefficients) influence bias.
Revised: Theorem 1 quantifies bias and in doing so reveals to us if and how various factors such as sample size and strength of connections (value of path coefficients) influence bias.

Original: This in turn allows us to use available information about the problem from prior experience, domain knowledge or external sources to determine if bias would be negligible or not.
Revised: This in turn allows us to use available information about the problem from prior experience, domain knowledge or external sources to determine if bias would be negligible or not.

Original: Specifically, bias becomes smaller as the number of bias-structure-free samples increases.
Revised: Specifically, bias becomes smaller as the number of bias-structure-free samples increases.

Original: In fact, if the numbers of deflecting and reflecting structures are fixed, the bias terms diminishes as n increases, indicated by the 1 /n for the reflecting bias term and 1 /n ( n − 1) for the deflecting bias term.
Revised: In fact, if the numbers of deflecting and reflecting structures are fixed, the bias terms diminishes as n increases, indicated by the 1 /n for the reflecting bias term and 1 /n ( n − 1) for the deflecting bias term.

Original: It is also evident that if the values of path coefficients are high, V al ( p ) would be high and this will result in increased bias.
Revised: It is also evident that if the values of path coefficients are high, V al ( p ) would be high and this will result in increased bias.

Original: Finally, if the interaction connections are sparse (fewer edges between units), the reduction in the total number of paths could potentially lower bias but more importantly the number of samples in the bias-free set B used in theorem 2 will tend to be larger, which in turn will help in computing better quality estimates.
Revised: Finally, if the interaction connections are sparse (fewer edges between units), the reduction in the total number of paths could potentially lower bias but more importantly the number of samples in the bias-free set B used in theorem 2 will tend to be larger, which in turn will help in computing better quality estimates.

Original:  Experiments 5.1 Simulations
Revised:  Experiments 5.1 Simulations

Original: Simulated Model We randomly generate balanced interaction network with n units (i.e., the sample size is n ), with C i
Revised: Simulated Model We randomly generate balanced interaction network with n units (i.e., the sample size is n ), with C i

Original: → X i → Y i and X i
Revised: → X i → Y i and X i

Original: → M i for all i = 1 , . . . , n .
Revised: → M i for all i = 1 , . . . , n .

Original: For all ordered pairs of distinct units i, j , we randomly add deflecting bias structures in the form of X
Revised: For all ordered pairs of distinct units i, j , we randomly add deflecting bias structures in the form of X

Original: i ← C i → Y j with probability dRate .
Revised: i ← C i → Y j with probability dRate .

Original: For all units i , we randomly add reflecting bias structures in the form of X i → M k → Y i with probability rRate for a random k ̸
Revised: For all units i , we randomly add reflecting bias structures in the form of X i → M k → Y i with probability rRate for a random k ̸

Original: = i .
Revised: = i .

Original: Experiment: Bias of REG It follows from theorem 1 that larger sample sizes and smaller path values on the bias structures result in smaller bias.
Revised: Experiment: Bias of REG It follows from theorem 1 that larger sample sizes and smaller path values on the bias structures result in smaller bias.

Original: We perform two simulations to show how bias varies as a function of sample size and path values.
Revised: We perform two simulations to show how bias varies as a function of sample size and path values.

Original: We simulate data such that for each variable, the exogenous error term follows a Gaussian distribution with mean 0 and standard deviation 1 .
Revised: We simulate data such that for each variable, the exogenous error term follows a Gaussian distribution with mean 0 and standard deviation 1 .

Original: For each set of parameters, we randomly generate an interaction network, and simulate the data 10000 times.
Revised: For each set of parameters, we randomly generate an interaction network, and simulate the data 10000 times.

Original: Each time, we record the result from a naive regression of Y on X (REG).
Revised: Each time, we record the result from a naive regression of Y on X (REG).

Original: As a comparison, we also record the result from Theorem 2 (THM-2).
Revised: As a comparison, we also record the result from Theorem 2 (THM-2).

Original: We run the algorithm (provided in the appendix) to randomly select bias-free subsets for 10 times and select the largest subset.
Revised: We run the algorithm (provided in the appendix) to randomly select bias-free subsets for 10 times and select the largest subset.

Original: Simulation 1: X i → Y i ’s edge coefficient is 100, the edge coefficients of C i
Revised: Simulation 1: X i → Y i ’s edge coefficient is 100, the edge coefficients of C i

Original: → X i , X i →
Revised: → X i , X i →

Original: M j , M j → Y i are all set to 10 , the numbers of deflecting bias structures and additional reflecting bias structures are both 100 .
Revised: M j , M j → Y i are all set to 10 , the numbers of deflecting bias structures and additional reflecting bias structures are both 100 .

Original: Simulation 2: Number of units n = 1000 , X i
Revised: Simulation 2: Number of units n = 1000 , X i

Original: → Y i ’s edge coefficient is 100, the numbers of deflecting bias structures and additional reflecting bias structures are both 100 .
Revised: → Y i ’s edge coefficient is 100, the numbers of deflecting bias structures and additional reflecting bias structures are both 100 .

Original: The results are plotted in Figure 9.
Revised: The results are plotted in Figure 9.

Original: As seen in the plots, as n increases or the path values on the bias structures decreases (both with all other parameters fixed), β Y X from a naive regression approaches TACE .
Revised: As seen in the plots, as n increases or the path values on the bias structures decreases (both with all other parameters fixed), β Y X from a naive regression approaches TACE .

Original: Such results coincide with Theorem 1.
Revised: Such results coincide with Theorem 1.

Original: The β Y X computed by THM-2 is very close to TACE and the two lines almost overlap.
Revised: The β Y X computed by THM-2 is very close to TACE and the two lines almost overlap.

