SlowGuess/NeurIPS_2023_OCR huggingface dataset:
By leveraging the definition of the nuisance in Def. 2, the nuisance composing  $A_0^1$  is  $\{\mu_{1,0}^2, \pi_{1,0}^1\}$  which are defined as follow:
$$
\mu_ {1, 0} ^ {2} (X, W) := \mathbb {E} _ {P} \left[ \mathbb {1} _ {z} (Z) | X, W \right],
$$
$$
\pi_ {1, 0} ^ {1} (X, W) := \mathbb {1} _ {x} (X) / P (X | W).
$$
The nuisance composing  $A_0^2$  is  $\{\mu_{2,0}^2\}$  which is  $\mu_{2,0}^2 \coloneqq \mathbb{E}_{P_z}[\mathbb{1}_y(Y)]$ .
A.3.2 Construction of Estimators
We apply the procedure in Def. 3 to construct estimators  $\hat{A}^1$  and  $\hat{A}^2$  for  $A_0^1$  and  $A_0^2$ . We choose  $L = 2$ . We first construct  $\hat{A}^1$  for the fixed  $\{z,x\} \in \mathfrak{D}_{Z,X}$ . We note that  $\hat{A}_{\ell}^{1}$  for  $\ell \in \{1,2\}$  is given as follows: For a fixed  $z,x$ ,
$$
\hat {A} _ {\ell} ^ {1} := \mathbb {E} _ {D _ {\ell}} \left[ \pi_ {\ell} ^ {1} (X, W) \left\{\mathbb {1} _ {z} (Z) - \mu_ {1, \ell} ^ {2} (X, W) \right\} + \mu_ {1, \ell} ^ {2} (x, W) \right], \tag {A.26}
$$
$$
\hat {A} ^ {1} = 1 / L \sum_ {\ell = 1} ^ {L} \hat {A} _ {\ell} ^ {1}, \tag {A.27}
$$
where  $\pi^1, \mu^2$  are nuisances estimated using  $D \backslash D_{\ell}$ . Specifically,  $\mu_{1,\ell}^2(X,W)$  is obtained by using the XGBoost [Chen and Guestrin, 2016] regression model which regresses  $\mathbb{1}_z(Z)$  onto the  $\{X,W\}$  using  $D \backslash D_{\ell}$ .  $\mu_{1,\ell}^2(x,W)$  is evaluated from  $D_{\ell}$  after fixing a column for  $X$  to  $x$ . In similar,  $\pi^1$  as follow: we first model  $P(X|W)$  by regressing  $X$  onto  $W$  from the data  $D \backslash D_{\ell}$  using the XGBoost [Chen and Guestrin, 2016]. Then, we evaluate  $\pi_{1,0}^1(X,W)$  by plugging in the trained  $P(X|W)$ .
We now construct  $\hat{A}^2$  for the fixed  $z$  and  $y$ . We first take the subsamples  $D_z$  from the experimental samples  $D_{\sigma (Z)}\in \mathbb{D}$ , where  $D_z$  is the sample where  $Z = z$ . Then, we compute the following:
$$
\hat {A} ^ {2} = \mu_ {2} ^ {2} = \mathbb {E} _ {D _ {z}} [ \mathbb {1} _ {y} (Y) ]. \tag {A.28}
$$
Then, following Def. 4, the MR-gID is constructed as follow:
$$
f \left(\hat {A} ^ {1}, \hat {A} ^ {2}\right) = \sum_ {z \in \mathfrak {D} _ {Z}} \hat {A} ^ {1} \hat {A} ^ {2}. \tag {A.29}
$$
A.4 Example 6
A.4.1 Specification of Nuisances
Recall that the topological order of the variable is  $X_{1} \prec_{G} X_{2} \prec_{G} R \prec_{G} W \prec_{G} Y$ . Also,
$$
