Original: C AUSAL E STIMATION FOR T EXT D ATA WITH (A PPARENT ) O VERLAP V IOLATIONS
Revised: C AUSAL E STIMATION FOR T EXT D ATA WITH (A PPARENT ) O VERLAP V IOLATIONS

Original: A BSTRACT
Revised: A BSTRACT

Original: Consider the problem of estimating the causal effect of some attribute of a text document; for example: what effect does writing a polite vs. rude email have on response time?
Revised: Consider the problem of estimating the causal effect of some attribute of a text document; for example: what effect does writing a polite vs. rude email have on response time?

Original: To estimate a causal effect from observational data, we need to adjust for confounding aspects of the text that affect both the treatment and outcome—e.g., the topic or writing level of the text.
Revised: To estimate a causal effect from observational data, we need to adjust for confounding aspects of the text that affect both the treatment and outcome—e.g., the topic or writing level of the text.

Original: These confounding aspects are unknown a priori, so it seems natural to adjust for the entirety of the text (e.g., using a transformer).
Revised: These confounding aspects are unknown a priori, so it seems natural to adjust for the entirety of the text (e.g., using a transformer).

Original: However, causal identiﬁcation and estimation procedures rely on the assumption of overlap: for all levels of the adjustment variables, there is randomness leftover so that every unit could have (not) received treatment.
Revised: However, causal identiﬁcation and estimation procedures rely on the assumption of overlap: for all levels of the adjustment variables, there is randomness leftover so that every unit could have (not) received treatment.

Original: Since the treatment here is itself an attribute of the text, it is perfectly determined, and overlap is apparently violated.
Revised: Since the treatment here is itself an attribute of the text, it is perfectly determined, and overlap is apparently violated.

Original: The purpose of this paper is to show how to handle causal identiﬁcation and obtain robust causal estimation in the presence of apparent overlap violations.
Revised: The purpose of this paper is to show how to handle causal identiﬁcation and obtain robust causal estimation in the presence of apparent overlap violations.

Original: In brief, the idea is to use supervised representation learning to produce a data representation that preserves confounding information while eliminating information that is only predictive of the treatment.
Revised: In brief, the idea is to use supervised representation learning to produce a data representation that preserves confounding information while eliminating information that is only predictive of the treatment.

Original: This representation then sufﬁces for adjustment and satisﬁes overlap.
Revised: This representation then sufﬁces for adjustment and satisﬁes overlap.

Original: Adapting results on non-parametric estimation, we ﬁnd that this procedure is robust to conditional outcome misestimation, yielding a low-absolute-bias estimator with valid uncertainty quantiﬁcation under weak conditions.
Revised: Adapting results on non-parametric estimation, we ﬁnd that this procedure is robust to conditional outcome misestimation, yielding a low-absolute-bias estimator with valid uncertainty quantiﬁcation under weak conditions.

Original: Empirical results show strong improvements in bias and uncertainty quantiﬁcation relative to the natural baseline.
Revised: Empirical results show strong improvements in bias and uncertainty quantiﬁcation relative to the natural baseline.

Original: Code, demo data and a tutorial are available at https://github.com/gl-ybnbxb/TI-estimator .
Revised: Code, demo data and a tutorial are available at https://github.com/gl-ybnbxb/TI-estimator .

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: We consider the problem of estimating the causal effect of an attribute of a passage of text on some downstream outcome.
Revised: We consider the problem of estimating the causal effect of an attribute of a passage of text on some downstream outcome.

Original: For example, what is the effect of writing a polite or rude email on the amount of time it takes to get a response?
Revised: For example, what is the effect of writing a polite or rude email on the amount of time it takes to get a response?

Original: In principle, we might hope to answer such questions with a randomized experiment.
Revised: In principle, we might hope to answer such questions with a randomized experiment.

Original: However, this can be difﬁcult in practice—e.g., if poor outcomes are costly or take long to gather.
Revised: However, this can be difﬁcult in practice—e.g., if poor outcomes are costly or take long to gather.

Original: Accordingly, in this paper, we will be interested in estimating such effects using observational data.
Revised: Accordingly, in this paper, we will be interested in estimating such effects using observational data.

Original: There are three steps to estimating causal effects using observational data (See Chapter
Revised: There are three steps to estimating causal effects using observational data (See Chapter

Original: Murphy (2023)).
Revised: Murphy (2023)).

Original: First, we need to specify a concrete causal quantity as our estimand.
Revised: First, we need to specify a concrete causal quantity as our estimand.

Original: That is, give a formal quantity target of estimation corresponding to the high-level question of interest.
Revised: That is, give a formal quantity target of estimation corresponding to the high-level question of interest.

Original: The next step is causal identiﬁcation: we need to prove that this causal estimator can, in principle, be estimated using only observational data.
Revised: The next step is causal identiﬁcation: we need to prove that this causal estimator can, in principle, be estimated using only observational data.

Original: The standard approach for identiﬁcation relies on adjusting for confounding variables that affect both the treatment and the outcome.
Revised: The standard approach for identiﬁcation relies on adjusting for confounding variables that affect both the treatment and the outcome.

Original: For identiﬁcation to hold, our adjustment variables must satisfy two conditions: unconfoundedness and overlap.
Revised: For identiﬁcation to hold, our adjustment variables must satisfy two conditions: unconfoundedness and overlap.

Original: The former requires the adjustment variables contain sufﬁcient information on all common causes.
Revised: The former requires the adjustment variables contain sufﬁcient information on all common causes.

Original: The latter requires that the adjustment variable does not contain enough information about treatment assignment to let us perfectly predict it.
Revised: The latter requires that the adjustment variable does not contain enough information about treatment assignment to let us perfectly predict it.

Original: Intuitively, to disentangle the effect of treatment from the effect of confounding, we must observe each treatment state at all levels of confounding.
Revised: Intuitively, to disentangle the effect of treatment from the effect of confounding, we must observe each treatment state at all levels of confounding.

Original: The ﬁnal step is estimation using a ﬁnite data sample.
Revised: The ﬁnal step is estimation using a ﬁnite data sample.

Original: Here, overlap also turns out to be critically important as a major determinant of the best possible accuracy (asymptotic variance) of the estimator Chernozhukov et al.
Revised: Here, overlap also turns out to be critically important as a major determinant of the best possible accuracy (asymptotic variance) of the estimator Chernozhukov et al.

Original: Since the treatment is a linguistic property, it is often reasonable to assume that text data has information about all common causes of the treatment and the outcome.
Revised: Since the treatment is a linguistic property, it is often reasonable to assume that text data has information about all common causes of the treatment and the outcome.

Original: Thus, we may aim to satisfy unconfoundedness in the text setting by adjusting for all the text as the confounding part.
Revised: Thus, we may aim to satisfy unconfoundedness in the text setting by adjusting for all the text as the confounding part.

Original: However, doing so brings about overlap violation.
Revised: However, doing so brings about overlap violation.

Original: Since the treatment is a linguistic property determined by the text, the probability of treatment given any text is either 0 or 1.
Revised: Since the treatment is a linguistic property determined by the text, the probability of treatment given any text is either 0 or 1.

Original: The polite / rude tone is determined by the text itself.
Revised: The polite / rude tone is determined by the text itself.

Original: Therefore, overlap does not hold if we naively adjust for all the text as the confounding part.
Revised: Therefore, overlap does not hold if we naively adjust for all the text as the confounding part.

Original: This problem is the main subject of this paper.
Revised: This problem is the main subject of this paper.

Original: Or, more precisely, our goal is to ﬁnd a causal estimand, causal identiﬁcation conditions, and a robust estimation procedure that will allow us to effectively estimate causal effects even in the presence of such (apparent) overlap violations.
Revised: Or, more precisely, our goal is to ﬁnd a causal estimand, causal identiﬁcation conditions, and a robust estimation procedure that will allow us to effectively estimate causal effects even in the presence of such (apparent) overlap violations.

Original: In fact, there is an obvious ﬁrst approach: simply use a standard plug-in estimation procedure that relies only on modeling the outcome from the text and treatment variables.
Revised: In fact, there is an obvious ﬁrst approach: simply use a standard plug-in estimation procedure that relies only on modeling the outcome from the text and treatment variables.

Original: In particular, do not make any explicit use of the propensity score, the probability each unit is treated.
Revised: In particular, do not make any explicit use of the propensity score, the probability each unit is treated.

Original: Pryzant et al.
Revised: Pryzant et al.

Original: (2020) use an approach of this kind and show it is reasonable in some situations.
Revised: (2020) use an approach of this kind and show it is reasonable in some situations.

Original: Indeed, we will see in Sections 3 and 4 that this procedure can be interpreted as a point estimator of a controlled causal effect.
Revised: Indeed, we will see in Sections 3 and 4 that this procedure can be interpreted as a point estimator of a controlled causal effect.

Original: Even once we understand what the implied causal estimand is, this approach has a major drawback: the estimator is only accurate when the text-outcome model converges at a very fast rate.
Revised: Even once we understand what the implied causal estimand is, this approach has a major drawback: the estimator is only accurate when the text-outcome model converges at a very fast rate.

Original: This is particularly an issue in the text setting, where we would like to use large, ﬂexible, deep learning models for this relationship.
Revised: This is particularly an issue in the text setting, where we would like to use large, ﬂexible, deep learning models for this relationship.

Original: In practice, we ﬁnd that this procedure works poorly: the estimator has signiﬁcant absolute bias and (the natural approach to) uncertainity quantiﬁcation almost never includes the estimand true value; see Section 5.
Revised: In practice, we ﬁnd that this procedure works poorly: the estimator has signiﬁcant absolute bias and (the natural approach to) uncertainty quantiﬁcation almost never includes the estimand true value; see Section 5.

Original: The contribution of this paper is a method for robustly estimating causal effects in text.
Revised: The contribution of this paper is a method for robustly estimating causal effects in text.

Original: The main idea is to break estimation into a two-stage procedure, where in the ﬁrst stage we learn a representation of the text that preserves enough information to account for confounding, but throws away enough information to avoid overlap issues.
Revised: The main idea is to break estimation into a two-stage procedure, where in the ﬁrst stage we learn a representation of the text that preserves enough information to account for confounding, but throws away enough information to avoid overlap issues.

Original: Then, we use this representation as the adjustment variables in a standard double machine-learning estimation procedure Chernozhukov et al.
Revised: Then, we use this representation as the adjustment variables in a standard double machine-learning estimation procedure Chernozhukov et al.

Original: (2016; 2017a).
Revised: (2016; 2017a).

Original: To establish this method, the contributions of this paper are:
Revised: To establish this method, the contributions of this paper are:

Original: We give a formal causal estimand corresponding to the text-attribute question.
Revised: We give a formal causal estimand corresponding to the text-attribute question.

Original: We show this estimand is causally identiﬁed under weak conditions, even in the presence of apparent overlap issues.
Revised: We show this estimand is causally identiﬁed under weak conditions, even in the presence of apparent overlap issues.

