Paper: Catch the Shadow: Automatic Shadow Variables Generation for Treatment Effect Estimation under Collider Bias (ICLR-2024)

Reviewer comment (Evaluation):
Insufficient evaluation. The selected baseline methods/datasets are not designed for collider bias evaluation. Please see questions for more details.

Author response:
**Response:** We thank the reviewer for the useful comments. 

-	For **selected datasets,** to the best of our knowledge, there are no public datasets with CATE ground truths designed for collider bias, thus we choose the widely-used datasets for evaluating the CATE estimation and manually introduce the varying levels of collider bias.
-	For **selected baseline methods,** to the best of our knowledge, we are the first approach to adopt representation learning for addressing the collider bias in CATE estimation (please kindly correct us if we were wrong). For the previous statistical methods tackling the collider bias, they all need prior knowledge of instrumental variables or shadow variables, resulting in different data forms.

Therefore, we compare our method with the baseline methods addressing confounding bias or sample selection bias in which $X$ and $T$ cause $S$, but not $Y$ cause $S$.

Reviewer comment (Novelty):
Limited technical contribution. The proposed method mainly follows the theoretical results outlined by previous works, especially those by Miao and d’Haultfoeuille. Although this is not a critical problem by itself, it is amplified by the fact that the designed algorithm is fragmented into three components. The first and second components (which forms the "Shadow-Catcher"), mandates multiple testing which is subsequently proposed to be addressed by p-value correction. Then the third component comes in and uses existing CATE estimators to obtain the final estimation. There is little to none discussions on the guarantee of the algorithm procedure when putting these three components together.

Author response:
**Response:** We thank the reviewer for pointing out this issue. We also notice that Reviewer 5YRg considered the proposed methods a bit hacky (a combination of multiple steps), but all the steps are well-motivated. To make the entire framework of ShadowCatcher and ShadowEstimator more connected and clearer, **we briefly summarize the overall process** of using ShadowCatcher and ShadowEstimator:

-	ShadowCatcher first takes the fully observed $X$ and $T$, and the observed $Y$ of $S=1$ units as inputs to generate shadow-variable representations $Z$, and then tests whether the generated $Z$ satisfies Assumption 1.
-	If the generated $Z$ does not pass the hypothesis test, ShadowCatcher should re-generate new shadow-variable representations until the generated $Z$ finally passes the test.
-	Finally, ShadowEstimator uses the generated $Z$ to estimate treatment effects with observational samples.

For a clearer presentation, **we add an overall flowchart as shown in Figure 4 on page 14.**

Moreover, **we have the following discussions on the guarantee of the algorithm procedure when putting these three components together,** including: (1) the generation phase of ShadowCatcher, (2) the hypothesis test phase of ShadowCatcher, and (3) ShadowEstimator.

- Putting together (1) the generation phase of ShadowCatcher and (2) the hypothesis test phase of ShadowCatcher is reasonable: as we wrote in Section 3.3, the hypothesis test is theoretically guaranteed (by Theorem 2) to be able to reject $Z$ that does not satisfy the shadow-variable assumption. Therefore, similar to the standard statistical process for finding the shadow variables with the validity hypothesis test, putting together (1) and (2) has sound theoretical guarantees, and one of our main contributions is the automatically generated $Z$.

- Putting together (1, 2) and (3) ShadowEstimator is also reasonable: as we wrote in Section 3.2, the valid $Z$ makes $f(Y | X,T,Z,S=0)$ identifiable, which is theoretically guaranteed (by Theorem 1). Since the generated $Z$ by ShadowCatcher passes the hypothesis test, as well as the ShadowEstimator strictly follows the theoretical results of using $Z$ to identify $f(Y | X,T,Z,S=0)$, the whole procedure has theoretical guarantees following the well-established shadow variable theory.

***

**We hope the above discussion will fully address your concerns about our work, and we would really appreciate it if you could be generous in raising your score.** We look forward to your insightful and constructive responses to further help us improve the quality of our work. Thank you!

***

Reviewer comment (Evaluation):
How is the CATE estimations evaluated on Twins dataset? The Twins dataset is usually used only for evaluating ATE estimations as there is no ground truth counterfactual outcomes.

