SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In this section, we present the structural causal models (SCMs) utilized for generating the dataset. Furthermore, we include a segment of the code employed to generate the dataset.
D.1.1 Example 1
We define the following structural causal models for Example 1:
$$
U _ {W}, U _ {W Z}, U _ {W Y}, U _ {X Y} \sim \text {n o r m a l} (0, 1),
$$
$$
\mathbf {W} := f _ {W} \left(U _ {W Z}, U _ {W Y}, U _ {W}\right),
$$
$$
X := f _ {X} (\mathbf {W}, U _ {X Y}),
$$
$$
Z := f _ {Z} (W, X, U _ {W Z}),
$$
$$
Y := f _ {Y} (Z, U _ {W Y}, U _ {X Y}),
$$
$$
f _ {W} \left(U _ {W Z}, U _ {W Y}\right) := \left\lfloor \frac {1}{1 + \exp \left(U _ {W} - U _ {W Z} + U _ {W Y}\right)} \right\rfloor ,
$$
$$
f _ {X} (\mathbf {W}, U _ {X Y}) := \left\lfloor \frac {1}{1 + \exp (\mathbf {c} _ {X} ^ {\intercal} \mathbf {W} + U _ {X Y})} \right\rfloor ,
$$
$$
f _ {Z} (W, X, U _ {W Z}) := \left\lfloor \frac {1}{1 + \exp (U _ {W Z} \cdot \mathbf {c} _ {Z} ^ {\intercal} \mathbf {W} + 4 X - 2 + U _ {W Z})} \right\rfloor
$$
$$
f _ {Y} (Z, U _ {W Y}, U _ {X Y}) := \left\lfloor \frac {1}{1 + \exp (4 Z - 2 + 0 . 5 U _ {W Y} - U _ {X Y})} \right\rfloor ,
$$
where the coefficient vector  $\mathbf{c}_X, \mathbf{c}_Z$  are such that their  $i$ th element is 0 if  $i$  is an even number and 1 otherwise.
D.1.2 Example 2
We define the following structural causal models for Example 2:
$$
U _ {X _ {1}, X _ {2}}, U _ {X _ {1}, W}, U _ {X _ {1}, R}, U _ {X _ {2}, W}, U _ {X _ {2}, Y} \sim \text {n o r m a l} (0, 1),
$$
$$
X _ {1} := f _ {X _ {1}} \left(U _ {X _ {1}, X _ {2}}\right),
$$
$$
X _ {2} := f _ {X _ {2}} \left(U _ {X _ {1}, X _ {2}}\right),
$$
$$
R := f _ {R} \left(U _ {X _ {1}, R}, X _ {1}, X _ {2}\right),
$$
$$
W := f _ {W} \left(U _ {X _ {1}, W}, U _ {X _ {2}, W}, X _ {1}, R\right),
$$
$$
Y := f _ {Y} \left(U _ {X _ {2}, Y}, X _ {2}, R, W\right),
$$
Detailed parametrization of parameters including learning rates, maximum depth of the trees, etc. are explained in https://xgboost.readthedocs.io/en/stable/python/python_api.html#xgboost.XGClassifier.
(a) An SCM  $\mathcal{M}$  for Project STAR
(b)  $G(\mathcal{M}_{x_1,x_2})$
Figure E.4: Example causal graphs for Section E. Nodes representing the treatment and outcome are marked in blue and red, respectively.
