SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \hat {A} _ {\ell} - A _ {0} \\ = \sum_ {i = 1} ^ {m - 1} \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {i + 1}, \ell} - P _ {\sigma (\mathbf {Z} _ {i + 1}) | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0}}} \left[ \bar {\pi} _ {0} ^ {i} \left\{\check {\mu} _ {0} ^ {i + 2} - \mu_ {0} ^ {i + 1} \right\} \right] + \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {1, \ell}} - P _ {\sigma (\mathbf {Z} _ {1}) | \mathbf {z} _ {1}, \mathbf {r} _ {0}}} \left[ \check {\mu} _ {0} ^ {2} \right] (B.79) \\ + \sum_ {i = 1} ^ {m - 1} \mathbb {E} _ {\overline {{D}} _ {\mathbf {z} _ {i + 1}, \ell} - P _ {\sigma (\mathbf {Z} _ {i + 1}) | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0}}} \left[ \bar {\pi} _ {\ell} ^ {i} \{\check {\mu} _ {\ell} ^ {i + 2} - \mu_ {\ell} ^ {i + 1} \} - \bar {\pi} _ {0} ^ {i} \{\check {\mu} _ {0} ^ {i + 2} - \mu_ {0} ^ {i + 1} \} \right] \\ + \mathbb {E} _ {\overline {{D}} _ {\mathbf {z} _ {1, \ell}} - P _ {\sigma (\mathbf {z} _ {1}) | \mathbf {z} _ {1}, \mathbf {r} _ {0}}} \left[ \breve {\mu} _ {\ell} ^ {2} - \breve {\mu} _ {0} ^ {2} \right] (B.80) \\ + \sum_ {i = 1} ^ {m - 1} \mathbb {E} _ {P _ {\sigma} \left(\mathbf {z} _ {i + 1}\right)} \left[ \bar {\pi} _ {\ell} ^ {i} \left\{\check {\mu} _ {\ell} ^ {i + 2} - \mu_ {\ell} ^ {i + 1} \right\} - \bar {\pi} _ {0} ^ {i} \left\{\check {\mu} _ {0} ^ {i + 2} - \mu_ {0} ^ {i + 1} \right\} \mid \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma} \left(\mathbf {z} _ {1}\right)} \left[ \check {\mu} _ {\ell} ^ {2} - \check {\mu} _ {0} ^ {2} \mid \mathbf {z} _ {1}, \mathbf {r} _ {0} \right]. (B.81) \\ \end{array}
$$
$$
R _ {1, \ell} ^ {a} := \mathbb {E} _ {\overline {{D}} _ {\mathbf {z} _ {1}, \ell} - P _ {\sigma (\mathbf {Z} _ {1}) | \mathbf {z} _ {1}}, \mathbf {r} _ {0}} \left[ \breve {\mu} _ {0} ^ {2} \right] \tag {B.82}
$$
and for  $i = 1,\dots ,m - 1$
$$
R _ {i + 1, \ell} ^ {a} := \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {i + 1}, \ell - P _ {\sigma (\mathbf {Z} _ {i + 1}) | \mathbf {z} _ {i + 1}}, \mathbf {r} _ {0}}} \left[ \bar {\pi} _ {0} ^ {i} \left\{\tilde {\mu} _ {0} ^ {i + 2} - \mu_ {0} ^ {i + 1} \right\} \right]. \tag {B.83}
$$
By the central limit theorem, we note that  $R_{i,\ell}^{a}$  for  $i = 1,\dots ,m$  is a random variable such that  $n_{i,\ell}^{1 / 2}R_{i,\ell}^{a}$  converges in distribution to a mean-zero normal random variable. Therefore,
$$
\text {E q .} (\mathrm {B}. 7 9) = \sum_ {i = 1} ^ {m} R _ {i, \ell} ^ {a}, \tag {B.84}
$$
also behaves the same.
