SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$\mathbf{R} \coloneqq (\mathbf{R}_1 \cup \mathbf{R}_2) \backslash \mathbf{W}$ . 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,
$$
\begin{array}{l} A _ {0} [ \mathbf {W}, \emptyset , \mathbf {R}; \mathbb {Z}, \operatorname {s e q} ] (\mathbf {w}, \mathbf {r}) (B.14) \\ = \prod_ {j: W _ {j} \in \mathbf {W}} P _ {\mathbf {z} _ {j}} \left(w _ {j} \mid \overline {{\mathbf {w}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {j}\right) (B.15) \\ = \prod_ {k: W _ {1, k} \in \mathbf {W} _ {1} \text {s . t} W _ {1, k} = W _ {j}} P _ {\mathbf {z} _ {k} ^ {1}} \left(w _ {1, k} | \overline {{\mathbf {w}}} _ {1} ^ {j - 1}, \overline {{\mathbf {r}}} _ {1} ^ {j - 1} \backslash \mathbf {z} _ {j} ^ {1}\right) \times \prod_ {k: W _ {2, k} \in \mathbf {W} _ {2} \text {s . t} W _ {2, k} = W _ {j}} P _ {\mathbf {z} _ {k} ^ {2}} \left(w _ {2, k} | \overline {{\mathbf {w}}} _ {2} ^ {j - 1}, \overline {{\mathbf {r}}} _ {2} ^ {j - 1} \backslash \right. (B.16) \\ = \prod_ {j = 1} ^ {m ^ {1}} P _ {\mathbf {z} _ {j} ^ {1}} \left(w _ {1, j} | \overline {{\mathbf {w}}} _ {1} ^ {j - 1}, \overline {{\mathbf {r}}} _ {1} ^ {j - 1} \backslash \mathbf {z} _ {j} ^ {1}\right) \times \prod_ {j = 1} ^ {m ^ {2}} P _ {\mathbf {z} _ {j} ^ {2}} \left(w _ {2, j} | \overline {{\mathbf {w}}} _ {2} ^ {j - 1}, \overline {{\mathbf {r}}} _ {2} ^ {j - 1} \backslash \mathbf {z} _ {j} ^ {2}\right) (B.17) \\ = A _ {0} ^ {1} \times A _ {0} ^ {2}. (B.18) \\ \end{array}
$$
B.4 Proof of Lemma 4
$$
A _ {0} ^ {1} / A _ {0} ^ {2} = A _ {0} \left[ \mathbf {W}, \emptyset , \mathbf {R}; \mathbb {Z} _ {1}, \operatorname {s e q} ^ {1} \right] (\mathbf {w}, \mathbf {r}) = \prod_ {j: W _ {j} \in \mathbf {W}} P _ {\mathbf {z} _ {j}} \left(w _ {j} | \overline {{\mathbf {w}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1} \backslash \mathbf {z} _ {j}\right). \tag {5}
$$
$$
\begin{array}{l} A _ {0} ^ {1} / A _ {0} ^ {2} = \frac {\prod_ {j = 1} ^ {m ^ {1}} P _ {\mathbf {z} _ {j} ^ {1}} \left(w _ {1 , j} \mid \overline {{\mathbf {w}}} _ {1} ^ {j - 1} , \overline {{\mathbf {r}}} _ {1} ^ {j - 1} \backslash \mathbf {z} _ {j} ^ {1}\right)}{\prod_ {j = 1} ^ {m ^ {2}} P _ {\mathbf {z} _ {j} ^ {2}} \left(w _ {2 , j} \mid \overline {{\mathbf {w}}} _ {2} ^ {j - 1} , \overline {{\mathbf {r}}} _ {2} ^ {j - 1} \backslash \mathbf {z} _ {j} ^ {2}\right)} (B.19) \\ = \prod_ {k: W _ {k} \in \mathbf {W} _ {1} \backslash \mathbf {W} _ {2}} P _ {\mathbf {z} _ {k} ^ {1}} \left(w _ {1, k} | \overline {{\mathbf {w} _ {1}}} ^ {k - 1}, \overline {{\mathbf {r} _ {1}}} ^ {k - 1} \backslash \mathbf {z} _ {k} ^ {1}\right) (B.20) \\ = \prod_ {k: W _ {k} \in \mathbf {W} _ {1} \backslash \mathbf {W} _ {2}} P _ {\mathbf {z} _ {k} ^ {1}} \left(w _ {1, k} | \overline {{\mathbf {w} _ {1}}} ^ {k - 1} \cap \mathbf {w}, \overline {{\mathbf {w} _ {1}}} ^ {k - 1} \backslash \mathbf {w}, \overline {{\mathbf {r} _ {1}}} ^ {k - 1} \backslash \mathbf {z} _ {k} ^ {1}\right). (B.21) \\ \end{array}