Original: 5.2 Case Study
Revised: 5.2 Case Study

Original: Settings We are interested in analyzing the effect of tutoring time on students’ grades.
Revised: Settings We are interested in analyzing the effect of tutoring time on students’ grades.

Original: In particular, we wish to compute the effect provided through the tutoring program only, but not through “side effects” from other units, such as learning from classmates, although such interactions are encouraged in this scenario.
Revised: In particular, we wish to compute the effect provided through the tutoring program only, but not through “side effects” from other units, such as learning from classmates, although such interactions are encouraged in this scenario.

Original: For instance, unit i might help unit j understand the course materials better which in turn might improve j ’s grade.
Revised: For instance, unit i might help unit j understand the course materials better which in turn might improve j ’s grade.

Original: If unit i helped unit j improve their understanding and unit j states this in the peer review, then it would boost i ’s grade.
Revised: If unit i helped unit j improve their understanding and unit j states this in the peer review, then it would boost i ’s grade.

Original: To construct an interaction network and apply our results, we ask the students to fill out a survey including 1) their tutoring time, 2) their grade, 3) whom they helped, 4) who helped them, 5) peer review score.
Revised: To construct an interaction network and apply our results, we ask the students to fill out a survey including 1) their tutoring time, 2) their grade, 3) whom they helped, 4) who helped them, 5) peer review score.

Original: Construction of the Interaction Network Three generic variables are T (tutoring time in hours), U (understanding of course materials), and R (grade).
Revised: Construction of the Interaction Network Three generic variables are T (tutoring time in hours), U (understanding of course materials), and R (grade).

Original: For each unit i , T i → U i
Revised: For each unit i , T i → U i

Original: → R i .
Revised: → R i .

Original: In addition, if i helped j , add U i
Revised: In addition, if i helped j , add U i

Original: → U j (deflecting bias structure).
Revised: → U j (deflecting bias structure).

Original: If i first helped j and j mentioned this in the peer review and thus boosted i ’s grade, add U i
Revised: If i first helped j and j mentioned this in the peer review and thus boosted i ’s grade, add U i

Original: → U j
Revised: → U j

Original: → R i (reflecting bias structure).
Revised: → R i (reflecting bias structure).

Original: We assume no additional back-and-forth help happens.
Revised: We assume no additional back-and-forth help happens.

Original: Simulation Let there be 500 students, assume each student on average help 5 other students, and the other student has a 0 .
Revised: Simulation Let there be 500 students, assume each student on average help 5 other students, and the other student has a 0 .

Original: 5 chance of helping back.
Revised: 5 chance of helping back.

Original: Let TACE = 2 , and the U i
Revised: Let TACE = 2 , and the U i

Original: → U j and U i → R j edges both have the value 2 .
Revised: → U j and U i → R j edges both have the value 2 .

Original: We randomly generate an interaction network and simulate data based on these parameters.
Revised: We randomly generate an interaction network and simulate data based on these parameters.

Original: Results We apply THM-2 to select a bias-free subset, and compute β GT using data from that subset.
Revised: Results We apply THM-2 to select a bias-free subset, and compute β GT using data from that subset.

Original: We get the result 1 . 963 , with the size of the subset 72 .
Revised: We get the result 1 . 963 , with the size of the subset 72 .

Original: As a result, the effect of tutoring time on students’ grades not through other units is estimated to be 1 . 963 , which is close to the ground truth
Revised: As a result, the effect of tutoring time on students’ grades not through other units is estimated to be 1 . 963 , which is close to the ground truth

Original: TACE (2).
Revised: TACE (2).

Original: We further repeat the experiment 1000 times to show the distribution of the results.
Revised: We further repeat the experiment 1000 times to show the distribution of the results.

Original: Each time a random structure is generated and random data are simulated.
Revised: Each time a random structure is generated and random data are simulated.

Original: THM-2 is on average able to select a bias-free subset of size 76 , and the average recovered TACE = 2 . 0002 .
Revised: THM-2 is on average able to select a bias-free subset of size 76 , and the average recovered TACE = 2 . 0002 .

Original: The result from
Revised: The result from

Original: REG had a significantly high bias with TACE averaging at 194 . 11 .
Revised: REG had a significantly high bias with TACE averaging at 194 . 11 .

Original: Also since every time the data are regenerated, the model is different, and REG uses all the data, it has a larger variance.
Revised: Also since every time the data are regenerated, the model is different, and REG uses all the data, it has a larger variance.

Original: The two plots in Figure 10 show the distribution of results from THM-2 and REG.
Revised: The two plots in Figure 10 show the distribution of results from THM-2 and REG.

Original: The histograms of the results of β Y X computed by THM-2 and REG are shown in Figure 10.
Revised: The histograms of the results of β Y X computed by THM-2 and REG are shown in Figure 10.

Original:  Related Work
Revised:  Related Work

Original: One of the most studied concepts related to interactions among units is interference [Cox, 1958].
Revised: One of the most studied concepts related to interactions among units is interference [Cox, 1958].

Original: Majority of literature in empirical fields assume no-interference.
Revised: Majority of literature in empirical fields assume no-interference.

Original: In fact, SUTVA is a common assumption in causal inference [Rubin, 1978].
Revised: In fact, SUTVA is a common assumption in causal inference [Rubin, 1978].

Original: Recent years have witnessed a rise in papers on interference that employ graphical models.
Revised: Recent years have witnessed a rise in papers on interference that employ graphical models.

Original: These include Ogburn and VanderWeele [2014] that was the first to model interference using DAGs, Sherman and Shpitser [2018] that modeled interference using chain graphs which permits modeling unknown interactions between units and Bhattacharya et al.
Revised: These include Ogburn and VanderWeele [2014] that was the first to model interference using DAGs, Sherman and Shpitser [2018] that modeled interference using chain graphs which permits modeling unknown interactions between units and Bhattacharya et al.