Author response:
**Response:** The reviewer raises an interesting concern. Following previous studies [1, 2], we use the Twins dataset to evaluate CATE estimation performance. Nonetheless, **we agree with the reviewer that “the Twins dataset is usually used only for evaluating ATE estimations as there is no ground truth counterfactual outcomes”.**

As suggested by the reviewer, **we conducted experiments on two additional datasets with more comprehensive DGPs,** namely **Jobs** and **ACIC 2016**, and **removed CATE evaluation results on Twins dataset** in Table 2. Since the Jobs dataset contains no counterfactual outcomes, we follow [3, 4] to use the policy risk for evaluating the CATE estimation performance, which measures the expected loss if the treatment is taken according to the CATE estimation. The results are shown as below.

| Method | ACIC within-sample | ACIC out-of-sample |Jobs within-sample | Jobs out-of-sample |
|:---------| :---------: | :---------: | :---------: | :---------: |
| Heckit | 3.106$\pm$0.444 | 3.340$\pm$0.111 | 0.328$\pm$0.050 | 0.331$\pm$0.052  |
| DR | 2.346$\pm$0.129 | 2.653$\pm$0.222  | 0.316$\pm$0.007 | 0.317$\pm$0.036 |
| IPSW    |  4.244$\pm$0.145 | 5.411$\pm$0.073 | 0.284$\pm$0.051  | 0.289$\pm$0.063 |
| BNN   | 2.173$\pm$0.150 | 2.586$\pm$0.486 | 0.303$\pm$0.025 | 0.304$\pm$0.041|  
| TARNet    | 2.275$\pm$0.756 | 2.805$\pm$0.766 | 0.315$\pm$0.012 | 0.316$\pm$0.050 |
| CFR   | 2.107$\pm$0.297 | 2.361$\pm$0.587 | 0.313$\pm$0.018 | 0.314$\pm$0.072|
| CForest     | 4.137$\pm$0.295 | 4.605$\pm$0.137  | 0.326$\pm$0.012 | 0.326$\pm$0.059|
| DR-CFR     | 2.240$\pm$0.691 | 2.340$\pm$0.663 | 0.322$\pm$0.022 | 0.323$\pm$0.099|
| TEDVAE     | 3.501$\pm$0.708 | 4.468$\pm$0.813  | 0.296$\pm$0.046 | 0.300$\pm$0.031  |
| DeR-CFR| 2.214$\pm$0.204| 2.246$\pm$0.598 | 0.309$\pm$0.023 | 0.311$\pm$0.029|
| DESCN   | 2.185$\pm$0.150 | 2.306$\pm$0.236 | 0.331$\pm$0.010 | 0.331$\pm$0.051 |
| ES-CFR     | 3.875$\pm$0.224 | 4.494$\pm$0.214   | 0.290$\pm$0.045   | 0.293$\pm$0.046 |
| Ours (New)    | **1.911$\pm$0.126** | **2.047$\pm$0.351** | **0.279$\pm$0.017** | **0.280$\pm$0.018**|  

From the above results, we observe that: (1) The performance of all the estimators is not as good as that on other datasets like the IHDP dataset because the data generation process of the ACIC 2016 datasets is more complex and comprehensive. (2) DR-CFR, DESCN, and ES-CFR show the most competitive performance among all baseline methods. (3) The proposed method stably outperforms all baselines addressing confounding bias or sample selection bias, since the previous methods are limited to the cases in which $X$ and $T$ cause $S$, and cannot handle the cases with addition $Y$ cause $S$. Instead, our ShadowCatcher and ShadowEstimator can address collider bias for accurate CATE estimation. (4) The proposed method on the Jobs dataset shows the lowest policy risk, which demonstrates the effectiveness of our methods in real-world applications.

Reviewer comment (Experiments):
Have the authors compared with more recently proposed CATE/ATE estimators? For example TEDVAE [1], DR-CFR [2]. As the currently compared methods are mainly not designed for collider and are a bit outdated, comparing with more recent methods can further demonstrate if the proposed approach is effective. As we are now at ICLR 2024, it seems insufficient to have most of the baselines proposed in 2016/2017.