We now analyze the second term. Define
$$
R _ {1, \ell} ^ {b} := \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {1}, \ell} - P _ {\sigma (\mathbf {z} _ {1}) | \mathbf {z} _ {1}}, \mathbf {r} _ {0}} \left[ \tilde {\mu} _ {\ell} ^ {2} - \check {\mu} _ {0} ^ {2} \right], \tag {B.85}
$$
and for  $i = 1,2,\dots ,m - 1$
$$
R _ {i + 1, \ell} ^ {b} := \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {i + 1}, \ell} - P _ {\sigma (\mathbf {Z} _ {i + 1}) | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0}}} \left[ \bar {\pi} _ {\ell} ^ {i} \left\{\check {\mu} _ {\ell} ^ {i + 2} - \mu_ {\ell} ^ {i + 1} \right\} - \bar {\pi} _ {0} ^ {i} \left\{\check {\mu} _ {0} ^ {i + 2} - \mu_ {0} ^ {i + 1} \right\} \right]. \tag {B.86}
$$
By [Kennedy et al., 2020, Lemma 2] and the continuous mapping theorem in Lemma S.8,
$$
R _ {1, \ell} ^ {b} = O _ {P _ {\sigma (\mathbf {Z} _ {1})}} \left(\frac {\| \breve {\mu} _ {\ell} ^ {2} - \breve {\mu} _ {0} ^ {2} \|}{\sqrt {n _ {1 , \ell}}}\right), \tag {B.87}
$$
$$
R _ {i + 1, \ell} ^ {b} = O _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left(\frac {\| \| \overline {{\pi}} _ {\ell} ^ {i} \{\tilde {\mu} _ {\ell} ^ {i + 2} - \mu_ {\ell} ^ {i + 1} \} - \bar {\pi} _ {0} ^ {i} \{\tilde {\mu} _ {0} ^ {i + 2} - \mu_ {0} ^ {i + 1} \} \| \|}{\sqrt {n _ {i + 1 , \ell}}}\right). \tag {B.88}
$$
Under the given assumption, for  $i = 1,2,\dots ,m$
$$
R _ {i, \ell} ^ {b} = o _ {P _ {\sigma (\mathbf {Z} _ {i})}} (1), \tag {B.89}
$$
$$
\text {E q .} (\mathrm {B}. 8 0) = \sum_ {i = 1} ^ {m} o _ {P _ {\sigma \left(\mathbf {Z} _ {i}\right)}} (1). \tag {B.90}
$$
$$
R _ {i, \ell} := R _ {i, \ell} ^ {a} + R _ {i, \ell} ^ {b}. \tag {B.91}
$$
Then,  $R_{i,\ell}$  is also a random variable such that  $n_{i,\ell}^{1/2} R_{i,\ell}$  converges in distribution to a mean-zero normal random variable, by Slutsky's theorem.
We now analyze the third term. By Lemma S.6, the third term can be analyzed as follow:
$$
\text {E q .} \left(\mathrm {B}. 8 1\right) = \sum_ {i = 1} ^ {m - 1} O _ {P _ {\sigma \left(\mathbf {Z} _ {i + 1}\right)}} \left(\| \mu_ {\ell} ^ {i + 1} - \mu_ {0} ^ {i + 1} \| \| \pi_ {\ell} ^ {i} - \pi_ {0} ^ {i} \|\right). \tag {B.92}
$$
$$
\begin{array}{l} \hat {A} _ {\ell} - A _ {0} = \text {E q .} (\mathrm {B}. 7 9) + \text {E q .} (\mathrm {B}. 8 0) + \text {E q .} (\mathrm {B}. 8 1) (B.93) \\ = \sum_ {i = 1} ^ {m} R _ {i, \ell} + \sum_ {i = 1} ^ {m - 1} O _ {P _ {\sigma \left(\mathbf {Z} _ {i + 1}\right)}} \left(\| \mu_ {\ell} ^ {i + 1} - \mu_ {0} ^ {i + 1} \| \| \pi_ {\ell} ^ {i} - \pi_ {0} ^ {i} \|\right). (B.94) \\ \end{array}
$$
Define  $R_{i} \coloneqq (1 / L)\sum_{\ell = 1}^{L}R_{i,\ell}$ . We note that  $R_{i}$  is a random variable such that  $n_i^{1 / 2}R_i$  converges in distribution to a mean-zero normal random variable. Then,
$$
\begin{array}{l} \hat {A} - A _ {0} = \frac {1}{L} \sum_ {\ell = 1} ^ {L} \left\{\hat {A} _ {\ell} - A _ {0} \right\} (B.95) \\ = \sum_ {i = 1} ^ {m} R _ {i} + \frac {1}{L} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m - 1} O _ {P _ {\sigma (\mathbf {Z} _ {i + 1})}} \left(\| \pi_ {\ell} ^ {i} - \pi_ {0} ^ {i} \| \| \mu_ {\ell} ^ {i + 1} - \mu_ {0} ^ {i + 1} \|\right). (B.96) \\ \end{array}
$$
B.7 Proof of Theorem 2
We first restate the definition of the MR-gID estimator, the theorem, and its corresponding assumptions.