Original: We show how to efﬁciently estimate this quantity using the adapted double-ML technique just described.
Revised: We show how to efﬁciently estimate this quantity using the adapted double-ML technique just described.

Original: We show that this estimator admits a central limit theorem at a fast ( p n ) rate under weak conditions on the rate at which the ML model learns the text-outcome relationship (namely, convergence at n 1 / 4 rate).
Revised: We show that this estimator admits a central limit theorem at a fast ( p n ) rate under weak conditions on the rate at which the ML model learns the text-outcome relationship (namely, convergence at n 1 / 4 rate).

Original: This implies absolute bias decreases rapidly, and an (asymptotically) valid procedure for uncertainity quantiﬁcation.
Revised: This implies absolute bias decreases rapidly, and an (asymptotically) valid procedure for uncertainty quantiﬁcation.

Original: We test the performance of this procedure empirically, ﬁnding signiﬁcant improvements in bias and uncertainty quantiﬁcation relative to the outcome-model-only baseline.
Revised: We test the performance of this procedure empirically, ﬁnding signiﬁcant improvements in bias and uncertainty quantiﬁcation relative to the outcome-model-only baseline.

Original: Related work The most related literature is on causal inference with text variables.
Revised: Related work The most related literature is on causal inference with text variables.

Original: Papers include treating text as treatment Pryzant et al.
Revised: Papers include treating text as treatment Pryzant et al.

Original: (2020); Wood-Doughty et al.
Revised: (2020); Wood-Doughty et al.

Original: Egami et al.
Revised: Egami et al.

Original: (2018); Fong & Grimmer (2016); Wang & Culotta (2019);
Revised: (2018); Fong & Grimmer (2016); Wang & Culotta (2019);

Original: Tan et al.
Revised: Tan et al.

Original: (2014)), as outcome Egami et al.
Revised: (2014)), as outcome Egami et al.

Original: ; Sridhar & Getoor (2019), as confounder Veitch et al.
Revised: ; Sridhar & Getoor (2019), as confounder Veitch et al.

Original: ; Roberts et al.
Revised: ; Roberts et al.

Original: ; Mozer et al.
Revised: ; Mozer et al.

Original: ; Keith et al.
Revised: ; Keith et al.

Original: (2020), and discovering or predicting causality from text del Prado Martin & Brendel (2016);
Revised: (2020), and discovering or predicting causality from text del Prado Martin & Brendel (2016);

Original: Tabari et al.
Revised: Tabari et al.

Original: Balashankar et al.
Revised: Balashankar et al.

Original: ; Mani & Cooper (2000).
Revised: ; Mani & Cooper (2000).

Original: There are also numerous applications using text to adjust for confounding (e.g., Olteanu et al., 2017; Hall, 2017; Kiciman et al., 2018; Sridhar et al., 2018; Sridhar & Getoor, 2019; Saha et al., 2019; Karell & Freedman, 2019; Zhang et al., 2020).
Revised: There are also numerous applications using text to adjust for confounding (e.g., Olteanu et al., 2017; Hall, 2017; Kiciman et al., 2018; Sridhar et al., 2018; Sridhar & Getoor, 2019; Saha et al., 2019; Karell & Freedman, 2019; Zhang et al., 2020).

Original: Of these, Pryzant et al.
Revised: Of these, Pryzant et al.

Original: (2020) also address non-parametric es- timation of the causal effect of text attributes.
Revised: (2020) also address non-parametric es- timation of the causal effect of text attributes.

Original: Their focus is primarily on mismeasurement of the treatments, while our motivation is robust estimation.
Revised: Their focus is primarily on mismeasurement of the treatments, while our motivation is robust estimation.

Original: This paper also relates to work on causal estimation with (near) overlap violations.
Revised: This paper also relates to work on causal estimation with (near) overlap violations.

Original: D’Amour et al.
Revised: D’Amour et al.

Original: (2021) points out high-dimensional adjustment (e.g., Rassen et al., 2011;
Revised: (2021) points out high-dimensional adjustment (e.g., Rassen et al., 2011;

Original: Louizos et al., 2017; Li et al., 2016; Athey et al., 2017) suffers from overlap issues.
Revised: Louizos et al., 2017; Li et al., 2016; Athey et al., 2017) suffers from overlap issues.

Original: Extra assumptions such as sparsity are often needed to meet the overlap condition.
Revised: Extra assumptions such as sparsity are often needed to meet the overlap condition.

Original: These results do not directly apply here because we assume there exists a low-dimensional summary that sufﬁces to handle confounding.
Revised: These results do not directly apply here because we assume there exists a low-dimensional summary that sufﬁces to handle confounding.

Original: D’Amour & Franks (2021) studies summary statistics that sufﬁce for identiﬁcation, which they call deconfounding scores.
Revised: D’Amour & Franks (2021) studies summary statistics that sufﬁce for identiﬁcation, which they call deconfounding scores.

Original: The supervised representation learning approach in this paper can be viewed as an extremal case of the deconfounding score.
Revised: The supervised representation learning approach in this paper can be viewed as an extremal case of the deconfounding score.

Original: However, they consider the case where ordinary overlap holds with all observed features, with the aim of using both the outcome model and propensity score to ﬁnd efﬁcient statistical estimation procedures (in a linear-gaussian setting).
Revised: However, they consider the case where ordinary overlap holds with all observed features, with the aim of using both the outcome model and propensity score to ﬁnd efﬁcient statistical estimation procedures (in a linear-gaussian setting).

Original: This does not make sense in the setting we consider.
Revised: This does not make sense in the setting we consider.

Original: Additionally, our main statistical result (robustness to outcome model estimation) is new.
Revised: Additionally, our main statistical result (robustness to outcome model estimation) is new.

Original:  N OTATION AND P ROBLEM S ETUP
Revised:  N OTATION AND P ROBLEM S ETUP

Original: [Table]
Revised: [Table]

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Revised: /VuqkJr/2MyyQ1KNniozAVxMRkdjUZcIXMiIkFyhS3uxI2oooyY7Mp2RC85ZNXoXVR9Sw3Liu1mzyOIpzAKZyDB1dQgzuoQxMYIDzDK7w5j86L

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Original: [Figure]
Revised: [Figure]

Original: We follow the causal setup of Pryzant et al.
Revised: We follow the causal setup of Pryzant et al.

Original: We are interested in estimating the causal effect of treatment A on outcome Y .
Revised: We are interested in estimating the causal effect of treatment A on outcome Y .

Original: For example, how does writing a negative sentiment ( A ) review ( X ) affect product sales ( Y )?
Revised: For example, how does writing a negative sentiment ( A ) review ( X ) affect product sales ( Y )?

Original: There are two immediate challenges to estimating such effects with observed text data.
Revised: There are two immediate challenges to estimating such effects with observed text data.

Original: First, we do not actually observe A , which is the intent of the writer.
Revised: First, we do not actually observe A , which is the intent of the writer.

Original: Instead, we only observe ˜ A , a version of A that is inferred from the text itself.
Revised: Instead, we only observe ˜ A , a version of A that is inferred from the text itself.

Original: In this paper, we will assume that
Revised: In this paper, we will assume that

Original: A = ˜
Revised: A = ˜

Original: A almost surely—e.g., a reader can always tell if a review was meant to be negative or positive.
Revised: A almost surely—e.g., a reader can always tell if a review was meant to be negative or positive.

Original: This assumption is often reasonable, and follows Pryzant et al.
Revised: This assumption is often reasonable, and follows Pryzant et al.

Original: The next challenge is that the treatment may be correlated with other aspects of the text ( Z ) that are also relevant to the outcome— e.g., the product category of the item being reviewed.
Revised: The next challenge is that the treatment may be correlated with other aspects of the text ( Z ) that are also relevant to the outcome— e.g., the product category of the item being reviewed.

Original: Such
Revised: Such

Original: Z can act as confounding variables, and must somehow be adjusted for in a causal estimation problem.
Revised: Z can act as confounding variables, and must somehow be adjusted for in a causal estimation problem.

Original: Each unit ( A i , Z i , X i , Y i ) is drawn independently and identically from an unknown distribution P .
Revised: Each unit ( A i , Z i , X i , Y i ) is drawn independently and identically from an unknown distribution P .

Original: Figure 1 shows the causal relationships among variables, where solid arrows represent causal relations, and the dotted line represents possible correlations between two variables.
Revised: Figure 1 shows the causal relationships among variables, where solid arrows represent causal relations, and the dotted line represents possible correlations between two variables.

Original: We assume that text
Revised: We assume that text

Original: X contains all common causes of ˜ A and the outcome Y .
Revised: X contains all common causes of ˜ A and the outcome Y .

Original:  I DENTIFICATION AND C AUSAL ESTIMAND
Revised:  I DENTIFICATION AND C AUSAL ESTIMAND

Original: The ﬁrst task is to translate the qualitative causal question of interest—what is the effect of A on Y —into a causal estimand.
Revised: The ﬁrst task is to translate the qualitative causal question of interest—what is the effect of A on Y —into a causal estimand.

Original: This estimand must both be faithful to the qualitative question and be identiﬁable from observational data under reasonable assumptions.
Revised: This estimand must both be faithful to the qualitative question and be identiﬁable from observational data under reasonable assumptions.

Original: The key challenges here are that we only observe ˜ A (not A itself), there are unknown confounding variables inﬂuencing the text, and ˜ A is a deterministic function of the text, leading to overlap violations if we naively adjust for all the text.
Revised: The key challenges here are that we only observe ˜ A (not A itself), there are unknown confounding variables inﬂuencing the text, and ˜ A is a deterministic function of the text, leading to overlap violations if we naively adjust for all the text.

Original: Our high-level idea is to split the text into abstract (unknown) parts depending on whether they are confounding—affect both ˜
Revised: Our high-level idea is to split the text into abstract (unknown) parts depending on whether they are confounding—affect both ˜

Original: A and Y —or whether they affect ˜
Revised: A and Y —or whether they affect ˜

Original: A alone.
Revised: A alone.

Original: The part of the text that affects only ˜
Revised: The part of the text that affects only ˜

Original: A is not necessary for causal adjustment, and can be thrown away.
Revised: A is not necessary for causal adjustment, and can be thrown away.

Original: If this part contains “enough” information about ˜ A , then throwing it away can eliminate our ability to perfectly predict ˜ A , thus ﬁxing the overlap issue.
Revised: If this part contains “enough” information about ˜ A , then throwing it away can eliminate our ability to perfectly predict ˜ A , thus ﬁxing the overlap issue.

Original: We now turn to formalizing this idea, showing how it can be used to deﬁne an estimand and to identify this estimand from observational data.
Revised: We now turn to formalizing this idea, showing how it can be used to deﬁne an estimand and to identify this estimand from observational data.