$$
We note that  $\overline{\mathbf{W}_1}^{k-1} \cap \mathbf{W} = \overline{\mathbf{W}}^{j-1}$  for some  $W_j \in \mathbf{W}$  s.t.  $W_j = W_{1,k}$ . Also,  $(\mathbf{W}_1 \backslash \mathbf{W}) \cup \mathbf{R}_1 = \mathbf{W}_2 \cup \mathbf{R}_1$ . Therefore,  $\cup_{k: W_k \in \mathbf{W}_1 \backslash \mathbf{W}_2} \{\overline{\mathbf{W}_1}^{k-1} \backslash \mathbf{W}, \overline{\mathbf{R}_1}^{k-1}\} = \mathbf{R} := (\mathbf{R}_1 \cup \mathbf{W}_2) \cap \operatorname{pre}(\mathbf{W}; G)$ . Therefore,
$$
\begin{array}{l} A _ {0} ^ {1} / A _ {0} ^ {2} = \prod_ {k: W _ {k} \in \mathbf {W} _ {1} \backslash \mathbf {W} _ {2}} P _ {\mathbf {z} _ {k} ^ {1}} \left(w _ {1, k} | \overline {{\mathbf {w} _ {1}}} ^ {k - 1} \cap \mathbf {w}, \overline {{\mathbf {w} _ {1}}} ^ {k - 1} \backslash \mathbf {w}, \overline {{\mathbf {r} _ {1}}} ^ {k - 1} \backslash \mathbf {z} _ {k} ^ {1}\right) (B.22) \\ = \prod_ {\ell : W _ {\ell} \in \mathbf {W}} P _ {\mathbf {z} _ {\ell}} \left(w _ {\ell} | \overline {{\mathbf {w}}} ^ {\ell - 1}, \overline {{\mathbf {r}}} ^ {\ell - 1}\right) (B.23) \\ = A _ {0} \left[ \mathbf {W}, \emptyset , \mathbf {R}; \mathbb {Z} _ {1}, \operatorname {s e q} ^ {1} \right] (\mathbf {w}, \mathbf {r}). (B.24) \\ \end{array}
$$
Proof of Theorem 1. Throughout the proof, we refer to the algorithm developed in [Lee et al., 2019, Algo. 1] as the "standard gID" algorithm, in comparison to our gID algorithm presented in Algo. 1. It is established that the standard gID algorithm is sound, as stated in [Lee et al., 2019, Theorem 2]. This means that if the algorithm returns an identification expression, it must be correct. Furthermore, the standard gID algorithm is proven to be complete [Lee et al., 2019, Theorem 3]. In other words, the causal effect  $P(\mathbf{y}|\mathrm{do}(\mathbf{x}))$  is identifiable from  $\mathbb{P}$  and the causal graph  $G$  if and only if the standard gID algorithm does not return FAIL.
In our proof, we will show the soundness and completeness of Algo. 1 based on the foundation provided by the standard gID algorithm.
Algo. 1 is sound - If Algo. 1 returns an expression  $f(\{A_0^k\}_{k=1}^K)$ , then it holds that  $f(\{A_0^k\}_{k=1}^K) = P(\mathbf{y}|\mathrm{do}(\mathbf{x}))$ . The soundness of Algo. 1 is derived from the soundness of Tian's c-factor operation, as demonstrated in [Tian and Pearl, 2003, Lemmas (3,4)] and Lemma 1.
