SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In Scenario 1, we aim to show that all estimators  $T^{\mathrm{reg}}, T^{\mathrm{pw}}, T^{\mathrm{mr}}$  are converging to the true causal quantity  $P(\mathbf{y}|do(\mathbf{x}))$ . In Scenario 2, we aim to show that the MR-gID estimator exhibits fast convergence behavior compared to competing estimators. In Scenario (3,4), our goal is to highlight the multiply robustness property of the MR-gID estimator.
(a) Graph
(b) Scenario 2
(c) Scenario 3
Figure 3: A graph and the AAE-plot for Project STAR.
(d) Scenario 4
Experimental Results. The AAE plots for all scenarios are presented in Fig. 2. All the estimators ('reg', 'pw', 'mr') converge in Scenario 1 as the sample size grows. In Scenario 2, where the estimated nuisances are controlled to be bounded in probability at  $n^{-1/4}$  rate, the proposed MR-gID  $\hat{\psi}$  outperforms the other two estimators by achieving fast convergence. This result corroborates the robustness property in Thm. 2. In Scenarios (3,4), where the estimated nuisances for  $\{\mu^i\}_{i=2}^m$  or  $\{\pi^i\}_{i=1}^{m-1}$  are wrongly specified, the MR-gID estimator converges while other estimators fail to converge. This result corroborates the multiply robustness property in Coro. 2.
4.2 Project STAR Dataset
This section provides an overview of the analysis using Project STAR dataset [Krueger and Whitmore, 2001, Schanzenbach, 2006]. Project STAR investigated the impact of teacher/student ratios on academic achievement for students in kindergarten through third grade. The dataset  $D$  includes class size  $(X_{1})$ , the academic outcome in kindergarten  $(W)$  for kindergarten, the academic outcome in second grade  $(R)$ , class size  $(X_{2})$ , and the academic outcome for the third grade  $(Y)$ . We assume that the SCM  $\mathcal{M}$  underlying Project STAR dataset  $D$  can be depicted in Figure 3a. The target quantity is  $\mathbb{E}[Y|do(x_1,x_2)]$  where  $P_{x_1,x_2}(y) = \sum_{r\in \mathfrak{D}_R}P_{x_1}(r)\sum_{x_1',w\in \mathfrak{D}_{X_1,W}}P_{x_2}(y|x_1',w,r,x_2)P_{x_2}(x_1',w)$ . The detailed procedures are in Appendix E.
Experimental Setup. We generate two datasets  $D_{1}$  and  $D_{2}$  from the original dataset  $D$  to demonstrate the gID estimation.  $D_{1}$  is a random subsample of  $D$  with only  $\{X_{1}, W, R\}$  and follows  $P_{\sigma(X_{1})}(X_{1}, W, R)$ .  $D_{2}$  is constructed by resampling from  $D$  in a way that the confounding bias between  $X_{1}$  and  $W$ , and  $X_{1}$  and  $Y$  presents, following  $P_{\sigma(X_{2})}(X_{1}, W, X_{2}, R, Y)$ . We conducted 100 simulations by generating new instances of  $D_{1}$  and  $D_{2}$  to create the AAE plot. Estimators were constructed solely from  $D_{1}$  and  $D_{2}$ , with  $D$  used exclusively to construct the ground-truth estimate.
Experimental Results. We evaluated the  $\mathrm{AAE^{est}}$  of estimators  $T^{\mathrm{est}}$  for  $\mathrm{est} \in \{\mathrm{reg}, \mathrm{pw}, \mathrm{mr}\}$ . The AAE plots for scenarios (2,3,4) are in Figs. (3b,3c,3d). Our findings indicate that the MR-gID estimator  $T^{\mathrm{mr}}$  consistently provided reliable estimates for the ground-truth quantity.