\begin{array}{l} A _ {0} ^ {2} := A _ {0} \left[ \left\{X _ {2}, W \right\}, \left\{X _ {2} \right\}, R; \mathbb {Z} ^ {2} = \left\{X _ {1} \right\}, \operatorname {s e q} ^ {2} = \left(x _ {1}, x _ {1}\right) \right] (w, r) (A.30) \\ = \sum_ {x _ {2} ^ {\prime} \in \mathfrak {S} _ {X _ {2}}} P _ {x _ {1}} \left(w | r, x _ {2} ^ {\prime}\right) P _ {x _ {1}} \left(x _ {2} ^ {\prime}\right) (A.31) \\ A _ {0} ^ {1 3} := A _ {0} \left[ \{R, Y \}, \emptyset , \left\{X _ {2}, W \right\}; \mathbb {Z} ^ {1 3} = \left\{X _ {1}, X _ {2} \right\}, \operatorname {s e q} ^ {1 3} = \left(x _ {1}, x _ {2}\right) \right] \left(\{r, y \}, \left\{x _ {2}, w \right\}\right) (A.32) \\ = P _ {x _ {1}} (r \mid x _ {2}) P _ {x _ {2}} (y \mid r, w). (A.33) \\ \end{array}
$$
$$
P \left(y \mid d o \left(x _ {1}, x _ {2}\right)\right) = \sum_ {r, w \in \mathfrak {S} _ {R, W}} A _ {0} ^ {2} A _ {0} ^ {1 3}. \tag {A.34}
$$
The nuisance composing  $A_0^2$  is  $\{\mu_{2,0}^2,\pi_{2,0}^1\}$  which are defined as follow:
$$
\mu_ {2, 0} ^ {2} (R, X _ {2}) := \mathbb {E} _ {P _ {x _ {1}}} [ \mathbb {1} _ {w} (W) | R, X _ {2} ], \tag {A.35}
$$
$$
\pi_ {2, 0} ^ {1} (R, X _ {2}) := \mathbb {1} _ {r} (R) / P _ {x _ {1}} (R | X _ {2}). \tag {A.36}
$$
The nuisance composing  $A_0^{13}$  is  $\{\mu_{2,0}^2,\pi_{2,0}^1\}$  which are defined as follow:
$$
\begin{array}{l} \mu_ {1 3, 0} ^ {2} (R, W) := \mathbb {E} _ {P _ {x _ {2}}} \left[ \mathbb {1} _ {r, y} (R, Y) | R, W \right] (A.37) \\ = \mathbb {E} _ {P _ {\sigma (x _ {2})}} [ \mathbb {1} _ {r, y} (R, Y) | R, W, x _ {2} ] (A.38) \\ = \mathbb {1} _ {r} (R) \mathbb {E} _ {P _ {\sigma \left(X _ {2}\right)}} \left[ \mathbb {1} _ {y} (Y) \mid R, W, x _ {2} \right], (A.39) \\ \end{array}
$$
$$
\pi_ {1 3, 0} ^ {1} \left(X _ {2}, W\right) := \frac {P _ {\sigma \left(X _ {1}\right)} \left(R \mid x _ {2} , x _ {1}\right)}{P _ {\sigma \left(X _ {2}\right)} \left(R \mid x _ {2}\right)} \frac {\mathbb {1} _ {w} (W)}{P _ {\sigma \left(X _ {2}\right)} \left(W \mid R , x _ {2}\right)}. \tag {A.40}
$$
A.4.2 Construction of Estimators
We apply the procedure in Def. 3 to construct estimators  $\hat{A}^2$  and  $\hat{A}^{13}$  for  $A_0^2$  and  $A_0^{13}$ . We choose  $L = 2$ . We first construct  $\hat{A}^2$  for the fixed  $\{w, r, x_1\}$ . We note that  $\hat{A}^2 \coloneqq (1 / L)\sum_{\ell=1}^{L}\hat{A}_{\ell}^2$  for  $\ell \in \{1,2\}$  where  $\hat{A}_{\ell}^2$  is given as follows:
$$
\hat {A} _ {\ell} ^ {2} := \mathbb {E} _ {D _ {x _ {1}, \ell}} \left[ \pi_ {2, \ell} ^ {1} (R, X _ {2}) \left\{\mathbb {1} _ {w} (W) - \mu_ {2, \ell} ^ {2} (R, X _ {2}) \right\} + \mu_ {2, \ell} ^ {2} (r, X _ {2}) \right], \tag {A.41}
$$
where  $D_{x_1}$  is a subsample of  $D_{\sigma(X_1)}$  fixing  $X_1 = x_1$ , and  $\pi_{2,\ell}^1, \mu_{2,\ell}^2$  are nuisances trained using  $D_{x_1} \backslash D_{x_1,\ell}$ . We note that  $\mu_{2,\ell}^2(R, X_2)$  is constructed by regressing  $\mathbb{1}_w(W)$  onto  $\{R, X_2\}$ . Also,  $\pi_{2,\ell}^1(R, X_2)$  is constructed by regressing  $R$  onto  $X_2$ .
