SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Lemma 3 (Multiplication). Let  $A_0^i \coloneqq A_0[\mathbf{W}_i, \emptyset, \mathbf{R}_i; \mathbb{Z}_i, \mathbf{seq}^i](\mathbf{w}_i, \mathbf{r}_i) \coloneqq \prod_{j=1}^{m^i} P_{\mathbf{z}_j^i}(w_{i,j} | \overline{\mathbf{w}}_i^{j-1}, \overline{\mathbf{r}}_i^{j-1} \backslash \mathbf{z}_j^i)$  for  $i \in \{1,2\}$  where  $\mathbf{seq}^i \coloneqq (\mathbf{z}_j^i)_{j=1}^{m^i}$ . Let  $\mathbf{W} \coloneqq \mathbf{W}_1 \cup \mathbf{W}_2$ . Let  $\mathbf{R} \coloneqq (\mathbf{R}_1 \cup \mathbf{R}_2) \backslash \mathbf{W}$ . Let  $(\mathbf{W}, \mathbf{R})$  be ordered by  $\prec_G$ . Let  $\mathbb{Z} \coloneqq \mathbb{Z}_1 \cup \mathbb{Z}_2$ . Assume the following: (1)  $\mathbf{W}_1 \cap \mathbf{W}_2 = \emptyset$ ; and (2)  $\forall W_j \in \mathbf{W}, \exists W_{i,k} \in \mathbf{W}_i$  such that  $(\overline{\mathbf{W}}^{j-1}, \overline{\mathbf{R}}^{j-1}) = (\overline{\mathbf{W}}_i^{k-1}, \overline{\mathbf{R}}_i^{k-1})$ . Let  $\mathbf{seq} \coloneqq (\mathbf{z}_j)_{j:W_j \in \mathbf{W}}$  where  $\mathbf{z}_j = \mathbf{z}_k^i$  for all  $j$ . Then,
$$
A _ {0} ^ {1} \times A _ {0} ^ {2} = A _ {0} [ \mathbf {W}, \emptyset , \mathbf {R}; \mathbb {Z}, \operatorname {s e q} ] (\mathbf {w}, \mathbf {r}) = \prod_ {j: W _ {j} \in \mathbf {W}} P _ {\mathbf {z} _ {j}} \left(w _ {j} | \overline {{\mathbf {w}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {j}\right). \tag {4}
$$
This lemma provides a graphical criterion where a product  $A_0^1 \times A_0^2$  is given as a g-mSBD operator.
Lemma 4 (Division). Let  $A_0^i \coloneqq A_0[\mathbf{W}_i, \emptyset, \mathbf{R}_i; \mathbb{Z}_i, \mathbf{seq}^i](\mathbf{w}_i, \mathbf{r}_i) \coloneqq \prod_{j=1}^{m^i} P_{\mathbf{z}_j^i}(w_{i,j} | \overline{\mathbf{w}}_i^{j-1}, \overline{\mathbf{r}}_i^{j-1} \backslash \mathbf{z}_j^i)$  for  $i \in \{1,2\}$  where  $\mathbf{seq}^i \coloneqq (\mathbf{z}_j^i)_{j=1}^{m^i}$ . Let  $\mathbf{W} \coloneqq \mathbf{W}_1 \backslash \mathbf{W}_2$ . Let  $\mathbf{R} \coloneqq (\mathbf{R}_1 \cup \mathbf{W}_2) \cap pre(\mathbf{W}; G)$ . Assume the following: (1)  $\mathbf{W}_2 \subseteq \mathbf{W}_1$ ; and (2)  $\forall W_j \in \mathbf{W}, \exists W_{1,k} \in \mathbf{W}_1$  such that  $(\overline{\mathbf{W}}^{j-1}, \overline{\mathbf{R}}^{j-1}) = (\overline{\mathbf{W}}_1^{k-1}, \overline{\mathbf{R}}_1^{k-1})$ ,  $\mathbf{Z}_{i,k} = \mathbf{Z}_j$  and  $\mathbf{z}_{i,k} = \mathbf{z}_j$ . Then,
$$
A _ {0} ^ {1} / A _ {0} ^ {2} = A _ {0} \left[ \mathbf {W}, \emptyset , \mathbf {R}; \mathbb {Z} _ {1}, \mathbf {s e q} ^ {1} \right] (\mathbf {w}, \mathbf {r}) = \prod_ {j: W _ {j} \in \mathbf {W}} P _ {\mathbf {z} _ {j}} \left(w _ {j} | \overline {{\mathbf {w}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {j}\right). \tag {5}
$$
This lemma provides a graphical criterion where a product  $A_0^1 / A_0^2$  is given as a g-mSBD operator.