Original: [2020]that proposed structure learning methods for chain graphs.
Revised: [2020]that proposed structure learning methods for chain graphs.

Original: These works rely on partial interference which divides units into equal-sized blocks under the assumption that interactions occur only within a block but not across different blocks.
Revised: These works rely on partial interference which divides units into equal-sized blocks under the assumption that interactions occur only within a block but not across different blocks.

Original: [Nabi et al., 2020] developed methods for identification and estimation of multiple queries under conditions of interference and homophily, and applied the results to the problem of ad-placements.
Revised: [Nabi et al., 2020] developed methods for identification and estimation of multiple queries under conditions of interference and homophily, and applied the results to the problem of ad-placements.

Original: Sobel [2006] was the first to notice the effect of interference in the housing mobility problem, and proposes causal estimands for this application.
Revised: Sobel [2006] was the first to notice the effect of interference in the housing mobility problem, and proposes causal estimands for this application.

Original: Aronow and Samii [2017], Sussman and Airoldi [2017] modeled general interference (without assuming partial inference) by constructing a function to define a unit’s exposure level on the number of treated neighbors they have.
Revised: Aronow and Samii [2017], Sussman and Airoldi [2017] modeled general interference (without assuming partial inference) by constructing a function to define a unit’s exposure level on the number of treated neighbors they have.

Original: The methods are less restricted than partial interference methods, and allow units to be affected by any number of neighbors.
Revised: The methods are less restricted than partial interference methods, and allow units to be affected by any number of neighbors.

Original: However, they are limited to interference and do not handle other forms of interactions.
Revised: However, they are limited to interference and do not handle other forms of interactions.

Original: Jagadeesan et al.
Revised: Jagadeesan et al.

Original: [2020] proposed a quasi-coloring method to estimate direct effect under interference using experimental data.
Revised: [2020] proposed a quasi-coloring method to estimate direct effect under interference using experimental data.

Original: However, it does not easily generalize to observational studies.
Revised: However, it does not easily generalize to observational studies.

Original: Other papers along a similar direction include Fatemi and Zheleva [2020], which proposed experiment design to minimize interaction bias and selection bias at the same time, and Liu and Hudgens [2014], which proposed a two-stage randomization design to minimize interference bias.
Revised: Other papers along a similar direction include Fatemi and Zheleva [2020], which proposed experiment design to minimize interaction bias and selection bias at the same time, and Liu and Hudgens [2014], which proposed a two-stage randomization design to minimize interference bias.

Original: Tchetgen Tchetgen et al.
Revised: Tchetgen Tchetgen et al.

Original: [2021] proposed a g-computation method, which is the first to model general interference using graphical models (chain-graphs), but requires the interference effects to be symmetrical between units.
Revised: [2021] proposed a g-computation method, which is the first to model general interference using graphical models (chain-graphs), but requires the interference effects to be symmetrical between units.

Original: Sävje et al.
Revised: Sävje et al.

Original: [2021] and Hudgens and Halloran [2008] defined queries similar to TACE, named
Revised: [2021] and Hudgens and Halloran [2008] defined queries similar to TACE, named

Original: EATE and PADE, respectively.
Revised: EATE and PADE, respectively.

Original: These queries generalize traditional ACE to allow a unit’s outcome to be affected by treatments of other units.
Revised: These queries generalize traditional ACE to allow a unit’s outcome to be affected by treatments of other units.

Original: However, they do not allow outcomes to be affected by other units’ variables other than treatments.
Revised: However, they do not allow outcomes to be affected by other units’ variables other than treatments.

Original: Hudgens and Halloran [2008] defined six types of queries in the problems involving interference.
Revised: Hudgens and Halloran [2008] defined six types of queries in the problems involving interference.

Original: Work in interference that focuses on different queries/problems include a few as follows.
Revised: Work in interference that focuses on different queries/problems include a few as follows.

Original: VanderWeele et al.
Revised: VanderWeele et al.

Original: [2012b] is the first to decompose the spillover effect (the effect of a unit’s treatment on another’s outcome (Quammen [2012])) to contagion and infectiousness effects using counterfactual mediation analysis.
Revised: [2012b] is the first to decompose the spillover effect (the effect of a unit’s treatment on another’s outcome (Quammen [2012])) to contagion and infectiousness effects using counterfactual mediation analysis.

Original: Shpitser et al.
Revised: Shpitser et al.

Original: [2017] presented decomposition for units with unknown and symmetrical interaction patterns and analyzed different interference paths.
Revised: [2017] presented decomposition for units with unknown and symmetrical interaction patterns and analyzed different interference paths.

Original: In linear models, the contagion and infectiousness effects reduce to the directed paths from X j to Y i .
Revised: In linear models, the contagion and infectiousness effects reduce to the directed paths from X j to Y i .

Original: Moreover, their work does not handle reflection bias.
Revised: Moreover, their work does not handle reflection bias.

Original: Hu et al.
Revised: Hu et al.

Original: [2021] was the first to define and provide estimands for the average indirect effect.
Revised: [2021] was the first to define and provide estimands for the average indirect effect.

Original: VanderWeele et al.
Revised: VanderWeele et al.

Original: [2014] developed methods for sensitivity analysis under interference.
Revised: [2014] developed methods for sensitivity analysis under interference.

Original: Other types of interactions include the contagion effects, which are defined as a unit’s outcome affecting another unit’s outcome [VanderWeele and An, 2013].
Revised: Other types of interactions include the contagion effects, which are defined as a unit’s outcome affecting another unit’s outcome [VanderWeele and An, 2013].

Original: Work on this line usually used longitudinal data, including Burt [1987], Lyons [2011], VanderWeele et al.
Revised: Work on this line usually used longitudinal data, including Burt [1987], Lyons [2011], VanderWeele et al.