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,
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  $\| \check{\mu}_{k,\ell}^{2} - \check{\mu}_{k,0}^{2}\|_{P_{\sigma (\mathbf{Z}_1^k)}} = o_{P_{\sigma (\mathbf{Z}_1^k)}}(1).$
$$
\hat {\psi} - \psi_ {0} = \sum_ {k = 1} ^ {K} \sum_ {i = 1} ^ {m ^ {k}} O _ {P _ {\sigma (\mathbf {Z} _ {i} ^ {k})}} \left(n _ {k, i} ^ {- 1 / 2}\right) + \frac {1}{L} \sum_ {k = 1} ^ {K} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m ^ {k} - 1} O _ {P _ {\sigma (\mathbf {z} _ {i} ^ {k})}} \left(\| \mu_ {k, \ell} ^ {i + 1} - \mu_ {k, 0} ^ {i + 1} \| \| \pi_ {k, \ell} ^ {i} - \pi_ {k, 0} ^ {i} \|\right). \tag {12}
$$
Proof of Theorem 2. We first define the following notation. For a map  $g(x)$ , we will use  $\nabla_{x}g(x_{0})[h] := \lim_{t\to 0}(g(x_{0} + th) - g(x_{0})) / t$ . We first note that by the definition of Fréchet Derivative-based Taylor expansion [Blanchard and Brüning, 2015, Def. 34.1] and the given assumption ('Twice differentiability'), the error  $\hat{\psi} - \psi_0$  can be represented as follow:
$$
\hat {\psi} - \psi_ {0} = \sum_ {k = 1} ^ {K} \nabla_ {A ^ {k}} f \left(\left\{A _ {0} ^ {j} \right\} _ {j = 1} ^ {K}\right) \left[ \hat {A} ^ {k} - A _ {0} ^ {k} \right] + o \left(\hat {A} ^ {k} - A _ {0} ^ {k}\right), \tag {B.97}
$$
where, by Prop. 1,
$$
\hat {A} ^ {k} - A _ {0} ^ {k} = \sum_ {i = 1} ^ {m ^ {k}} O _ {P _ {\sigma (\mathbf {Z} _ {i} ^ {k})}} \left(n _ {k, i} ^ {- 1 / 2}\right) + \frac {1}{L} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m ^ {k} - 1} O _ {P _ {\sigma (\mathbf {Z} _ {i + 1} ^ {k})}} \left(\| \mu_ {k, \ell} ^ {i + 1} - \mu_ {k, 0} ^ {i + 1} \| \| \pi_ {k, \ell} ^ {i} - \pi_ {k, 0} ^ {i} \|\right). \tag {B.98}
$$
Therefore, by the big O in probability calculus [Van der Vaart, 2000, Chap. 2],
$$
\begin{array}{l} o \left(\hat {A} ^ {k} - A _ {0} ^ {k}\right) = o \left(\sum_ {i = 1} ^ {m ^ {k}} O _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} \left(n _ {k, i} ^ {- 1 / 2}\right) + \frac {1}{L} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m ^ {k} - 1} O _ {\bar {P} _ {\mathbf {z} _ {i + 1} ^ {k}}} \left(\| \mu_ {k, \ell} ^ {i + 1} - \mu_ {k, 0} ^ {i + 1} \| \| \pi_ {k, \ell} ^ {i} - \pi_ {k, 0} ^ {i} \|\right).\right) (B.99) \\ = \sum_ {i = 1} ^ {m ^ {k}} o _ {P _ {\sigma (\mathbf {Z} _ {i} ^ {k})}} \left(n _ {k, i} ^ {- 1 / 2}\right) + \frac {1}{L} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m ^ {k} - 1} o _ {P _ {\sigma (\mathbf {Z} _ {i + 1} ^ {k})}} \left(\left\| \mu_ {k, \ell} ^ {i + 1} - \mu_ {k, 0} ^ {i + 1} \right\| \left\| \pi_ {k, \ell} ^ {i} - \pi_ {k, 0} ^ {i} \right\|\right). (B.100) \\ \end{array}