We present a framework for estimating the causal effect  $P(\mathbf{y} \mid do(\mathbf{x}))$  by combining multiple observational and experimental datasets and a causal graph  $G$ . We introduce the generalized multi-outcome sequential back-door adjustment (g-mSBD) operator (Def. 1) and its operations. We show that any g-identifiable causal effects can be expressed as a function of the g-mSBD operators as specified in Algo. 1 (Thm. 1). We then develop an estimator called DR-g-mSBD (Def. 3) for the g-mSBD operator and analyze its statistical properties in Prop. 1. Based on the DR-g-mSBD estimator, we develop the MR-gID estimator (Def. 4) and analyze its statistical properties (Thm. 2 and Coro. 2) which exhibits fast convergence and multiply-robustness. Our experimental results demonstrate that the MR-gID estimator is a consistent and robust estimator of  $P(\mathbf{y} \mid do(\mathbf{x}))$  against model misspecification and slow convergence.
This research was supported in part by the NSF, ONR, AFOSR, DoE, Amazon, JP Morgan, and The Alfred P. Sloan Foundation.
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Supplement to "Estimating Causal Effects Identifiable from a Combination of Observations and Experiments"
1.1 Preliminaries 3
1.2 Problem Statement 4
2 Expressing Causal Effects as a Combination of mSBD Adjustments 4
3 Estimating g-Identifiable Causal Effects 6
4 Experiments 8
4.1 Synthetic Dataset Analysis 9
4.2 Project STAR Dataset 10
5 Conclusions 10
A Further Details 16
A.1 Example 3 16
A.2 Example 4 17
A.3 Example 5 19
A.3.1 Specification of Nuisances 19
A.3.2 Construction of Estimators 19
A.4 Example 6 20
A.4.1 Specification of Nuisances 20
A.4.2 Construction of Estimators 20
A.5 Details on Regression-based (REG) and Probability Weighting-based (PW) estimators. 20
A.5.1 Regression-based Estimator 21
A.5.2 Probability-weighting based Estimator 22
B Proofs 23
B.1 Proof of Lemma 1 23
B.2 Proof of Lemma 2 24
B.3 Proof of Lemma 3 24
B.4 Proof of Lemma 4 25
B.5 Proof of Theorem 1 26
B.6 Proof of Proposition 1 26
B.7 Proof of Theorem 2 35
B.8 Proof of Corollary 2 36
C Discussion 38
C.1 Relaxation of Discreteness Assumption 38
C.2 Sequential Doubly Robustness:  $2^{m - 1}$  robustness versus  $m$  -robustness 39
D Details of Experiments 39
D.1 Designs of Simulations 40
D.1.1 Example 1 40
D.1.2 Example 2 40
E Project STAR: Estimating Joint Effects of Class Sizes to Academic Outcomes 41
A Further Details
We restate the notation here. To clarify the relationship between the experimental samples, where randomization is applied to  $\mathbf{Z} \subseteq \mathbf{V}$ , and the distribution  $P_{\mathbf{z}}(\mathbf{V} \backslash \mathbf{z})$ , we introduce the notation  $P_{\sigma(\mathbf{Z})}(\mathbf{V})$ , where  $\sigma(\mathbf{Z})$  indicates that  $\mathbf{Z}$  has been randomized. The distribution  $P_{\sigma(\mathbf{Z})}(\mathbf{V})$  is derived from the Structural Causal Model (SCM), where the original equation  $Z \gets f_Z(pa_z, u_z)$  for  $Z \in \mathbf{Z}$  is replaced by a function that assigns a value to  $Z = z$  randomly, independent of other endogenous variables. For example, assigning  $Z = 1$  and 0 with a probability of 0.5 each.