Author response:
**Response:** We thank the reviewer for pointing out this issue. Following the reviewer’s suggestion, for both synthetic and real-world experiments, **we added comparisons with more recent CATE estimation baselines**, including: DeR-CFR [1], DR-CFR [5], TEDVAE [6], DESCN [7], and ES-CFR [8]. Among them, **ES-CFR [8] is the state-of-the-art CATE estimation method published in NeurIPS 23,** and the results are as below.

| Method | $\beta=1$ selected |  $\beta=1$ unselected |  $\beta=3$ selected | $\beta=3$ unselected | $\beta=5$ selected | $\beta=5$ unselected |
|:---------| :---------: | :---------: | :---------: | :---------: | :---------: | :---------: |
DeR-CFR | 0.291$\pm$0.010 | 0.309$\pm$0.014 | 0.323$\pm$0.015 | 0.348$\pm$0.017 | 0.358$\pm$0.011 | 0.439$\pm$0.013|
DR-CFR  | 0.284$\pm$0.038 | 0.307$\pm$0.040 | 0.340$\pm$0.055 | 0.355$\pm$0.064 | 0.366$\pm$0.051 | 0.435$\pm$0.060|
TEDVAE  | 0.281$\pm$0.056 | 0.419$\pm$0.070 | 0.378$\pm$0.063 | 0.420$\pm$0.059 | 0.394$\pm$0.054 | 0.431$\pm$0.067|
DESCN | 0.295$\pm$0.002 | 0.312$\pm$0.002 | 0.326$\pm$0.003 | 0.357$\pm$0.004 | 0.365$\pm$0.003 | 0.449$\pm$0.011|
ES-CFR  | 0.289$\pm$0.003 | 0.305$\pm$0.004 | 0.331$\pm$0.003 | 0.359$\pm$0.003 | 0.369$\pm$0.003 | 0.448$\pm$0.005|
Ours (New)    | **0.227$\pm$0.001** | **0.229$\pm$0.001** | **0.249$\pm$0.013** | **0.255$\pm$0.021** | **0.299$\pm$0.008** | **0.300$\pm$0.008** |
|||||||

| Method | IHDP within-sample |  IHDP out-of-sample |  ACIC within-sample | ACIC out-of-sample | Jobs within-sample | Jobs out-of-sample |
|:---------| :---------: | :---------: | :---------: | :---------: | :---------: | :---------: |
|DR-CFR |  1.299$\pm$0.087 | 1.399$\pm$0.171 | 2.240$\pm$0.691 | 2.340$\pm$0.663 | 0.322$\pm$0.022 | 0.323$\pm$0.099|
|TEDVAE     | 4.246$\pm$0.394 | 4.347$\pm$0.563 | 3.501$\pm$0.708 | 4.468$\pm$0.813  | 0.296$\pm$0.046 | 0.300$\pm$0.031  |
|DeR-CFR | 1.446$\pm$0.345 | 1.571$\pm$0.371 | 2.214$\pm$0.204 | 2.246$\pm$0.598  | 0.309$\pm$0.023 | 0.311$\pm$0.029|
|DESCN   | 1.193$\pm$0.057 | 1.665$\pm$0.246 | 2.185$\pm$0.150 | 2.306$\pm$0.236 | 0.331$\pm$0.010 | 0.331$\pm$0.051 |
|ES-CFR     | 1.499$\pm$0.096 | 1.436$\pm$0.095 | 3.875$\pm$0.224 | 4.494$\pm$0.214   | 0.290$\pm$0.045   | 0.293$\pm$0.046 |
|Ours (New)    | **0.703$\pm$0.106** | **0.723$\pm$0.102** | **1.911$\pm$0.126** | **2.047$\pm$0.351** | **0.279$\pm$0.017** | **0.280$\pm$0.018** |
||||||||

The results show that our method outperforms all these more recent CATE estimation baselines, demonstrating the effectiveness of our method.

Reviewer comment (Experiments):
Also is there any reason for selecting the datasets used in the manuscript? IHDP itself is a semi-synthetic dataset with only one data generation process, which only covers rather limited real-world scenarios. Twins is commonly used for ATE estimation instead of CATE estimation.

Author response:
**Response:** Thank you for the comment.

-	Since IHDP is a semi-synthetic dataset with only one data generation process, **we added experiments on **ACIC 2016** dataset with more comprehensive DGPs.**
-	Since Twins is commonly used for ATE estimation instead of CATE estimation, **we added experiments on a new real-world dataset,** namely the **Jobs** dataset, and **removed CATE evaluation results on Twins dataset** in Table 2.

Please kindly refer to our response to [Q1] and Table 2 in the revised manuscript.

Reviewer comment (Theory):
The assumption that X and T are observed when the individual is not sampled seems quite strong and probably not true in many real-world use cases (see Question 1 below to address this).