$$
By applying the given assumption ('Boundedness'), we have the following:
$$
\begin{array}{l} \nabla_ {A ^ {k}} f (\{A _ {0} ^ {j} \} _ {j = 1} ^ {m}) [ \hat {A} ^ {k} - A _ {0} ^ {k} ] = \sum_ {i = 1} ^ {m ^ {k}} O _ {P _ {\sigma (\mathbf {Z} _ {i} ^ {k})}} (n _ {k, i} ^ {- 1 / 2}) \\ + \frac {1}{L} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m ^ {k} - 1} O _ {P _ {\sigma \left(\mathbf {Z} _ {i + 1} ^ {k}\right)}} \left(\| \mu_ {k, \ell} ^ {i + 1} - \mu_ {k, 0} ^ {i + 1} \| \| \pi_ {k, \ell} ^ {i} - \pi_ {k, 0} ^ {i} \|\right). \tag {B.101} \\ \end{array}
$$
$$
\begin{array}{l} \hat {\psi} - \psi_ {0} = \sum_ {k = 1} ^ {K} \nabla_ {A ^ {k}} f \left(\left\{A _ {0} ^ {j} \right\} _ {j = 1} ^ {K}\right) \left[ \hat {A} ^ {k} - A _ {0} ^ {k} \right] + o \left(\hat {A} ^ {k} - A _ {0} ^ {k}\right) (B.102) \\ = \sum_ {k = 1} ^ {K} \sum_ {i = 1} ^ {m ^ {k}} O _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} \left(n _ {k, i} ^ {- 1 / 2}\right) + \frac {1}{L} \sum_ {k = 1} ^ {K} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m ^ {k} - 1} O _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} \left(\| \mu_ {k, \ell} ^ {i + 1} - \mu_ {k, 0} ^ {i + 1} \| \| \pi_ {k, \ell} ^ {i} - \pi_ {k, 0} ^ {i} \|\right) (B.103) \\ + \sum_ {k = 1} ^ {K} \sum_ {i = 1} ^ {m ^ {k}} o _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} \left(n _ {k, i} ^ {- 1 / 2}\right) + \frac {1}{L} \sum_ {k = 1} ^ {K} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m ^ {k} - 1} o _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} \left(\| \mu_ {k, \ell} ^ {i + 1} - \mu_ {k, 0} ^ {i + 1} \| \| \pi_ {k, \ell} ^ {i} - \pi_ {k, 0} ^ {i} \|\right) (B.104) \\ = \sum_ {k = 1} ^ {K} \sum_ {i = 1} ^ {m ^ {k}} O _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} \left(n _ {k, i} ^ {- 1 / 2}\right) + \frac {1}{L} \sum_ {k = 1} ^ {K} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m ^ {k} - 1} O _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} \left(\| \mu_ {k, \ell} ^ {i + 1} - \mu_ {k, 0} ^ {i + 1} \| \| \pi_ {k, \ell} ^ {i} - \pi_ {k, 0} ^ {i} \|\right). (B.105) \\ \end{array}
$$
B.8 Proof of Corollary 2
Proof of Corollary 2. We first note that  $f$  is a continuous function under the twice differentiability condition in Assumption 2. Suppose each  $\hat{A}^k$  is a consistent estimator of  $A_0^k$  under given conditions. Then, by the continuous mapping theorem,  $f(\{\hat{A}^k\}_{k=1}^K)$  is consistent to  $P_{\mathbf{x}}(\mathbf{y}) = f(\{A_0^k\}_{k=1}^K)$ . Therefore, it suffices to show that each  $\hat{A}^k$  is a consistent estimator of  $A_0^k$ .