Original: [2012a].
Revised: [2012a].

Original: Homophily effects are defined as the behavior of connected units are similar
Revised: Homophily effects are defined as the behavior of connected units are similar

Original: [Jagadeesan et al., 2020].
Revised: [Jagadeesan et al., 2020].

Original: Work in this line include McPherson et al.
Revised: Work in this line include McPherson et al.

Original: [2001], Jagadeesan et al.
Revised: [2001], Jagadeesan et al.

Original: The existing work above does not model interactions using graphical models.
Revised: The existing work above does not model interactions using graphical models.

Original:  Conclusions
Revised:  Conclusions

Original: In this paper, we represent interactions among units using causal graphical models.
Revised: In this paper, we represent interactions among units using causal graphical models.

Original: We derive theorems to quantify the interaction bias for average treatment effects in linear models.
Revised: We derive theorems to quantify the interaction bias for average treatment effects in linear models.

Original: We provide sufficient and necessary graphical conditions to detect interaction bias.
Revised: We provide sufficient and necessary graphical conditions to detect interaction bias.

Original: Additionally, we develop a method to compute an unbiased estimate of causal effect in cases where blindly assuming IID is expected to yield a significant bias.
Revised: Additionally, we develop a method to compute an unbiased estimate of causal effect in cases where blindly assuming IID is expected to yield a significant bias.

Original: Finally, we discuss the performance of our method through simulation studies.
Revised: Finally, we discuss the performance of our method through simulation studies.

Original: Acknowledgments and Disclosure of Funding
Revised: Acknowledgments and Disclosure of Funding

Original: We would like to thank anonymous reviewers for their comments, and Totte Harinen and Rumen Iliev for helpful discussions.
Revised: We would like to thank anonymous reviewers for their comments, and Totte Harinen and Rumen Iliev for helpful discussions.

Original: This research was supported in parts by grants from the National Science
Revised: This research was supported in parts by grants from the National Science

Original: Foundation [#IIS-2106908 and #2231798], Office of Naval Research [#N00014-21-1-2351], andToyota Research Institute of North America [#PO-000897]. Checklist
Revised: Foundation [#IIS-2106908 and #2231798], Office of Naval Research [#N00014-21-1-2351], and

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

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

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

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Original: 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: [Yes] (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 experiments multiple times)?

Original: [No] (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)?

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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?

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Original: 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?

Original: [N/A] 5.

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

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

Original: [N/A]

Original: A

Original: X

Original: Y

Original: X

Original: Y

Original: X

Original: Y

Original: X

Original: Y

Original: C

Original: X

Original: Y

Original: Example 2.

Original: Input: t = 3 , interaction network in Figure 11.

Original: Iteration 1: Units =

Original: B = { 5 , 2 } .

Original: Iteration 2: Units =

Original: B = { 5 , 3 } .

Original: Iteration 3: Units =

Original: B = { 5 , 2 } .

Original: The three choices of B all have the same size 2.

Original: So the output is any of the three choices of B .

Original: Note that the subnetwork formed by 5 and 3 contains a bidirected path between Y 3 and Y 5 (due to the path Y 3 ← C 4 → X 4 → Y 4 → Y 5 ), and this does not constitute a bias structure.

Original: Complexity Analysis The time complexity is O ( tn 2 d p ) .

Original: d is the maximum degree of each node (how many other nodes a node is directly connected to), and p is the length (number of edges) of the longest simple path.

Original: This is polynomial if the degree is bounded.

Original: Lemma 2.

Original: The following two statements are equivalent.

Original: The first statement is used in this algorithm for simpler computation, and the second statement is used in the main text for easier understanding.

Original: For each individual i in B , i has no deflecting bias structure in G ∗ with another individual j in B . 2.

Original: For each individual i in B , i has no deflecting bias structure in the latent projection of G ∗ on B .

Original: The definition of latent projection is by Pearl [2009], as follows.

Original: Definition 9 (Projection[Pearl, 2009]) .

Original: A latent structure L [ O ] = <

Original: D [ O ] , O > is a projection of another latent structure L if and only if:

Original: every unobservable variable of D [ O ] is a parentless common cause of exactly two nonadjacent observable variables; and 2.

Original: for every stable distribution P generated by L , there exists a stable distribution P ′ generated by L

Original: [ O ] such

Original: that I ( P [ O ] )

Original: =

Original: I ( P ′

Original: [ O ] ) .

Original: Proof of Lemma 2.

Original: Proof.

Original: If statement 1 is false, then there exists an open path between X i and Y j in G ∗ , where i, j ∈ B .

Original: The latent projection contains both i and j so the open path still exists, which imply a deflecting bias structure in the latent projection.

Original: If statement 2 is false, then there exists an open path between X i and Y j in the latent projection.

Original: This implies a deflecting bias structure in G ∗ .

Original: B An Additional Simulation

Original: [-Experiment: Subset Size of THM-
-]
Revised: [-Experiment: Subset Size of THM-
-]

Original: We use same parameter settings as the previous experiment, except that we let dRate and rRate vary in 0 .

Original: The subset sizes selected by THMare in Table 1. Observe that as the graph gets denser (larger dRate and rRate ), THM-2 is unable to use most of the input samples.

Original: However, for the tests with samples ≥ 3 , THM-2 yields very accurate estimates.

Original: Given that the ground truth is 100 , the estimates of THM-2 range between 99 . 96 and 100 . 06 .

Original: [Table]

Original: C Proof of the Theorems

Original: All lemmas and proofs are attached in Section D of the appendix.

Original: Theorem 1.

Original: Let M ∗ ( G ∗ , S ∗ ) be a balanced interaction model in which treatment variable X i and outcome variable Y i are not confounded by any variable in V i , ∀ i .

Original: Let D be the available data generated by M ∗ and let G † be the approximate graph constructed using D .