We will now show that Algo. 1 is complete. Suppose there exists an input  $(\mathbf{x},\mathbf{y},\mathbb{Z},\mathbb{P},G)$  for which the standard gID algorithm does not return FAIL while Algo. 1 does return FAIL. This implies the existence of  $\mathbf{D}_j$  such that  $Q[\mathbf{D}_j]$  is not identifiable from all  $Q[\mathbf{S}_j^i ]$  where  $\mathbf{D}_j$  is a c-component in  $G(\mathbf{D})$ , and  $\mathbf{S}_j^i$  is the c-component in  $G(\mathbf{V}\backslash \mathbf{Z}_i)$  that contains  $\mathbf{D}_j$ . This observation is a consequence of the soundness and completeness of the SUBID procedure, as established in [Huang and Valtorta, 2006, Theorem 1].
It should be noted that  $Q[\mathbf{D}_j]$  is not identifiable from  $\mathbf{S}_j^i$  for all  $\{i : \mathbf{Z}_i \in \mathbb{Z}\}$  only when there exists a c-component  $\mathbf{T}_j^i$  in  $G(\mathbf{S}_j^i)$  that serves as an ancestral set of  $\mathbf{D}_j$  and includes  $\mathbf{D}_j$ . However, in such a scenario, the standard gID algorithm fails due to lines 12 and 13 of the algorithm. This contradicts the initial assumption that the standard gID algorithm does not return FAIL. Consequently, Algo. 1 returns FAIL whenever the standard gID algorithm does so. The completeness of the standard gID algorithm implies that Algo. 1 is complete in the g-identification task.
The fact that  $f(\cdot)$  is a function involving marginalization, multiplications, and divisions of g-mSBD (generalized modified single back-door) operators is a consequence of applying Lemmas (2, 3, 4) within the algorithm. These lemmas establish the properties and operations of the g-mSBD operators, which are then utilized in the construction of  $f(\cdot)$  in Algo. 1.
B.6 Proof of Proposition 1
We first restate the g-mSBD adjustment, its nuisances, and the g-mSBD estimator here:
Definition 1 (generalized-mSBD adjustment (g-mSBD)). Let  $(\mathbf{W},\mathbf{R})$  be a disjoint pair in  $\mathbf{V}$  topologically ordered as  $(\mathbf{W},\mathbf{R}) = \{\mathbf{R}_0,W_1,\dots ,\mathbf{R}_{m - 1},W_m,\mathbf{R}_m\}$  by  $\prec_{G}$ , where  $\mathbf{R}_i$  can be empty. Let  $\overline{\mathbf{W}}^{i - 1} := \{W_j\}_{j = 1}^{i - 1}$  and  $\overline{\mathbf{R}}^{i - 1} := \{\mathbf{R}_j\}_{j = 0}^{i - 1}$  for  $\forall i\in [m]$ . Let  $\mathbf{C}\subseteq \mathbf{W}$ . Let  $\mathbb{Z}_0\subseteq \mathbb{Z}$  be some set such that  $\forall \mathbf{Z}\in \mathbb{Z}_0,\mathbf{W}\cap \mathbf{Z} = \emptyset$ . Let  $\operatorname {seq}(\mathbb{Z}_0)$  denote a sequence  $(\mathbf{z}_1,\dots ,\mathbf{z}_m)$  where  $\mathbf{z}_i$  denotes some realization of  $\mathbf{Z}_i\in \mathbb{Z}_0$  (same  $\mathbf{z}_i$  could appear multiple times in the sequence). Then, the g-mSBD adjustment is expressed as an operator  $A_0[\mathbf{W},\mathbf{C},\mathbf{R};\mathbb{Z}_0,\operatorname {seq},G](\mathbf{w}\backslash \mathbf{c},\mathbf{r})$  defined by
Definition 2 (Nuisances for g-mSBD). Nuisances for g-mSBD  $A_0$  in Eq. (1) are  $\{\mu_0^{i+1}, \pi_0^i\}_{i=1}^{m-1}$  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  $i = m-1, \dots, 1$ ,
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  $A_0[\mathbf{W}, \mathbf{C}, \mathbf{R}; \mathbb{Z}_0 := \{\mathbf{Z}_i\}_{i=1}^m, \mathbf{seq} := (\mathbf{z}_i)_{i=1}^m](\mathbf{w} \backslash \mathbf{c}, \mathbf{r})$  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
$$
\hat {A} _ {\ell} := \hat {A} _ {\ell} \left(\left\{\mu_ {\ell} ^ {j + 1}, \pi_ {\ell} ^ {j} \right\} _ {j \in [ m - 1 ]}\right) := \sum_ {j = 1} ^ {m - 1} \mathbb {E} _ {\overline {{D}} _ {\mathbf {z} _ {j + 1, \ell}}} \left[ \bar {\pi} _ {\ell} ^ {j} \left\{\check {\mu} _ {\ell} ^ {j + 2} - \mu_ {\ell} ^ {j + 1} \right\} \right] + \mathbb {E} _ {\overline {{D}} _ {\mathbf {z} _ {1, \ell}}} \left[ \check {\mu} _ {\ell} ^ {2} \right], \tag {9}