We have rewritten the identification algorithm proposed by Lee et al. [2019] as Algo. 1 to express the g-identifiable causal effect as a combination of marginalizations, multiplications, and divisions of g-mSBD. It's worth of noting that the identification algorithm proposed by Lee et al. [2019] and Algo. 1 are equivalent.
Theorem 1 (Expression of g-Identifiable Causal Effects). Algo. 1 returns any g-identifiable causal effects as a function of a set  $\{A_0^k\}$  of g-mSBD adjustment operators in the form
$$
P (\mathbf {y} \mid d o (\mathbf {x})) = f \left(\left\{A _ {0} ^ {k} \right\} _ {k = 1} ^ {K}\right), \tag {6}
$$
where the function  $f(\cdot)$  applies marginalization, multiplication, or division over  $g$ -mSBD operators in  $\{A_0^k\}$  as specified by Algo. 1.
For concreteness, we demonstrate the application of Algo. 1 for Figs. (1a,1b), where the effects  $P(\mathbf{y}|do(\mathbf{x}))$  are g-identifiable. Detailed and visually friendly demonstrations are described in Appendix A.
Example 3 (Application of Algo. 1 to Example 1). Note  $\mathbb{Z} = \{\emptyset, Z\}$ . Line 3-4:  $\mathbf{D} = \{Z, Y\}$  where  $\mathbf{D}_1 := \{Z\}$  and  $\mathbf{D}_2 := \{Y\}$ . Line 5-13: Identify  $Q[\mathbf{D}_1]$  from  $\mathbf{Z}_1 = \emptyset$  as follow. Note  $\mathbf{D}_1 \subseteq \mathbf{S}^0$ , where  $\mathbf{S}^0 := \mathbf{V}$  where  $Q[\mathbf{S}^0] = A_0[\mathbf{S}^0, \emptyset, \emptyset; \mathbb{Z}^0 := \emptyset, \emptyset](\mathbf{s}^0, \emptyset) = P(\mathbf{v})$ . Run  $Q[\mathbf{D}_1] = \mathrm{SUBID}(\mathbf{D}_1, \mathbf{S}^0, Q[\mathbf{S}^0], G)$  and obtain  $Q[\mathbf{D}_1] = A_0^1 := A_0[\{W, Z\}, W, X; \emptyset, \emptyset](z, x) = \sum_{w \in \mathfrak{S}_W} P(z|x, w)P(w)$ . Lemmas (2,3,4) are used in running the sub-procedure. Now, identify  $Q[\mathbf{D}_2]$  from  $\mathbf{Z}_2 = \{Z\}$  as follow. Note  $\mathbf{D}_2 \subseteq \mathbf{S}^1 := \{W, X, Y\}$ , the c-component in  $G(\mathbf{V} \backslash Z)$ . Note  $Q[\mathbf{S}^1] = A_0[\mathbf{S}^1, \emptyset, \emptyset; \mathbb{Z}^1 := \{Z\}, \mathrm{seq}^1](\mathbf{s}^1, \emptyset) = P_z(w, x, y)$ , where  $\mathrm{seq}^1 = (z, z, z)$ . We run  $Q[\mathbf{D}_2] = \mathrm{SUBID}(\mathbf{D}_2, \mathbf{S}^1, Q[\mathbf{S}^1], G(\mathbf{S}^1))$ , and obtain  $Q[\mathbf{D}_2] = A_0^2 := A_0[Y, \emptyset, \emptyset; \mathbb{Z}^1, \mathrm{seq}^1](y, \emptyset) = P_z(y)$ . Lemma 2 is used in the sub-procedure. Line 14-15:  $P(y|do(x)) = \sum_{z \in \mathfrak{S}_Z} A_0^1 A_0^2$ .