Original: Let TACE XY be identifiable in G † and be given by β Y X , the regression coefficient of Y on X .

Original: Let α denote the true value of TACE X,Y in M ∗ .

Original: If X satisfies ASDC then the interaction bias is given by,

Original: [Equation]

Original: [Table]

Original: where P [ iji ] is the set of reflecting bias structures between X i and Y i through any explicit variable

Original: W j of unit j with i ̸

Original: = j , P [ ji ] is the set of deflecting bias structures between X j and Y i with i ̸

Original: = j , and R p is the root of path p .

Original: Proof.

Original: By Lemma 9,

Original: [Equation]

Original: where P is the set of directed paths from X i to Y i for any i passing through an intermediate node

Original: W j ∈ V ( j ) , i ̸

Original: = j , R [ iji ] is the set of roots of the open paths between X i and Y i through some W j with j ̸

Original: = i , R [ ji ] is the set of roots of the open paths between X j and Y i for j ̸

Original: = i , and c R is the sum of values of the directed paths from a variable R ( ∈ ( R [ iji ] \ { X i } ) or ∈ R [ ji ] ) to Y i not passing through any variable in R [ iji ] ∪ R [ ji ] for any j ̸

Original: = i .

Original: We prove this is equivalent to

Original: [Equation]

Original: where P [ iji ] is the set of open paths between X i and Y i through any W j ∈ V ( j ) with i ̸

Original: = j , P [ ji ] is the set of open paths between X j and Y i through any W j ∈ V ( j ) with i ̸

Original: = j , and R p is the root of path p .

Original: We first check the term (cid:80) R ∈R [ ji ] c R β RX .

Original: For an R that is the root of a path between X j and Y i , since X satisfies ASDC, we must have R ∈ V ( j ) .

Original: Rename it as R j .

Original: We also have β RX =

Original: σ RX /σ 2 X .

Original: By Wright’s Rules, σ RX is equal to the sum of open path values between R and X times the variance of the root of that path.

Original: Recall that R ∈ Anc ( X ) , X satisfies ASDC, so R satisfies ASDC.

Original: So σ RX is equal to the sum of open path values between R j and X j times the variance of the root of that path.

Original: We prove that each term that appears in A = (cid:80)

Original: [Equation]

Original: ∈

Original: [Equation]

Original: ji ]

Original: V al (

Original: [Equation]

Original: X also appears in B =

Original: (cid:80)

Original: [Equation]

Original: c

Original: [Equation]

Original: β RX , and there is no extra term.

Original: Each R p in A is a root between X j and Y i for some j ̸

Original: = i , and must be included if it is a root.

Original: So we just have to check all the roots between X j and Y i for some j ̸

Original: = i .

Original: For each root R p , we check where in B will σ 2 R p /σ

Original: 2 X exist.

Original: When R in B is R p , the term containing σ 2 R p /σ 2 X in β RX is the sum of paths from R p to X j where R p is the root, so is the sum of directed paths from R p to X j .

Original: So the term containing σ 2 R p /σ

Original: 2 X in c R β RX is the sum of paths between Y i and X j through R p with 1) R p being the root and 2) the sub-path from R p to Y i does not go through any variable in R [ iki ] ∪ R [ ki ] for any k ̸

Original: = i .

Original: The terms that are left in V al ( p ) σ 2 Rp σ 2 X to cover in B are the X j − R p − Y i paths

Original: whose sub-path from

Original: R p to Y i go through some variable in R [ iki ] ∪ R [ ki ] for any k ̸

Original: = i .

Original: We just have to go over all types of R in B , and see which ones contain σ 2 R p /σ

Original: 2 X .

Original: Case 1: R ∈

Original: Anc ( R p ) .

Original: There is no such a path in c R or β RX .

Original: c R does not go through R since

Original: R ∈ R [ ji ] c R β RX .

Original: β RX also does not contain σ 2 R p since R ∈

Original: Anc ( R p ) , so R p is never a root on any paths between R and X j .

Original: Hence c R β RX does not contain such a path.

Original: Case 2: R ∈ Desc ( R p ) .

Original: Again, c R does not contain R p .

Original: However β RX contains σ 2 R p .

Original: R p can be a root on some paths between R and X j .

Original: Those paths are from R p to R and R p to X j .

Original: Recall that c R denotes directed paths from R to Y i .

Original: The term that contains σ 2 R p in c R β RX are the paths between X j and Y i , that pass through some variable in R [ iki ] ∪ R [ ki ] ( R ), with R p being the root.

Original: As a result, this case completely cover the missing term.

Original: Case 3: R ⊥⊥ R p .

Original: It is easy to derive that in this case, c R β RX does not contain a path that goes through R p .

Original: Otherwise R and R p would be dependent.

Original: Case 4: R and R p are only connected through common ancestors.

Original: In this case, in any path that contains both R and R p , R p will not be the root.

Original: Their common ancestors will be the roots.

Original: So this case also does not provide any term containing σ 2 R p /σ

Original: 2 X .

Original: We have proved that for every R p in A , the coefficient of σ 2 R p /σ

Original: 2 X (equal to a sum of those paths in

Original: P [ ji ] with R p being the root) is equal to the the coefficient of σ 2 R p /σ

Original: 2 X in B .

Original: As stated before, A and

Original: B have the same set of roots, so they have the same σ 2 R p /σ

Original: 2 X terms.

Original: So the sum of those terms are equal.

Original: Next, we prove the reflecting bias terms are also equal.

Original: Observe that (cid:83) 1 ≤ i ≤ n P [ iji ] =

Original: P , so we just have to prove that (cid:80) p ∈ P [ iji ] V al ( p ) + (cid:80) R ∈ ( R [ iji ] \

Original: { X i } )

Original: c R β RX is equivalent to (cid:80) p ∈ P [ iji ] V al ( p ) σ 2 Rp σ 2 X .