Author response:
**Response:** Thank you for the comment. We agree with the reviewer that there are also cases "if the individual is not sampled, it's not only $Y$ that is not observed but also $X$ (and $T$), in which case the shadow variable approach would not be applicable". **Instead, the proposed method (and most methods on sample selection bias) is applicable when the target population is determined, which also covers a wide range of real-world scenarios.**

For example, when studying the effect of vaccines ($T$) on disease prevention ($Y$), the target population is determined as, for example, the population of a country or region. Given the target population, governments or institutions would collect all information in terms of their covariates ($X$), and vaccine status ($T$). However, only a portion of the population will remain in follow-up ($S=1$), making the outcome ($Y$) able to be collected, while others will be lost ($S=0$), due to emigration, death, etc.

Another example is semi-supervised learning, in which one of the covariates is regarded as the treatment ($T$), and the rest of the covariates are denoted as $X$. Then the labeled samples are marked as $S=1$ with fully observed $X$, $T$, and $Y$, whereas the unlabeled samples are marked as $S=0$ with only $X$ and $T$ observed.

Reviewer comment (Novelty):
Overall, the method appears a bit hacky (a combination of multiple steps), but all the steps are well-motivated.

Author response:
**Response:** Thank you for the comment. We agree that the proposed method is a bit hacky but well-motivated. To make the entire framework of ShadowCatcher and ShadowEstimator more connected and clearer, **we briefly summarize the overall process** of using ShadowCatcher and ShadowEstimator:

-	ShadowCatcher first takes the fully observed $X$ and $T$, and the observed $Y$ of $S=1$ units as inputs to generate shadow-variable representations $Z$, and then tests whether the generated $Z$ satisfies Assumption 1.
-	If the generated $Z$ does not pass the hypothesis test, ShadowCatcher should re-generate new shadow-variable representations until the generated $Z$ finally passes the test.
-	Finally, ShadowEstimator uses the generated $Z$ to estimate treatment effects with observational samples.

**We also add an overall flowchart as shown in Figure 4 on page 14.**

Reviewer comment (Presentation):
When printed out, fonts in Figures 2 and 3 are barely readable.

Author response:
**Response:** We thank the reviewer for pointing out this issue. **We have adjusted them in our revised version** on page 5 and Figure 3 on page 7 to make them clearer.

Reviewer comment (Evaluation):
Quite often if the individual is not sampled, it’s not only Y that is not observed but also X (and T), in which case the shadow variable approach would not be applicable. Could the authors discuss how this affects the usefulness in practice, and give representative real-world examples about situations in which they expect the method to be useful and where not?

Author response:
**Response:** Thank you for the comment. We agree with the reviewer that there are also cases "if the individual is not sampled, it's not only $Y$ that is not observed but also $X$ (and $T$), in which case the shadow variable approach would not be applicable". **Instead, the proposed method (and most methods on sample selection bias) is applicable when the target population is determined, which also covers a wide range of real-world scenarios.**

For example, when studying the effect of vaccines ($T$) on disease prevention ($Y$), the target population is determined as, for example, the population of a country or region. Given the target population, governments or institutions would collect all information in terms of their covariates ($X$), and vaccine status ($T$). However, only a portion of the population will remain in follow-up ($S=1$), making the outcome ($Y$) able to be collected, while others will be lost ($S=0$), due to emigration, death, etc.

Another example is semi-supervised learning, in which one of the covariates is regarded as the treatment ($T$), and the rest of the covariates are denoted as $X$. Then the labeled samples are marked as $S=1$ with fully observed $X$, $T$, and $Y$, whereas the unlabeled samples are marked as $S=0$ with only $X$ and $T$ observed.

Reviewer comment (Theory):
The equation for the loss function in the “Hypothesis Test Phase” paragraph is not understood by the reviewer. Could there be a typo?

Author response:
**Response:** We thank the reviewer for pointing out typos here. **The correct equation should be $L_q = 1/n * \sum_{i=1}^n||(s_i / q(x_i, t_i, y_i) - 1) \cdot (x_i, z_i, t_i)||_2^2$, where $(x_i, z_i, t_i)$ is a vector and $||.||_2$ is the L2-norm.** $L_q$ aims to learn a solution q of Q by Theorem 2 in Equation (6). Specifically, if $E[S / Q(X, T, Y) - 1 | X, Z, T] = 0$ (by Theorem 2 in Equation (6)), then $E[E[S / Q(X, T, Y) - 1 | X, Z, T] * (X, Z, T) ]=0$. The left hand side equals to $E[(S / Q(X, T, Y) - 1) * (X, Z, T)] = 0$. Then $L_q$ is just to minimize the square of the L2-norm of that Equation. We have corrected that on page 6, and added a more detailed explanation of $L_q$ in Appendix A.3.3 on page 18.