Original: Causal model
Revised: Causal model

Original: The ﬁrst idea is to decompose the text into three parts: one part affected by only A , one part affected interactively by A and Z , and another part affected only by Z .
Revised: The ﬁrst idea is to decompose the text into three parts: one part affected by only A , one part affected interactively by A and Z , and another part affected only by Z .

Original: We use X A , X A ^ Z and X Z to denote them, respectively; see Figure 2 for the corresponding causal model.
Revised: We use X A , X A ^ Z and X Z to denote them, respectively; see Figure 2 for the corresponding causal model.

Original: Note that there could be additional information in the text in addition to these three parts.
Revised: Note that there could be additional information in the text in addition to these three parts.

Original: However, since they are irrelevant to both A and Z , we do not need to consider them in the model.
Revised: However, since they are irrelevant to both A and Z , we do not need to consider them in the model.

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

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Revised: aKo4lQwbLBaxbHtUoeARNjTXAtuJRBp6Alve8GZab41QKh5H93qcoBvSfsQDzqg2ltvuZVfdJ/T7SB4mvXLFrtozkWVwcqhArnqv/NX1Y5aGGGkmqFIdx060m1GpORM4KXVThQllQ9rHjsGIhqjcbHb0hJwYxydBLM2LNJm5vycyGio1Dj3TGVI9UIu1qflfrZPq4NLNeJSkGiM2XxSkguiYTBMgPpfItBgboExycythAyop0yankgnBWfzyMjTPqo7hu/NK7TqPowhHcAyn4MAF1OAW6tAABo/wDK/wZo2sF+vd+pi3Fqx85hD+yPr8AVeCkc

Original: [Figure]
Revised: [Figure]

Revised: </

Revised: [Figure]

Original: and are properties
Revised: and are properties

Original: When investigating the causal relationship between 
Revised: When investigating the causal relationship between and satisfying

Original: Controlled direct effect (CDE)
Revised: Controlled direct effect (CDE)

Original: The treatment A affects the outcome through two paths.
Revised: The treatment A affects the outcome through two paths.

Original: Both “directly” through X A —the part of the text determined just by the treatment—and also through a path going through X A ^
Revised: Both “directly” through X A —the part of the text determined just by the treatment—and also through a path going through X A ^

Original: Z —the part of the text that relies on interaction effects with other factors.
Revised: Z —the part of the text that relies on interaction effects with other factors.

Original: Our formal causal effect aims at capturing the effect of A through only the ﬁrst, direct, path.
Revised: Our formal causal effect aims at capturing the effect of A through only the ﬁrst, direct, path.

Original: [Equation]
Revised: [Equation]

Original: Here, do is Pearl’s do notation, and the estimand is a variant of the controlled direct effect (Pearl, 2009).
Revised: Here, do is Pearl’s do notation, and the estimand is a variant of the controlled direct effect (Pearl, 2009).

Original: Intuitively, it can be interpreted as the expected change in the outcome induced by changing the treatment from 1 to 0 while keeping part of the text affected by Z the same as it would have been had we set A = 1.
Revised: Intuitively, it can be interpreted as the expected change in the outcome induced by changing the treatment from 1 to 0 while keeping part of the text affected by Z the same as it would have been had we set A = 1.

Original: This is a reasonable formalization of the qualitative “effect of A on Y ”.
Revised: This is a reasonable formalization of the qualitative “effect of A on Y ”.

Original: Of course, it is not the only possible formalization.
Revised: Of course, it is not the only possible formalization.

Original: Its advantage is that, as we will see, it can be identiﬁed and estimated under reasonable conditions.
Revised: Its advantage is that, as we will see, it can be identiﬁed and estimated under reasonable conditions.

Original: Identiﬁcation To identify CDE we must rewrite the expression in terms of observable quantities.
Revised: Identiﬁcation To identify CDE we must rewrite the expression in terms of observable quantities.

Original: There are three challenges: we need to get rid of the do operator, we don’t observe
Revised: There are three challenges: we need to get rid of the do operator, we don’t observe

Original: A (only ˜ A ), and the variables X
Revised: A (only ˜ A ), and the variables X

Original: A ^
Revised: A ^

Original: Z , X Z are unknown (they are latent parts of X ).
Revised: Z , X Z are unknown (they are latent parts of X ).

Original: Informally, the identiﬁcation argument is as follows.
Revised: Informally, the identiﬁcation argument is as follows.

Original: First, X A
Revised: First, X A

Original: ^ Z , X Z block all backdoor paths (common causes) in Figure 2.
Revised: ^ Z , X Z block all backdoor paths (common causes) in Figure 2.

Original: Moreover, because we have thrown away X A , we now satisfy overlap.
Revised: Moreover, because we have thrown away X A , we now satisfy overlap.

Original: Accordingly, the do operator can be replaced by conditioning following the usual causal-adjustment argument.
Revised: Accordingly, the do operator can be replaced by conditioning following the usual causal-adjustment argument.

Original: Next, A = ˜
Revised: Next, A = ˜

Original: A almost surely, so we can just replace A with ˜
Revised: A almost surely, so we can just replace A with ˜

Original: A .
Revised: A .

Original: Now, our estimand has been reduced to:
Revised: Now, our estimand has been reduced to:

Original: [Equation]
Revised: [Equation]

Original: The ﬁnal step is to deal with the unknown X A
Revised: The ﬁnal step is to deal with the unknown X A

Original: ^ Z , X Z .
Revised: ^ Z , X Z .

Original: To ﬁx this issue, we ﬁrst deﬁne the conditional outcome Q according to:
Revised: To ﬁx this issue, we ﬁrst deﬁne the conditional outcome Q according to:

Original: [Equation]
Revised: [Equation]

Original: A key insight here is that, subject to the causal model in Figure 2, we have Q ( ˜ A , X ) =
Revised: A key insight here is that, subject to the causal model in Figure 2, we have Q ( ˜ A , X ) =

Original: E (
Revised: E (

Original: Y j ˜ A , X A
Revised: Y j ˜ A , X A

Original: ^ Z , X Z ) .
Revised: ^ Z , X Z ) .

Original: But this is exactly the quantity in (3.2).
Revised: But this is exactly the quantity in (3.2).

Original: Moreover, Q ( ˜ A , X ) is an observable data quantity (it depends only on the distribution of the observed quantities).
Revised: Moreover, Q ( ˜ A , X ) is an observable data quantity (it depends only on the distribution of the observed quantities).

Original: In summary:
Revised: In summary:

Original: Theorem 1.
Revised: Theorem 1.

Original: Assume the following:
Revised: Assume the following:

Original: (Causal structure)
Revised: (Causal structure)

Original: The causal relationships among A, ˜ A, Z, Y, and X satisfy the causal DAG in Figure 2; 2.
Revised: The causal relationships among A, ˜ A, Z, Y, and X satisfy the causal DAG in Figure 2; 2.

Original: (Overlap) 0 < P ( A = 1 j X A
Revised: (Overlap) 0 < P ( A = 1 j X A

Original: ^ Z , X Z ) < 1 ; 3.
Revised: ^ Z , X Z ) < 1 ; 3.

Original: (Intention equals perception)
Revised: (Intention equals perception)

Original: A = ˜
Revised: A = ˜

Original: A almost surely with respect to all interventional distributions.
Revised: A almost surely with respect to all interventional distributions.

Original: Then, the CDE is identiﬁed from observational data as
Revised: Then, the CDE is identiﬁed from observational data as

Original: [Equation]
Revised: [Equation]

Original: where η ( X ) :
Revised: where η ( X ) :

Original: = ( Q ( 0, X ) , Q ( 1, X )) .
Revised: = ( Q ( 0, X ) , Q ( 1, X )) .

Revised: The proof is in Appendix B.

Original: We give the result in terms of an abstract sufﬁcient statistic η ( X ) to emphasize that the actual conditional expectation model is not required, only some statistic that is informationally equivalent.
Revised: We give the result in terms of an abstract sufﬁcient statistic η ( X ) to emphasize that the actual conditional expectation model is not required, only some statistic that is informationally equivalent.

Original: We emphasize that, regardless of whether the overlap condition holds or not, the propensity score of η ( X ) is accessible and meaningful.
Revised: We emphasize that, regardless of whether the overlap condition holds or not, the propensity score of η ( X ) is accessible and meaningful.

Original: Therefore, we can easily identify when identiﬁcation fails as long as η ( X ) is well-estimated.
Revised: Therefore, we can easily identify when identiﬁcation fails as long as η ( X ) is well-estimated.

Original:  M ETHOD
Revised:  M ETHOD

Original: Our ultimate goal is to draw a conclusion about whether the treatment has a causal effect on the outcome.
Revised: Our ultimate goal is to draw a conclusion about whether the treatment has a causal effect on the outcome.

Original: Following  previous section, we have reduced this problem to estimating τ CDE , deﬁned in Theorem 1.
Revised: Following the previous section, we have reduced this problem to estimating τ CDE , deﬁned in Theorem 1.

Original: The task now is to develop an estimation procedure, including uncertainty quantiﬁcation.
Revised: The task now is to develop an estimation procedure, including uncertainty quantiﬁcation.

Original: 4.1 O UTCOME ONLY ESTIMATOR
Revised: 4.1 O UTCOME ONLY ESTIMATOR

Original: We start by introducing the naive outcome only estimator as a ﬁrst approach to CDE estimation.
Revised: We start by introducing the naive outcome only estimator as a ﬁrst approach to CDE estimation.

Original: The estimator is adapted from Pryzant et al.
Revised: The estimator is adapted from Pryzant et al.

Original: The observation here is that, taking η ( X ) =
Revised: The observation here is that, taking η ( X ) =

Original: ( Q ( 0, X ) , Q ( 1, X )) in (3.4), we have
Revised: ( Q ( 0, X ) , Q ( 1, X )) in (3.4), we have

Original: [Equation]
Revised: [Equation]

Original: Since Q ( A , X ) is a function of the whole text data X , it is estimable from observational data.
Revised: Since Q ( A , X ) is a function of the whole text data X , it is estimable from observational data.

Original: Namely, it is the solution to the square error risk:
Revised: Namely, it is the solution to the square error risk:

Original: [Equation]
Revised: [Equation]

Original: With a ﬁnite sample, we can estimate Q as ˆ Q by ﬁtting a machine-learning model to minimize the (possibly regularized) square error empirical risk.
Revised: With a ﬁnite sample, we can estimate Q as ˆ Q by ﬁtting a machine-learning model to minimize the (possibly regularized) square error empirical risk.

Original: That is, ﬁt a model using mean square error as the objective function.
Revised: That is, ﬁt a model using mean square error as the objective function.

Original: Then, a straightforward estimator is:
Revised: Then, a straightforward estimator is:

Original: [Equation]
Revised: [Equation]

Original: where n 1 is the number of treated units.
Revised: where n 1 is the number of treated units.

Original: It should be noted that the model for Q is not arbitrary.
Revised: It should be noted that the model for Q is not arbitrary.