It should be noted that when considering observational data,  $P \coloneqq P_{\sigma(\emptyset)}$ . For any sets  $\mathbf{A}$ ,  $\mathbf{B}$ , and  $\mathbf{Z} \subseteq \mathbf{V}$ , the interventional distribution can be represented as  $P(\mathbf{A}|\mathrm{do}(\mathbf{z}), \mathbf{B}) = P_{\sigma(\mathbf{Z})}(\mathbf{A}|\mathbf{Z} = \mathbf{z}, \mathbf{B})$  according to the definition of the do-operator and the  $P_{\sigma(\mathbf{Z})}$  distribution. To emphasize that the distribution is induced from randomization and conditioning on  $\mathbf{Z} = \mathbf{z}$ , we use  $P_{\mathbf{z}}(\mathbf{A}|\mathbf{B}) \coloneqq P_{\sigma(\mathbf{Z})}(\mathbf{A}|\mathbf{Z} = \mathbf{z}, \mathbf{B})$ . The experimental samples obtained from randomization  $\sigma(\mathbf{Z})$  lead to samples  $D_{\sigma(\mathbf{Z})}$  that follow  $P_{\sigma(\mathbf{Z})}(\mathbf{V})$ . We denote the subsample of  $D_{\sigma(\mathbf{Z})}$ , where  $\mathbf{Z} = \mathbf{z}$  is fixed, as  $D_{\mathbf{z}}$  which follows  $P_{\mathbf{z}}(\mathbf{V})$ .
A.1 Example 3
We provide a detailed illustration of Example 3, demonstrating the application of Lemmas (2,3,4).
Input:  $\mathbf{x} = \{x\}$ ,  $\mathbf{y} := \{y\}$ ,  $\mathbb{Z} := \{\emptyset, Z\}$ . The goal is to identify  $P(y|do(x))$  from  $\mathbb{P}$  which contains  $P$  and  $P_{\sigma(Z)}(\mathbf{V})$ . In the identification,  $P(\mathbf{V})$  and  $P_z(\mathbf{V} \backslash Z) := P_{\sigma(Z)}(\mathbf{V}|z)$  for  $z \in \mathfrak{D}_Z$  are used.
Line 3-4: Since  $\mathbf{V} \backslash \mathbf{X} = \{Z, Y\}$ ,  $\mathbf{D} = a n(Y)_{G(Z, Y)} = \{Z, Y\}$ . Let  $\mathbf{D}_1 := \{Z\}$  and  $\mathbf{D}_2 := \{Y\}$ .
We now run Line 5-13. We first run  $\mathbf{D}_1 = \{Z\}$  and  $\mathbf{Z}_1 = \emptyset$ . Then,
1. Line 7: The c-component  $\mathbf{S}_1^1 = \mathbf{V} = \{W, X, Z, Y\}$  includes  $\mathbf{D}_1$ .
2. Line 8: The c-factor  $Q[\mathbf{S}_1^1]$  is identified as
$$
Q \left[ \mathbf {S} _ {1} ^ {1} \right] = A _ {0} \left[ \mathbf {S} _ {1} ^ {1}, \emptyset , \emptyset ; \mathbb {Z} _ {1} ^ {1} := \emptyset , \emptyset \right] \left(\mathbf {s} _ {1} ^ {1}, \emptyset\right) = P (w) P (x | w) P (z | x, w) P (y | w, x, z).
$$
3. Line 9: Run  $Q[\mathbf{D}_1] = \mathrm{SUBID}(\mathbf{D}_1, \mathbf{S}_1^1, Q[\mathbf{S}_1^1], G(\mathbf{S}_1^1))$ .
(a) Line a.(2-3):  $\mathbf{A} = an(Z)_{G(\mathbf{V})} = \{W,X,Z\}$ . Then, by Lemma 2,
$$
Q [ \mathbf {A} ] = \sum_ {y \in \mathfrak {S} _ {Y}} Q [ \mathbf {S} ] = A _ {0} [ \{W, X, Z \}, \emptyset , \emptyset ; \emptyset , \emptyset ] (\{w, x, z \}, \emptyset) = P (w) P (x | w) P (z | x, w).
$$
(b) Line a.(7-8): Note  $\mathbf{S} = \{W,Z\}$  is a c-component in  $G(\mathbf{A})$  containing  $\mathbf{D}_1 = \{Z\}$ . Then,
$$
Q [ \mathbf {S} ] = (\sum_ {x, z \in \mathfrak {S} _ {X, Z}} Q [ \mathbf {A} ]) \times \frac {Q [ \mathbf {A} ]}{\sum_ {z \in \mathfrak {D} _ {Z}} Q [ \mathbf {A} ]}.