We first recall that  $\hat{A}^k \coloneqq (1 / L)\sum_{\ell = 1}^{L}\hat{A}_\ell^k$  by Def. 4. By applying Lemma S.7,  $\hat{A}_\ell^k - A_0^k$  can be rewritten as follow:
$$
\begin{array}{l} \hat {A} _ {\ell} ^ {k} - A _ {0} \\ = \sum_ {i = 1} ^ {m - 1} \mathbb {E} _ {\overline {{\mathbf {Z}}} _ {\mathbf {z} _ {i + 1} ^ {k}, \ell} ^ {k} - P _ {\sigma (\mathbf {Z} _ {i + 1} ^ {k}) | \mathbf {z} _ {i + 1} ^ {k}, \mathbf {r} _ {0}}} \left[ \bar {\pi} _ {k, 0} ^ {i} \left\{\tilde {\mu} _ {k, 0} ^ {i + 2} - \mu_ {k, 0} ^ {i + 1} \right\} \right] + \mathbb {E} _ {\overline {{\mathbf {Z}}} _ {\mathbf {z} _ {1, \ell} ^ {k}} ^ {k} - P _ {\sigma (\mathbf {Z} _ {1} ^ {k}) | \mathbf {z} _ {1} ^ {k}, \mathbf {r} _ {0}} \left[ \tilde {\mu} _ {k, 0} ^ {2} \right]} (B.106) \\ + \sum_ {i = 1} ^ {m - 1} \mathbb {E} _ {\overline {{\mathbf {D}}} _ {\mathbf {z} _ {i + 1} ^ {k}, \ell} ^ {k} - P _ {\sigma (\mathbf {Z} _ {i + 1} ^ {k}) | \mathbf {z} _ {i + 1} ^ {k}, \mathbf {r} _ {0}}} \left[ \bar {\pi} _ {k, \ell} ^ {i} \{\check {\mu} _ {k, \ell} ^ {i + 2} - \mu_ {k, \ell} ^ {i + 1} \} - \bar {\pi} _ {k, 0} ^ {i} \{\check {\mu} _ {k, 0} ^ {i + 2} - \mu_ {k, 0} ^ {i + 1} \} \right] \\ + \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {1, \ell} ^ {k}} - P _ {\sigma (\mathbf {z} _ {1} ^ {k}) | \mathbf {z} _ {1} ^ {k}}, \mathbf {r} _ {0}} \left[ \check {\mu} _ {k, \ell} ^ {2} - \check {\mu} _ {k, 0} ^ {2} \right] (B.107) \\ + \sum_ {i = 1} ^ {m - 1} \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1} ^ {k})}} \left[ \overline {{\pi}} _ {k, \ell} ^ {i} \{\breve {\mu} _ {k, \ell} ^ {i + 2} - \mu_ {k, \ell} ^ {i + 1} \} - \overline {{\pi}} _ {k, 0} ^ {i} \{\breve {\mu} _ {k, 0} ^ {i + 2} - \mu_ {k, 0} ^ {i + 1} \} | \mathbf {r} _ {0}, \mathbf {z} _ {i + 1} ^ {k} \right] \\ + \mathbb {E} _ {P _ {\sigma \left(\mathbf {z} _ {1} ^ {k}\right)}} \left[ \check {\mu} _ {k, \ell} ^ {2} - \check {\mu} _ {k, 0} ^ {2} \mid \mathbf {r} _ {0}, \mathbf {z} _ {i + 1} ^ {k} \right] (B.108) \\ \end{array}
$$
We first note that all term in Eq. (B.106) converges in the mean-zero normal distribution and is bounded in probability at  $n_{i+1,\ell,k}^{-1/2}$  rate. Therefore,
$$
\text {E q .} (\mathrm {B}. 1 0 6) = \sum_ {i = 1} ^ {m ^ {k}} O _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} \left(n _ {i, \ell , k} ^ {- 1 / 2}\right). \tag {B.109}
$$
We now analyze the second term. We note that, by [Kennedy et al., 2020, Lemma 2],
$$
\begin{array}{l} \mathbb {E} _ {\bar {D} _ {\mathbf {z} _ {i + 1} ^ {k}, \ell} ^ {k} - P _ {\sigma (\mathbf {Z} _ {i + 1} ^ {k}) | \mathbf {z} _ {i + 1} ^ {k}}, \mathbf {r} _ {0}} \left[ \bar {\pi} _ {k, \ell} ^ {i} \left\{\tilde {\mu} _ {k, \ell} ^ {i + 2} - \mu_ {k, \ell} ^ {i + 1} \right\} - \bar {\pi} _ {k, 0} ^ {i} \left\{\tilde {\mu} _ {k, 0} ^ {i + 2} - \mu_ {k, 0} ^ {i + 1} \right\} \right] (B.110) \\ = O _ {P _ {\sigma (\mathbf {Z} _ {i + 1} ^ {k})}} \left(\frac {\| \overline {{\pi}} _ {k , \ell} ^ {i} \{\check {\mu} _ {k , \ell} ^ {i + 2} - \mu_ {k , \ell} ^ {i + 1} \} - \overline {{\pi}} _ {k , 0} ^ {i} \{\check {\mu} _ {k , 0} ^ {i + 2} - \mu_ {k , 0} ^ {i + 1} \} \|}{\sqrt {n _ {i + 1 , \ell , k}}}\right). (B.111) \\ \end{array}