Original: One signiﬁcant issue for those models which directly regress Y on A and X is when overlap does not hold, the model could ignore A and only use X as the covariate.
Revised: One signiﬁcant issue for those models which directly regress Y on A and X is when overlap does not hold, the model could ignore A and only use X as the covariate.

Original: As a result, we need to choose a class of models that force the use of the treatment A .
Revised: As a result, we need to choose a class of models that force the use of the treatment A .

Original: To address this, we use a two-headed model that regress Y on X for A = 0 / 1 separately in the conditional outcome learning model (See
Revised: To address this, we use a two-headed model that regress Y on X for A = 0 / 1 separately in the conditional outcome learning model (See

Original: Section 4.2 and Figure 3).
Revised: Section 4.2 and Figure 3).

Original: As discussed in the introduction Section 1, this estimator yields a consistent point estimate, but does not offer a simple approach for uncertainty quantiﬁcation.
Revised: As discussed in the introduction Section 1, this estimator yields a consistent point estimate, but does not offer a simple approach for uncertainty quantiﬁcation.

Original: A natural guess for an estimate of its variance is:
Revised: A natural guess for an estimate of its variance is:

Original: [Equation]
Revised: [Equation]

Original: That is, just compute the variance of the mean conditional on the ﬁtted model.
Revised: That is, just compute the variance of the mean conditional on the ﬁtted model.

Original: However, this procedure yields asymptotically valid conﬁdence intervals only if the outcome model converges extremely quickly; i.e., if E [( ˆ Q (cid:0) Q ) 2 ] 12 = o ( n (cid:0) 12 ) .
Revised: However, this procedure yields asymptotically valid conﬁdence intervals only if the outcome model converges extremely quickly; i.e., if E [( ˆ Q (cid:0) Q ) 2 ] 12 = o ( n (cid:0) 12 ) .

Original: We could instead bootstrap, reﬁtting ˆ Q on each bootstrap sample.
Revised: We could instead bootstrap, reﬁtting ˆ Q on each bootstrap sample.

Original: However, with modern language models, this can be prohibitively computationally expensive.
Revised: However, with modern language models, this can be prohibitively computationally expensive.

Original: 4.2 T REATMENT I GNORANT E FFECT E STIMATION (TI- ESTIMATOR )
Revised: 4.2 T REATMENT I GNORANT E FFECT E STIMATION (TI- ESTIMATOR )

Original: Following Theorem 1, it sufﬁces to adjust for η ( X ) =
Revised: Following Theorem 1, it sufﬁces to adjust for η ( X ) =

Original: ( Q ( 0, X ) , Q ( 1, X )) .
Revised: ( Q ( 0, X ) , Q ( 1, X )) .

Original: Accordingly, we use the following pipeline.
Revised: Accordingly, we use the following pipeline.

Original: We ﬁrst estimate ˆ Q 0 ( X ) and ˆ Q 1 ( X ) (using a neural language model), as with the outcome-only estimator.
Revised: We ﬁrst estimate ˆ Q 0 ( X ) and ˆ Q 1 ( X ) (using a neural language model), as with the outcome-only estimator.

Original: Then, we take ˆ η ( X ) :
Revised: Then, we take ˆ η ( X ) :

Original: = ( ˆ Q 0 ( X ) , ˆ Q 1 ( X )) and estimate ˆ g η (cid:25) P ( A = 1 j ˆ η ) .
Revised: = ( ˆ Q 0 ( X ) , ˆ Q 1 ( X )) and estimate ˆ g η (cid:25) P ( A = 1 j ˆ η ) .

Original: That is, we estimate the propensity score corresponding to the estimated representation.
Revised: That is, we estimate the propensity score corresponding to the estimated representation.

Original: Finally, we plug the estimated ˆ Q and ˆ g η into a standard double machine learning estimator (Chernozhukov et al., 2016).
Revised: Finally, we plug the estimated ˆ Q and ˆ g η into a standard double machine learning estimator (Chernozhukov et al., 2016).

Original: We describe the three steps in detail.
Revised: We describe the three steps in detail.

Original: Q-Net
Revised: Q-Net

Original: In the ﬁrst stage, we estimate the conditional outcomes and hence obtain the estimated two-dimensional confounding vector ˆ η ( X ) .
Revised: In the ﬁrst stage, we estimate the conditional outcomes and hence obtain the estimated two-dimensional confounding vector ˆ η ( X ) .

Original: For concreteness, we will use the dragonnet architecture of Shi et al.
Revised: For concreteness, we will use the dragonnet architecture of Shi et al.

Original: Speciﬁcally, we train DistilBERT (Sanh et al., 2019) modiﬁed to include three heads, as shown in Figure 3.
Revised: Speciﬁcally, we train DistilBERT (Sanh et al., 2019) modiﬁed to include three heads, as shown in Figure 3.

Original: Two of the heads correspond to ˆ Q 

Revised: Two of the heads correspond to ˆ Q

Original: ( X ) and ˆ Q 1 ( X ) respectively.
Revised: ( X ) and ˆ Q 1 ( X ) respectively.

Original: As discussed in the Section 4.1, applying two heads can force the model to use the treatment A .
Revised: As discussed in the Section 4.1, applying two heads can force the model to use the treatment A .

Original: The ﬁnal head is a single linear layer predicting the treatment.
Revised: The ﬁnal head is a single linear layer predicting the treatment.

Original: This propensity score prediction head can help prevent (implicit) regularization of the model from throwing away X A
Revised: This propensity score prediction head can help prevent (implicit) regularization of the model from throwing away X A

Original: ^
Revised: ^

Original: Z information that is necessary for identiﬁcation.
Revised: Z information that is necessary for identiﬁcation.

Original: The output of this head is not used for the estimation since its purpose is to force the DistilBERT representation to preserve all confounding information.
Revised: The output of this head is not used for the estimation since its purpose is to force the DistilBERT representation to preserve all confounding information.

Original: This has been shown to improve causal estimation (Shi et al., 2019; Veitch et al., 2019).
Revised: This has been shown to improve causal estimation (Shi et al., 2019; Veitch et al., 2019).

Original: We train the model by minimizing the objective function
Revised: We train the model by minimizing the objective function

Original: [Equation]
Revised: [Equation]

Original: where θ are the model parameters, α , β are hyperparameters and L mlm ( (cid:1) ) is the masked language modeling objective of DistilBERT.
Revised: where θ are the model parameters, α , β are hyperparameters and L mlm ( (cid:1) ) is the masked language modeling objective of DistilBERT.

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: MLP MLP
Revised: MLP MLP

Original: Linear
Revised: Linear

Original: [Table]
Revised: [Table]

Original: /xpv/xm2bg7a+sPDwzgw78waJFAZd99spbGxube8Ud0t7+weHR+Xjk7aJU814i8Uy1t2AGi6F4i0UKHk30ZxGgeSdYHI/r3eeuDYiVo84Tbgf0ZESoWAUrdXrjymS5sCtdi8H5Ypbcxci6+DlUIFcjUH5qz+MWRpxhUxSY3qem6CfUY2CST4r9VPDE8omdMR7FhWNuPGzxcozcmGdIQljbZ9CsnB/T2Q0MmYaBbYzojg2q7W5+V+tl2J462dCJSlyxZYfhakkGJP5/WQoNGcopxYo08LuStiYasrQplSyIXirJ69D+6rmWW5eV+p3eRxFOINzqIIHN1CHB2hACxjE8Ayv8Oag8+K8Ox/L1oKTz5zCHzmfP6R8kCo=</latexit>
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Original: [Figure]
Revised: [Figure]

Original: There is a ﬁnal nuance.
Revised: There is a ﬁnal nuance.

Original: In practice, we split the data into K -folds.
Revised: In practice, we split the data into K -folds.

Original: For each fold j , we train a model ˆ Q (cid:0) j on the other K (cid:0) 1 folds.
Revised: For each fold j , we train a model ˆ Q (cid:0) j on the other K (cid:0) 1 folds.

Original: Then, we make predictions for the data points in fold j using ˆ Q (cid:0) j .
Revised: Then, we make predictions for the data points in fold j using ˆ Q (cid:0) j .

Original: Slightly abusing notation, we use ˆ Q a ( x ) to denote the predictions obtained in this manner.
Revised: Slightly abusing notation, we use ˆ Q a ( x ) to denote the predictions obtained in this manner.

Original: Propensity score estimation Next, we deﬁne ˆ η ( x ) :
Revised: Propensity score estimation Next, we deﬁne ˆ η ( x ) :

Original: = ( ˆ Q 0 ( x ) , ˆ Q 1 ( x )) and estimate the propensity score ˆ g η ( x ) (cid:25) P ( A = 1 j ˆ η ( x )) .
Revised: = ( ˆ Q 0 ( x ) , ˆ Q 1 ( x )) and estimate the propensity score ˆ g η ( x ) (cid:25) P ( A = 1 j ˆ η ( x )) .

Original: To do this, we ﬁt a nonparametric estimator to the binary classiﬁcation task of predicting A from ˆ η ( X ) in a cross ﬁtting or K-fold fashion.
Revised: To do this, we ﬁt a nonparametric estimator to the binary classiﬁcation task of predicting A from ˆ η ( X ) in a cross ﬁtting or K-fold fashion.

Original: The important insight here is that since ˆ η ( X ) is 2-dimensional, non-parametric estimation is possible at a fast rate.
Revised: The important insight here is that since ˆ η ( X ) is 2-dimensional, non-parametric estimation is possible at a fast rate.

Original: In Section 5, we try several methods and ﬁnd that kernel regression usually works well.
Revised: In Section 5, we try several methods and ﬁnd that kernel regression usually works well.

Original: We also deﬁne g η ( X ) :
Revised: We also deﬁne g η ( X ) :

Original: = P ( A = 1 j η ( X )) as the idealized propensity score.
Revised: = P ( A = 1 j η ( X )) as the idealized propensity score.

Original: The idea is that
Revised: The idea is that

Original: as ˆ η !
Revised: as ˆ η !

Original: η , we will also have ˆ g η !
Revised: η , we will also have ˆ g η !

Original: g η so long as we have a valid non-parametric estimate.
Revised: g η so long as we have a valid non-parametric estimate.

Original: CDE estimation The ﬁnal stage is to combine the estimated outcome model and propensity score into a CDE estimator.
Revised: CDE estimation The ﬁnal stage is to combine the estimated outcome model and propensity score into a CDE estimator.

Original: To that end, we deﬁne the inﬂuence curve of τ CDE as follows:
Revised: To that end, we deﬁne the inﬂuence curve of τ CDE as follows:

Original: [Equation]
Revised: [Equation]

Original: where p = P ( A = 1 )
Revised: where p = P ( A = 1 )

Original: Then, the standard double machine learning estimator of τ CDE
Revised: Then, the standard double machine learning estimator of τ CDE

Original: Chernozhukov et al.
Revised: Chernozhukov et al.