(c)  $G(\mathcal{M}_{x_2})$
$$
f _ {X _ {1}} \left(U _ {X _ {1}, X _ {2}}\right) := \left\lfloor \frac {1}{1 + \exp \left(2 U _ {X _ {1} , X _ {2}} - 1\right)} \right\rfloor ,
$$
$$
f _ {X _ {2}} (U _ {X _ {1}, X _ {2}}) := \left\lfloor \frac {1}{1 + \exp (3 U _ {X _ {1} , X _ {2}} + 1)} \right\rfloor ,
$$
$$
f _ {R} \left(U _ {X _ {1}, R}, X _ {1}, X _ {2}\right) := \left\lfloor \frac {1}{1 + \exp \left(U _ {X 1 , R} \left(2 X _ {2} - 1\right) + 2 X _ {1} + 3 X _ {2} + U _ {X 1 , R} - 4\right)} \right\rfloor ,
$$
$$
\begin{array}{l} f _ {W} (U _ {X _ {1}, W}, U _ {X _ {2}, W}, X _ {1}, R) := \left\lfloor \frac {1}{1 + \exp (U _ {X 1 _ {W}} (2 X _ {1} - 1) + (4 R - 2) + U _ {X _ {2} , W}} \right\rfloor , \\ f _ {Y} \left(U _ {X _ {2}, Y}, X _ {2}, R, W\right) := \left\lfloor \frac {1}{1 + \exp \left(0 . 5 U _ {X _ {2} , Y} (2 R - 1) - 2 X _ {2} + 2 W + U _ {X _ {2} , Y} - 2\right)} \right\rfloor . \\ \end{array}
$$
E Project STAR: Estimating Joint Effects of Class Sizes to Academic Outcomes
We applied the proposed estimators to the Project STAR dataset [Krueger and Whitmore, 2001, Schanzenbach, 2006]. Project STAR is an experimental study investigating teacher/student ratios' impact on academic achievement for kindergarten through third-grade students. In the study, students were randomly assigned to three different class sizes: small-size classes, regular classes, and large-size classes. The objective was to evaluate how class size affects academic outcomes [Schanzenbach, 2006]. In our analysis, we used the dataset introduced in the online complement of Stock et al. [2003].
Project STAR Dataset. We denote the Project STAR dataset as  $D$ . The dataset  $D$  includes the following information: class size for kindergarten  $(X_{1})$ , the academic outcome in kindergarten  $(W)$ , the academic outcome in second grade  $(R)$ , class size for third grade  $(X_{2})$ , the academic outcome in the third grade  $(Y)$ , free lunch receiving for kindergarten  $(C_{1})$ , gender  $(C_{2})$ , ethnicity  $(C_{3})$  and free lunch receiving for the third grade  $(C_{4})$ . We will use  $\mathbf{C}_1 \coloneqq (C_1, C_2)$  and  $\mathbf{C}_2 \coloneqq (C_3, C_4)$ .
Assumption on Dataset. We assume that the SCM  $\mathcal{M}$  generating the variables  $(X_{1},W,R,X_{2},Y,\mathbf{C}_{1},\mathbf{C}_{2})$  induces a causal graph depicted in Figure E.4a. Here, we mark  $\mathbf{C}_1,\mathbf{C}_2$  as gray to denote that these variables will be considered latent; i.e., these variables will not be used in the data analysis.
Project STAR dataset  $D$  is a longitudinal experimental study randomizing  $X_{1}$  and  $X_{2}$ ; i.e., the dataset is induced by the submodel  $M_{x_1,x_2}$  for  $x_{1},x_{2} \in \mathfrak{D}_{X_{1},X_{2}}$ , represented in Fig. E.4b. The samples for variables  $\{X_1,W\}$  follow a distribution  $P_{\sigma (X_1)}(X_1,W,R) = P_{\sigma (X_1,X_2)}(X_1,W,R)$ , and the samples for variables  $\{X_1,W,R,X_2,Y\}$  follow a distribution  $P_{\sigma (X_1,X_2)}(X_1,W,R,X_2,Y)$ . To demonstrate, we will describe how to generate the sample fol
lowing a distribution  $P_{\sigma (X_2)}(X_1,W,R,X_2,Y)$  by creating unmeasured confounding bias between  $X_{1}$  and  $W$  , and  $X_{1}$  and  $Y$  , which is depicted by Fig. E.4c.
Creation of Datasets from Marginal Experiments. In this empirical study, we create two datasets from this dataset:  $D_{1}$  and  $D_{2}$ . The dataset  $D_{1}$  is a random subsample of  $D$  only including  $\{X_{1}, W\}$ . Then,  $D_{1}$  follows  $P_{\sigma(X_{1})}(X_{1}, W, R)$ .
To construct the dataset  $D_{2}$  following the marginal experimental distribution  $P_{\sigma (X_2)}(X_1,W,R,X_2,Y)$ , the confounding bias between  $X_{1}$  and  $W$  and  $X_{1}$  and  $Y$  should be introduced. To do so, we follow a standard procedure for introducing confounding bias from experimental studies used in Hill [2011], Louizos et al. [2017], Zhang and Baireinboim [2019], Gentzel et al. [2021].