$$
By Lemma 2,
$$
\sum_ {x, z \in \mathfrak {S} _ {X, Z}} Q [ \mathbf {A} ] = A _ {0} [ W, \emptyset , \emptyset ; \emptyset , \emptyset ] (w, \emptyset) = P (w),
$$
$$
\sum_ {z \in \mathfrak {S} _ {Z}} Q [ \mathbf {A} ] = A _ {0} [ \{W, X \}, \emptyset , \emptyset ; \emptyset , \emptyset ] (\{w, x \}, \emptyset , \emptyset) = P (w) P (x | w).
$$
By Lemma 4,
$$
\begin{array}{l} \frac {Q [ \mathbf {A} ]}{\sum_ {z \in \mathfrak {S} _ {Z}} Q [ \mathbf {A} ]} = \frac {A _ {0} \left[ \{W , X , Z \} , \emptyset , \emptyset ; \emptyset , \emptyset \right] (\{w , x , z \} , \emptyset)}{A _ {0} \left[ \{W , X \} , \emptyset , \emptyset ; \emptyset , \emptyset \right] (\{w , x \} , \emptyset , \emptyset)} \\ = A _ {0} [ Z, \emptyset , \{W, X \}; \emptyset , \emptyset ] (z, \{w, x \}) \\ = P (z | w, x). \\ \end{array}
$$
By Lemma 3,
$$
\begin{array}{l} Q [ \mathbf {S} ] = A _ {0} [ W, \emptyset , \emptyset , \emptyset ; \emptyset , \emptyset ] (w, \emptyset) \times A _ {0} [ Z, \emptyset , \{W, X \}; \emptyset , \emptyset ] (z, \{w, x \}) \\ = A _ {0} [ \{W, Z \}, \emptyset , X; \emptyset , \emptyset ] (\{w, z \}, x) \\ = P (w) P (z \mid w, x), \\ \end{array}
$$
because the order is  $W\prec_{G}X\prec_{G}Z$
(c) Line a.9: Run  $Q[\mathbf{D}_1] = \mathrm{SUBID}(\mathbf{D}_1, \mathbf{S}, Q[\mathbf{S}], G(\mathbf{S}))$ .
(d) Line a.(2-3):  $\mathbf{A} = an(\mathbf{D}_1)_{G(\mathbf{S})} = \{Z\} = \mathbf{D}_1$ . Then, by Lemma 2,
$$
\begin{array}{l} Q \left[ \mathbf {D} _ {1} \right] = \sum_ {w \in \mathcal {W}} Q [ \mathbf {S} ] \\ = A _ {0} ^ {1} := A _ {0} [ \{W, Z \}, W, X; \emptyset , \emptyset ] (z, x) \\ = \sum_ {w \in \mathfrak {S} _ {W}} P (w) P (z | w, x). \\ \end{array}
$$
We now run Line 5-13. We first run  $\mathbf{D}_2 = \{Y\}$  and  $\mathbf{Z}_1 = \emptyset$ . We note that it fails since the sub-procedure  $\operatorname{subID}(\mathbf{D}_2, \mathbf{S}_1^1, Q[\mathbf{S}_1^1], G(\mathbf{S}_1^1))$  fails. Specifically,  $\mathbf{A} := an(\mathbf{D}_2)_{G(\mathbf{S}_1^1)} = \mathbf{V} = \mathbf{S}_1^1$ . Therefore, by Line a.5, the procedure fails.
We now run  $\mathbf{D}_2$  with  $\mathbf{Z}_2 = \{Z\}$ .
1. Line 7: The c-component  $\mathbf{S}_2^1 = \mathbf{V} \backslash Z = \{W, X, Y\}$ .
2. Line 8:  $Q[\mathbf{S}_2^1] = A_0[\mathbf{S}_2^1, \emptyset, \emptyset,; \mathbb{Z}_2^1 = \{Z\}, \mathbf{seq}_2^1](\mathbf{s}_2^1, \emptyset) = P_z(w)P_z(x|w)P_z(y|x, w)$ , where  $\mathbf{seq}_2^1(W_j) = (z, z, z)$ .