Original: (2016), and the α -level conﬁdence interval of this estimator, is given by
Revised: (2016), and the α -level conﬁdence interval of this estimator, is given by

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: ˆ p = 1 n P n i =
Revised: ˆ p = 1 n P n i =

Original: 1 A i , z 1 (cid:0) α/ 2 is the α/ 2-upper quantile of the standard normal, and ˆ sd ( (cid:1) ) is the sample standard deviation.
Revised: 1 A i , z 1 (cid:0) α/ 2 is the α/ 2-upper quantile of the standard normal, and ˆ sd ( (cid:1) ) is the sample standard deviation.

Original: Validity We now have an estimation procedure.
Revised: Validity We now have an estimation procedure.

Original: It remains to give conditions under which this procedure is valid.
Revised: It remains to give conditions under which this procedure is valid.

Original: In particular, we require that it should yield a consistent estimate and asymptotically correct conﬁdence intervals.
Revised: In particular, we require that it should yield a consistent estimate and asymptotically correct conﬁdence intervals.

Original: Theorem 2.
Revised: Theorem 2.

Original: Assume the following.
Revised: Assume the following.

Original: The mis-estimation of conditional outcomes can be bounded as follows
Revised: The mis-estimation of conditional outcomes can be bounded as follows

Original: [Equation]
Revised: [Equation]

Original: The propensity score function P ( A = 1 j(cid:1) , (cid:1) ) is Lipschitz continuous on R 2 , and 9 ϵ > 0 ,
Revised: The propensity score function P ( A = 1 j(cid:1) , (cid:1) ) is Lipschitz continuous on R 2 , and 9 ϵ > 0 ,

Original: [Equation]
Revised: [Equation]

Original: The propensity score estimate converges at least as quickly as k nearest neighbor;
Revised: The propensity score estimate converges at least as quickly as k nearest neighbor;

Original: [Equation]
Revised: [Equation]

Original:  There exist positive constants C 1 , C 2 , c, and q > 2 such that[Equation]
Revised: [Equation] There exist positive constants C 1 , C 2 , c, and q > 2 such that

Original: Then

Original: [Equation]
Revised: [Equation]

Original:  the estimator ˆ τ TI is consistent and
Revised: Then, the estimator ˆ τ TI is consistent and

Original: [Equation]
Revised: [Equation]

Original: (4.10) where σ 2 = E (cid:0) ϕ
Revised: (4.10) where σ 2 = E (cid:0) ϕ

Original: ( X ; Q , g η , τ CDE ) (cid:1) 2 .
Revised: ( X ; Q , g η , τ CDE ) (cid:1) 2 .

Original: The proof is provided in Appendix A.
Revised: The proof is provided in Appendix A.

Original: The key point from this theorem is that we get asymptotic normality at the (fast) p nrate while requiring only a slow ( n 1 / 4 ) convergence rate of Q .
Revised: The key point from this theorem is that we get asymptotic normality at the (fast) p nrate while requiring only a slow ( n 1 / 4 ) convergence rate of Q .

Original: Intuitively, the reason is simply that, because ˆ η ( X ) is only 2-dimensional, it is always possible to nonparametrically estimate the propensity score from ˆ η at a fast rate—even naive KNN works!
Revised: Intuitively, the reason is simply that, because ˆ η ( X ) is only 2-dimensional, it is always possible to nonparametrically estimate the propensity score from ˆ η at a fast rate—even naive KNN works!

Original: Effectively, this means the rate at which we estimate the true propensity score g η ( X ) =
Revised: Effectively, this means the rate at which we estimate the true propensity score g η ( X ) =

Original: P ( A = 1 j η ( X )) is dominated by the rate at which we estimate η ( X ) , which is in turn determined by the rate for ˆ Q .
Revised: P ( A = 1 j η ( X )) is dominated by the rate at which we estimate η ( X ) , which is in turn determined by the rate for ˆ Q .

Original: Now, the key property of the double ML estimator is that convergence only depends on the product of the convergence rates of ˆ Q and ˆ g .
Revised: Now, the key property of the double ML estimator is that convergence only depends on the product of the convergence rates of ˆ Q and ˆ g .

Original: Accordingly, this procedure is robust in the sense that we only need to estimate ˆ Q at the square root of the rate we needed for the naive Q -only procedure.
Revised: Accordingly, this procedure is robust in the sense that we only need to estimate ˆ Q at the square root of the rate we needed for the naive Q -only procedure.

Original: This is much more plausible in practice.
Revised: This is much more plausible in practice.

Original: As we will see in
Revised: As we will see in

Original: Section 5, the TI-estimator dramatically improves the quality of the estimated conﬁdence intervals and reduces the absolute bias of estimation.
Revised: Section 5, the TI-estimator dramatically improves the quality of the estimated conﬁdence intervals and reduces the absolute bias of estimation.

Original: Remark 3 .
Revised: Remark 3 .

Original: In addition to robustness to noisy estimation of Q , there are some other advantages this estimation procedure inherits from the double ML estimator.
Revised: In addition to robustness to noisy estimation of Q , there are some other advantages this estimation procedure inherits from the double ML estimator.

Original: If ˆ Q is consistent, then the estimator is nonparametrically efﬁcient in the sense that no other non-parametric estimator has a smaller asymptotic variance.
Revised: If ˆ Q is consistent, then the estimator is nonparametrically efﬁcient in the sense that no other non-parametric estimator has a smaller asymptotic variance.

Original: That is, the procedure using the data as efﬁciently as possible.
Revised: That is, the procedure using the data as efﬁciently as possible.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: We empirically study the method’s capability to provide accurate causal estimates with good uncertainty quantiﬁcation Testing using semi-synthetic data (where ground truth causal effects are known), we ﬁnd that the estimation procedure yields accurate causal estimates and conﬁdence intervals.
Revised: We empirically study the method’s capability to provide accurate causal estimates with good uncertainty quantiﬁcation Testing using semi-synthetic data (where ground truth causal effects are known), we ﬁnd that the estimation procedure yields accurate causal estimates and conﬁdence intervals.

Original: In particular, the TI-estimator has signiﬁcantly lower absolute bias and vastly better uncertainty quantiﬁcation than the Q -only method.
Revised: In particular, the TI-estimator has signiﬁcantly lower absolute bias and vastly better uncertainty quantiﬁcation than the Q -only method.

Original: Additionally, we study the effect of the choice of nonparametric propensity score estimator and the choice of double machine-learning estimator, and the method’s robustness in regard to ˆ Q ’s miscalibration.
Revised: Additionally, we study the effect of the choice of nonparametric propensity score estimator and the choice of double machine-learning estimator, and the method’s robustness in regard to ˆ Q ’s miscalibration.

Original: These results are reported in Appendices C and D.
Revised: These results are reported in Appendices C and D.

Original: Although these
Revised: Although these

Original: [Table]
Revised: [Table]

Original: choices do not matter asymptotically, we ﬁnd they have a signiﬁcant impact in actual ﬁnite sample estimation.
Revised: choices do not matter asymptotically, we ﬁnd they have a signiﬁcant impact in actual ﬁnite sample estimation.

Original: We ﬁnd that, in general, kernel regression works well for propensity score estimation and the vanilla the Augmented Inverse Probability of Treatment weighted
Revised: We ﬁnd that, in general, kernel regression works well for propensity score estimation and the vanilla the Augmented Inverse Probability of Treatment weighted

Original: Estimator (AIPTW) corresponding to the CDE works well.
Revised: Estimator (AIPTW) corresponding to the CDE works well.

Original: Finally
Revised: Finally

Original: [Table]
Revised: [Table]

Original: we reproduce the real-data analysis from Pryzant et al.
Revised: we reproduce the real-data analysis from Pryzant et al.

Original: We ﬁnd that politeness has a positive effect on reducing email response time.
Revised: We ﬁnd that politeness has a positive effect on reducing email response time.

Original: 5.1 A MAZON R EVIEWS
Revised: 5.1 A MAZON R EVIEWS

Original: Dataset We closely follow the setup of Pryzant et al.
Revised: Dataset We closely follow the setup of Pryzant et al.

Original: We use publicly available
Revised: We use publicly available

Original: Amazon reviews for music products as the basis for our semi-synthetic data.
Revised: Amazon reviews for music products as the basis for our semi-synthetic data.

Original: We include reviews for mp3, CD and vinyl, and among these exclude reviews for products costing more than $100 or shorter than 5 words.
Revised: We include reviews for mp3, CD and vinyl, and among these exclude reviews for products costing more than $100 or shorter than 5 words.

Original: The treatment A is whether the review is ﬁve stars ( A = 1) or one / two stars ( A = 0).
Revised: The treatment A is whether the review is ﬁve stars ( A = 1) or one / two stars ( A = 0).

Original: To have a ground truth causal effect, we must now simulate the outcome.
Revised: To have a ground truth causal effect, we must now simulate the outcome.

Original: To produce a realistic dataset, we choose a real variable as the confounder.
Revised: To produce a realistic dataset, we choose a real variable as the confounder.

Original: Namely, the confounder
Revised: Namely, the confounder

Original: C is whether the product is a CD ( C = 1) or not ( C = 0).
Revised: C is whether the product is a CD ( C = 1) or not ( C = 0).

Original: Then, outcome Y is generated according to Y
Revised: Then, outcome Y is generated according to Y

Original: [Equation]
Revised: [Equation]

Original: The true causal effect is controlled by
Revised: true causal effect is controlled by

Original: [Equation]
Revised: [Equation]

Original: We choose β a = 1.0, 0.0 to generate data with and without causal effects.
Revised: We choose β a = 1.0, 0.0 to generate data with and without causal effects.

Original: In this setting, β a is the oracle value of our causal estimand.
Revised: In this setting, β a is the oracle value of our causal estimand.

Original: The strength of confounding is controlled by β c .
Revised: The strength of confounding is controlled by β c .

Original: We choose β c = 50.0, 100.0.
Revised: We choose β c = 50.0, 100.0.

Original: The ground-truth propensity score is π ( C ) =
Revised: The ground-truth propensity score is π ( C ) =

Original: P ( A = 1 j C ) .
Revised: P ( A = 1 j C ) .

Original: We set it to have the value π ( 0 ) =
Revised: We set it to have the value π ( 0 ) =

Original: 0.8 and π ( 1 ) =
Revised: 0.8 and π ( 1 ) =

Original: 0.6 (by subsampling the data).
Revised: 0.6 (by subsampling the data).

Original: β o is an offset E [ π ( C )]
Revised: β o is an offset E [ π ( C )]

Original: = π ( 0 ) P ( C = 0 )
Revised: = π ( 0 ) P ( C = 0 )

Original: +
Revised: +

Original: π ( 1 ) P ( C = 1 ) , where P ( C = a ) , a = 0,1 are estimated from data.
Revised: π ( 1 ) P ( C = 1 ) , where P ( C = a ) , a = 0,1 are estimated from data.