Original: This can be proven using the exact same reasoning above, so we omit the proof.

Original: Thus, the two expressions for E [ ˆ β Y X ] are equivalent.

Original: Corollary 1.

Original: Let M ∗∗ ( G ∗∗ , S ) be a balanced interaction model in which X satisfies ASDC and

Original: TACE is identified as β Y X =

Original: α in the approximate graph, then interaction bias exists iff G ∗∗ contains a reflecting or deflecting bias structure.

Original: Proof.

Original: (if part) Follows from theorem 1.There are two terms that cause bias in theorem 1 and they can be attributed to the two bias structures.

Original: (only if part) Had there been additional structures that caused bias, then theorem 1 would have had additional terms to account for it.

Original: Since theorem 1 has only two bias terms fully accounted for by the two structures, there exist no other structure that creates bias.

Original: Theorem 2.

Original: Let G ∗ be an interaction network.

Original: Given the conditions in Theorem 1 and ‘ B ’ a bias-free subset for G ∗ , TACE XY =

Original: E [ ˆ β Y X ] where the regression coefficient is calculated using only samples in set B .

Original: Proof.

Original: We check the interaction network G ∗ S formed by B , by treating any variable from V ( j ) where j / ∈ S as unobserved.

Original: Next, we calculate E [ ˆ β Y X ] for G ∗ S .

Original: By Theorem 1,

Original: [Equation]

Original: The second term is obtained by summing over paths of the form: X i − · · · − W j − · · · − Y i , and the third term is obtained by summing over paths of the form: X i − · · · − Y j .

Original: These paths do not exist in

Original: G ∗ S .

Original: Hence, the two bias terms are 0 , and E [ ˆ β Y X ]

Original: = α .

Original: D Lemmas

Original: Lemma 1.

Original: If W satisfies ASDC, then any two explicit variables W i and W j are IID (Independent and Identically Distributed.)

Original: Proof.

Original: If W satisfies ASDC, and W i is the root for some i , then from the third property of ASDC,

Original: W i must be the root for all i .

Original: The roots are only caused by their error terms, the error terms are IID (identically distributed and independent), so W is IID.

Original: If W i is not the root for any i , W satisfies ASDC, and all its parents are IID, then we have for any

Original: i

Original: [Equation]

Original: (cid:88)

Original: [Equation]

Original: where c V i is the coefficient of the variable V i on the edge V i

Original: → W i .

Original: Each term is IID for any i ̸

Original: = j .

Original: So W i and W j are IID.

Original: If W i is not the root for any i , W satisfies ASDC, and there exists a parent of W , V such that V i and

Original: V j are not IID.

Original: Then from our previous derivation, there exists a parent of V , V ′ , such that V

Original: ′ i and V ′

Original: j are not IID.

Original: Keep tracing up until a root variable R , such that R i and R j are not IID.

Original: However, this violates our derivation in the beginning, that if a variable is the root and satisfies ASDC, it must be

Original: IID.

Original: We reach a contradiction.

Original: Hence, if W i is not the root for any i , W satisfies ASDC, then all its parents are IID, and W is thus IID.

Original: Lemma 3.

Original: Let X

Original: =

Original: X 1 , . .

Original: , X n } be n IID random variables

Original: where the σ 2 X > 0 , and a random variable W i .

Original: Among X , W i is dependent of X i only, and W i =

Original: aX i + b where a and b are constants.

Original: Then the following expectation exists.

Original: [Equation]

Original: Proof.

Original: We have to prove that the function f

Original: ( X 1 , . . .

Original: , X n , W i ) inside of the expectation is bounded.

Original: For convenience, rewrite it by plugging in W i

Original: =

Original: aX i + b .

Original: [Equation]

Original: For any X j with j ̸ = i , the denominator is a quadratic function on X j , and the numerator is a linear function of X j from the term ¯ X .

Original: For X i , the denominator is a quadratic function on X i , and the numerator is a quadratic function on X i .

Original: Since σ 2 X ̸ = 0 , X 1 , . . .

Original: , X n cannot take on the same value, so the denominator is always positive.

Original: When considering X i as the variable, f might only go to infinity when X i goes to infinity or negative infinity, and same with X j .

Original: When considering X i as the variable, and X j for all other j as constants, the denominator can be written in the form of AX 2 i + BX i + C , with A, B, C being constants.

Original: Hence, the order (of the polynomial) of the denominator is 2 , and the order of the numerator is 2 .

Original: So the limit of f when X i goes to ∞ or −∞ is a finite value equal to the ratio of the coefficient of X 2 i in the numerator divided by the coefficient of X 2 i in the denominator.

Original: When considering X j as the variable, the order of the denominator is 2 , and the order of the numerator is 1 .

Original: So the limit of f when X i goes to ∞ or −∞ is 0 .

Original: Hence, f is bounded.

Original: Lemma 4.

Original: Given a balanced interaction model M ∗∗ ( G ∗∗ , S ∗∗ ) , if generic variables V and X both satisfy ASDC, and dSep ( V i , X i |∅ ) for all i in G ∗∗ , then

Original: [Equation]

Original: Proof.

Original: The d-separation condition implies X i ⊥⊥ V i .

Original: V and X are IID implies that we can treat all

Original: X i ’s as the same variable X , and treat all

Original: V i ’s as the same variable V .

Original: Hence, X ⊥⊥ V and σ XV = 0 , which gives β V X = σ XV

Original: σ 2 X = 0 . Also note that [Equation] Since the ordinary least squares estimator is unbiased, we have E [ ˆ β V X ] =
Revised: Toyota Research Institute of North America [#PO-000897].

Original: β V X

Original: = 0 .

Original: Lemma 5.