3. Line 9: We run  $Q[\mathbf{D}_2] = \mathrm{SUBID}(\mathbf{D}_2, \mathbf{S}_2^1, Q[\mathbf{S}_2^1], G(\mathbf{S}_2^1))$ .
4. Line a.(2-3):  $\mathbf{A} = an(\mathbf{D}_2)_{G(\mathbf{S}_2^1)} = \{Y\} = \mathbf{D}_2$ . Then, by Lemma 2
$$
\begin{array}{l} Q \left[ \mathbf {D} _ {2} \right] = \sum_ {w, x \in \mathfrak {S} _ {W, X}} Q \left[ \mathbf {S} _ {2} ^ {1} \right] (A.1) \\ = \sum_ {w, x \in \mathfrak {S} _ {W, X}} A _ {0} [ \{W, X, Y \}, \emptyset , \emptyset ; \mathbb {Z} _ {2} ^ {1} = \{Z \}, \operatorname {s e q} _ {2} ^ {1} ] (y; \emptyset) (A.2) \\ = A _ {0} [ Y, \emptyset , \emptyset ; \mathbb {Z} _ {2} ^ {1} = \{Z \}, \operatorname {s e q} _ {2} ^ {1} ] (\{y \}; \emptyset) (A.3) \\ = P _ {z} (y). (A.4) \\ \end{array}
$$
$$
Q \left[ \mathbf {D} _ {1} \right] = A _ {0} ^ {1} := A _ {0} \left[ \{W, Z \}, W, X; \emptyset , \emptyset \right] (z, x) \tag {A.5}
$$
$$
Q \left[ \mathbf {D} _ {2} \right] = A _ {0} ^ {2} := A _ {0} [ Y, \emptyset , \emptyset ; \mathbb {Z} _ {2} ^ {1} = \{Z \}, \operatorname {s e q} _ {2} ^ {1} ] (\{y \}; \emptyset). \tag {A.6}
$$
By Line 14,
$$
P (y | d o (x)) = \sum_ {z \in \mathfrak {S} _ {Z}} Q [ \mathbf {D} _ {1} ] Q [ \mathbf {D} _ {2} ] = \sum_ {z \in \mathfrak {S} _ {Z}} A _ {0} ^ {1} A _ {0} ^ {2}. \tag {A.7}
$$
A.2 Example 4
We provide a detailed illustration of Example 4, demonstrating the application of Lemmas (2,3,4).
Input:  $\mathbf{x} = \{x_1, x_2\}$ ,  $\mathbf{y} := \{y\}$ ,  $\mathbb{Z} := \{X_1, X_2\}$ . The goal is to identify  $P(y|do(x_1, x_2))$  from  $\mathbb{P} := \{P_{\sigma(X_1)}(\mathbf{V}), P_{\sigma(X_2)}(\mathbf{V})\}$ . Specifically, two distributions  $P_{x_1}(\mathbf{V} \backslash X_1)$  and  $P_{x_2}(\mathbf{V} \backslash X_2)$  will be used in the identification task.
Line 3-4:  $\mathbf{D} = an(Y)_{G(R,W,Y)} = \{R,W,Y\}$ . Let  $\mathbf{D}_1 \coloneqq \{R\}$ ,  $\mathbf{D}_2 \coloneqq \{W\}$  and  $\mathbf{D}_3 = \{Y\}$ .
Line 5-13: Consider  $\mathbf{D}_1 = \{R\}$  and  $\mathbf{Z}_1 \coloneqq \{X_1\}$ . Note  $\mathbf{S}_1^1 \coloneqq \{R\} = \mathbf{D}_1$  is a c-component in  $G(\mathbf{V} \backslash X_1)$ . Therefore,  $Q[\mathbf{D}_1] = Q[\mathbf{S}_1^1]$ , where
$$
Q \left[ \mathbf {D} _ {1} ^ {1} \right] = A _ {0} ^ {1} := A _ {0} [ R, \emptyset , X _ {2}; \mathbb {Z} _ {1} ^ {1} := \{X _ {1} \}, \operatorname {s e q} _ {1} ^ {1} ] (r, x _ {2}) = P _ {x _ {1}} (r | x _ {2}), \tag {A.8}
$$
where  $\mathbf{seq}_1^1 \coloneqq (x_1)$ .