We now construct  $\hat{A}^{13} \coloneqq (1 / L)\sum_{\ell = 1}^{L}\hat{A}_{\ell}^{13}$  for  $\ell \in \{1,2\}$  where  $\hat{A}_{\ell}^{13}$  is given as follows:
$$
\hat {A} _ {\ell} ^ {1 3} := \mathbb {E} _ {D _ {x _ {2}, \ell}} \left[ \pi_ {1 3, \ell} ^ {1} (X _ {2}, W) \left\{\mathbb {1} _ {r, y} (R, Y) - \mu_ {1 3, \ell} ^ {2} (R, W) \right\} \right] + \mathbb {E} _ {\bar {D} _ {x _ {1}, \ell}} \left[ \mu_ {1 3, \ell} ^ {2} (R, w) \right\rbrace , \tag {A.42}
$$
where  $D_{x_1}$  is a subsample of  $D_{\sigma(X_1)}$  fixing  $X_1 = x_1$ , and  $\overline{D}_{x_1}$  is a subsample of  $D_{x_1}$  fixing  $X_2 = x_2$ .  $\pi_{2,\ell}^1, \mu_{2,\ell}^2$  are nuisances trained using  $\overline{D}_{x_1} \backslash \overline{D}_{x_1,\ell}$  and  $\overline{D}_{x_2} \backslash \overline{D}_{x_2,\ell}$ .
A.5 Details on Regression-based (REG) and Probability Weighting-based (PW) estimators.
In this section, we provide details on two alternative g-ID estimators used in Sec. 4:  $T^{\mathrm{reg}} \coloneqq f(\{\hat{A}^{k,\mathrm{reg}}\}_{k=1}^{K})$  ('regression-based estimators') where  $\hat{A}^{k,\mathrm{reg}}$  denotes the regression-based estimate
tor for the g-mSBD operator, and  $T^{\mathrm{pw}} = f(\{\hat{A}^{k,\mathrm{pw}}\}_{k=1}^{K})$  ('probability weighting-based estimators) where  $\hat{A}^{k,\mathrm{reg}}$  denotes the probability weighting-based estimator for the g-mSBD operator.
A.5.1 Regression-based Estimator
The regression-based g-mSBD estimator is defined as follows:
Definition A.1 (Regression-based g-mSBD Estimator). 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  $\mathbf{Z}_i = \mathbf{z}_i$ . A regression-based estimator  $\hat{A}^{\mathrm{reg}}$  for the g-mSBD adjustment  $A_0[\mathbf{W}, \mathbf{C}, \mathbf{R}; \mathbb{Z}_0 := \{\mathbf{Z}_i\}_{i=1}^m, \mathbf{seq} := (\mathbf{z}_i)_{i=1}^m](\mathbf{w} \backslash \mathbf{c}, \mathbf{r})$  is given 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  $i = m,\dots ,2$ . Define  $\check{\mu}_{\ell}^{i + 1}\coloneqq \mu_{\ell}^{i + 1}(\overline{\mathbf{W}}^i,\mathbf{r}_i,\overline{\mathbf{R}}^{1:i - 1})$
3. Estimate  $\hat{A}^{\mathrm{reg}}\coloneqq \hat{A}^{\mathrm{reg}}(\{\mu_{\ell}^{j + 1}\}_{j\in [m - 1],\ell \in [L]})\coloneqq (1 / L)\sum_{\ell = 1}^{L}\hat{A}_{\ell}^{\mathrm{reg}}(\{\mu_{\ell}^{j + 1}\}_{j\in [m - 1]})$  where
$$
\hat {A} _ {\ell} ^ {\text {r e g}} := \hat {A} _ {\ell} ^ {\text {r e g}} \left(\left\{\mu_ {\ell} ^ {j + 1} \right\} _ {j \in [ m - 1 ]}\right) := \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {1, \ell}}} \left[ \breve {\mu} _ {\ell} ^ {2} \right]. \tag {A.43}
$$
The error of the regression-based estimator is given as follows:
Proposition A.1 (Error Analysis of the regression-based g-mSBD estimator). Suppose  $\| \mu_{\ell}^{2} - \mu_0^2 \|_{P_{\sigma (\mathbf{Z}_1)}} = o_{P_{\sigma (\mathbf{Z}_1)}}(1)$ . Then,