$$
We note that  $\|\overline{\pi}_{k,\ell}^{i}\{\check{\mu}_{k,\ell}^{i+2} - \mu_{k,\ell}^{i+1}\} - \overline{\pi}_{k,0}^{i}\{\check{\mu}_{k,0}^{i+2} - \mu_{k,0}^{i+1}\} \|$  is bounded by some constant by the given condition. Therefore, it is bounded in probability at  $n_{i+1,\ell,k}^{-1/2}$  rate. By the same analysis,  $\mathbb{E}_{\overline{D}_{\mathbf{z}_{1,\ell}^k}}[-P_{\sigma(\mathbf{Z}_1^k)|\mathbf{z}_1^k,\mathbf{r}_0}[\check{\mu}_{k,\ell}^2 - \check{\mu}_{k,0}^2]$  is bounded in probability at  $1/\sqrt{n_{1,\ell,k}}$ -rates. Therefore,
$$
\text {E q .} (\mathrm {B}. 1 0 7) = \sum_ {i = 1} ^ {m ^ {k}} O _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} \left(n _ {i, \ell , k} ^ {- 1 / 2}\right). \tag {B.112}
$$
Finally, under the given assumption, the third term can be analyzed by Lemma S.6 and is zero:
$$
\text {E q .} (B. 1 0 8) = \sum_ {i = 1} ^ {m ^ {k} - 1} O _ {P _ {\sigma \left(\mathbf {Z} _ {i + 1} ^ {k}\right)}} \left(\| \mu_ {k, \ell} ^ {i + 1} - \mu_ {k, 0} ^ {i + 1} \| \| \pi_ {k, \ell} ^ {i} - \pi_ {k, 0} ^ {i} \|\right) = 0. \tag {B.113}
$$
$$
\hat {A} _ {\ell} ^ {k} - A _ {0} = \operatorname {E q.} (\mathrm {B}. 1 0 6) + \operatorname {E q.} (\mathrm {B}. 1 0 7) + \operatorname {E q.} (\mathrm {B}. 1 0 8) = \sum_ {i = 1} ^ {m} \sum_ {i = 1} ^ {m ^ {k}} O _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} \left(n _ {i, \ell , k} ^ {- 1 / 2}\right), \tag {B.114}
$$
and, finally,
$$
\begin{array}{l} \hat {A} - A _ {0} = \frac {1}{L} \sum_ {\ell = 1} ^ {L} \left\{\hat {A} _ {\ell} ^ {k} - A _ {0} \right\} (B.115) \\ = \frac {1}{L} \sum_ {\ell = 1} ^ {L} \sum_ {i = 1} ^ {m ^ {k}} O _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} \left(n _ {i, \ell , k} ^ {- 1 / 2}\right) (B.116) \\ = \sum_ {i = 1} ^ {m ^ {k}} O _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} \left(n _ {i, k} ^ {- 1 / 2}\right) (B.117) \\ = \sum_ {i = 1} ^ {m} o _ {P _ {\sigma \left(\mathbf {Z} _ {i} ^ {k}\right)}} (1), (B.118) \\ \end{array}
$$
where the last equation holds since  $O_P(n^{-\alpha}) = o_P(1)$  when  $\alpha > 0$ .
C.1 Relaxation of Discreteness Assumption
In this paper, we made the strong assumption that all variables are discrete. However, this assumption does not hold in general. In this section, we relax the assumption to a certain degree while ensuring that the proposed estimators and corresponding error analysis remain applicable without sacrificing generality.
First, we define a set of variables, denoted as disc, which must be discrete in order to apply the proposed estimators and leverage the error analyses presented in the paper.