Line 5-13: We now consider  $\mathbf{D}_3 = \{Y\}$ . Note that  $Q[\mathbf{D}_3]$  is not identifiable from  $P_{x_1}(\mathbf{V} \backslash X_1)$ . To witness, consider the c-component  $\mathbf{S}_3^1 := \{W, X_2, Y\}$  in  $G(\mathbf{V} \backslash X_1)$ . Then, the sub-procedure  $\operatorname{subID}(\mathbf{D}_3, \mathbf{S}_3^1, Q[\mathbf{S}_3^1], G(\mathbf{S}_3^1))$  fails because failure condition in line a.5 is triggered. Specifically,  $an(Y)_{G(\mathbf{S}_3^1)} = \mathbf{S}_3^1$ .
Therefore, we consider  $\mathbf{D}_3 = \{Y\}$  with  $\mathbf{Z}_3 \coloneqq x_2$ . Note  $\mathbf{S}_3^2 \coloneqq \{Y\} = \mathbf{D}_3$  is a c-component in  $G(\mathbf{V} \backslash X_2)$ . Therefore,  $Q[\mathbf{D}_3] = Q[\mathbf{S}_3^2]$  which is given by line 8:
$$
\begin{array}{l} Q \left[ \mathbf {D} _ {3} \right] = A _ {0} [ Y, \emptyset , \{R, W \}; \mathbb {Z} _ {3} ^ {2} := \{X _ {2} \}, \operatorname {s e q} _ {3} ^ {2} = (x _ {2}) ] (y, \{r, w \}) (A.9) \\ = P _ {x _ {2}} (y \mid r, w). (A.10) \\ \end{array}
$$
Line 5-13: Consider  $\mathbf{D}_2 = \{W\}$  and  $\mathbf{Z}_1 := X_1$ . Note that  $Q[\mathbf{D}_2]$  is not identifiable from  $P_{x_2}(\mathbf{V} \backslash X_2)$ . To witness, consider the c-component  $\mathbf{S}_2^2 := \{X_1, W\}$  in  $G(\mathbf{V} \backslash X_2)$ . Then, the sub-procedure  $\operatorname{subID}(\mathbf{D}_2, \mathbf{S}_2^2, Q[\mathbf{S}_2^2], G(\mathbf{S}_2^2))$  fails because failure condition in line a.5 is triggered. Specifically,  $an(W)_{G(\mathbf{S}_2^2)} = \mathbf{S}_2^2$ .
Therefore, we consider  $\mathbf{D}_2 = \{W\}$  with  $\mathbf{Z}_1 \coloneqq X_1$ .
1. Note  $\mathbf{S}_2^1 = \{X_2, W, Y\}$  is a c-component in  $G(\mathbf{V} \backslash X_1)$  containing  $\mathbf{D}_2$ . Then,
$$
\begin{array}{l} Q \left[ \mathbf {S} _ {2} ^ {1} \right] = A _ {0} \left[ \left\{X _ {2}, W, Y \right\}, \emptyset , R; \mathbb {Z} _ {2} ^ {1} := \left\{X _ {1} \right\}, \operatorname {s e q} _ {2} ^ {1} = \left(x _ {1}, x _ {1}, x _ {1}\right) \right] \left(\left\{x _ {2}, w, y \right\}, r\right) (A.11) \\ = P _ {x _ {1}} \left(x _ {2}\right) P _ {x _ {1}} \left(w | r, x _ {2}\right) P _ {x _ {1}} \left(y | x _ {2}, w, r\right). (A.12) \\ \end{array}