$$
\hat {A} ^ {r e g} - A _ {0} = O _ {P _ {\sigma \left(\mathbf {z} _ {1}\right)}} \left(n _ {1} ^ {- 1 / 2}\right) + \frac {1}{L} \sum_ {\ell = 1} ^ {L} O _ {P _ {\sigma \left(\mathbf {z} _ {1}\right)}} \left(\| \mu_ {\ell} ^ {2} - \mu_ {0} ^ {2} \|\right). \tag {A.44}
$$
Proof of Proposition A.1. We note that
$$
A _ {0} = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {1})}} \left[ \check {\mu} _ {0} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right] \tag {A.45}
$$
by the analysis in Lemma S.2. Therefore, by Lemma S.7,
$$
\hat {A} _ {\ell} ^ {\text {r e g}} - A _ {0} = \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {1}, \ell} - P _ {\sigma (\mathbf {Z} _ {1}) | \mathbf {r} _ {0}, \mathbf {z} _ {1}}} \left[ \check {\mu} _ {0} ^ {2} \right] \tag {A.46}
$$
$$
+ \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {1}, \ell - P _ {\sigma (\mathbf {Z} _ {1})} | \mathbf {r} _ {0}, \mathbf {z} _ {1}}} \left[ \breve {\mu} _ {\ell} ^ {2} - \breve {\mu} _ {0} ^ {2} \right] \tag {A.47}
$$
$$
+ \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {1})}} \left[ \check {\mu} _ {\ell} ^ {2} - \check {\mu} _ {0} ^ {2} | \mathbf {r} _ {0}, \mathbf {z} _ {1} \right]. \tag {A.48}
$$
By the central limit theorem,
$$
\text {E q .} (A. 4 6) = O _ {P _ {\sigma \left(\mathbf {z} _ {1}\right)}} \left(n _ {1, \ell} ^ {- 1 / 2}\right), \tag {A.49}
$$
where  $n_{1,\ell}\coloneqq |\overline{D}_{\mathbf{z}_1,\ell}|$
By [Kennedy et al., 2020, Lemma 2] and the given assumption that  $\| \mu_{\ell}^{2} - \mu_{0}^{2}\|_{P_{\sigma (Z_{1})}} = o_{P_{\sigma (Z_{1})}}(1)$ ,
$$
\text {E q .} (A. 4 7) = O _ {P _ {\sigma \left(\mathbf {Z} _ {1}\right)}} \left(1 / n _ {1, \ell} ^ {- 1 / 2}\right). \tag {A.50}
$$
Finally, by applying Cauchy-Schwarz inequality,
$$
\text {E q .} (A. 4 8) = O _ {P _ {\sigma \left(\mathbf {z} _ {1}\right)}} \left(\| \check {\mu} _ {\ell} ^ {2} - \check {\mu} _ {0} ^ {2} \|\right). \tag {A.51}
$$
$$
\begin{array}{l} \hat {A} ^ {\text {r e g}} - A _ {0} = \frac {1}{L} \sum_ {\ell = 1} ^ {L} \left(\hat {A} _ {\ell} ^ {\text {r e g}} - A _ {0}\right) (A.52) \\ = \frac {1}{L} \sum_ {\ell = 1} ^ {L} \left(O _ {P _ {\sigma (\mathbf {z} _ {1})}} \left(n ^ {- 1 / 2 _ {1, \ell}}\right) + O _ {P _ {\sigma (\mathbf {z} _ {1})}} \left(\| \check {\mu} _ {\ell} ^ {2} - \check {\mu} _ {0} ^ {2} \|\right)\right) (A.53) \\ = O _ {P _ {\sigma \left(\mathbf {z} _ {1}\right)}} \left(n ^ {- 1 / 2 _ {1}}\right) + \frac {1}{L} \sum_ {\ell = 1} ^ {L} O _ {P _ {\sigma \left(\mathbf {z} _ {1}\right)}} \left(\| \mu_ {\ell} ^ {2} - \mu_ {0} ^ {2} \|\right). (A.54) \\ \end{array}
$$
A.5.2 Probability-weighting based Estimator
In this section, we define and analyze the probability weighting-based g-mSBD estimator. The probability-weighting-based estimator is defined as follows:
Definition A.2 (Probability-weighting-based g-mSBD Estimator). 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  $\mathbf{Z}_i = \mathbf{z}_i$ . A probability weighting-based estimator  $\hat{A}^{\mathrm{pw}}$  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 given 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  $\pi_{\ell}^{i}$  denote learned  $\pi_0^i$  using  $\overline{D}_{\mathbf{z}_i}\backslash \overline{D}_{\mathbf{z}_i,\ell}$  for  $i = m - 1,\dots ,1$ . Let  $\overline{\pi}_{\ell}^{m - 1}:= \prod_{i = 1}^{m - 1}\pi_{\ell}^{i}$ .
3. Estimate  $\hat{A}^{\mathrm{pw}}\coloneqq \hat{A}^{\mathrm{pw}}(\{\pi_{\ell}^{j}\}_{j\in [m - 1],\ell \in [L]})\coloneqq (1 / L)\sum_{\ell = 1}^{L}\hat{A}_{\ell}^{\mathrm{pw}}(\{\pi_{\ell}^{j}\}_{j\in [m - 1]})$  where
$$
\hat {A} _ {\ell} ^ {\mathrm {p w}} := \hat {A} _ {\ell} ^ {\mathrm {p w}} \left(\left\{\pi_ {\ell} ^ {j} \right\} _ {j \in [ m - 1 ]}\right) := \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {m, \ell}}} \left[ \bar {\pi} _ {\ell} ^ {m - 1} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} (\mathbf {W} \backslash \mathbf {C}) \right]. \tag {A.55}
$$
Lemma S.1 (Representation of the g-mSBD operator using Probability Weighting). The  $g$ -mSBD adjustment  $A_0$  in Def. 1 can be represented as
$$
A _ {0} = \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {m})}} \left[ \bar {\pi} _ {0} ^ {m - 1} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} (\mathbf {W} \backslash \mathbf {C}) | \mathbf {z} _ {m}, \mathbf {r} _ {0} \right]. \tag {A.56}
$$
Proof of Lemma S.1. It suffices to show that, for  $k = m - 1,\dots ,1$
$$
\mathbb {E} _ {P _ {\sigma \left(\mathbf {z} _ {k + 1}\right)}} \left[ \bar {\pi} _ {0} ^ {k} \breve {\mu} _ {0} ^ {k + 2} \mid \mathbf {z} _ {k + 1}, \mathbf {r} _ {0} \right] = \mathbb {E} _ {P _ {\sigma \left(\mathbf {z} _ {k}\right)}} \left[ \bar {\pi} _ {0} ^ {k} \breve {\mu} _ {0} ^ {k + 1} \mid \mathbf {z} _ {k}, \mathbf {r} _ {0} \right]. \tag {A.57}
$$
If this holds, Lemma S.1 can be shown as follows:
$$
\begin{array}{l} A _ {0} = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {1})}} \left[ \breve {\mu} _ {0} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {2})}} \left[ \overline {{\boldsymbol {\pi}}} _ {0} ^ {1} \breve {\mu} _ {0} ^ {3} | \mathbf {z} _ {2}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {m})}} \left[ \overline {{\boldsymbol {\pi}}} _ {0} ^ {m - 1} \breve {\mu} _ {0} ^ {m + 1} | \mathbf {z} _ {m}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {m})}} \left[ \overline {{\pi}} _ {0} ^ {m - 1} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} (\mathbf {W} \backslash \mathbf {C}) | \mathbf {z} _ {m}, \mathbf {r} _ {0} \right]. \\ \end{array}
$$
Eq. (A.57) holds as follows:
$$