Original: Finally, the noise level is controlled by γ ; we choose 1.0 and 4.0 to simulate data with small and large noise.
Revised: Finally, the noise level is controlled by γ ; we choose 1.0 and 4.0 to simulate data with small and large noise.

Original: The ﬁnal dataset has 10,685 data entries.
Revised: The ﬁnal dataset has 10,685 data entries.

Original: Protocol For the language model, we use  pretrained distilbert-base-uncased model provided by the transformers package.
Revised: Protocol For the language model, we use the pretrained distilbert-base-uncased model provided by the transformers package.

Original: The model is trained in the k-folding fashion withfolds.
Revised: The model is trained in the k-folding fashion withfolds.

Original: We apply the Adam optimizer (Kingma & Ba, 2014) with a learning rate of 2 e (cid:0) 5 and a batch size of 64.
Revised: We apply the Adam optimizer (Kingma & Ba, 2014) with a learning rate of 2 e (cid:0) 5 and a batch size of 64.

Original: The maximum number of epochs is set as 20, with early stopping based on validation loss with a patience of 6.
Revised: The maximum number of epochs is set as 20, with early stopping based on validation loss with a patience of 6.

Original: Each experiment is replicated with ﬁve different seeds and the ﬁnal ˆ Q ( a , x i ) predictions are obtained by averaging the predictions from the 5 resulting models.
Revised: Each experiment is replicated with ﬁve different seeds and the ﬁnal ˆ Q ( a , x i ) predictions are obtained by averaging the predictions from the 5 resulting models.

Original: The propensity model is implemented by running the Gaussian process regression using GaussianProcessClassiﬁer in the sklearn package with DotProduct +
Revised: The propensity model is implemented by running the Gaussian process regression using GaussianProcessClassiﬁer in the sklearn package with DotProduct +

Original: WhiteKernel kernel.
Revised: WhiteKernel kernel.

Original: (We choose different random state for the GPR to guarantee the convergence of the GPR.)
Revised: (We choose different random state for the GPR to guarantee the convergence of the GPR.)

Original: The coverage experiment uses 100 replicates.
Revised: The coverage experiment uses 100 replicates.

Original: [Table]
Revised: [Table]

Original: Results The main question here is the efﬁcacy of the estimation procedure.
Revised: Results The main question here is the efﬁcacy of the estimation procedure.

Original: Table 1 compares the outcome-only estimator ˆ τ Q and the estimator ˆ τ TI .
Revised: Table 1 compares the outcome-only estimator ˆ τ Q and the estimator ˆ τ TI .

Original: First, the absolute bias of the new method is signiﬁcantly lower than the absolute bias of the outcome-only estimator.
Revised: First, the absolute bias of the new method is signiﬁcantly lower than the absolute bias of the outcome-only estimator.

Original: This is particularly true where there is moderate to high levels of confounding.
Revised: This is particularly true where there is moderate to high levels of confounding.

Original: Next, we check actual coverage rates over 100 replicates of the experiment.
Revised: Next, we check actual coverage rates over 100 replicates of the experiment.

Original: First, we ﬁnd that the naive approach for the outcome-only estimator fails completely.
Revised: First, we ﬁnd that the naive approach for the outcome-only estimator fails completely.

Original: The nominal conﬁdence interval almost never actually includes the true effect.
Revised: The nominal conﬁdence interval almost never actually includes the true effect.

Original: It is wildly optimistic.
Revised: It is wildly optimistic.

Original: By contrast, the conﬁdence intervals from the new method often cover the true value.
Revised: By contrast, the conﬁdence intervals from the new method often cover the true value.

Original: This is an enormous improvement over the baseline.
Revised: This is an enormous improvement over the baseline.

Original: Nevertheless, they still do not actually achieve their nominal (95%) coverage.
Revised: Nevertheless, they still do not actually achieve their nominal (95%) coverage.

Original: This may be because the ˆ Q estimate is still not good enough for the asymptotics to kick in, and we are not yet justiﬁed in ignoring the uncertainty from model ﬁtting.
Revised: This may be because the ˆ Q estimate is still not good enough for the asymptotics to kick in, and we are not yet justiﬁed in ignoring the uncertainty from model ﬁtting.

Original: 5.2 A PPLICATION : C ONSUMER C OMPLAINTS TO THE F INANCIAL P ROTECTION B UREAU
Revised: 5.2 A PPLICATION : C ONSUMER C OMPLAINTS TO THE F INANCIAL P ROTECTION B UREAU

Original: We follow the same pipeline of the real data experiment in (Pryzant et al., 2020, §6.2).
Revised: We follow the same pipeline of the real data experiment in (Pryzant et al., 2020, §6.2).

Original: The dataset is consumers complaints made to the ﬁnancial protection.
Revised: The dataset is consumers complaints made to the ﬁnancial protection.

Original: Treatment A is politeness (measured using Yeomans et al.
Revised: Treatment A is politeness (measured using Yeomans et al.

Original: and the outcome Y is a binary indicator of whether complaints receive a response within 15 days.
Revised: and the outcome Y is a binary indicator of whether complaints receive a response within 15 days.

Original: We use the same training procedure as for the simulation data.
Revised: We use the same training procedure as for the simulation data.

Original: Table 2 shows point estimates and their 95% conﬁdence intervals.
Revised: Table 2 shows point estimates and their 95% conﬁdence intervals.

Original: Notice that the naive estimator show a signiﬁcant negative effect of politeness on reducing response time.
Revised: Notice that the naive estimator show a signiﬁcant negative effect of politeness on reducing response time.

Original: On the other hand, the more accurate AIPTW method as well as the outcome-only estimator have conﬁdence intervals that cover only positive values, so we conclude that consumers’ politeness has a positive effect on response time.
Revised: On the other hand, the more accurate AIPTW method as well as the outcome-only estimator have conﬁdence intervals that cover only positive values, so we conclude that consumers’ politeness has a positive effect on response time.

Original: This matches our intuitions that being more polite should increase the probability of receiving a timely reply.
Revised: This matches our intuitions that being more polite should increase the probability of receiving a timely reply.

Original:  D ISCUSSION
Revised:  D ISCUSSION

Original: In this paper, we address the estimation of the causal effect of a text document attribute using observational data.
Revised: In this paper, we address the estimation of the causal effect of a text document attribute using observational data.

Original: The key challenge is that we must adjust for the text—to handle confounding—but adjusting for all of the text violates overlap.
Revised: The key challenge is that we must adjust for the text—to handle confounding—but adjusting for all of the text violates overlap.

Original: We saw that this issue could be effectively circumvented with a suitable choice of estimand and estimation procedure.
Revised: We saw that this issue could be effectively circumvented with a suitable choice of estimand and estimation procedure.

Original: In particular, we have seen an estimand that corresponds to the qualitative causal question, and an estimator that is valid even when the outcome model is learned slowly.
Revised: In particular, we have seen an estimand that corresponds to the qualitative causal question, and an estimator that is valid even when the outcome model is learned slowly.

Original: The procedure also circumvents the need for bootstrapping, which is prohibitively expensive in our setting.
Revised: The procedure also circumvents the need for bootstrapping, which is prohibitively expensive in our setting.

Original: There are some limitations.
Revised: There are some limitations.

Original: The actual coverage proportion of our estimator is below the nominal level.
Revised: The actual coverage proportion of our estimator is below the nominal level.

Original: This is presumably due to the imperfect ﬁt of the conditional outcome model.
Revised: This is presumably due to the imperfect ﬁt of the conditional outcome model.

Original: Diagnostics (see Appendix D) show that as conditional outcome estimations become more accurate, the TI estimator becomes less biased, and its coverage increases.
Revised: Diagnostics (see Appendix D) show that as conditional outcome estimations become more accurate, the TI estimator becomes less biased, and its coverage increases.

Original: It seems plausible that the issue could be resolved by using more powerful language models.
Revised: It seems plausible that the issue could be resolved by using more powerful language models.

Original: Although we have focused on text in this paper, the problem of causal estimation with apparent overlap violation exists in any problem where we must adjust for unstructured and high-dimensional covariates.
Revised: Although we have focused on text in this paper, the problem of causal estimation with apparent overlap violation exists in any problem where we must adjust for unstructured and high-dimensional covariates.

Original: Another interesting direction for future work is to understand how analogous procedures work outside the text setting.
Revised: Another interesting direction for future work is to understand how analogous procedures work outside the text setting.

Original: A CKNOWLEDGEMENT
Revised: A CKNOWLEDGEMENT

Original: Thanks to Alexander D’Amour for feedback on an earlier draft.
Revised: Thanks to Alexander D’Amour for feedback on an earlier draft.

Original: We acknowledge the University of Chicagos Research Computing Center for providing computing resources.
Revised: We acknowledge the University of Chicago’s Research Computing Center for providing computing resources.

Original: This work was partially supported by Open Philanthropy.
Revised: This work was partially supported by Open Philanthropy.

Original: A P ROOF OF A SYMPTOTIC N ORMALITY
Revised: A P ROOF OF A SYMPTOTIC N ORMALITY

Original: Theorem 2.
Revised: Theorem 2.

Original: Assume the following.
Revised: Assume the following.

Original: The mis-estimation of conditional outcomes can be bounded as follows
Revised: The mis-estimation of conditional outcomes can be bounded as follows

Original: [Equation]
Revised: [Equation]

Original: The propensity score function P ( A = 1 j(cid:1) , (cid:1) ) is Lipschitz continuous on R 2 , and 9 ϵ > 0 , P
Revised: The propensity score function P ( A = 1 j(cid:1) , (cid:1) ) is Lipschitz continuous on R 2 , and 9 ϵ > 0 , P

Original: [Equation]
Revised: [Equation]

Original: The propensity score estimate converges at least as quickly as k nearest neighbor;
Revised: The propensity score estimate converges at least as quickly as k nearest neighbor;

Original: [Equation]
Revised: [Equation]

Original: There exist positive constants C 1 , C 2 , c, and q > 2 such that
Revised: There exist positive constants C 1 , C 2 , c, and q > 2 such that

Original: [Equation]
Revised: [Equation]

Original: Then, the estimator ˆ τ TI is consistent and
Revised: Then, the estimator ˆ τ TI is consistent and

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: where σ 2 = E (cid:0) ϕ
Revised: where σ 2 = E (cid:0) ϕ

Original: ( X ; Q , g η , τ CDE ) (cid:1) 2 .
Revised: ( X ; Q , g η , τ CDE ) (cid:1) 2 .

Original: Proof.
Revised: Proof.

Original: We ﬁrst prove that misestimation of propensity score has  rate n (cid:0) 14 .
Revised: We ﬁrst prove that misestimation of propensity score has the rate n (cid:0) 14 .

Original: For simplicity, we use f g , ˆ f g : ( u , v ) 2 R 2 !
Revised: For simplicity, we use f g , ˆ f g : ( u , v ) 2 R 2 !