Reviewer comment (Evaluation):
Since the conditional independence assumption is not strictly constrained due to missing data, the authors use a hypothesis test phase to test whether this assumption is satisfied and only the generated representations that pass the test can ShadowCatcher finish learning. As the authors also state in the paper, the choice of $alpha$ will significantly affect the efficiency of ShadowCatcher. As the results in Table 5 and Figure 4 show, the smaller $alpha$ is (which means the test is more strict), the better the performance is. But the number of iterations also gets bigger when the test is too strict. Therefore, the tradeoff between the efficiency and performance of the proposed method should be considered carefully in real-world applications.

Author response:
Thank you for your constructive suggestion. **We have conducted additional experiments to explore the impact of different choices of $\alpha$ on the efficiency and performance**. The results are shown as below.

| Reject threshold | $\sqrt{\mathrm{PEHE}}$ selected data |  $\sqrt{\mathrm{PEHE}}$ unselected data | mean number of iterations |
|:---------| :---------: | :---------: | :---------: |
|$1e-4$| 0.235$\pm$0.003 | 0.235$\pm$0.008 | **1.2** |
| $5e-5$| 0.228$\pm$0.003 | 0.231$\pm$0.004 | 2 |
| $1e-5$| 0.228$\pm$0.002 | 0.231$\pm$0.001 | 4.6 |
| $5e-6$| 0.227$\pm$0.002 | 0.230$\pm$0.002 | 7.9 |
| $1e-6$| **0.227$\pm$0.001** | **0.229$\pm$0.001** | 10 |
| | | | | 

The above results show that the performance of ShadowCatcher improves as the reject threshold decreases, because the hypothesis test gets more strict, and the constraint gets more reliable. However, the number of iterations required for ShadowCatcher to pass the hypothesis test also increases rapidly, reducing the efficiency of ShadowCatcher. Therefore, choosing an appropriate $\alpha$ is a trade-off between efficiency and performance, which depends on the real-world application scenarios.

Reviewer comment (Presentation):
A minor concern is that the figure size is somewhat small.

Author response:
We thank the reviewer for pointing out this issue. **We have adjusted them in our revised version on page 5 and Figure 3 on page 7 to make them clearer.**

Reviewer comment (Theory):
I found myself getting just a little lost in the preliminaries of the shadow variable, particularly at the bottom of page 4. I found it confusing to say that f(Z | X, S) was identifiable from the observed data, and then to still say that we needed to find \tilde(OR) - I thought the f(Z|...) functions eliminated the need to calculate \tilde(OR) according to equation 4. I also don't quite see why Eq 5 is true and think this could use more explanation.

Author response:
> **[W1.1] Why do the $f(Z|...)$ functions not eliminate the need to calculate $\widetilde{OR}$ according to Equation (4)?**

**Response:** Thank you for the comment. The necessity for further calculating $\widetilde{OR}(X, T, Y)$ given $f(Z | X, T, S)$ comes from that: for identification of $f(Y | X, T, S=0)$, we need to obtain **both $OR(X, T, Y)$ and $E[\widetilde{OR}(X, T, Y) | X, Z, T, S=1]$** by Equation (2). The latter can be easily calculated by Equation (4) using $f(Z | X, T, S)$. However, **the former still needs $\widetilde{OR}(X, T, Y)$ and $\widetilde{OR}(X, T, Y=0)$** by Equation (5). Thus, it is still necessary to know what $\widetilde{OR}(X, T, Y)$ as a function of $X, T, Y$ is to calculate $\widetilde{OR}(X, T, Y=0)$, and we need to **further solve for $\widetilde{OR}(X, T, Y)$ as a function of $X, T, Y$ from the calculated $E[\widetilde{OR}(X, T, Y) | X, Z, T, S=1]$** by Equation (4).