A setting for the standard procedure is the following. For any arbitrary random variable  $X, Y, Z, W$  such that  $X \to Y$ ,  $Z \to Y$ , and there are no arrows into  $X$ , the dataset  $D := \{(X_{(i)}, Y_{(i)}, Z_{(i)}, W_{(i)} : i = 1, \dots, n\}$  is given. In  $D$ ,  $Z$  is not a confounding variable since  $Z \perp X$ . The goal is to generate a new dataset  $D' := \{(X_{(j)}, Y_{(j)}, Z_{(j)} : j = 1, \dots, n'\}$  where  $Z$  serves as a confounding variable between  $X$  and  $Y$ . The procedure named IntroduceConfounding  $(X, Y; Z, D)$  is given as follows: Initialize  $D' = \{\}$ . For  $i = 1, \dots, n$ , do the followings:
1. Generate the Bernoulli Random  $B_{(i)}$  with parameter  $P(X_{(i)}|Z_{(i)})$ .
2. If  $B_{(i)} = 1$ , include  $(X_{(i)},Y_{(i)},Z_{(i)},W_{(i)})$  in  $D^{\prime}$ .
Finally, we exclude  $Z$  is removed from  $D'$ . By doing so, we introduce unmeasured confounding bias between  $X$  and  $Y$  in  $D'$ ; i.e.,  $D' = \text{IntroduceConfounding}(X, Y; Z, D)$ .
To generate the dataset  $D_{2} \sim P_{\sigma(X_{2})}(X_{1}, W, R, X_{2}, Y, \mathbf{C}_{1}, \mathbf{C}_{2})$ , we have to introduce the unmeasured confounding bias between  $X_{1}$  and  $W$ , and  $X_{1}$  and  $Y$ . We do this by  $D_{2}' = \text{IntroduceConfounding}(X_{1}, W; \mathbf{C}_{1}, D)$ . Then,  $D_{2}'$  is a variable containing  $(X_{1}, W, R, X_{2}, \mathbf{C}_{2}, Y)$  and there is a confounding bias between  $X_{1}$  and  $W$ . Then, we set  $D_{2} = \text{IntroduceConfounding}(X_{1}, Y; \mathbf{C}_{2}, D_{2}')$ . Then,  $D_{2}$  is a variable containing  $(X_{1}, W, R, X_{2}, Y)$  and there is a confounding bias between  $X_{1}$  and  $Y$ , and  $X_{1}$  and  $W$ . A causal graph Fig. E.4c depicted the dependencies in  $D_{2}$ .
Goal. In this empirical study, we aim to study the joint effect of the class size for kindergarten  $(X_{1})$  and the third grade  $(X_{2})$  on the third grade's academic outcome  $(Y)$ ; i.e.,  $\mathbb{E}[Y|do(x_1,x_2)]$ . Since  $D$  is a longitudinal experimental dataset following  $P_{\mathrm{rand}(X_1,X_2)}(C,X_1,W,X_2,Y)$ , the ground-truth  $\mathbb{E}[Y|do(x_1,x_2)]$  is estimated as  $\mathbb{E}_D[Y\mathbb{1}_{x_1,x_2}(X_1,X_2)] / \mathbb{E}_D[\mathbb{1}_{x_1,x_2}(X_1,X_2)]$ .
Causal Effect Identification. We identify  $P(y|do(x_1, x_2))$  through Algo. 1.
1. Line 3: Set  $\mathbf{D} \coloneqq a n(Y)_{G(\mathbf{V} \backslash \{X_1, X_2\})} = \{Y, R, W\}$ .
2. Line 4: Set  $\mathbf{D}_1\coloneqq \{R\}$ $\mathbf{D}_2\coloneqq \{W\}$ $\mathbf{D}_3\coloneqq \{Y\}$
Each  $Q[\mathbf{D}_1] = Q[W]$ ,  $Q[\mathbf{D}_2] = Q[R]$ ,  $Q[\mathbf{D}_3] = Q[Y]$  are identified as follows: For  $Q[\mathbf{D}_1]$ ,
1. Line 7: For  $\mathbf{D}_1, \mathbf{Z}_1 \coloneqq \{X_1\}, \mathbf{z}_1 \coloneqq \{x_1\}, \mathbf{S}_1^1 \coloneqq \{W\}$ .
