Paper: Bounds on Representation-Induced Confounding Bias for Treatment Effect Estimation (ICLR-2024)

Reviewer comment (Novelty):
As discussed in the Summary part of this review, for me it was hard to understand why we need a new approach of choosing X. The concept of RICB is, in essence, equivalent to confounder bias but formulated in a different choice of X. For example, in the traditional way of choosing X as only unobservables and talking about $\phi(X)$ not being a balancing score, potential effect of unobservable covariates not being included as X can be discussed. So I am not sure about the potential benefit of choosing X to include unobservable covariates.

Author response:
Thank you! We appreciate your valuable review and that you found our paper well-written, and experiments comprehensive. In the following, we will respond to your questions.

**Response to Weaknesses and Questions.**



1. **Motivation of our approach.** We are more than happy to clarify the motivation behind our approach.  

  + _What is the motivation behind our approach?_ In our paper, we build upon the task of CATE estimation. That is, we follow previous literature [1-7] and assume that **all the confounders** are **observed** inside of $X$. Hence, when using representation learning methods, one maps an $X$ onto lower-dimensional representation $\Phi(X)$. Formally, $X$ is $n$-dimensional and $\Phi(X)$ is $p$-dimensional with $p &lt; n$. Then, by learning the representation $\Phi(X)$, we face an information loss as not all information about $X$ can be stored in the representation. In simple words, $X$ is now split into (i) the main component $\Phi(X)$ which is used for the downstream analysis and (ii) an implicit (complementary) component $\Omega(X)$ that captures everything else. Importantly, not only $X$ is observed, but also both components $\Phi(X)$ and $\Omega(X)$ are observed or could be inferred. However, CATE estimation using representation learning is only based on component (i) and not component (ii), (which is discarded). In the language of our paper, **component (ii) is all the information, discarded by the representation, which could contain ground-truth confounders and, thus, lead to bias**. 


   +  _How is our approach related to traditional, hidden confounders?_ An alternative view of component (ii) is to build upon causal sensitivity analysis and simply say that these are essentially ‘non-observable’ confounders. The reason is that, even though they are observed, they are actually not used in existing CATE estimation methods. As a result, **CATE estimation with a lower-dimensional representation leads to a confounding bias**. We highlight the differences between (a) traditional, hidden confounding and (b) our setting in the following table.
  | Setting / Application            | Observed variables | Unobserved variables | Ignorability            | Confounding induced by the usage of |
  |----------------------------------|--------------------|----------------------|-------------------------|-------------------------------------|
  | Hidden confounding               | $X$                | $U$                  | $Y[a] \mathrel{\unicode{x2AEB}} A \mid X, U$ | $X$                                 |
  | Representation learning for CATE | $X, \Phi(X)$       | —                    | $Y[a] \mathrel{\unicode{x2AEB}} A \mid X$    | $\Phi(X)$                           |

   + _Why is our approach relevant and general?_ As we show in our paper, such bias is inherent to **almost all** existing CATE methods that build upon representation learning. This just happens naturally because learning a lower-dimensional representation leads to information loss and thus confounding bias. Oftentimes, such confounding bias is reinforced by the learning objective (e.g., balancing), which presents further sources of bias. Of note, there is no ‘easy fix’ (or ‘no free lunch’) to the confounding bias from representation learning. On the one hand, CATE based on representation learning is actively searching to learn a lower-dimensional representation as this has lower variance and generalizes better. So, we actively aim for $p &lt; n$ (and not $p \approx n$). Yet, this leads to confounding bias. On the other hand, we could remove confounding bias by setting $p \approx n$, yet this would preclude any gains in prediction performance from representation learning. 

       To avoid confusion, we would also like to clarify the following. The bidirectional edge in Fig. 1 between $Y$ and $X$ is not used to indicate hidden confounding wrt. to treatment $A$, but rather to denote the general causal diagram, implied by assumptions (1)-(3). 
   
&nbsp; &nbsp; &nbsp;  **Action:** We added clarifications along the above lines to our paper.

Reviewer comment (Theory):
Theoretical bounds provided should be appreciated, but I cannot be sure how strong this theoretical bound is only from current version of the manuscript.

Author response:
2. **Bounds.** Thank you for raising this important question. We added a **new Corrolary 1** in **Appendix B,** where we state that bounds are **valid and sharp** (in the sense of sensitivity models). This result was formally proved in [8]. This implies the following: 
   * Valid means that our bounds always contain the ground-truth representation CATE (given the ground-truth representation-conditional distribution of the outcome). 
   * Sharp means that the bounds only include the observational distributions complying with a sensitivity constraint from Eq. 10.  