Original: R to denote conditional probability P
Revised: R to denote conditional probability P

Original: ( A = 1 j u , v ) =
Revised: ( A = 1 j u , v ) =

Original: f g ( u , v ) and the estimated propensity function by running the nonparametric regression ˆP ( A = 1 j u , v ) =
Revised: f g ( u , v ) and the estimated propensity function by running the nonparametric regression ˆP ( A = 1 j u , v ) =

Original: ˆ f g ( u , v ) .
Revised: ˆ f g ( u , v ) .

Original: Speciﬁcally, we have f g ( Q ( 0, X ) , Q ( 1, X ))
Revised: Speciﬁcally, we have f g ( Q ( 0, X ) , Q ( 1, X ))

Original: =
Revised: =

Original: g η ( X ) and ˆ f g ( ˆ Q 0 ( X ) , ˆ Q 1 ( X )) =
Revised: g η ( X ) and ˆ f g ( ˆ Q 0 ( X ) , ˆ Q 1 ( X )) =

Original: ˆP ( A = 1 j ˆ Q 

Revised: ˆP ( A = 1 j ˆ Q

Original: ( X ) , ˆ Q 1 ( X ))
Revised: ( X ) , ˆ Q 1 ( X ))

Original: =
Revised: =

Original: ˆ g η ( X ) .
Revised: ˆ g η ( X ) .

Original: Since E [ (cid:0) ˆ Q 0 ( X ) (cid:0) Q ( 0, X ) (cid:1) 2 ] 12 ,
Revised: Since E [ (cid:0) ˆ Q 0 ( X ) (cid:0) Q ( 0, X ) (cid:1) 2 ] 12 ,

Original: E [ (cid:0) ˆ Q 1 ( X ) (cid:0) Q ( 1, X ) (cid:1) 2 ] 12 = o ( n (cid:0) 1 / 4 ) and f g is Lipschitz continuous, we have
Revised: E [ (cid:0) ˆ Q 1 ( X ) (cid:0) Q ( 1, X ) (cid:1) 2 ] 12 = o ( n (cid:0) 1 / 4 ) and f g is Lipschitz continuous, we have

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: =
Revised: =

Original: [Equation]

Original: ƒ

Original: [Equation]
Revised: [Equation]

Original: Since the true propensity function f g is Lipschitz continuous on R 2 , the mean squared error rate of the k nearest neighbor is O ( n (cid:0) 1 / 2 )
Revised: Since the true propensity function f g is Lipschitz continuous on R 2 , the mean squared error rate of the k nearest neighbor is O ( n (cid:0) 1 / 2 )

Original: Györﬁ et al.
Revised: Györﬁ et al.

Original: In addition, since the propensity score function and its estimation are bounded under
Revised: In addition, since the propensity score function and its estimation are bounded under

Original: [Equation]
Revised: [Equation]

Original: we have the following equation
Revised: we have the following equation

Original: [Equation]
Revised: [Equation]

Original: due to the dominated convergence theorem.
Revised: due to the dominated convergence theorem.

Original: By (A.1) and (A.2), we can bound the mean squared error of estimated propensity score in the following form:
Revised: By (A.1) and (A.2), we can bound the mean squared error of estimated propensity score in the following form:

Original: [Equation]
Revised: [Equation]

Original: that is E
Revised: that is E

Original: [Equation]
Revised: [Equation]

Original: Before we apply the conclusion of Theorem 5.1 in (Chernozhukov et al., 2017b), we need to check all assumptions in Assumption 5.1 hold in Chernozhukov et al.
Revised: Before we apply the conclusion of Theorem 5.1 in (Chernozhukov et al., 2017b), we need to check all assumptions in Assumption 5.1 hold in Chernozhukov et al.

Original: (2017b).
Revised: (2017b).

Original: Let
Revised: Let

Revised: C : = max

Original: [Equation]
Revised: [Equation]

Original: q 1 + 2 q ) 1 q , C' .
Revised: q 1 + 2 q ) 1 q , C' .

Original: (a) E [ Y (cid:0) Q ( A , X ) j η ( X ) , A ] = 0, E
Revised: (a) E [ Y (cid:0) Q ( A , X ) j η ( X ) , A ] = 0, E

Original: [ A (cid:0) g η ( X ) j η ( X )]
Revised: [ A (cid:0) g η ( X ) j η ( X )]

Original: = 0 are easily checked by invoking deﬁnitions of Q and g η .
Revised: = 0 are easily checked by invoking deﬁnitions of Q and g η .

Original: (b) E [ j Y j q ] 1 q (cid:20) C , E
Revised: (b) E [ j Y j q ] 1 q (cid:20) C , E

Original: [( Y (cid:0) Q ( A , X )) 2 ](cid:21) c  and sup η 2 supp ( η ( X ))
Revised: [( Y (cid:0) Q ( A , X )) 2 ] 12 (cid:21) c , and sup η 2 supp ( η ( X ))

Original: E [( Y (cid:0) Q ( A , X ))
Revised: E [( Y (cid:0) Q ( A , X ))

Original: 2 j η ( X )
Revised: 2 j η ( X )

Original: =
Revised: =

Original: η ] (cid:20) C are guaranteed by the fourth condition in the theorem.
Revised: η ] (cid:20) C are guaranteed by the fourth condition in the theorem.

Original: (c) P (cid:0) ϵ (cid:20) g η
Revised: (c) P [Equation]

Original: ( X ) (cid:20) 1 (cid:0) ϵ (cid:1) = 1 is the second condition in the theorem.
Revised: X ) (cid:20) 1 (cid:0) ϵ (cid:1) = 1 is the second condition in the theorem.

Original: (d) Since propensity score function and its estimation are bounded under 1, we have(cid:0) E
Revised: (d) Since propensity score function and its estimation are bounded under 1, we have

Original: [Equation]
Revised: [Equation]

Original: ˆ

Original: [Equation]

Original: [Equation] 1, X ) [Equation] 1 q (cid:20) ( C q 1 + C q 1 + 2 q ) 1 q (cid:20) C (e) Based on (A.3) and condition 1 in the theorem, we have (cid:128) E
Revised: C (e) Based on (A.3) and condition 1 in the theorem

Original: [Equation]
Revised: [Equation]

Original: ( X )

Original: [Equation] ( 1, X ) (cid:1) [Equation] X ) (cid:0) Q ( 0, X ) (cid:1)
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: (f) Based on condition 3 in the theorem [Equation] we have
Revised: (f) Based on condition 3 in the theorem, we have

Original: [Equation]
Revised: [Equation]

Original: We consider a smaller positive constant ˜ ϵ instead of ϵ .
Revised: We consider a smaller positive constant ˜ ϵ instead of ϵ .

Original: Note that for ˜ ϵ <
Revised: Note that for ˜ ϵ <

Original: ϵ , we still have P ( ˜ ϵ (cid:20) g
Revised: ϵ , we still have P ( ˜ ϵ (cid:20) g

Original: η
Revised: η

Original: ( X ) (cid:20) 1 (cid:0) ˜ ϵ )
Revised: ( X ) (cid:20) 1 (cid:0) ˜ ϵ )

Original: = 1.
Revised: = 1.

Original: Then,
Revised: Then,

Original: [Equation]
Revised: [Equation]

Original: supp
Revised: supp

Original: [Equation]
Revised: [Equation]

Original: [Table] 
Revised: [Table] [Equation] [Table] [Equation]

Revised: [Table]

Revised: [Equation]

Original:  [Equation]
Revised: [Table] [Equation] [Table] [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table][Equation] [Table] [Equation]
Revised: [Table]

Original: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table][Equation] [Table] [Equation]
Revised: [Table]

Original: [Table]

Original: [Equation]
Revised: [Equation]

Original: = O ( n (cid:0) 12 )
Revised: = O ( n (cid:0) 12 )

Original: Hence, P ( sup x
Revised: Hence, P ( sup x

Original: [Equation]
Revised: [Equation]

Original: g
Revised: g

Original: [Equation]
Revised: [Equation]

Original: With (a)-(f), we can invoke the conclusion in Theorem 5.1 in (Chernozhukov et al., 2017b), and get the asymptotic normality of the TI estimator.
Revised: With (a)-(f), we can invoke the conclusion in Theorem 5.1 in (Chernozhukov et al., 2017b), and get the asymptotic normality of the TI estimator.

Original: B P ROOF OF C AUSAL I DENTIFICATION
Revised: B P ROOF OF C AUSAL I DENTIFICATION

Original: Theorem 1.
Revised: Theorem 1.

Original: Assume the following:
Revised: Assume the following:

Original: (Causal structure)
Revised: (Causal structure)

Original: The causal relationships among A, ˜ A, Z, Y, and X satisfy the causal DAG in Figure 2; 2.
Revised: The causal relationships among A, ˜ A, Z, Y, and X satisfy the causal DAG in Figure 2; 2.

Original: (Overlap) 0 < P ( A = 1 j X A
Revised: (Overlap) 0 < P ( A = 1 j X A

Original: ^ Z , X Z ) < 1 ; 3.
Revised: ^ Z , X Z ) < 1 ; 3.

Original: (Intention equals perception)
Revised: (Intention equals perception)

Original: A = ˜
Revised: A = ˜

Original: A almost surely with respect to all interventional distributions.
Revised: A almost surely with respect to all interventional distributions.

Original: Then, the CDE is identiﬁed from observational data as
Revised: Then, the CDE is identiﬁed from observational data as

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: η ( X ) :
Revised: η ( X ) :

Original: = ( Q ( 0, X ) , Q ( 1, X ))
Revised: = ( Q ( 0, X ) , Q ( 1, X ))

Original: Proof.
Revised: Proof.

Original: We ﬁrst prove that this two-dimensional confounding part η ( X ) satisﬁes positivity.
Revised: We ﬁrst prove that this two-dimensional confounding part η ( X ) satisﬁes positivity.

Original: Since ( Q ( 0, X ) , Q ( 1, X ))
Revised: Since ( Q ( 0, X ) , Q ( 1, X ))

Original: =
Revised: =

Original: ( E [ Y j A = 1, X A
Revised: ( E [ Y j A = 1, X A

Original: ^
Revised: ^

Original: Z , X Z ] , E [
Revised: Z , X Z ] , E [

Original: Y j A = 0, X A
Revised: Y j A = 0, X A

Original: ^
Revised: ^

Original: Z , X Z ]) is a function of ( X A ^ Z , X Z ) , the following equations hold:
Revised: Z , X Z ]) is a function of ( X A ^ Z , X Z ) , the following equations hold:

Original: [Equation]
Revised: [Equation]

Original: As 0 < P ( A = 1 j X A
Revised: As 0 < P ( A = 1 j X A

Original: ^ Z , X Z ) < 1, we have 0 < P ( A = 1 j Q ( 0, X ) , Q ( 1, X )) < 1.
Revised: ^ Z , X Z ) < 1, we have 0 < P ( A = 1 j Q ( 0, X ) , Q ( 1, X )) < 1.