2. Line 8: Set  $Q[\mathbf{S}_1^1] = A_0[\mathbf{S}_1^1, \emptyset, \mathbf{R}_1^1; \mathbb{Z}_1^1 = \{X_1\}, \mathbf{seq}_1^1 = (x_1)](w, x_1)$  where  $\mathbf{R}_1^1 \coloneqq \emptyset$ .
3. Line a.4: Since  $\mathbf{S}_1^1 = \mathbf{D}_1$ , we set
$$
\begin{array}{l} Q [ \mathbf {D} _ {1} ] = Q [ \mathbf {S} _ {1} ^ {1} ] \\ = A _ {0} ^ {W} \\ := A _ {0} [ W, \emptyset , \emptyset ; \mathbb {Z} _ {1} ^ {1} = \{X _ {1} \}, \operatorname {s e q} _ {1} ^ {1} = (x _ {1}) ] (w, \emptyset) \\ = P _ {\sigma (X _ {1})} (w | x _ {1}) = P _ {x _ {1}} (w). \\ \end{array}
$$
For  $Q[\mathbf{D}_2] = Q[R]$ ,
1. Line 7: For  $\mathbf{D}_2, \mathbf{Z}_2 \coloneqq \{X_1\}, \mathbf{z}_2 \coloneqq \{x_1\}, \mathbf{S}_2^1 \coloneqq \{R\}$ .
2. Line 8: Set  $Q[\mathbf{S}_2^1] = A_0[\mathbf{S}_2^1, \emptyset, \mathbf{R}_2^1; \mathbb{Z}_2^1 = \{X_1\}, \mathbf{seq}_2^1 = (x_1)](r, x_1)$  where  $\mathbf{R}_1^1 \coloneqq \{W\}$ .
3. Line a.4: Since  $\mathbf{S}_2^1 = \mathbf{D}_2$ , we set
$$
\begin{array}{l} Q [ \mathbf {D} _ {2} ] = Q [ \mathbf {S} _ {2} ^ {1} ] \\ = A _ {0} ^ {R} \\ := A _ {0} \left[ \mathbf {S} _ {2} ^ {1}, \emptyset , \{W \}; \mathbb {Z} _ {2} ^ {1} = \{X _ {1} \}, \operatorname {s e q} _ {2} ^ {1} = (x _ {1}) \right] (w, r) \\ = P _ {\sigma \left(X _ {1}\right)} (r | r, x _ {1}) = P _ {x _ {1}} (r | w). \\ \end{array}
$$
For  $Q[\mathbf{D}_3] = Q[Y]$ ,
1. Line 7: For  $\mathbf{D}_3, \mathbf{Z}_3 \coloneqq \{X_2\}, \mathbf{z}_3 \coloneqq \{x_2\}, \mathbf{S}_1^2 \coloneqq \{Y, X_1, W\}$ .
2. Line 8: Set  $Q[\mathbf{S}_1^2] = A_0[\mathbf{S}_1^2, \emptyset, \mathbf{R}_1^2; \mathbb{Z}_1^2 = \{X_2\}, \mathbf{seq}_1^2 = (x_2)](y, r)$  where  $\mathbf{R}_1^2 \coloneqq \{W\}$ ; i.e.,
$$
\begin{array}{l} Q \left[ \mathbf {S} _ {1} ^ {2} \right] = A _ {0} \left[ \left\{X _ {1}, W, Y \right\}, \emptyset , \{R \}; \mathbb {Z} _ {1} ^ {2} = \left\{X _ {2} \right\}, \mathbf {s e q} _ {1} ^ {2} = (x _ {2}) \right] \left(\left(x _ {1}, w, y\right), r\right) \\ = P _ {\sigma \left(X _ {2}\right)} \left(y | x _ {1}, w, r, x _ {2}\right) P _ {\sigma \left(X _ {2}\right)} \left(x _ {1}, w | x _ {2}\right). \\ \end{array}
$$
3. Line a.2: Set  $\mathbf{A} \coloneqq an(\mathbf{D}_3)_{G(\mathbf{S}_1^2)} = \{Y\}$ .
4. Line a.3:  $Q[\mathbf{A}] = A_0[\{Y\}, \{X_1, W\}, \{R\}; \mathbb{Z}_1^2 = \{X_2\}, \mathbf{seq}_1^2 = (x_2)](y, r)$ .