Original: Given a balanced interaction model, with the following conditions: 1) X i and Y

Original: i are not confounded by a path containing only variables in V i , ∀ i , and 2)

Original: X i satisfies ASDC.

Original: Then there exists a set S consisting of the following three subsets of explicit variables:

Original: S 1 : X i , 2.

Original: S 2 : the root variables (excluding X i ) of each open path between X j and Y i ( j can be the same as i ), 3.

Original: S 3 : the root variables of this interaction network that are in Anc ( Y i ) and d-separated (by an empty set) from X j for all j ,

Original: such that Y i can be expressed as a linear function of the variables in S i.e.,

Original: [Equation]

Original: where c W t is equal to the sum of the values of the directed paths from W t to Y i that do not go through any variable in S .

Original: Proof.

Original: Consider the following protocol.

Original: • Start from the initial structural equation of Y i , Y i =

Original: f ( Pa ( Y i )) , denoted SE ( Y i ) .

Original: • For each variable

Original: A q in the r.h.s.

Original: of SE ( Y i ) , – if A q ∈ S , keep it.

Original: – if A q / ∈ S and not a root of the network, replace it with its structural equation, A q = g ( Pa ( A q )) and plug it into SE ( Y i ) .

Original: – if A q / ∈ S and is a root of the network, keep it.

Original: • Keep replacing until no more replacement can be done in the r.h.s.

Original: of SE ( Y i ) .

Original: Denote the final SE ( Y i ) as SE f ( Y i ) .

Original: We prove SE f ( Y i ) is

Original: [Equation]

Original: where c W t is equal to the sum of the product of path coefficients of the directed paths from W t to Y i that do not go through any variable in S .

Original: First, we prove that the r.h.s.

Original: of SE f ( Y i ) contains only variables in S .

Original: If it contains a variable,

Original: A r / ∈ S , then A r must be a root variable of the network.

Original: Otherwise it would have been replaced by its parents according to the protocol.

Original: A r / ∈ S , so A r / ∈ S 3 , hence A r must be d-connected (given an empty set) to at least one X j for some j .

Original: Since A r is a root of the network, A r must be the ancestor of X j .

Original: We next discuss if it is X j for j = i or j ̸ = i .

Original: • j = i , i.e., A r is an ancestor of X i .

Original: Since X is ASDC, X i cannot be caused by a variable belonging to another unit.

Original: Hence, we have r = i .

Original: If all directed paths from A r to Y i pass through variables in S , then A r cannot be replaced into the r.h.s.

Original: of SE f ( Y i ) .

Original: Hence, there exists at least one directed path from A r to Y i that does not pass through any variable in S , which we denote as p d .

Original: Since A r is an ancestor of X i and A r to Y i is a directed path not through S (including X i ), there exists a confounding path between X i and Y i through A r .

Original: Since X i and Y i are not confounded by only variables of i , p d must go through a variable of a different unit, and is the root of that confounding path.

Original: However, then A r ∈ S 2 by definition, which contradicts the assumption that A r / ∈ S .

Original: • j ̸

Original: = i , i.e., A r is an ancestor of X j for some j ̸

Original: = i .

Original: Again, there exists at least one directed path from A r to Y i that does not pass through any variable in S , which we denote as p d .

Original: Since A r is ancestor to both X j and Y i , there is a confounding path between X j and Y i through A r .

Original: A r is the root on this path, which implies A r ∈ S 3 , and contradicts the assumption that A r / ∈ S .

Original: Thus, our counterproof assumption is wrong, which means the r.h.s.

Original: of SE f ( Y i ) generated by the above protocol contains only variables in S .

Original: Next we prove that the coefficients C W t for each W t S in the linear combination is equal to the sum of the values of the directed paths from W t to Y i that do not go through any variable in S .

Original: In the protocol above, every time a variable is replaced by its parents, there is a multiplier equal to the directed edge between each parent and the variable.

Original: For example, in SE Y i , a term is γC i .

Original: If C i is replaced by its parents, D j and E k , where C i =

Original: δD j + θE k , then the term in SE Y i becomes γ ( δD j + θE k ) .

Original: So the coefficient of D j is C

Original: i ’s coefficient γ multiplied by δ , the edge D j → C i .

Original: Since replacements of a variable stops if it is in S , we have that the final coefficient of a variable is equal to the sum of all directed paths from that variable to Y i , which do not pass through any other variable in S .

Original: Lemma 6.

Original: Given n IID random variables X 1 , . . .

Original: , X n , and n IID random variables R 1 , . .

Original: R n .

Original: For each i , R i is not independent of X i only.

Original: Then we have

Original: [Equation]

Original: and β RX is the OLS regression coefficient of R on X , treating X 1 , . . .

Original: , X n as a single variable X , and R 1 , . . .

Original: , R n as a single variable R .

Original: Proof.

Original: The above expression only depends on i , and from the property of IID, it is the same for any i .

Original: We sum over i for that expression, and get

Original: [Equation]

Original: Divided by n on both sides, we have the equation in the lemma.

Original: Lemma 7.

Original: Given n IID random variables X 1 , . . .

Original: , X n , and n IID random variables R 1 , . .

Original: R n .

Original: For each i , R i is not independent of X i only.

Original: Then we have

Original: [Equation]

Original: for i ̸

Original: = j , and β RX is the OLS regression coefficient of R on X , treating X 1 , . . .

Original: , X n as a single variable X , and R 1 , . . .

Original: , R n as a single variable R .

Original: Proof.

Original: Denote the expectation of interest as E ij .

Original: X and R are both IID regarding different units, and

Original: X i and R j are independent for i ̸

Original: = j .

Original: Thus, E ij = E i ′

Original: j , for any i ′ ̸

Original: = j .

Original: Below when the sum is over i ̸

Original: = j , it means summing over i ∈ { 1 , . . . , n } \ { j } .