\begin{array}{l} \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {k + 1})}} \left[ \overline {{\boldsymbol {\pi}}} _ {0} ^ {k} \breve {\boldsymbol {\mu}} _ {0} ^ {k + 2} | \mathbf {z} _ {k + 1}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {k + 1})}} \left[ \overline {{\pi}} _ {0} ^ {k} \mu_ {0} ^ {k + 1} | \mathbf {z} _ {k + 1}, \mathbf {r} _ {0} \right] \\ = \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {k})}} \left[ \overline {{\pi}} _ {0} ^ {k - 1} \breve {\mu} _ {0} ^ {k + 1} | \mathbf {z} _ {k}, \mathbf {r} _ {0} \right]. \\ \end{array}
$$
Equipped with Lemma S.1, we analyze the error of the probability-weighting-based estimator as follow:
Proposition A.2 (Error Analysis of the probability weighting-based g-mSBD estimator). Suppose  $\left\| \{\pi_{\ell}^{1}\tau_{\ell}^{2} - \pi_{0}^{1}\tau_{0}^{2}\} \right\|_{P_{\sigma (\mathbf{Z}_2)}} = o_{P_{\sigma (\mathbf{Z}_2)}}(1)$ . Then,
$$
\hat {A} ^ {p w} - A _ {0} = O _ {P _ {\sigma (\mathbf {z}, \cdot)}} \left(n _ {m} ^ {- 1 / 2}\right) + \frac {1}{L} \sum_ {\ell = 1} ^ {L} O _ {P _ {\sigma (\mathbf {z} _ {m})}} \left(\| \bar {\pi} _ {\ell} ^ {m} - \bar {\pi} _ {0} ^ {m} \|\right). \tag {A.58}
$$
Proof of Proposition A.2. By Lemma S.7 and Assumption 1,
$$
\begin{array}{l} \hat {A} _ {\ell} ^ {\mathrm {p w}} - A _ {0} = \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {m}, \ell} - P _ {\sigma (\mathbf {Z} _ {m}) | \mathbf {z} _ {m}, \mathbf {r} _ {0}}} \left[ \bar {\pi} _ {0} ^ {m} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} (\mathbf {W} \backslash \mathbf {C}) \right] (A.59) \\ + \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {m}, \ell} - P _ {\sigma (\mathbf {Z} _ {m}) | \mathbf {z} _ {m}, \mathbf {r} _ {0}}} \left[ \left(\bar {\pi} ^ {m} - \bar {\pi} _ {0} ^ {m}\right) \mathbb {I} _ {\mathbf {w} \backslash \mathbf {c}} (\mathbf {W} \backslash \mathbf {C}) \right] (A.60) \\ + \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {m}) | \mathbf {z} _ {m}, \mathbf {r} _ {0}}} \left[ \left(\bar {\pi} ^ {m} - \bar {\pi} _ {0} ^ {m}\right) \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} (\mathbf {W} \backslash \mathbf {C}) \right]. (A.61) \\ \end{array}
$$
$$
\text {E q .} (A. 5 9) = O _ {P _ {\sigma \left(\mathbf {Z} _ {m}\right)}} \left(n _ {m, \ell} ^ {- 1 / 2}\right). \tag {A.62}
$$
By [Kennedy et al., 2020, Lemma 2] and the given assumption,
$$
\text {E q .} (A. 6 0) = O _ {P _ {\sigma \left(\mathbf {Z} _ {m}\right)}} \left(n _ {m, \ell} ^ {- 1 / 2}\right). \tag {A.63}
$$
$$
\text {E q .} (A. 6 1) = O _ {P _ {\sigma \left(\mathbf {Z} _ {m}\right)}} \left(\left\| \left\{\bar {\pi} ^ {m} - \bar {\pi} _ {0} ^ {m} \right\} \right\}\right). \tag {A.64}
$$