Definition 4 (Discreteness set disc). For some given inputs  $(\mathbf{x},\mathbf{y},\mathbb{Z},\mathbb{P},G)$ , suppose
$$
f \left(\left\{A _ {0} ^ {k} \right\} _ {k = 1} ^ {K}\right) = \operatorname {G I D} (\mathbf {x}, \mathbf {y}, \mathbb {Z}, \mathbb {P}, G), \tag {C.1}
$$
where each  $A_0^k$  is specified as
$$
A _ {0} ^ {k} := A _ {0} \left[ \mathbf {W} ^ {k}, \mathbf {C} ^ {k}, \mathbf {R} ^ {k}; \mathbb {Z} ^ {k}, \operatorname {s e q} ^ {k} \right] \left(\mathbf {w} ^ {k} \backslash \mathbf {c} ^ {k}, \mathbf {r} _ {k}\right), \tag {C.2}
$$
for  $\mathbf{W}^k, \mathbf{R}^k \subseteq \mathbf{V}$  and  $\mathbb{Z}^k \subseteq \mathbb{Z}$ . Then, the discreteness set  $\mathrm{disc}(\{A_0^k\}_{k=1}^K)$  is defined as follows:
$$
\operatorname {d i s c} \left(\left\{A _ {0} ^ {k} \right\} _ {k = 1} ^ {K}\right) := \bigcup_ {k = 1} ^ {K} \left\{\left(\mathbf {W} ^ {k} \backslash \mathbf {C} ^ {k}\right) \cup \mathbf {R} ^ {k} \cup \mathbb {Z} ^ {k} \right\}. \tag {C.3}
$$
For Example 1, the discreteness set is given as
$$
\operatorname {d i s c} \left(\left\{A _ {0} ^ {1}, A _ {0} ^ {2} \right\}\right) = (Z, X) \cup (Y, Z) = \{Z, X, Y \} = \mathbf {V} \backslash \{W \}. \tag {C.4}
$$
For Example 2, the discreteness set is given as
$$
\operatorname {d i s c} \left(\left\{A _ {0} ^ {2}, A _ {0} ^ {1 3} \right\}\right) = (W, R, X _ {1}) \cup (R, Y, X _ {2}, W, X _ {1}) = \left\{X _ {1}, X _ {2}, R, W, Y \right\} = \mathbf {V} \tag {C.5}
$$
Equipped with the discreteness set, we relax the assumption as follows:
Assumption 1.1 (Relaxed Regularity). For variables  $\mathbf{V}$  and the Radon-Nikodym derivative  $p_{\sigma (\mathbf{Z})}$  of  $P_{\sigma (\mathbf{Z})}$  for  $\mathbf{Z}\in \mathbb{Z}$ , the following conditions hold:
1. All variables in disc  $(\{A_0^k\})$  are discrete;
2.  $p_{\sigma(\mathbf{Z})}(\mathbf{v}) > c, \forall \mathbf{v} \in \mathfrak{D}_{\mathbf{V}}$  for some  $c \in (0,1)$ .
We note that the proposed estimator is well-defined and corresponding error analyses Theorem 2 and Corollary 2 hold true under the relaxed assumption:
Lemma S.9 (Well-defined MR-gID Estimator under Relaxed Regularity). The MR-gID estimator in Definition 4 is pathwise-differentiable under Assumption 1.1 and Assumption 2.
Proof of Lemma S.9. For  $k = 1,\dots ,K$  , define
$$
\overline {{A}} ^ {k} := \sum_ {i = 1} ^ {m ^ {k} - 1} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {i + 1} ^ {k})}} \left[ \overline {{\pi}} _ {\mathbf {k}} ^ {i} \left\{\breve {\mu} _ {k} ^ {i + 2} - \mu_ {k} ^ {i + 1} \right\} | \mathbf {z} _ {i + 1} ^ {k}, \mathbf {r} _ {0} ^ {k} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {1} ^ {k})}} \left[ \breve {\mu} _ {k} ^ {2} | \mathbf {z} _ {1} ^ {k}, \mathbf {r} _ {0} ^ {k} \right]. \tag {C.6}
$$
To establish the pathwise-differentiability of the MR-gID estimator  $f(\{\hat{A}^k\}_{k=1}^K)$  as defined in Definition 4, it is sufficient to ensure the pathwise-differentiability of individual  $\overline{A}^k$ . Under Assumption 1.1,  $\mu_k^{m^k+1} := \mathbb{1}_{\mathbf{w}^k \setminus \mathbf{c}^k}(\mathbf{W}^k \setminus \mathbf{C}^k)$  is well-defined since  $(\mathbf{W}^k \setminus \mathbf{C}^k) \in \mathrm{disc}(\{A_0^k\}_{k=1}^K)$  are discrete. Also, each  $\mathbb{1}_{\mathbf{r}_i^k}(\mathbf{R}_i^k)$  in each  $\pi_k^i$  are well-defined since  $\mathbf{R}_i^k \in \mathrm{disc}(\{A_0^k\}_{k=1}^K)$  are discrete. Finally, the conditional expectation  $\mathbb{E}_{P_{\sigma(\mathbf{Z}_i^k)}}[\cdot|\mathbf{z}_i^k, \mathbf{r}_0^k]$  is well-defined since  $\mathbf{Z}_i^k \in \mathrm{disc}(\{A_0^k\}_{k=1}^K)$  are discrete. Also, under the positivity condition stated in Assumption 1.1,  $\overline{A}^k$  in Eq. (C.6) is pathwise-differentiable. By combining this with Assumption 2, we conclude that the MR-gID estimator is pathwise-differentiable.