$$
We analyze the bias of the g-mSBD estimator using the following results:
Lemma S.2 (Representation of g-mSBD). The  $g$ -mSBD adjustment  $A_0$  in Def. 1 can be represented as
$$
A _ {0} = \sum_ {i = 1} ^ {m - 1} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {i + 1})}} \left[ \bar {\pi} _ {0} ^ {i} \left\{\check {\mu} _ {0} ^ {i + 2} - \mu_ {0} ^ {i + 1} \right\} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] + \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {1})}} \left[ \check {\mu} _ {0} ^ {2} | \mathbf {z} _ {1}, \mathbf {r} _ {0} \right], \tag {B.25}
$$
where  $\breve{\mu}_{\ell}^{i+1}(\overline{\mathbf{W}}^i, \overline{\mathbf{R}}^{1:i-1}) := \mu_{\ell}^{i+1}(\overline{\mathbf{W}}^i, \mathbf{r}_i, \overline{\mathbf{R}}^{1:i-1})$  and  $\overline{\pi}^i := \prod_{j=1}^i \pi^j$  as defined in Def. 3.
Proof of Lemma S.2. Throughout the proof, we will use  $\mathbf{w}'\backslash \mathbf{c}'$  as some realization of W\C. Recall that  $\mathbb{1}_{\mathbf{w}\backslash \mathbf{c}}(\mathbf{w}'\backslash \mathbf{c}') = 1$  when  $\mathbf{w}^{\prime}\backslash \mathbf{c}^{\prime} = \mathbf{w}\backslash \mathbf{c}$  and zero otherwise. We first recall that
$$
A _ {0} = \sum_ {\mathbf {w} ^ {\prime} \in \mathfrak {S} _ {\mathbf {W}}} \prod_ {i: W _ {i} \in \mathbf {W}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}\right) P _ {\sigma \left(\mathbf {Z} _ {i}\right)} \left(w _ {i} ^ {\prime} \mid \overline {{\mathbf {w} ^ {\prime}}} ^ {i - 1}, \overline {{\mathbf {r}}} ^ {i - 1}, \mathbf {z} _ {i}\right), \tag {B.26}
$$
by the definition of the experimental distribution  $P_{\sigma (\mathbf{Z}_i)}$
For all  $i = 1,\dots ,m - 1$
$$
\begin{array}{l} \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {i + 1})}} \left[ \overline {{\boldsymbol {\pi}}} _ {0} ^ {i} \left(\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i}\right) \{\check {\mu} _ {0} ^ {i + 2} \left(\overline {{\mathbf {W}}} ^ {i + 1}, \overline {{\mathbf {R}}} ^ {1: i}\right) - \mu_ {0} ^ {i + 1} \left(\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i}\right) \} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] (B.27) \\ \stackrel {1} {=} \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \overline {{\pi}} _ {0} ^ {i} \left(\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i}\right) \{\mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \check {\mu} _ {0} ^ {i + 2} \left(\overline {{\mathbf {W}}} ^ {i + 1}, \overline {{\mathbf {R}}} ^ {1: i}\right) | \overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i}, \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] - \mu_ {0} ^ {i + 1} \left(\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i}\right) \} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] (B.28) \\ \stackrel {2} {=} \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {i + 1})}} \left[ \overline {{\boldsymbol {\pi}}} _ {0} ^ {i} \left(\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i}\right) \left\{\mu_ {0} ^ {i + 1} \left(\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i}\right) - \mu_ {0} ^ {i + 1} \left(\overline {{\mathbf {W}}} ^ {i}, \overline {{\mathbf {R}}} ^ {1: i}\right) \right\} | \mathbf {z} _ {i + 1}, \mathbf {r} _ {0} \right] (B.29) \\ = 0, (B.30) \\ \end{array}
$$
where the equation  $\stackrel{1}{=}$  holds by the total law of expectation, and  $\stackrel{2}{=}$  holds by the definition of  $\check{\mu}_0^{i+1}$ .