5. Line a.4: Since  $\mathbf{A} = \mathbf{D}_2$ , we set
$$
\begin{array}{l} Q [ \mathbf {D} _ {2} ] = Q [ \mathbf {A} ] \\ = A _ {0} ^ {Y} \\ := A _ {0} \left[ \left\{Y \right\}, \left\{X _ {1}, W \right\}, \left\{R \right\}; \mathbb {Z} _ {1} ^ {2} = \left\{X _ {2} \right\}, \operatorname {s e q} _ {1} ^ {2} = \left(x _ {2}\right) \right] (y, r) \tag {E.1} \\ = \sum_ {x _ {1} ^ {\prime}, w \in \mathfrak {D} _ {X _ {1}, W}} P _ {\sigma (X _ {2})} (y | x _ {1} ^ {\prime}, w, r, x _ {2}) P _ {\sigma (X _ {2})} (x _ {1} ^ {\prime}, w | x _ {2}). \\ \end{array}
$$
Then, by Line 14,
$$
\begin{array}{l} P (y | d o (x _ {1}, x _ {2})) = \sum_ {r, w \in \mathfrak {D} _ {R, W}} Q [ W ] Q [ R ] Q [ Y ] \\ = \sum_ {r, w \in \mathfrak {D} _ {R, W}} A _ {0} ^ {W} A _ {0} ^ {R} A _ {0} ^ {Y}. \\ \end{array}
$$
$$
\begin{array}{l} A _ {0} ^ {W R} := A _ {0} ^ {W} A _ {0} ^ {R} \\ = A _ {0} \left[ \left\{R, W \right\}, \emptyset , \emptyset ; \mathbb {Z} _ {2} ^ {1} = \left\{X _ {1} \right\}, \operatorname {s e q} _ {2} ^ {1} = \left(x _ {1}, x _ {1}\right) \right] \left(\left(w, r\right), \emptyset\right) \\ = P _ {\sigma \left(X _ {1}\right)} (w, r | x _ {1}) = P _ {x _ {1}} (w, r). \\ \end{array}
$$
$$
\begin{array}{l} P(y|d o(x_{1},x_{2})) = \sum_{r,w\in \mathfrak{D}_{R,W}}A_{0}^{W}A_{0}^{R}A_{0}^{Y} \\ = \sum_ {r, w \in \mathfrak {D} _ {R, W}} A _ {0} ^ {W R} A _ {0} ^ {Y} \\ = \sum_ {r \in \mathfrak {D} _ {R}} \left(\sum_ {w \in \mathfrak {D} _ {W}} A _ {0} ^ {W R}\right) A _ {0} ^ {Y}, \\ \end{array}
$$
where the last equation holds since  $A_0^Y$  is not a function of  $W$ . By Lemma 2
$$
A _ {0} ^ {R ^ {\prime}} := \sum_ {w \in \mathfrak {D} _ {W}} A _ {0} ^ {W R} = A _ {0} [ R, \emptyset , \emptyset ; \{X _ {1} \}, (x _ {1}) ] (r, \emptyset) = P _ {\sigma (X _ {1})} (r | x _ {1}) = P _ {x _ {1}} (r).
$$
Scenario 1	Scenario 2	Scenario 3	Scenario 4
Fig. E.3a(Project STAR)	0.25
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Figure E.5: AAE Plot for Project STAR dataset analysis for Scenarios  $\{1,2,3,4\}$ .