Example 4 (Application of Algo. 1 to Example 2). Note  $\mathbb{Z} = \{X_1, X_2\}$ . Line 3-4:  $\mathbf{D} = \{R, W, Y\}$  where  $\mathbf{D}_1 := \{R\}$ ,  $\mathbf{D}_2 := \{W\}$ , and  $\mathbf{D}_3 = \{Y\}$ . Line 5-13: In  $G(\mathbf{V} \backslash X_1)$ ,  $\mathbf{D}_1 = \mathbf{S}_1^1 := \{R\}$ .  $Q[\mathbf{D}_1] = Q[\mathbf{S}_1^1] = A_0^1 := A_0[R, \emptyset, X_2; \mathbb{Z}^1 := \{X_1\}, \mathbf{seq}^1](r, x_2) = P(r|do(x_1), x_2)$ , where  $\mathbf{seq}^1 = (x_1)$ . In  $G(\mathbf{V} \backslash X_1)$ ,  $\mathbf{D}_2 \subseteq \mathbf{S}_2^1 := \{X_2, W, Y\}$ .  $Q[\mathbf{S}_2^1] = A_0[\mathbf{S}_2^1, \emptyset, R; \mathbb{Z}^2 := \{X_1\}, \mathbf{seq}^2][\mathbf{s}_2^1, r] = P_{x_1}(x_2)P_{x_1}(w|x_2, r), P_{x_1}(y|x_2, w, r)$  where  $\mathbf{seq}^2 = (x_1, x_1, x_1)$ . Run  $Q[\mathbf{D}_2] = \mathrm{SUBID}(\mathbf{D}_2, \mathbf{S}_2^1, Q[\mathbf{S}_2^1], G(\mathbf{S}_2^1)) = A_0^2 := A_0[\{X_2, W\}, X_2, R; \mathbb{Z}^2, \mathbf{seq}^2](w, r) = \sum_{x_2' \in \mathfrak{S}_{X_2}} P_{x_1}(w|r, x_2', P_{x_1}(x_2')$  where  $\mathbf{seq}^2 = (x_1, x_1)$ . Lemma 2 is used in the sub-procedure. Since  $\mathrm{SUBID}(\mathbf{D}_3, \mathbf{S}_3^1, Q[\mathbf{S}_3^1], G(\mathbf{S}_3^1))$  return FAIL, we find the c-component  $\mathbf{S}_1^2 := \{Y\}$  where  $\mathbf{D}_3 = \mathbf{S}_3^2$ . Note  $Q[\mathbf{D}_3] = Q[\mathbf{S}_3^2] = A_0^3 := A_0[Y, \emptyset, \{R, W\}; \mathbb{Z}^3 := \{X_2\}, \mathbf{seq}^3](y, \{r, w\}) = P_{x_2}(y|w, r)$ , where  $\mathbf{seq}^3 = (x_2)$ . Line 14-15: Applying Lemma 3,  $A_0^{13} := A_0^1 \times A_0^3 = A_0[\{R, Y\}, \emptyset, \{X_2, W\}; \mathbb{Z}^{13}] := \{X_1, X_2\}, \mathbf{seq}^{13}[(r, y), \{x_2, w\}) = P_{x_1}(r|x_2,)P_{x_2}(y|r, w)$ , where  $\mathbf{seq}^{13} = (x_1, x_2)$ . Then,  $P(y|do(x_1, x_2)) = \sum_{r, w \in \mathfrak{S}_{R,W}} A_0^{13}A_0^2$ .
3 Estimating g-Identifiable Causal Effects
In this section, we develop an estimator for  $P(\mathbf{y} | \text{do}(\mathbf{x}))$  using samples  $\mathbb{D} \coloneqq \{D_{\sigma(\mathbf{Z}_i)} \sim P_{\sigma(\mathbf{Z}_i)}(\mathbf{V}) \in \mathbb{P}\}$  obtained from randomized experiments and observations (where  $\mathbf{Z}_i = \emptyset$ ). We use  $P_{\sigma(\mathbf{Z})}$  instead of  $P_{\mathbf{z}}$  to highlight the distribution  $D_{\sigma(\mathbf{Z}_i)} \in \mathbb{D}$  follows.