Lemma 1 (c-component Identification [Jung et al., 2021b]). Let  $\mathbf{S}$  denote a c-component in  $G_{i}:= G(\mathbf{V}\backslash \mathbf{Z}_{i})$  for some  $\mathbf{Z}_i\in \mathbb{Z}$ . Let  $\mathbf{R}:= pa(\mathbf{S})_{G_i}\backslash \mathbf{S}$ . Let  $(\mathbf{S},\mathbf{R})$  be ordered as  $(\mathbf{R}_0,S_1,\dots ,\mathbf{R}_{m - 1},S_m)$  by  $\prec_{G}$ . Let  $\mathbf{A}\subseteq \mathbf{S}$  denote a set satisfying  $\mathbf{A} = an(\mathbf{A})_{G_i(\mathbf{S})}$ . Let  $\mathbf{C}:= (\mathbf{S}\backslash \mathbf{A})$ . Let  $\mathbb{Z}_0\coloneqq \{\mathbf{Z}_i\}$  and  $\operatorname {seq}(\mathbb{Z}_0)$  be a sequence of  $\mathbf{z}_i$  repeating  $m$  times. Then, the  $c$ -factor  $Q[\mathbf{A}]$  is  $g$ -identifiable as follows:
Proof of Lemma 1. Let  $\mathbf{C}_0\coloneqq \mathrm{pre}(\mathbf{A};G(\mathbf{S}))\cap \mathbf{C}$ . Let  $\mathbf{C}_1\coloneqq \mathbf{C}\backslash \mathbf{C}_0$ . We first note that, by [Jung et al., 2021b, Lemma 1],
$$
Q [ \mathbf {A} ] = \sum_ {\mathbf {c} _ {0} \in \mathbf {S} _ {\mathbf {C} _ {0}}} \prod_ {j: V _ {j} \in \mathbf {A} \cup \mathbf {C} _ {0}} P _ {\mathbf {z} _ {i}} \left(v _ {j} | \overline {{\mathbf {s}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {i}\right). \tag {B.1}
$$
Therefore, it suffices to show that
$$
\text {E q .} (B. 1) = A _ {0} [ \mathbf {S}, \mathbf {R}; \mathbb {Z} _ {0} := \{\mathbf {Z} _ {i} \}, \operatorname {s e q} ] (\mathbf {s} \backslash \mathbf {c}, \mathbf {r}). \tag {B.2}
$$
It holds as follows:
$$
\begin{array}{l} A _ {0} [ \mathbf {S}, \mathbf {R}; \mathbb {Z} _ {0} := \left\{\mathbf {Z} _ {i} \right\}, \operatorname {s e q} ] (\mathbf {s} \backslash \mathbf {c}, \mathbf {r}) = \sum_ {\mathbf {c} \in \mathfrak {S} _ {\mathbf {C}}} \prod_ {V _ {j} \in \mathbf {S}} P _ {\mathbf {z} _ {i}} \left(v _ {j} | \overline {{\mathbf {s}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {i}\right) (B.3) \\ = \sum_ {\mathbf {c} _ {0} \in \mathfrak {S} _ {\mathbf {C} _ {0}}} \sum_ {\mathbf {c} _ {1} \in \mathfrak {S} _ {\mathbf {C} _ {1}}} \prod_ {V _ {j} \in \mathbf {S}} P _ {\mathbf {z} _ {i}} \left(v _ {j} | \overline {{\mathbf {s}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {i}\right) (B.4) \\ = \sum_ {\mathbf {c} _ {0} \in \mathfrak {S} _ {\mathbf {C} _ {0}}} \prod_ {V _ {j} \in \mathbf {S} \backslash \mathbf {C} _ {1}} P _ {\mathbf {z} _ {i}} \left(v _ {j} | \overline {{\mathbf {s}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {i}\right) (B.5) \\ = \sum_ {\mathbf {c} _ {0} \in \mathfrak {S} _ {\mathbf {C} _ {0}}} \prod_ {V _ {j} \in \mathbf {A} \cup \mathbf {C} _ {0}} P _ {\mathbf {z} _ {i}} \left(v _ {j} | \overline {{\mathbf {s}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {i}\right) (B.6) \\ = \text {E q .} (\mathrm {B}. 1). (B.7) \\ \end{array}
$$
We note that the third equation holds since the all  $P_{\mathbf{z}_i}(v_j|\overline{\mathbf{s}}^{j - 1},\overline{\mathbf{r}}^{j - 1}\backslash \mathbf{z}_i)$  will be marginalized out if  $V_{j}\in \mathbf{C}_{1}$ . The fourth equation holds since  $\mathbf{S}:= \mathbf{A}\cup \mathbf{C}_0\cup \mathbf{C}_1$ , which implies that  $\mathbf{S}\backslash \mathbf{C}_1 = \mathbf{A}\cup \mathbf{C}_0$ .