Original: Furthermore, we have 0 < P ( ˜ A
Revised: Furthermore, we have 0 < P ( ˜ A

Original: = 1 j Q ( 0, X ) , Q ( 1, X )) <
Revised: = 1 j Q ( 0, X ) , Q ( 1, X )) <

Original: 1 due to almost everywhere equivalence of A and ˜
Revised: 1 due to almost everywhere equivalence of A and ˜

Original: A .
Revised: A .

Original: Since A =
Revised: Since A =

Original: ˜
Revised: ˜

Original: A
Revised: A

Original: [Equation]
Revised: [Equation]

Original: we can rewrite (3.1) by replacing A with ˜ A in the following form:
Revised: we can rewrite (3.1) by replacing A with ˜ A in the following form:

Original: [Equation]
Revised: [Equation]

Original: The equivalence of the ﬁrst and the second line is because X A ^ Z , X Z block all backdoor paths between ˜ A and Y (See Figure 2) and 0 < P ( ˜ A
Revised: The equivalence of the ﬁrst and the second line is because X A ^ Z , X Z block all backdoor paths between ˜ A and Y (See Figure 2) and 0 < P ( ˜ A

Original: = 1 j Q ( 0, X ) , Q ( 1, X ))
Revised: = 1 j Q ( 0, X ) , Q ( 1, X ))

Original: < 1.
Revised: < 1.

Original: Thus, the “ do-operation ” in the ﬁrst line can be safely removed.
Revised: Thus, the “ do-operation ” in the ﬁrst line can be safely removed.

Original: Equivalence of the second line and the third line is due to Q ( ˜ A , X ) =
Revised: Equivalence of the second line and the third line is due to Q ( ˜ A , X ) =

Original: E (cid:0) Y j ˜ A , X A
Revised: E (cid:0) Y j ˜ A , X A

Original: ^ Z , X Z (cid:1) , which is subject to the causal model in Figure 2.
Revised: ^ Z , X Z (cid:1) , which is subject to the causal model in Figure 2.

Original: The last equation is based on the fact that η ( X ) is a function of only X A ^
Revised: The last equation is based on the fact that η ( X ) is a function of only X A ^

Original: Z
Revised: Z

Original: and
Revised: and

Original: X Z .
Revised: X Z .

Original: (It can be easily checked by using the deﬁnition of the expectation.)
Revised: (It can be easily checked by using the deﬁnition of the expectation.)

Original: (B.2) shows that ( Q ( 0, X ) , Q ( 1, X )) is a two-dimensional confounding variable such that
Revised: (B.2) shows that ( Q ( 0, X ) , Q ( 1, X )) is a two-dimensional confounding variable such that

Original: CDE is identiﬁable when we adjust for it as the confounding part.
Revised: CDE is identiﬁable when we adjust for it as the confounding part.

Original: Note that if f and h are two invertible functions on R , ( f ( Q ( 0, X )) , h ( Q ( 1, X )))
Revised: Note that if f and h are two invertible functions on R , ( f ( Q ( 0, X )) , h ( Q ( 1, X )))

Original: also sufﬁces the identiﬁcation for CDE.
Revised: also sufﬁces the identiﬁcation for CDE.

Original: Since the sigma algebra should be the same for ( Q ( 0, X ) , Q ( 1, X )) and f ( Q ( 0, X )) , h ( Q ( 1, X )) , i.e.,
Revised: Since the sigma algebra should be the same for ( Q ( 0, X ) , Q ( 1, X )) and f ( Q ( 0, X )) , h ( Q ( 1, X )) , i.e.,

Original: [Equation]
Revised: [Equation]

Original: Hence, we have
Revised: Hence, we have

Original: [Equation]
Revised: [Equation]

Original: C A DDITIONAL E XPERIMENTS
Revised: C A DDITIONAL E XPERIMENTS

Original: We conduct additional experiments to show how the estimation of causal effect changes 1) over different nonparametric models for the propensity score estimation, and 2) when using different double machine learning estimators on causal estimation.
Revised: We conduct additional experiments to show how the estimation of causal effect changes 1) over different nonparametric models for the propensity score estimation, and 2) when using different double machine learning estimators on causal estimation.

Original: Speciﬁcally, for the ﬁrst study, we apply different nonparametric models and the logistic regression to the estimated confounding part ˆ η
Revised: Speciﬁcally, for the ﬁrst study, we apply different nonparametric models and the logistic regression to the estimated confounding part ˆ η

Original: X ) = (cid:0) ˆ Q
Revised: X ) = (cid:0) ˆ Q

Original: X )  ˆ Q  X ) (cid:1) to obtain propensity scores.
Revised: X ) , ˆ Q 1 ( X ) (cid:1) to obtain propensity scores.

Original: We use
Revised: We use

Original: ATT AIPTW in all above cases for causal effect estimation.
Revised: ATT AIPTW in all above cases for causal effect estimation.

Original: For the second study , we ﬁx the ﬁrst two stages of the TI estimator, i e
Revised: For the second study, we ﬁx the ﬁrst two stages of the TI estimator i .e.

Original: we apply Q-Net for the conditional outcomes and compute propensity scores with the Gaussian process regression where the kernel function is the summation of dot product and white noise.
Revised: we apply Q-Net for the conditional outcomes and compute propensity scores with the Gaussian process regression where the kernel function is the summation of dot product and white noise.

Original: Estimated conditional outcomes and propensity scores are plugged into different double machine learning estimators.
Revised: Estimated conditional outcomes and propensity scores are plugged into different double machine learning estimators.

Original: We make the following conclusions with results of above experiments.
Revised: We make the following conclusions with results of above experiments.

Original: The choice of nonparametric models is signiﬁcant.
Revised: The choice of nonparametric models is signiﬁcant.

Original: Table 3 summarizes results with applying different regression models for the propensity estimation.
Revised: Table 3 summarizes results with applying different regression models for the propensity estimation.

Original: We can see that suitable nonparametric models will strongly increase the coverage proportion over true causal estimand.
Revised: We can see that suitable nonparametric models will strongly increase the coverage proportion over true causal estimand.

Original: Therefore, we conclude that the accuracy in causal estimation is highly dependent on the choice of nonparametric models.
Revised: Therefore, we conclude that the accuracy in causal estimation is highly dependent on the choice of nonparametric models.

Original: In practice, when there is some prior information about the propensity score function, we should apply the most suitable nonparametric model to increase the reliability of our causal estimation.
Revised: In practice, when there is some prior information about the propensity score function, we should apply the most suitable nonparametric model to increase the reliability of our causal estimation.

Original: The ATT AIPTW is consistently the best double machine learning estimator.
Revised: The ATT AIPTW is consistently the best double machine learning estimator.

Original: Tableshows results by applying different double machine learning estimators.
Revised: Tableshows results by applying different double machine learning estimators.

Original: We apply both estimators for the average treatment effect (ATE) and the controlled direct effect (CDE).
Revised: We apply both estimators for the average treatment effect (ATE) and the controlled direct effect (CDE).

Original: The bias of “unadjusted” estimator ˆ τ naive is also included in Table 4 (a).
Revised: The bias of “unadjusted” estimator ˆ τ naive is also included in Table 4 (a).

Original: For  bias,ATT
Revised: For absolute bias,

Original:  AIPTW ˆ τ TI has comparable results with other double machine learning estimators in most cases.
Revised: ATT AIPTW ˆ τ TI has comparable results with other double machine learning estimators in most cases.

Original: For coverage proportion of conﬁdence intervals, though it has lower rates in some cases, ˆ τ TI has consistently the best performance.
Revised: For coverage proportion of conﬁdence intervals, though it has lower rates in some cases, ˆ τ TI has consistently the best performance.

Original: Especially in high confounding situations, the advantage of ˆ τ TI is obvious.
Revised: Especially in high confounding situations, the advantage of ˆ τ TI is obvious.

Original: Estimator For each dataset, we compute estimators as follows.
Revised: Estimator For each dataset, we compute estimators as follows.

Original: n 1 and n 0 stands for the number of individuals in the treated and controlled group.
Revised: n 1 and n 0 stands for the number of individuals in the treated and controlled group.

Original: n = n 1 + n 0 is the total number of individuals.
Revised: n = n 1 + n 0 is the total number of individuals.

Original: • “Unadjusted” baseline estimator: ˆ τ naive
Revised: • “Unadjusted” baseline estimator: ˆ τ naive

Original: [Equation]
Revised: [Equation]

Original: • “Outcome-only” estimator: ˆ τQ
Revised: • “Outcome-only” estimator: ˆ τ

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: • ATT [Equation] ˆ
Revised: • ATT AIPTW: ˆ

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: D D ISCUSSION OF L OW C OVERAGE
Revised: D D ISCUSSION OF L OW C OVERAGE

Original: In this section, we discuss why the conﬁdence intervals we get (See Table 1) have lower coverage than the nominated level 95%.
Revised: In this section, we discuss why the conﬁdence intervals we get (See Table 1) have lower coverage than the nominated level 95%.

Original: We conduct diagnostics and ﬁnd that the inaccuracy of Q ’s estimations is responsible for the low coverage.
Revised: We conduct diagnostics and ﬁnd that the inaccuracy of Q ’s estimations is responsible for the low coverage.

Original: We compute  biases, variances, and coverages of τ TI ’s with different mean squared errors ˆ E [( Q (cid:0) ˆ Q ) 2 ] by using different numbers of datasets.
Revised: We compute absolute biases, variances, and coverages of τ TI ’s with different mean squared errors ˆ E [( Q (cid:0) ˆ Q ) 2 ] by using different numbers of datasets.

Original: According to Figure 4–Figure 5, as the mean squared error of Q increases, the bias of τ TI grows and the coverage of τ TI drops.
Revised: According to Figure 4–Figure 5, as the mean squared error of Q increases, the bias of τ TI grows and the coverage of τ TI drops.

Original: Speciﬁcally, the highest coverage of each setting is almost 95% (use 50 datasets with most accurate conditional outcome estimations).
Revised: Speciﬁcally, the highest coverage of each setting is almost 95% (use 50 datasets with most accurate conditional outcome estimations).

Original: In practice, one direct way to improve the TI estimator’s accuracy is to apply better NLP models so that more accurate conditional outcome estimations can be obtained.
Revised: In practice, one direct way to improve the TI estimator’s accuracy is to apply better NLP models so that more accurate conditional outcome estimations can be obtained.

Original: [Table]
Revised: [Table]

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

Original: [Figure]
Revised: [Figure]

Original: Q loss
Revised: Q loss

Original: [Figure]
Revised: [Figure]

Original: i a s 5.
Revised: i a [Figure]

Original: Q loss
Revised: Q loss

Original: V a r i a n c e5.
Revised: V a r i a n c e

Original: [Figure]
Revised: [Figure]

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

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

Original: [Figure]
Revised: [Figure]