$$
\begin{array}{l} P(y|do(x_{1},x_{2})) = \sum_{r\in \mathfrak{D}_{R}}(\sum_{w\in \mathfrak{D}_{W}}A_{0}^{WR})A_{0}^{Y} \\ = \sum_ {r \in \mathfrak {D} _ {R}} A _ {0} ^ {R ^ {\prime}} A _ {0} ^ {Y}. \\ \end{array}
$$
Causal Effect Estimation. Here, we only describe the nuisance for  $A_0^Y$  in Eq. (E.1), since estimating  $A_0^{R'} = P_{x_1}(r)$  is trivial. We define the nuisance as follows: For the fixed  $x_2 \in \mathfrak{D}_{X_2}$ ,
$$
\mu_ {0} \left(X _ {1}, W, R\right) := \mathbb {E} _ {P _ {x _ {2}}} [ Y | X _ {1}, W _ {1}, C ], \tag {E.2}
$$
$$
\pi_ {0} (R | X _ {1}, W) := \frac {\mathbb {1} _ {r} (R)}{P _ {x _ {2}} (R | X _ {1} , W)}. \tag {E.3}
$$
Then,  $A_0^Y$  in Eq. (E.1) can be expressed as follows:
$$
\begin{array}{l} \text {E q .} (E.4) \\ = \mathbb {E} _ {P _ {x _ {2}}} \left[ Y \frac {\mathbb {1} _ {r} (R)}{\pi_ {0} (R | X _ {1} , W)} \right], \text {o r}, (E.5) \\ = \mathbb {E} _ {P _ {x _ {2}}} \left[ \mu_ {0} \left(X _ {1}, W, r\right) \right], \text {o r}, (E.6) \\ = \mathbb {E} _ {P _ {x _ {2}}} \left[ \frac {\mathbb {1} _ {r} (R)}{\pi_ {0} (R | X _ {1} , W)} \{Y - \mu_ {0} (X _ {1}, W, R) \} + \mu_ {0} (X _ {1}, W, r) \right]. (E.7) \\ \end{array}
$$
We then construct the regression-based, probability weighting-based, and MR-gID (MR)  $T^{\mathrm{reg}}, T^{\mathrm{pw}}, T^{\mathrm{mr}}$  using the following procedure.
1. For each fixed  $x_2 \in \mathfrak{D}_{X_2}$  and a sample set  $D_{x_2}$  for  $i \in \{1, 2\}$ , randomly split the sample as  $D_{x_2, t}$  and  $D_{x_2, e}$ .
2. Use  $D_{x_2,t}$  to train the model for learning nuisances in Eq. (E.2) and Eq. (E.3). Let  $\mu(X_1, W, R)$  and  $\pi(R|X_1, W)$  denote the learnt models. We use the XGBoost [Chen and Guestrin, 2016] to learn the model.
3. Then, each estimator is defined as follows:
$$
T ^ {\text {r e g}} := \mathbb {E} _ {D _ {x _ {2}, e}} [ \mu (X _ {1}, W, r) ] \tag {E.8}
$$
$$
T ^ {\mathrm {p w}} := \mathbb {E} _ {D _ {x _ {2}, e}} \left[ \frac {\mathbb {1} _ {r} (R)}{\pi (R | X _ {1} , W)} Y \right] \tag {E.9}
$$
$$
T ^ {\mathrm {m r}} := \mathbb {E} _ {D _ {x _ {2}, e}} \left[ \frac {\mathbb {1} _ {r} (R)}{\pi (R | X _ {1} , W)} \{Y - \mu (X _ {1}, W, R) \} \right] + \mathbb {E} _ {D _ {x _ {2}, e}} \left[ \mu (X _ {1}, W, r) \right]. \tag {E.10}
$$
Experimental Results As described in the Experimental Setup section (Sec. 4), we evaluated the AAE $^{\text{est}}$  of estimators  $T^{\text{est}}$  for  $\text{est} \in \{\text{reg}, \text{pw}, \text{mr}\}$  in Scenarios  $\{1, 2, 3, 4\}$ . The AAE plots for all
cases can be seen in Fig. E.5. In this particular scenario, the sample size was not varied since the sample itself was externally given.
In Case 2, we introduced variation by adjusting the size of the converging noise  $\epsilon$ , which follows a normal distribution  $\mathrm{Normal}(n^{-\alpha}, n^{-2\alpha})$  for  $n \in \{200, 400, 600, 800, 1000\}$ . It was observed that the MR-gID estimator  $T^{\mathrm{mr}}$  outperformed the other two estimators by achieving fast convergence, as demonstrated in Theorem 2. For Scenarios  $\{3, 4\}$ , the DML estimator  $T^{\mathrm{mr}}$  exhibited doubly robust properties, as illustrated in Corollary 2.