Proof of Lemma 2. Let  $\mathbf{W}_{\mathrm{mar}}^c \coloneqq \{\mathbf{W}_0, \mathbf{C}\} \backslash \mathbf{W}_{\mathrm{mar}}$ . We note that
$$
\begin{array}{l} \sum_ {\mathbf {w} _ {0} \in \mathfrak {S} _ {W _ {0}}} A _ {0} [ \mathbf {W}, \mathbf {C}, \mathbf {R}; \mathbb {Z} _ {0}, \operatorname {s e q} ] (\mathbf {w} \backslash \mathbf {c}, \mathbf {r}) (B.8) \\ = \sum_ {\mathbf {w} _ {0}, \mathbf {c} _ {0} \in \mathfrak {S} _ {\mathbf {W} _ {0}}, \mathbf {c} _ {0}} \prod_ {j: W _ {j} \in \mathbf {W}} P _ {\mathbf {z} _ {j}} \left(w _ {j} \mid \overline {{\mathbf {w}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {j}\right) (B.9) \\ = \sum_ {\mathbf {w} _ {\operatorname {m a r}} ^ {c} \mathfrak {S} _ {\mathbf {W} _ {\operatorname {m a r}} ^ {c}}} \sum_ {\mathbf {w} _ {\operatorname {m a r}} \mathfrak {S} _ {\mathbf {W} _ {\operatorname {m a r}}}} \prod_ {i = k + 1} ^ {m} P _ {\mathbf {z} _ {i}} \left(w _ {i} | \overline {{\mathbf {w}}} ^ {i - 1}, \overline {{\mathbf {r}}} ^ {i - 1} \backslash \mathbf {z} _ {i}\right) P _ {\mathbf {z} _ {j}} \left(w _ {j}, \dots , w _ {k} | \overline {{\mathbf {w}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {j}\right) \prod_ {\ell = 1} ^ {j - 1} P _ {\mathbf {z} _ {\ell}} \left(w _ {\ell} | \overline {{\mathbf {w}}} ^ {\ell - 1}, \overline {{\mathbf {r}}} ^ {\ell - 1} \backslash \mathbf {z} _ {\ell}\right) (B.10) \\ = \sum_ {\mathbf {c} ^ {\prime} \in \mathfrak {S} _ {\mathbf {C} ^ {\prime}}} P _ {\mathbf {z} _ {j}} \left(\left\{w _ {j}, \dots , w _ {k} \right\} \backslash \mathbf {w} _ {\operatorname {m a r}} \mid \overline {{\mathbf {w}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {j}\right) \prod_ {\ell = 1} ^ {j - 1} P _ {\mathbf {z} _ {\ell}} \left(w _ {\ell} \mid \overline {{\mathbf {w}}} ^ {\ell - 1}, \overline {{\mathbf {r}}} ^ {\ell - 1} \backslash \mathbf {z} _ {\ell}\right) (B.11) \\ = \sum_ {\mathbf {c} ^ {\prime} \in \mathfrak {S} _ {\mathbf {C} ^ {\prime}}} \prod_ {W _ {j} \in \mathbf {W} \backslash \mathbf {W} _ {\operatorname {m a r}}} P _ {\mathbf {z} _ {j}} \left(w _ {j} | \overline {{\mathbf {w}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {j}\right) (B.12) \\ = A _ {0} \left[ \mathbf {W} ^ {\prime}, \mathbf {C} ^ {\prime}, \mathbf {R} ^ {\prime}; \mathbb {Z} ^ {\prime}, \operatorname {s e q} ^ {\prime} \right] \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}, \mathbf {r} ^ {\prime}\right). (B.13) \\ \end{array}
$$
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