> **[W1.2] Why is Equation (5) true?**

**Response:** By Equation (2), $OR(X, T, Y=0) = 1$ (just substitute $Y=0$ into Equation (2)). Therefore, by the definition of $\widetilde{OR}(X, T, Y)$ that $\widetilde {OR}(X, T, Y) = OR(X, T, Y) / E[OR(X, T, Y) | X, T, S=1]$, **the right-hand side of Equation (5) equals to $OR(X, T, Y) / OR(X, T, Y=0)$**. Since $OR(X, T, Y=0) = 1$, we have **$OR(X, T, Y) / OR(X, T, Y=0)$ equals the left-hand side of Equation (5) (i.e., $OR(X, T, Y)$)**. To make this clearer, **we make a detailed explanation of Equation (5) in Appendix A.3.2 on page 18**.

Reviewer comment (Theory):
I found some of the loss functions through 3.3 and 3.4 a little unintuitive. 1) I found it a little odd to try to look at -Z to maximize MSE - this means that the function will behave differently for Z close to 0 since Z and -Z are near each other. Would it be reasonable or more sensible to take random Z instead? or why is -Z the best idea? 2) I'm not sure why h_r, a function that aims to predict Z well, will also help to move the Z0 and Z1 distributions towards each other. 3) what is the notation \dot (x_i, z_i, t_i) in the loss function for Q? it's not clear if there's a typo here, since this just looks like a tuple. 4) I'm not sure why distilling h_z0 / h_z1 into a \tilde(or) function is necessary.

Author response:
> **[W2.1] Would it be reasonable or more sensible to take random $Z$ as the implementation of $Z^-$ instead?**

**Response:** Thank you for such insightful advice. **It is absolutely reasonable and we have incorporated this suggestion throughout our paper.** In fact, the original description of $Z^-$ in our article that "the opposite value of $Z$" is not accurate since we misuse the word "opposite." **We have corrected it in our revised version (on Page 6) that $Z^-$ indeed denotes a value that differs from $Z$ to constrain the conditional dependence assumption**, which any strategy can implement. We choose $-Z$ as $Z^-$ for continuous $Z$s just because this is easy to implement. We do agree that **taking a random Z instead is a better idea** since when $Z$ is close to 0, $-Z$ does not satisfy the requirement of $Z^-$. **We use this new implementation to re-run all experiments and achieve obvious performance improvements**. The results are shown as below.

| Method | $\beta=1$ selected |  $\beta=1$ unselected |  $\beta=3$ selected | $\beta=3$ unselected | $\beta=5$ selected | $\beta=5$ unselected |
|:---------| :---------: | :---------: | :---------: | :---------: | :---------: | :---------: |
| Ours (Old) | 0.241$\pm$0.014 | 0.248$\pm$0.009 | 0.305$\pm$0.013 | 0.326$\pm$0.015 | 0.333$\pm$0.040 | 0.404$\pm$0.053 |
| Ours (New) | **0.227$\pm$0.001** | **0.229$\pm$0.001** | **0.249$\pm$0.013** | **0.255$\pm$0.021** | **0.299$\pm$0.008** | **0.300$\pm$0.008** |
||||||||

| Method | IHDP within-sample |  IHDP out-of-sample |  ACIC within-sample | ACIC out-of-sample | Jobs within-sample | Jobs out-of-sample |
|:---------| :---------: | :---------: | :---------: | :---------: | :---------: | :---------: |
| Ours (Old) | 1.039$\pm$0.069 | 1.065$\pm$0.099 | 2.078$\pm$0.333 | 2.142$\pm$0.390 | 0.283$\pm$0.018 | 0.284$\pm$0.080 |
| Ours (New) | **0.703$\pm$0.106** | **0.723$\pm$0.102** | **1.911$\pm$0.126** | **2.047$\pm$0.351** | **0.279$\pm$0.017** | **0.280$\pm$0.018** |
||||||||

The original implementation of our method is denoted as Ours (Old), whereas the new implementation of our method is denoted as Ours (New). Note that we add an additional real-world dataset - Jobs to better evaluate the performance, where the evaluation metric is the policy risk $\hat{R}_{\mathrm{Pol}}=1-(\mathbb{E}[Y(1)\mid \tau(\mathbf{x}) > 0,T=1] \cdot \mathbb{P}(\tau(\mathbf{x}) > 0) + \mathbb{E}[Y(0)\mid \tau(\mathbf{x}) \leq 0,T=0]\cdot \mathbb{P}(\tau(\mathbf{x}) \leq 0)$ instead of $\sqrt{\mathrm{PEHE}}$ for other datasets. **The new implementation (taking a random $Z$ instead of $-Z$ as $Z^{-}$) shows obvious performance improvements on all datasets.** We really appreciate your great advice!