We first introduce an estimator for the g-mSBD adjustment that exhibits the doubly robust property. The nuisance parameters for the g-mSBD adjustment are defined as follows:
Definition 2 (Nuisances for g-mSBD). Nuisances for  $g$ -mSBD  $A_0$  in Eq. (1) are  $\{\mu_0^{i+1}, \pi_0^i\}_{i=1}^m$  defined as follows. Let  $\mu_0^{m+1} = \mu^{m+1} \coloneqq \mathbb{1}_{\mathbf{w} \setminus \mathbf{c}}(\mathbf{W} \setminus \mathbf{C})$ . For  $i = m-1, \dots, 1$ ,
$$
\mu_ {0} ^ {i + 1} \left(\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i}\right) := \mathbb {E} _ {P _ {\sigma \left(\mathbf {z} _ {i + 1}\right)}} \left[ \mu_ {0} ^ {i + 2} \left(\overline {{\mathbf {W}}} ^ {i + 1}, \mathbf {r} _ {i + 1}, \overline {{\mathbf {R}}} ^ {1: i}\right) \mid \overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i}, \mathbf {r} _ {0}, \mathbf {z} _ {i + 1} \right] \tag {7}
$$
$$
\pi_ {0} ^ {i} \left(\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i}\right) := \frac {P _ {\sigma \left(\mathbf {Z} _ {i}\right)} \left(\overline {{\mathbf {W}}} ^ {i} , \overline {{\mathbf {R}}} ^ {1 : i - 1} \mid \mathbf {z} _ {i} , \mathbf {r} _ {0}\right)}{P _ {\sigma \left(\mathbf {Z} _ {i + 1}\right)} \left(\overline {{\mathbf {W}}} ^ {i} , \overline {{\mathbf {R}}} ^ {1 : i - 1} \mid \mathbf {z} _ {i + 1} , \mathbf {r} _ {0}\right)} \frac {\mathbb {1} _ {\mathbf {r} _ {i}} \left(\mathbf {R} _ {i}\right)}{P _ {\sigma \left(\mathbf {Z} _ {i + 1}\right)} \left(\mathbf {R} _ {i} \mid \overline {{\mathbf {W}}} ^ {i} , \overline {{\mathbf {R}}} ^ {1 : i - 1} , \mathbf {z} _ {i + 1} , \mathbf {r} _ {0}\right)}. \tag {8}
$$
Remark 1 (Simplification of Nuisances). Although the nuisances  $\pi_0^i$  may seem complicated, they can be simplified in several important special cases. For example,  $\pi_0^i = \mathbb{1}_{\mathbf{r}_i}(\mathbf{R}_i) / P_{\sigma (\mathbf{Z})}(\mathbf{R}_i|\overline{\mathbf{W}}^i,\overline{\mathbf{R}}^{i - 1},\mathbf{z},\mathbf{r}_0)$  if  $\mathbb{Z} = \{\mathbf{Z}\}$  for any  $\mathbf{Z}\subseteq \mathbf{V}$  where  $\mathbf{Z}$  is possibly empty.
In general, employing off-the-shelf classification methods for density ratio estimation is feasible, leveraging the techniques outlined in Section 5.4 of Díaz et al. [2021].
We now introduce a g-mSBD estimator exhibiting the robustness properties using these nuisances. This estimator is motivated by the double/debiased machine-learning style estimators [Chernozhukov et al., 2018, 2022]:
Definition 3 (DR-g-mSBD Estimators). Let  $D_{\sigma(\mathbf{Z}_i)}$  for  $\mathbf{Z}_i \in \mathbb{Z}$  denote the experimental samples from randomizing the variable  $\mathbf{Z}_i$ . Let  $\overline{D}_{\mathbf{z}_i}$  for  $\mathbf{z}_i \in \mathfrak{D}_{\mathbf{Z}_i}$  denote the subsamples of  $D_{\sigma(\mathbf{Z}_i)}$  fixing  $\mathbf{R}_0 \backslash \mathbf{Z}_i = \mathbf{r}_0 \backslash \mathbf{z}_i$  and  $\mathbf{Z}_i = \mathbf{z}_i$ . The DR-g-mSBD estimator  $\hat{A}$  for the g-mSBD adjustment  $A_0[\mathbf{W}, \mathbf{C}, \mathbf{R}; \mathbb{Z}_0 := \{\mathbf{Z}_i\}_{i=1}^m, \text{seq} := (\mathbf{z}_i)_{i=1}^m](\mathbf{w} \backslash \mathbf{c}, \mathbf{r})$  is defined as follows:
1. Randomly partition  $\overline{D}_{\mathbf{z}_i}$  into  $\{\overline{D}_{\mathbf{z}_i,\ell}\}_{\ell \in [L]}$ ; i.e.,  $\overline{D}_{\mathbf{z}_i} = \cup_{\ell = 1}^{L}\overline{D}_{\mathbf{z}_i,\ell}$ ,  $\forall \mathbf{Z}_i\in \mathbb{Z}$  and  $\mathbf{z}_i\in \mathfrak{D}_{\mathbf{Z}_i}$ .
2. For each fold  $\ell \in [L]$ , let  $\mu_{\ell}^{i+1}$  denote learned  $\mu_0^{i+1}$  using  $\overline{D}_{\mathbf{z}_{i+1}} \backslash \overline{D}_{\mathbf{z}_{i+1},\ell}$  for  $i = m, \dots, 2$ ; and  $\pi_{\ell}^{i}$  learned  $\pi_0^i$  for  $i = 1, \dots, m-1$ . Define  $\breve{\mu}_{\ell}^{i+1} := \mu_{\ell}^{i+1}(\overline{\mathbf{W}}^i, \mathbf{r}_i, \overline{\mathbf{R}}^{1:i-1})$  and  $\overline{\pi}_{\ell}^i := \prod_{j=1}^i \pi_\ell^j$ .