    We empirically evaluated validity and sharpness jointly. E.g., the reported results in Tables 2-4 and Figures 4-6 showed improvements over the baselines due to our framework. Therein, a drop in error rates implies validity, and the fact that we defer not too many observations/individuals leads to sharpness.


    A further question is how tight our bounds are, yet tightness is an empirical quantity that depends on both the fitted CATE estimator and a chosen sensitivity model. Hence, we cannot assess tightness theoretically but only through empirical analysis. To do so, we added new **Figures 7 and 8,** where we plot the ground-truth decision boundaries and decision boundaries based on the estimated bounds for the synthetic benchmark. We see that estimated bounds are valid (they contain the ground-truth decision boundary of CATE wrt. representations) and sufficiently tight, especially, when the representation is of the same dimensionality as the covariates (so there is less chance of the presence of the RICB). This confirms the effectiveness of our bounds.  

    **Action:** We added a new **Corrolary 1 **in** Appendix B** on the properties of the MSM bounds to the revised version of the paper. We added **Figures 7-8** where we visualize the bounds (ground-truth and estimated decision boundaries) to show that our bounds are valid and tight.

Reviewer comment (Theory):
In terms of Weakness 1: Could you please give a clear motivation of choosing X to include unobservable but considering $\phi(X)$ as observable, and then reframing the confounder bias we know as RICB in the newly proposed setting? Does the fact that we are dealing with representation learning make some difference? I just want to try to understand.

Author response:
Thank you! We appreciate your valuable review and that you found our paper well-written, and experiments comprehensive. In the following, we will respond to your questions.

**Response to Weaknesses and Questions.**



1. **Motivation of our approach.** We are more than happy to clarify the motivation behind our approach.  

  + _What is the motivation behind our approach?_ In our paper, we build upon the task of CATE estimation. That is, we follow previous literature [1-7] and assume that **all the confounders** are **observed** inside of $X$. Hence, when using representation learning methods, one maps an $X$ onto lower-dimensional representation $\Phi(X)$. Formally, $X$ is $n$-dimensional and $\Phi(X)$ is $p$-dimensional with $p &lt; n$. Then, by learning the representation $\Phi(X)$, we face an information loss as not all information about $X$ can be stored in the representation. In simple words, $X$ is now split into (i) the main component $\Phi(X)$ which is used for the downstream analysis and (ii) an implicit (complementary) component $\Omega(X)$ that captures everything else. Importantly, not only $X$ is observed, but also both components $\Phi(X)$ and $\Omega(X)$ are observed or could be inferred. However, CATE estimation using representation learning is only based on component (i) and not component (ii), (which is discarded). In the language of our paper, **component (ii) is all the information, discarded by the representation, which could contain ground-truth confounders and, thus, lead to bias**. 


   +  _How is our approach related to traditional, hidden confounders?_ An alternative view of component (ii) is to build upon causal sensitivity analysis and simply say that these are essentially ‘non-observable’ confounders. The reason is that, even though they are observed, they are actually not used in existing CATE estimation methods. As a result, **CATE estimation with a lower-dimensional representation leads to a confounding bias**. We highlight the differences between (a) traditional, hidden confounding and (b) our setting in the following table.
  | Setting / Application            | Observed variables | Unobserved variables | Ignorability            | Confounding induced by the usage of |
  |----------------------------------|--------------------|----------------------|-------------------------|-------------------------------------|
  | Hidden confounding               | $X$                | $U$                  | $Y[a] \mathrel{\unicode{x2AEB}} A \mid X, U$ | $X$                                 |
  | Representation learning for CATE | $X, \Phi(X)$       | —                    | $Y[a] \mathrel{\unicode{x2AEB}} A \mid X$    | $\Phi(X)$                           |

   + _Why is our approach relevant and general?_ As we show in our paper, such bias is inherent to **almost all** existing CATE methods that build upon representation learning. This just happens naturally because learning a lower-dimensional representation leads to information loss and thus confounding bias. Oftentimes, such confounding bias is reinforced by the learning objective (e.g., balancing), which presents further sources of bias. Of note, there is no ‘easy fix’ (or ‘no free lunch’) to the confounding bias from representation learning. On the one hand, CATE based on representation learning is actively searching to learn a lower-dimensional representation as this has lower variance and generalizes better. So, we actively aim for $p &lt; n$ (and not $p \approx n$). Yet, this leads to confounding bias. On the other hand, we could remove confounding bias by setting $p \approx n$, yet this would preclude any gains in prediction performance from representation learning. 