> **[W2.2] Why will $h_r$, a function that aims to predict Z well, also help to move the $Z_0$ and $Z_1$ distributions towards each other?**

**Response:** The reviewer raises an interesting concern. Please kindly note that $h_r$ itself is not the key to moving the $Z_0$ and $Z_1$ distributions towards each other. Instead, **as $h_r$ is the estimate of $f(Z | X, T, Y, S=1)$, if $h_r$ can also predict $Z$ well for $S=0$ data, then $f(Z | X, T, Y, S=1) = f(Z | X, T, Y, S=0)$** and the assumption that $Z$ is independent of $S$ given $X, T, Y$ is satisfied.

> **[W2.3] What is the notation $\cdot (x_i, z_i, t_i)$ in the loss function for $Q$? it's not clear if there's a typo here.**

**Response:** We thank the reviewer for pointing out typos here. **The correct equation should be $L_q = 1/n * \sum_{i=1}^n||(s_i / q(x_i, t_i, y_i) - 1) \cdot (x_i, z_i, t_i)||_2^2$, where $(x_i, z_i, t_i)$ is a vector and $||.||_2$ is the L2-norm.** $L_q$ aims to learn a solution q of Q by Theorem 2 in Equation (6). Specifically, if $E[S / Q(X, T, Y) - 1 | X, Z, T] = 0$ (by Theorem 2 in Equation (6)), then $E[E[S / Q(X, T, Y) - 1 | X, Z, T] * (X, Z, T) ]=0$. The left hand side equals to $E[(S / Q(X, T, Y) - 1) * (X, Z, T)] = 0$. Then $L_q$ is just to minimize the square of the L2-norm of that Equation. We have corrected that on page 6, and added a more detailed explanation of $L_q$ in Appendix A.3.3 on page 18.

> **[W2.4] why is distilling $h_{z_0} / h_{z_1}$ into a $\widetilde{or}$ function is necessary?**

**Response:** Thank you for the comment, but we believe this question comes from [W1.1] in which Proposition 1 and Equation (5) confuse you. Please kindly refer to our **response to [W1.1]**, it is necessary because we need to **further solve for $\widetilde{OR}(X, T, Y)$ as a function of $X, T, Y$ from the calculated $E[\widetilde{OR}(X, T, Y) | X, Z, T, S=1]$ by Equation (4)**, so that we can obtain $OR(X, T, Y)$ from $\widetilde{OR}(X, T, Y)$ by Equation (5) and further identify $f(Y | X, Z, T, S=0)$ by Equation (2).

Reviewer comment (Theory):
I'm confused by the comment right at the end of 3 around deconfounding methods: I thought there was an assumption around unconfoundedness. Is this still a fair comparison to other methods if deconfounding methods are used?

Author response:
> **[W3] I thought there was an assumption around unconfoundedness. Is this still a fair comparison to other methods if de-confounding methods are used?**

**Response:** The reviewer raises an interesting concern. However, the de-confounding way is just an implementation of the outcome estimators in ShadowCatcher and ShadowEstimator, which **will not affect (either positive or negative) the ability to address collider bias**. As you may noticed, we do need the unconfoundedness assumption on the target population (but not on only $S=1$ data because this assumption can be violated due to collider bias). This means we assume that there are no unobserved confounders, but there can possibly be confounding bias caused by fully observed covariates. The de-confounding method (an IPM to balance treated and controlled representations following [1]) we use is just to **avoid the impact of such confounding bias on CATE estimation, thus we suppose the evaluation is fair to see each method's ability to address collider bias**.

Reviewer comment (Evaluation):
What are the failure modes of this method - under what circumstances will learning a shadow variable be harder than others?

Author response:
> **[Q1] What are the failure modes of this method - under what circumstances will learning a shadow variable be harder than others?**

**Response:** Thank you for the comment. We believe **it depends on the extent to which covariates are involved in sample selection** in the data. If all covariates are highly involved in sample selection, it may be more difficult to extract representations that satisfy the assumption of shadow variables, that is, the conditional independence with $S$ given all covariates, treatment, and outcome.