3. Estimate  $\hat{A} \coloneqq \hat{A} (\{\mu_{\ell}^{j + 1}, \pi_{\ell}^{j}\}_{j \in [m - 1], \ell \in [L]}) \coloneqq (1 / L) \sum_{\ell = 1}^{L} \hat{A}_{\ell} (\{\mu_{\ell}^{j + 1}, \pi_{\ell}^{j}\}_{j \in [m - 1]})$  where
$$
\hat {A} _ {\ell} := \hat {A} _ {\ell} \left(\left\{\mu_ {\ell} ^ {j + 1}, \pi_ {\ell} ^ {j} \right\} _ {j \in [ m - 1 ]}\right) := \sum_ {j = 1} ^ {m - 1} \mathbb {E} _ {\overline {{D}} _ {\mathbf {z} _ {j + 1, \ell}}} \left[ \bar {\pi} _ {\ell} ^ {j} \left\{\tilde {\mu} _ {\ell} ^ {j + 2} - \mu_ {\ell} ^ {j + 1} \right\} \right] + \mathbb {E} _ {\overline {{D}} _ {\mathbf {z} _ {1, \ell}}} \left[ \tilde {\mu} _ {\ell} ^ {2} \right], \tag {9}
$$
where  $\mathbb{E}_{\overline{D}_{\mathbf{z}_j,\ell}}[\cdot ]$  is an empirical average over samples  $\overline{D}_{\mathbf{z}_j,\ell}$
We now analyze the doubly robustness property of this estimator.
Proposition 1 (Asymptotic Analysis of g-mSBD Estimators). Assume that the nuisance estimates  $\mu_{\ell}^{i}$  and  $\pi_{\ell}^{i}$  are  $L_{2}$ -consistent; i.e.,  $\| \mu_{\ell}^{i + 1} - \mu_0^{i + 1}\|_{P_{\sigma (\mathbf{Z}_{i + 1})}} = o_{P_{\sigma (\mathbf{Z}_{i + 1})}}(1),\| \breve{\mu}_{\ell}^{i + 2} - \breve{\mu}_0^{i + 2}\|_{P_{\sigma (\mathbf{Z}_{i + 1})}} = o_{P_{\sigma (\mathbf{Z}_{i + 1})}}(1)$  and  $\| \pi_{\ell}^{i} - \pi_{0}^{i}\|_{P_{\sigma (\mathbf{Z}_{i + 1})}} = o_{P_{\sigma (\mathbf{Z}_{i + 1})}}(1)$  for  $i = 1,\dots ,m - 1$  and  $\| \breve{\mu}_{\ell}^{2} - \breve{\mu}_{0}^{2}\|_{P_{\sigma (\mathbf{Z}_{1})}} = o_{P_{\sigma (\mathbf{Z}_{1})}}(1)$ . Let  $n_i\coloneqq |\overline{D}_{\mathbf{z}_i}|$  for  $i\in \{1,\dots ,m\}$ . Then,
$$
\hat {A} - A _ {0} = \sum_ {i = 1} ^ {m} R _ {i} + \frac {1}{L} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m - 1} O _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left(\| \mu_ {\ell} ^ {i + 1} - \mu_ {0} ^ {i + 1} \| \| \pi_ {\ell} ^ {i} - \pi_ {0} ^ {i} \|\right), \tag {10}
$$
where  $R_{i}$  is a random variable such that  $n_i^{1/2}R_i$  converges in distribution to a mean-zero normal random variable.
This estimator possesses a doubly robustness property since the estimator is bounded in probability at rate  $n^{-1/2}$  (for  $n := \min\{n_1, \dots, n_m\}$ , whenever  $O_{P_{\sigma(\mathbf{Z}_{i+1})}}\left(\|\mu_\ell^{i+1} - \mu_0^{i+1}\| \|\pi_\ell^i - \pi_0^i\|\right) = O_P(n_{i+1}^{-1/2})$  for all  $i$ ).
We now construct an estimator for the g-identification expression using the DR-g-mSBD estimator defined in Def. 3. The resulting estimator is called the MR-gID estimator:
Definition 4 (MR-gID Estimator). The MR-gID estimator  $\hat{\psi}$  for the identification expression of the causal effect  $\psi_0\coloneqq f(\{A_0^k\}_{k = 1}^K)$  in Theorem 1 is given as follows: For each  $A_0^k$  composing  $f(\{A_0^k\}_{k = 1}^K)$ , let  $\hat{A}^{k}\coloneqq \hat{A}^{k}(\{\mu_{k,\ell}^{j + 1},\pi_{k,\ell}^{j}\}_{j\in [m^{k} - 1],\ell \in [L]})$  denote the DR-g-mSBD estimator with nuisance estimates  $\{\mu_{k,\ell}^{j + 1},\pi_{k,\ell}^{j}\}$  for the true nuisances  $\{\mu_{k,0}^{j + 1},\pi_{k,0}^{j}\}$ . Then,
$$
\hat {\psi} := f \left(\left\{\hat {A} ^ {k} \right\} _ {k = 1} ^ {K}\right). \tag {11}
$$
We impose assumptions on the identification expression and its nuisances for further analysis.