       To avoid confusion, we would also like to clarify the following. The bidirectional edge in Fig. 1 between $Y$ and $X$ is not used to indicate hidden confounding wrt. to treatment $A$, but rather to denote the general causal diagram, implied by assumptions (1)-(3). 
   
&nbsp; &nbsp; &nbsp;  **Action:** We added clarifications along the above lines to our paper.

Reviewer comment (Theory):
In terms of Weakness 2: How tight is the bound for some popular special cases, especially for the settings you did your experiment? Are they practically good?

Author response:
2. **Bounds.** Thank you for raising this important question. We added a **new Corrolary 1** in **Appendix B,** where we state that bounds are **valid and sharp** (in the sense of sensitivity models). This result was formally proved in [8]. This implies the following: 
   * Valid means that our bounds always contain the ground-truth representation CATE (given the ground-truth representation-conditional distribution of the outcome). 
   * Sharp means that the bounds only include the observational distributions complying with a sensitivity constraint from Eq. 10.  

    We empirically evaluated validity and sharpness jointly. E.g., the reported results in Tables 2-4 and Figures 4-6 showed improvements over the baselines due to our framework. Therein, a drop in error rates implies validity, and the fact that we defer not too many observations/individuals leads to sharpness.


    A further question is how tight our bounds are, yet tightness is an empirical quantity that depends on both the fitted CATE estimator and a chosen sensitivity model. Hence, we cannot assess tightness theoretically but only through empirical analysis. To do so, we added new **Figures 7 and 8,** where we plot the ground-truth decision boundaries and decision boundaries based on the estimated bounds for the synthetic benchmark. We see that estimated bounds are valid (they contain the ground-truth decision boundary of CATE wrt. representations) and sufficiently tight, especially, when the representation is of the same dimensionality as the covariates (so there is less chance of the presence of the RICB). This confirms the effectiveness of our bounds.  

    **Action:** We added a new **Corrolary 1 **in** Appendix B** on the properties of the MSM bounds to the revised version of the paper. We added **Figures 7-8** where we visualize the bounds (ground-truth and estimated decision boundaries) to show that our bounds are valid and tight.

Reviewer comment (Theory):
No theoretical proof of the proposed bounds.

Author response:
Thank you for suggesting to make the derivation of the bounds more explicit. For our bounds, we adapted the theoretic results, provided in [3,4]. We followed your suggestion and added a more rigorous derivation (see new **Lemma 3** and the theoretical proof).

    **Action:** We added a new **Lemma 3** to **Appendix B** where we detail the derivation of the bounds.

Reviewer comment (Writing):
It seems that the notations/symbols are not defined correctly. For example, in the section of notations, the authors claim that $\mu_a^x(x)=\mathbb{E}(Y|A=1,X=x)$, but $\mu_a^x(x)$ should be $\mathbb{E}(Y|A=a,X=x)$. In the same paragraph, the authors claim that $\mu_a^\phi(\phi)=\mathbb{E}(Y|A=1,\Phi(X)=\phi)$, but $\mu_a^\phi(\phi)$ should be $\mathbb{E}(Y|A=a,\Phi(X)=\phi)$.

Author response:
We apologize for the small errors in notation (e.g., in the definitions of $\mu$) and thank you for spotting them. **Action:** We corrected the errors in the revised version of the paper (e.g., we corrected the definitions for $\mu_a^x(x)$ and $\mu_a^\phi(\phi)$).

Reviewer comment (Writing):
The authors define $\pi_a^x(x)= \mathbb{P}(A=a|X=x)$. I wonder why the authors do not simply use $\pi_a^x$ or $\pi_a(x)$. Problem arises when the authors introduce the overlap assumption. The authors claim that $\mathbb{P}(0<\pi_a^x(X)<1)=1$, but I cannot obtain $\pi_a^x(X)$ from the definition. Indeed, in the definition of $\pi_a^x(x)= \mathbb{P}(A=a|X=x)$, the two “x”s in $\pi_a^x(x)$ should be mapped to “x” in $\mathbb{P}(A=a|X=x)$. Nevertheless, when $\pi_a^x(x)$ is changed to $\pi_a^x(X)$ or $\pi_a^X(x)$, the mapping procedure is not clear.