It suffices to show that
$$
\mathbb {E} _ {P _ {\sigma \left(\mathbf {z} _ {1}\right)}} \left[ \mu_ {0} ^ {2} \left(\overline {{\mathbf {W}}} ^ {1}, \overline {{\mathbf {r}}} ^ {1}\right) \mid \mathbf {z} _ {1}, \mathbf {r} _ {0} \right] = A _ {0} = \sum_ {\mathbf {w} ^ {\prime} \in \mathfrak {D} _ {\mathbf {W}}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}\right) \prod_ {j = 1} ^ {m} P _ {\sigma \left(\mathbf {Z} _ {j}\right)} \left(w _ {j} ^ {\prime} \mid \overline {{\mathbf {w} ^ {\prime}}} ^ {j - 1}, \overline {{\mathbf {r}}} ^ {j - 1}, \mathbf {z} _ {j}\right). \tag {B.31}
$$
To prove the equation, we show that, for all  $k = m,m - 1,\dots ,2$
$$
\mu_ {0} ^ {k} \left(\overline {{\mathbf {W}}} ^ {k - 1}, \overline {{\mathbf {R}}} ^ {1: k - 1}\right) = \sum_ {\overline {{\mathbf {w}}} ^ {\prime} ^ {k: m} \in \mathfrak {S} _ {\overline {{\mathbf {W}}} ^ {k: m}}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}\right) \prod_ {j = k} ^ {m} P _ {\sigma (\mathbf {Z} _ {j})} \left(w _ {j} ^ {\prime} | \overline {{\mathbf {W}}} ^ {k - 1}, \overline {{\mathbf {R}}} ^ {1: k - 1}, \overline {{\mathbf {w} ^ {\prime}}} ^ {k: j - 1}, \overline {{\mathbf {r}}} ^ {k: j - 1}, \mathbf {r} _ {0}, \mathbf {z} _ {j}\right).