Assumption 2 (Analysis of MR-gID). The identification function  $f(\{A^k\}_{k=1}^K)$  in Thm. 1 and each nuisances  $\{\mu_{k,\ell}^{i+1}, \pi_{k,\ell}^i\}_{k,\ell}$  for  $\hat{A}^k$  satisfy the following properties:
1. Twice differentiability:  $f(\{A^k\}_{k=1}^K)$  is twice continuously Frechet differentiable w.r.t.  $\{A^k\}_{k=1}^K$ . w.r.t.  $\{A^k\}_{k=1}^K$ .
2. Boundedness:  $\forall k\in [K]$  and  $\forall \mathbf{Z}_i\in \mathbb{Z}$ ,  $\nabla_{A^k}f(\{A_0^j\}_{j = 1}^K)[\hat{A}^k -A_0^k ] = O_{P_{\sigma (\mathbf{Z}_i)}}(\hat{A}^k -A_0^k)$ .
3.  $L_{2}$  Consistency:  $\| \mu_{k,\ell}^{i + 1} - \mu_{k,0}^{i + 1}\|_{P_{\sigma (\mathbf{Z}_{i + 1}^k)}} = o_{P_{\sigma (\mathbf{Z}_{i + 1}^k)}}(1),\| \check{\mu}_{k,\ell}^{i + 2} - \check{\mu}_{k,0}^{i + 2}\|_{P_{\sigma (\mathbf{Z}_{i + 1}^k)}} = o_{P_{\sigma (\mathbf{Z}_{i + 1}^k)}}(1)$ $\| \pi_{k,\ell}^{i} - \pi_{k,0}^{i}\|_{P_{\sigma (\mathbf{Z}_{i + 1}^{k})}} = o_{P_{\sigma (\mathbf{Z}_{i + 1}^{k})}}(1),$  and  $\| \check{\mu}_{k,\ell}^{2} - \check{\mu}_{k,0}^{2}\|_{P_{\sigma (\mathbf{Z}_1^k)}} = o_{P_{\sigma (\mathbf{Z}_1^k)}}(1).$
Assumption 2 is imposed to limit the error of the MR-gID, which is a linear function of the errors of each DR-g-mSBD estimator.
Theorem 2 (Asymptotic Analysis of MR-gID). Suppose Assumption 2 holds. Let  $n_{k,i} \coloneqq |\overline{D}_{\mathbf{z}_i^k}|$  for  $\mathbf{Z}_i^k \in \mathbb{Z}$  and  $\mathbf{z}_i^k \in \mathfrak{D}_{\mathbf{Z}_i^k}$ . Let  $\hat{\psi}$  denote the MR-gID estimator in Def. 4 for the causal effect  $\psi_0 \coloneqq f(\{A_0^k\}_{k=1}^K)$  in Theorem 1. Then, the error of  $\hat{\psi}$  is given as
$$
\hat {\psi} - \psi_ {0} = \sum_ {k = 1} ^ {K} \sum_ {i = 1} ^ {m ^ {k}} O _ {P _ {\sigma (\mathbf {Z} _ {i} ^ {k})}} \left(n _ {k, i} ^ {- 1 / 2}\right) + \frac {1}{L} \sum_ {k = 1} ^ {K} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m ^ {k} - 1} O _ {P _ {\sigma (\mathbf {Z} _ {i} ^ {k})}} \left(\| \mu_ {k, \ell} ^ {i + 1} - \mu_ {k, 0} ^ {i + 1} \| \| \pi_ {k, \ell} ^ {i} - \pi_ {k, 0} ^ {i} \|\right). \tag {12}
$$
We highlight that the MR-gID  $\hat{\psi}$  exhibits robustness property since  $\hat{\psi} -\psi_0$  for  $\psi_0 = P(\mathbf{y}|do(\mathbf{x}))$  is bounded at rate  $n^{-1 / 2}$  (for  $n = \min \{n_{k,i}\}$  and  $P\in \mathbb{P}$ ) even when all nuisances  $\{\mu_{k,\ell}^{i + 1},\pi_{k,\ell}^i\}$  are bounded at slower  $n^{-1 / 4}$  rate. Furthermore, the MR-gID estimator exhibits multiply robustness, as the error of the MR-gID is a linear function of the error of DR-g-mSBD, which demonstrates the doubly robustness property. Formally,
Corollary 2 (Multiply Robustness (Corollary of Thm. 2)). Suppose (1) Assumption 2 holds; (2) Either  $\pi_{k,\ell}^{i} = \pi_{k,0}^{i}$  or  $\mu_{k,\ell}^{j} = \mu_{k,0}^{j}$  for  $j = i + 1,\dots ,m^k$  for all  $i,\ell ,k$ ; and (3) all nuisances  $\{\pi_{k,\ell}^{i},\mu_{k,\ell}^{i + 1}\}_{i,\ell ,k}$  are bounded by some constant. Then, the MR-gID  $\hat{\psi}$  (Def. 4) is consistent to  $\psi_0$ .