Author response:
Regarding $\pi_a^x(\cdot)$, **the upper index is not a variable or argument but an indicator** that this propensity relates to the original covariate space (in contrast to $\pi_a^\phi(\cdot)$, which relates to the representation propensity). Therefore, the overlap assumption is defined properly, as only the $x$ inside parenthesis is considered as an argument, namely, $\pi_a^x(X)$. Hence,  $\pi_a^x(\cdot)$ serves as a measurable function, and the probabilistic statement maps a random variable $X$ to the (random) propensity score. Notably, our overlap assumption is equivalent to the $0 < \pi_a^x(x) < 1$ for all $x$ s.t. $\mathbb{P}(X = x) > 0$. **Action:** We clarified the notation around $\pi_a^x(\cdot)$.

Reviewer comment (Theory):
According to the definition of $X$, $X=\{X^\emptyset,X^a,X^y,X^\bigtriangleup\}$. At the same time, $X$ is independent of $X^\emptyset$, $X^a$, $X^y$, $X^\bigtriangleup$ conditioning on $\Phi(X)$. It is strange to claim Eqn. (4).

Author response:
Whenever $\Phi(X)$ is invertible wrt. to some sub-covariate, the conditional independence statement contains (partially) random $X$ and deterministic (constant) sub-covariate (due to invertibility). $X$ and its deterministic sub-covariate are then independent by the definition ([https://en.wikipedia.org/wiki/Independence_(probability_theory)#For_real_valued_random_vectors](https://en.wikipedia.org/wiki/Independence_(probability_theory)#For_real_valued_random_vectors)). 

    Eq. (4) is a way to formalize that the full information about all four sub-covariates is preserved in the representation. This means that the representation is an invertible function wrt. sub-covariates. Conversely, if the information is partially or fully removed in the representation, the non-independence statement will hold (e.g., in Fig. 1). This notation is e.g. consistent with the literature on the prognostic scores [1].

    **Action:** We clarified in the revised version of the paper why Eq. 4 holds.

Reviewer comment (Theory):
In the example “Representations with removed noise and instruments”, the authors claim that under Eqn. (5), the validity follows from the d-separation in clustered causal diagram and Appendix B. In Appendix B, only investigations related to the example “Invertible representations” are presented.

Author response:
Thank you for spotting a missing reference. 

    **Action:** We added a missing reference to **Lemma 2 (Removal of noise and instruments)** in the revised version of our paper.

Reviewer comment (Theory):
I suspect the equality of $\mathbb{E}(Y[1]-Y[0]|X=x)=\mathbb{E}(Y[1]-Y[0]|X^\bigtriangleup=x^\bigtriangleup, X^y=x^y)$ and $\mathbb{E}(Y[1]-Y[0]|X^\bigtriangleup=x^\bigtriangleup, X^y=x^y)= \mathbb{E}(Y[1]-Y[0]|\Phi(X)=\Phi(x) )$ in Eqn. (6) under Eqn. (5). Could the authors provide more details?

Author response:
Thank you for asking this question. Eq. 6 holds under the assumptions of the clustered causal diagram and invertibility of $\Phi(\cdot)$ wrt. $X^\Delta$ and $X^y$.

    **Action:** We provide the derivation of Eq. 6 in a newly added **Lemma 2 (Removal of noise and instruments)** in a revised version of the paper.

Reviewer comment (Theory):
The largest weakness I see is the same as what is commonly shared throughout the sensitivity analysis literature, namely that practitioners must place assumptions on the extent of confounding.

Author response:
**Answer to Weaknesses.** Thank you. As you nicely point out, a common limitation in the use of MSMs is that the sensitivity parameter (which guides the amount of hidden confounding) must be chosen through expert knowledge. Upon reading your comment, we realized that we should explain more clearly that we do **not** have this limitation but that we can learn the sensitivity parameter from data. The reason is that our framework aims at partial identification of CATE wrt. representations but not sensitivity analysis in the classical sense. Therefore, we employ the marginal sensitivity model (MSM) in an unconventional way. Specifically, we do not require practitioners to pre-specificy the extent of confounding through expert knowledge. Instead, our framework works in a data-driven manner and can infer the sensitivity parameters from the data (see Sec. 4.2, Stage 1). This is a crucial difference between usual MSM applications to detect hidden confounding and our setting where all the confounders are observed.

**Action:** We stress this key difference more clearly in the revised version of the paper. In particular, we highlighted that we do **not** place assumptions on the extent of confounding (see **Section 2**).