$$
This equation holds when  $k = m$ , because
$$
\begin{array}{l} \mu_ {0} ^ {m} \left(\overline {{\mathbf {W}}} ^ {m - 1}, \overline {{\mathbf {R}}} ^ {1: m - 1}\right) := \mathbb {E} _ {P _ {\sigma (\mathbf {Z} _ {m})}} \left[ \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}\right) | \overline {{\mathbf {W}}} ^ {m - 1}, \overline {{\mathbf {R}}} ^ {1: m - 1}, \mathbf {r} _ {0}, \mathbf {z} _ {m} \right] \tag {B.33} \\ = \sum_ {w _ {m} ^ {\prime} \in \mathfrak {S} _ {w _ {m}}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}\right) P _ {\sigma \left(\mathbf {Z} _ {m}\right)} \left(w _ {m} ^ {\prime} \mid \overline {{\mathbf {W}}} ^ {m - 1}, \overline {{\mathbf {R}}} ^ {1: m - 1}, \mathbf {r} _ {0}, \mathbf {z} _ {m}\right). \quad (B. 3 4) \\ \end{array}
$$
For  $k = m - 1$ ,
$$
\begin{array}{l} \mu_ {0} ^ {m - 1} \left(\overline {{\mathbf {W}}} ^ {m - 2}, \overline {{\mathbf {R}}} ^ {1: m - 2}\right) (B.35) \\ := \mathbb {E} _ {P _ {\sigma} (\mathbf {Z} _ {m - 1})} \left[ \breve {\mu} _ {0} ^ {m} \left(\overline {{\mathbf {W}}} ^ {m - 1}, \mathbf {r} _ {m - 1}, \overline {{\mathbf {R}}} ^ {1: m - 2}\right) | \overline {{\mathbf {W}}} ^ {m - 2}, \overline {{\mathbf {R}}} ^ {1: m - 2}, \mathbf {z} _ {m - 1}, \mathbf {r} _ {0} \right] (B.36) \\ = \sum_ {\overline {{\mathbf {w}}} ^ {\prime} ^ {m - 1: m} \in \mathfrak {S} _ {\overline {{\mathbf {W}}} ^ {m - 1: m}}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}\right) \prod_ {j = m - 1} ^ {m} P _ {\sigma \left(\mathbf {Z} _ {j}\right)} \left(w _ {j} ^ {\prime} \mid \overline {{\mathbf {w}}} ^ {\prime} ^ {m - 1: j - 1}, \overline {{\mathbf {r}}} ^ {m - 1: j - 1}, \overline {{\mathbf {W}}} ^ {m - 2}, \overline {{\mathbf {R}}} ^ {1: m - 2}, \mathbf {z} _ {j}, \right. (B.37) \\ \end{array}
$$
Based on this observation, we make the following induction hypothesis: Suppose, for a fixed  $k \in \{2, \dots, m\}$ , the following holds:
$$
\mu_ {0} ^ {k + 1} \left(\overline {{\mathbf {W}}} ^ {k}, \overline {{\mathbf {R}}} ^ {1: k}\right) \stackrel {\text {i n d u c t i o n}} {=} \sum_ {\overline {{\mathbf {w}}} ^ {j ^ {k + 1: m}} \in \mathfrak {S} _ {\overline {{\mathbf {W}}} ^ {k + 1: m}}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}\right) \prod_ {j = k + 1} ^ {m} P _ {\sigma (\mathbf {Z} _ {j})} \left(w _ {j} ^ {\prime} \mid \overline {{\mathbf {W}}} ^ {k}, \overline {{\mathbf {R}}} ^ {1: k}, \overline {{\mathbf {w}}} ^ {\prime} ^ {k + 1: j - 1}, \overline {{\mathbf {r}}} ^ {k + 1: j - 1}, \mathbf {z} _ {j}, \mathbf {r} _ {0}\right).
$$
Then, the induction hypothesis holds for  $k - 1$  as follows:
$$
\begin{array}{l} \mu_ {0} ^ {k} \left(\overline {{\mathbf {W}}} ^ {k - 1}, \overline {{\mathbf {R}}} ^ {1: k - 1}\right) (B.39) \\ := \mathbb {E} _ {P _ {\sigma (\mathbf {z} _ {k + 1})}} \left[ \breve {\mu} ^ {k + 1} \left(\overline {{\mathbf {W}}} ^ {k}, \overline {{\mathbf {R}}} ^ {1: k - 1}, \mathbf {r} _ {k}\right) \mid \overline {{\mathbf {W}}} ^ {k - 1}, \overline {{\mathbf {R}}} ^ {1: k - 1}, \mathbf {z} _ {k + 1}, \mathbf {r} _ {0} \right] (B.40) \\ = \sum_ {\overline {{\mathbf {w}}} ^ {\prime} ^ {k: m} \in \mathfrak {S} _ {\overline {{\mathbf {W}}} ^ {k: m}}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}\right) \prod_ {j = k + 1} ^ {m} P _ {\sigma \left(\mathbf {Z} _ {j}\right)} \left(w _ {j} ^ {\prime} | \overline {{\mathbf {W}}} ^ {k - 1}, \overline {{\mathbf {R}}} ^ {1: k - 1}, \overline {{\mathbf {w}}} ^ {\prime} ^ {k: j - 1}, \overline {{\mathbf {r}}} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} _ {0}\right) P _ {\sigma \left(\mathbf {Z} _ {k}\right)} \left(w _ {k} ^ {\prime} | \overline {{\mathbf {W}}} ^ {k - 1}, \overline {{\mathbf {R}}} ^ {1: k - 1}, \mathbf {z} _ {k}, \mathbf {r} _ {0}\right) (B.41) \\ = \sum_ {\overline {{\mathbf {w} ^ {\prime}}} ^ {k: m} \in \mathfrak {S} _ {\overline {{\mathbf {W}}} ^ {k: m}}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}\right) \prod_ {j = k} ^ {m} P _ {\sigma \left(\mathbf {Z} _ {j}\right)} \left(w _ {j} ^ {\prime} | \overline {{\mathbf {W}}} ^ {k - 1}, \overline {{\mathbf {R}}} ^ {1: k - 1}, \overline {{\mathbf {w} ^ {\prime}}} ^ {k: j - 1}, \overline {{\mathbf {r}}} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} _ {0}\right). (B.42) \\ \end{array}
$$
Also, we already checked that the induction hypothesis holds for  $k = m$ . Therefore, the hypothesis holds for all  $k = 2,\dots ,m$ :
$$
\mu_ {0} ^ {k} \left(\overline {{\mathbf {W}}} ^ {k - 1}, \overline {{\mathbf {R}}} ^ {1: k - 1}\right) = \sum_ {\overline {{\mathbf {w}}} ^ {\prime} ^ {k: m} \in \mathfrak {S} _ {\overline {{\mathbf {W}}} ^ {k: m}}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}\right) \prod_ {j = k} ^ {m} P _ {\sigma \left(\mathbf {Z} _ {j}\right)} \left( \right.w _ {j} ^ {\prime} \mid \overline {{\mathbf {W}}} ^ {k - 1}, \overline {{\mathbf {R}}} ^ {1: k - 1}, \overline {{\mathbf {w} ^ {\prime}}} ^ {k: j - 1}, \overline {{\mathbf {r}}} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {i}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {i}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {i}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {i}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {i}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {i}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} ^ {k: j - 1}, \mathbf {z} _ {j}, \mathbf {r} ^ {k: j - 1}, \mathbf {\Delta}
$$
$$
\mu_ {0} ^ {2} \left(W _ {1}, \mathbf {R} _ {1}\right) = \sum_ {\overline {{\mathbf {w} ^ {\prime}}} ^ {2: m} \in \mathfrak {S} _ {\overline {{\mathbf {W}}} ^ {2: m}}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} \left(\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}\right) \prod_ {j = 2} ^ {m} P _ {\sigma \left(\mathbf {Z} _ {j}\right)} \left(w _ {j} ^ {\prime} \mid W _ {1}, \mathbf {R} _ {1}, \overline {{\mathbf {w} ^ {\prime}}} ^ {2: j - 1}, \overline {{\mathbf {r}}} ^ {2: j - 1}, \mathbf {z} _ {j}, \mathbf {r} _ {0}\right), \tag {B.44}
$$
$$
\mu_ {0} ^ {2} (W _ {1}, \mathbf {r} _ {1}) = \sum_ {\overline {{\mathbf {w} ^ {\prime}}} ^ {2: m} \in \mathfrak {S} _ {\overline {{\mathbf {W} ^ {2 : m}}}}} \mathbb {1} _ {\mathbf {w} \backslash \mathbf {c}} (\mathbf {w} ^ {\prime} \backslash \mathbf {c} ^ {\prime}) \prod_ {j = 2} ^ {m} P _ {\sigma (\mathbf {Z} _ {j})} (w _ {j} ^ {\prime} | W _ {1}, \overline {{\mathbf {w} ^ {\prime}}} ^ {2: j - 1}, \overline {{\mathbf {r}}} ^ {j - 1}, \mathbf {z} _ {j}) \qquad \mathrm {(B . 4 5)}
$$