For concreteness, we illustrate the application of Thm. 2 for Examples (1, 2). Detailed procedures are provided in Appendix A.
Example 5 (Application of Thm. 2 to Example 1). Recall that  $P(y|do(x)) = f(\{A_0^1, A_0^2\}) \coloneqq \sum_{z \in \mathfrak{S}_Z} A_0^1 A_0^2$ . The nuisance set for  $A_0^1$  is  $\mu_{1,0}^2(X, W) \coloneqq \mathbb{E}_P[\mathbb{1}_z(Z)|X, W]$  and  $\pi_{1,0}^1(X, W) \coloneqq \mathbb{1}_x(X)/P(X|W)$ . Then, the estimator for  $A_0^1$  is  $\hat{A}^1 \coloneqq \hat{A}^1(\{\mu_{1,\ell}^2, \pi_{1,\ell}^1\}_{\ell \in [L]})$  defined in Def. 3. The nuisance set for  $A_0^2$  is  $\mu_{2,0}^2 \coloneqq \mathbb{E}_{P_{\sigma(Z)}}[\mathbb{1}_y(Y)]$ . Then, the estimator for  $A_0^2$  is  $\hat{A}^2 \coloneqq \hat{A}^2(\{\mu_{2,\ell}^2\}_{\ell \in [L]})$ . Then, the estimator is constructed as Def. 4, as  $f(\{\hat{A}^1, \hat{A}^2\})$ . By Thm. 2, the error of the estimator is  $O_P(n_0^{-1/2}) + O_P(n_z^{-1/2}) + (1/L)\sum_{\ell=1}^{L}O_P(\|\mu_{1,\ell}^2 - \mu_{1,0}^2\|\|\pi_{1,\ell}^1 - \pi_{1,0}^1\|)$ , where  $n_0 \coloneqq |D|$  and  $n_z \coloneqq |D_z|$  where  $D \sim P$  and  $D_z \sim P_z$ .
Example 6 (Application of Thm. 2 to Example 2). Recall that  $P(y|do(x_1,x_2)) = f(\{A_0^2,A_0^{13}\}) = \sum_{r,w\in \mathfrak{S}_{R,W}}A_0^2 A_0^{13}$ . The nuisance set for  $A_0^2$  is  $\mu_{2,0}^2 \coloneqq \mathbb{E}_{P_{x_1}}[\mathbb{1}_w(W)|R,X_2]$  and  $\pi_{1,0}^1 \coloneqq \mathbb{1}_r(R) / P_{x_1}(R|X_2)$ . Then, the estimator for  $A_0^2$  is  $\hat{A}^2 \coloneqq \hat{A}^2 (\{\mu_{2,\ell}^2,\pi_{1,\ell}^1\}_{\ell \in [L]})$ . The nuisance set for  $A_0^{13}$  is  $\mu_{13,0}^2 \coloneqq \mathbb{E}_{P_{x_2}}[\mathbb{1}_{r,y}(R,Y)|R,W]$  and  $\pi_{13,0}^1 \coloneqq \frac{P_{\sigma(X_1)}(R|x_2,x_1)}{P_{\sigma(X_2)}(R|x_2)}\frac{1_w(W)}{P_{\sigma(X_2)}(W|R,x_2)}$ . Then, the estimator is  $\hat{A}^{13} = \hat{A}^{13}(\{\mu_{13,0}^2,\pi_{13,0}^1\}_{\ell \in [L]})$ . Then, the estimator is constructed as Def. 4, as  $f(\{\hat{A}^{13},\hat{A}^2\})$ . By Thm. 2, the error of the estimator is  $O_{P_{\sigma (X_1)}}(n_1^{-1 / 2}) + O_{P_{\sigma (X_2)}}(n_2^{-1 / 2}) + (1 / L)\sum_{\ell = 1}^{L}\{O_{P_{\sigma (X_1)}}(\| \mu_{2,\ell}^2 -\mu_{2,0}^2 ||\| \pi_{2,\ell}^1 -\pi_{2,0}^1 ||) + O_{P_{\sigma (X_2)}}(\| \mu_{13,\ell}^2 -\mu_{13,0}^2 ||\| \pi_{13,\ell}^1 -\pi_{13,0}^1 ||)\}$ , where  $n_1\coloneqq |D_1|$  and  $n_2\coloneqq |D_2|$  where  $D_{1}\sim P_{x_{1}}$  and  $D_{2}\sim P_{x_{2}}$ .