C.2 Sequential Doubly Robustness:  $2^{m-1}$  robustness versus  $m$ -robustness
In this section, we discuss the practical properties of the proposed doubly robust g-mSBD estimator in Def. 3. We recall that the estimator is doubly robust by the analysis in Lemma S.6 as follows:
$$
\bar {A} - A _ {0} = \sum_ {r = 1} ^ {m - 1} O _ {P _ {\sigma (\mathbf {z} _ {r + 1})}} \left(\| \pi^ {r} - \pi_ {0} ^ {r} \| \| \mu^ {r + 1} - \mu_ {0} ^ {r + 1} \|\right). \tag {B.69}
$$
The term in Eq. (B.69) exhibits doubly robustness, becoming zero when either  $\pi^r = \pi_0^r$  or  $\mu^{r + 1} = \mu_0^{r + 1}$  hold for all  $r = 1,2,\ldots ,m - 1$ . This phenomenon is referred to as the sequential doubly robustness [Luedtke et al., 2017], or  $2^{m - 1}$  robustness in the sense that there are  $2^{m - 1}$  ways to make Eq. (B.69) zero [Vansteelandt et al., 2007, Rotnitzky et al., 2017].
While the proposed doubly robust g-mSBD estimator defined in Definition 3 exhibits doubly robustness, as shown in Proposition 1, it does not satisfy  $2^{m-1}$  robustness. This is due to the dependencies between  $\mu_{\ell}^{r}$  and  $\mu_{\ell}^{s}$  for  $s \in \{r+1, \cdots, m\}$ . Specifically, if  $\mu_{\ell}^{s}$  is misspecified for some  $s > r$ , it renders the case  $\mu_{\ell}^{r} = \mu_{0}^{r}$  impossible. Consequently, instead of having  $2^{m-1}$  possibilities, there are only  $m$  ways to make Eq. (B.69) equal to zero. For each  $r = 1, \cdots, m-1$ , this requires either  $\pi_{\ell}^{r} = \pi_{0}^{r}$  or  $\mu_{\ell}^{s} = \mu_{0}^{s}$  for  $s > r$ . This condition is referred to as  $m$ -robustness. In summary, the doubly robust g-mSBD estimator achieves  $m$ -robustness instead of  $2^{m-1}$  robustness. We acknowledge that an interesting open direction is to explore ways to enhance the doubly robust g-mSBD estimator to attain  $2^{m-1}$  robustness, building upon the findings presented in Luedtke et al. [2017].
D Details of Experiments
As described in Sec. 4, we used the XGBoost [Chen and Guestrin, 2016] as a model for estimating nuisances  $\mu, \pi, \{\mu^i\}_{i=2}^m, \{\pi^i\}_{i=1}^m$ . We implemented the model using Python. In modeling nuisance using the XGBoost, we used the command
xgboost.XGBClassifier.eval_metric='logloss')<sup>1</sup> to use the XGBoost with the default parameter settings. In estimating the weight, we set the weight  $\pi_{\ell}^{i} = 10$  whenever the estimated weight is over 10 [Crump et al., 2009]. For Example 1, the dimension of  $W$  is set as  $|W| = 10$ . We chose  $L = 2$ . All variables are set to be binary. We compute the effect of  $P(Y = 1|do(x))$ .
D.1 Designs of Simulations