$$
2. Run  $Q[\mathbf{D}_2] = \mathrm{SUBID}(\mathbf{D}_2, \mathbf{S}_2^1, Q[\mathbf{S}_2^1], G(\mathbf{S}_2^1))$ .
3. Line a.(2-3):  $\mathbf{A} = an(W)_{G(\mathbf{S}_2^1)} = \{W\} = \mathbf{D}_2$ . Then,
$$
\begin{array}{l} Q \left[ \mathbf {D} _ {2} \right] = \sum_ {y, x _ {2} \in \mathfrak {S} _ {Y, X _ {2}}} Q \left[ \mathbf {S} _ {2} ^ {1} \right] (A.13) \\ = \sum_ {y, x _ {2} \in \mathfrak {S} _ {Y}, X _ {2}} A _ {0} \left[ \left\{X _ {2}, W, Y \right\}, \emptyset , R; \mathbb {Z} _ {2} ^ {1}, \mathbf {s e q} _ {2} ^ {1} \right] \left(\left\{x _ {2}, w, y \right\}, r\right) (A.14) \\ = A _ {0} \left[ \left\{X _ {2}, W \right\}, X _ {2}, R; \mathbb {Z} _ {2} ^ {1}, \mathbf {s e q} _ {2} ^ {1} = \left(x _ {1}, x _ {1}\right) \right] (w, r) (A.15) \\ = \sum_ {x _ {2} ^ {\prime} \in \mathcal {X} _ {2}} P _ {x _ {1}} \left(x _ {2}\right) P _ {x _ {1}} \left(w | r, x _ {2} ^ {\prime}\right). (A.16) \\ \end{array}
$$
Also, by Lemma 3,
$$
\begin{array}{l} Q \left[ \mathbf {D} _ {1} \right] Q \left[ \mathbf {D} _ {3} \right] (A.17) \\ = A _ {0} ^ {1 3} (A.18) \\ := A _ {0} [ R, \emptyset , X _ {2}; \mathbb {Z} _ {1} ^ {1} := \{X _ {1} \}, \operatorname {s e q} _ {1} ^ {1} ] (r, x _ {2}) \times A _ {0} [ Y, \emptyset , \{R, W \}; \mathbb {Z} _ {3} ^ {2} := \{X _ {2} \}, \operatorname {s e q} _ {3} ^ {2} ] (y, \{r, w \}) (A.19) \\ = A _ {0} \left[ \left\{R, Y \right\}, \emptyset , \left\{X _ {2}, W \right\}; Z ^ {1 3} := \left\{X _ {1}, X _ {2} \right\}, \operatorname {s e q} ^ {1 3} = \left(x _ {1}, x _ {2}\right), G \right] \left(\left\{r, y \right\}, \left\{x _ {2}, w \right\}\right) (A.20) \\ = P _ {x _ {1}} (r \mid x _ {2}) P _ {x _ {2}} (y \mid r, w). (A.21) \\ \end{array}
$$
$$
P \left(y \mid d o \left(x _ {1}, x _ {2}\right)\right) = \sum_ {r, w \in \mathfrak {S} _ {R, W}} A _ {0} ^ {2} A _ {0} ^ {1 3}. \tag {A.22}
$$
A.3 Example 5
A.3.1 Specification of Nuisances
Recall that the topological order of the variable is  $W \prec_{G} X \prec_{G} Z \prec_{G} Y$ . Also,  $P(y|do(x)) = \sum_{z \in \mathfrak{S}_Z} A_0^1 A_0^2$  where
$$
A _ {0} ^ {1} := A _ {0} \left[ \{W, Z \}, W, X; \emptyset , \emptyset \right] (z, x) = \sum_ {w \in \mathfrak {S} _ {W}} P (z | x, w) P (w) \tag {A.23}
$$
$$
A _ {0} ^ {2} := A _ {0} [ Y, \emptyset , \emptyset ; \{Z \}, (z) ] (y, \emptyset) = P _ {z} (y) := P _ {\sigma (Z)} (y | z). \tag {A.24}
$$
$$
P (y | d o (x)) = f \left(A _ {0} ^ {1}, A _ {0} ^ {2}\right) := \sum_ {z \in \mathfrak {S} _ {Z}} A _ {0} ^ {1} A _ {0} ^ {2}. \tag {A.25}
$$