In this section, we demonstrate the MR-gID estimator from Definition (4) through Examples (1,2) and Project STAR dataset[Krueger and Whitmore, 2001, Schanzenbach, 2006]. For each example, the proposed estimator is constructed using a dataset  $\mathbb{D} \coloneqq \{D_{\mathbf{Z}_i}, \mathbf{Z}_i \in \mathbb{Z}\}$  simulated from an underlying SCM. Our goal is to provide empirical evidence of the fast convergence behavior and the robustness property of the proposed estimator compared to competing baseline estimators. We consider two standard baselines in the literature: the 'regression-based estimator (reg)' only uses the
Scenario 1	Scenario 2	Scenario 3	Scenario 4
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Figure 2: AAE Plots for Examples (1,2) for Scenarios  $\{1,2,3,4\}$  depicted in the Experimental Setup section. The  $x$ -axis and  $y$ -axis are the number of samples and AAE, respectively.
regression nuisance parameters  $\mu^2$  for  $\mu_0^2$  defined in Def. 2, and the 'probability weighting-based estimator (pw)' only uses the probability weighting parameters  $\pi^{m-1}$  for  $\pi_0^{m-1}$  defined in Def. 2, while our MR-gID uses both in estimating the g-mSBD operators  $A^k$  composing  $f(\{A^k\})$  in Thm. 1. Details of the regression-based ('reg') and the probability weighting-based ('pw') estimators are provided in Appendix A. The details of the simulation are in Appendix (D, E).
4.1 Synthetic Dataset Analysis
Accuracy Measure. We compare the proposed estimator ('mr') in Def. 4 to the regression-based estimator ('reg') and the probability weighting-based estimator ('pw'). In particular, we use  $T^{\mathrm{est}}(\mathbf{x})$  for  $\mathrm{est} \in \{\mathrm{reg}, \mathrm{pw}, \mathrm{mr}\}$  to denote the g-ID estimators that leverage regression-based ('reg'), probability weighting-based ('pw'), and MR-gID in estimating each operator  $A^k$  in the identification expression  $f(\{A^k\})$  of the causal effect  $P(\mathbf{y}|do(\mathbf{x}))$ . We assess the quality of the estimators by computing the average absolute error  $\mathrm{AAE}^{\mathrm{est}} \coloneqq \frac{1}{|\mathfrak{D}_{\mathbf{X}}|} \sum_{\mathbf{x} \in \mathfrak{D}_{\mathbf{X}}} |T^{\mathrm{est}}(\mathbf{x}) - P(\mathbf{y}|do(\mathbf{x}))|$  where  $|\mathfrak{D}_{\mathbf{X}}|$  is the cardinality of  $\mathfrak{D}_{\mathbf{X}}$ . Nuisance functions are estimated using gradient boosting models called XGBoost [Chen and Guestrin, 2016]. We ran 100 simulations for each  $n = \{200, 400, 600, 800, 1000\}$  for  $n \coloneqq |D_{\mathbf{Z}}|$  for  $\forall \mathbf{Z} \subseteq \mathbb{Z}$ . We label the box-plot for these AAEs as 'AAE-plot'.
Experimental Setup. We evaluate the  $\mathrm{AAE^{est}}$  for Examples (1,2) in four scenarios:
- (Scenario 1) There were no noises in estimating nuisances.
- (Scenario 2) We introduced a converging noise  $\epsilon$  in estimating the nuisance, decaying at a  $n^{-\alpha}$  rate (i.e.,  $\epsilon \sim \mathrm{Normal}(n^{-\alpha}, N^{-2\alpha})$ ) for  $\alpha = 1/4$  to emphasize the errors induced by the finiteness of samples. This scenario is inspired by the experimental design discussed in [Kennedy, 2020].
- (Scenario 3) Nuisance  $\{\mu_{k,\ell}^{i+1}\}_{\ell,k,i}$  are estimated incorrectly - simulated by training the model with a random matrix having the same dimension as the input matrix.
- (Scenario 4) Nuisance  $\{\pi_{k,\ell}^i\}_{\ell ,k,i}$  are estimated incorrectly as in Scenario 3.