Original: We have

Original: [Equation]

Original: By Lemma 6, we have

Original: [Equation]

Original: Divided by ( n − 1) on both sides, we get the equation we wanted to prove.

Original: Lemma 8.

Original: Given n IID random variables X 1 , . . .

Original: , X n , and a variable L t independent of X 1 , . . .

Original: , X n .

Original: Then we have

Original: [Equation]

Original: Proof.

Original: Denote the expectation of interest as E i , then E i =

Original: E j for any i, j , since X i and X j are IID.

Original: So we have

Original: [Equation]

Original: To prove Theorem 1, we first prove a slightly different version of it, Lemma 9.

Original: Lemma 9.

Original: Given the interaction network G ∗ of a balanced linear interaction model, with X i and Y i not confounded by any variable in V i , ∀ i .

Original: Given that X satisfies ASDC, then the expected value of the OLS estimator ˆ β Y X is given by

Original: [Equation]

Original: where P is the set of directed paths from X i to Y i for all i through any W j ∈ V ( j ) with i ̸

Original: = j , R [ iji ] is the set of roots of the open paths between X i and Y i through some W j with j ̸

Original: = i , R [ ji ] is the set of roots of the open paths between X j and Y i for j ̸

Original: = i , and c R is the sum of values of the directed paths from a variable R ( ∈ ( R [ iji ] \ { X i } ) or ∈ R [ ji ] ) to Y i not passing through any variable in

Original: R [ iji ] ∪ R

Original: [ ji ] for any j ̸ = i .

Original: Proof.

Original: [Equation]

Original: 

Original: Y i is can be written as a linear combination of the set in Lemma 5, S .

Original: By Lemma 5, S is composed of

Original: X i , 2.

Original: the root variables (excluding X i ) of each open path between X j and Y i , and 3.

Original: the root variables of this interaction network that are in Anc ( Y i ) and d-separated (by an empty set) from X j for all j , denoted by L i

Original: The second component can be further divided into two sub-components as follows.

Original: R [ iji ] \ { X i } , the set of roots of the open paths between X i and Y i through some W j with j ̸

Original: = i , with X i excluded, and 2.

Original: R [ ji ] , the set of roots of the open paths between X j and Y i for i ̸

Original: = j .

Original: We have

Original: [Equation]

Original: where c i , c R , and c L denote coefficients for the linear combination.

Original: The variables in the above expression are S , i.e., S = R [ iji ] ∪ R [ ji ] ∪ L i .

Original: Next, we compute the coefficients

Original: c i , c R , c L .

Original: c i is the sum of the directed path values from X i to Y i not passing through any variable in S .

Original: There are three types of directed paths from X i to Y i :

Original: the directed edge X i

Original: → Y i , 2. directed paths

Original: X i → · · · → V i → · · · → Y i , and 3.

Original: directed paths

Original: X i → · · · → V j → · · · → Y i for j ̸

Original: = i .

Original: The first two types belong to TACE by definition.

Original: So c i = α

Original: + c i 3 , where c i 3 is the coefficient contributed by the third type of directed paths.

Original: Note that V j cannot be a root of another path between

Original: X k and Y l for some k ̸ = l .

Original: This is because V j is caused by X i , so V cannot be ASDC, so X cannot be ASDC since X k is caused by V j , which violates the assumption that X is ASDC.

Original: Hence, c i 3 is equal to the sum of all directed paths from X i to Y i through some variable V j for any j , which is equal to (cid:80) p ∈P in the lemma statement.

Original: For the second and third components in Y i , each c R is the sum of the directed paths (multiplications of edge coefficients) from R to Y i not through variables in S .

Original: This follows from Lemma 5.

Original: We have

Original: [Equation]

Original: For the first term: similar to the way ¯ Y is removed before, in the first term, we can change X i to

Original: X i − ¯ X .

Original: The numerator and the denominator are the same in the expectation.

Original: So the first term is α .

Original: The second term is equal to

Original: [Equation]

Original: By Lemma 6, it becomes

Original: [Equation]

Original: where c i 3 is the sum of directed paths from X i to Y i through V j for any j ̸

Original: =

Original: i and any V .

Original: For the third term: we look at one single R first.

Original: R is the root variable of an open path between X i and Y i through some W j with j ̸

Original: = i , so R causes X i .

Original: Then R must belong to unit i since X satisfies

Original: ASDC.

Original: Since R is the root,

Original: R ∈

Original: Anc ( X ) , so R satisfies ASDC, and is IID for different units.

Original: So we relabel this R as R i , and we have IID R 1 , . . .

Original: , R n .

Original: Applying Lemma 6, we have the expectation term is equal to β RX /n .

Original: c R is the sum of the directed paths from R i to Y i , not through variables in S .

Original: So the third term is equal to

Original: [Equation]

Original: For the fourth term: we look at one single R first.

Original: R is the root variable of an open path between X j and Y i , for some j ̸

Original: = i , so either R causes X j or R = X j .

Original: If R causes X j , then R must belong to unit j , because X satisfies ASDC.

Original: So either case R belongs to unit j .

Original: Since R is the root,

Original: R ∈

Original: Anc ( X ) , so R satisfies ASDC, and is IID for different units.

Original: So we relabel this R as R j , and we have IID

Original: R 1 , . . .

Original: , R n .

Original: Applying Lemma 7, we have the expectation term is equal to − β RX / ( n ( n − 1)) .

Original: c R is the sum of the directed paths from R j to Y i , not through variables in S .

Original: So the fourth term is equal to

Original: [Equation]

Original: The fifth term is 0 by Lemma 4.

Original: Finally, recall that V al ( p ) denotes the value of an open path p .

Original: Plugging the above values back into the expression for E [ ˆ β Y X ] , we have the results as in Lemma 9.