SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \frac {\delta}{T} \sum_ {t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) (23) \\ = \frac {\delta}{T} \left(\sum_ {t \leq t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) + \sum_ {t > t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right)\right) (24) \\ = \frac {\delta}{T} \sum_ {t \leq t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) + \frac {\delta}{T} \sum_ {t > t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) (25) \\ = \frac {\delta}{T} \sum_ {t \leq t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) + \frac {\delta}{T - t ^ {\prime}} \frac {T - t ^ {\prime}}{\delta} \frac {\delta}{T} \sum_ {t > t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) (26) \\ = \frac {\delta}{T} \sum_ {t \leq t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) + \frac {\delta}{T - t ^ {\prime}} \frac {T - t ^ {\prime}}{\delta} \frac {\delta}{T} \sum_ {t > t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) (27) \\ = \frac {\delta}{T} \sum_ {t \leq t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) + \frac {T - t ^ {\prime}}{T} \left(\frac {\delta}{T - t ^ {\prime}} \sum_ {t > t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right)\right) (28) \\ \end{array}
$$
Take a limit of Eq.(23), we have:
$$
\begin{array}{l} \lim  _ {T \rightarrow \infty} \frac {\delta}{T} \sum_ {t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) (30) \\ = \lim  _ {T \rightarrow \infty} \frac {\delta}{T} \sum_ {t \leq t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) + \lim  _ {T \rightarrow \infty} \frac {T - t ^ {\prime}}{T} \left(\frac {\delta}{T - t ^ {\prime}} \sum_ {t > t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right)\right) (31) \\ = 0 + \lim  _ {T \rightarrow \infty} \frac {T - t ^ {\prime}}{T} \left(\frac {1}{n _ {1} (T) - n _ {1} \left(t ^ {\prime}\right)} \sum_ {n _ {1} (t) > n _ {1} \left(t ^ {\prime}\right)} ^ {n _ {1} (T)} f \left(Z _ {n _ {1} (t)} ^ {q (t)}\right)\right), \text {w h e r e} t > t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j} (32) \\ \xlongequal {\text {B i r k h o f f ' s E r g o d i c T h e o r e m}} 0 + E \left(f \left(Z _ {n _ {1} (t)} ^ {q (t)}\right)\right) (33) \\ = E \left(\mathbb {1} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right)\right), \text {w h e r e} t > t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j} (34) \\ = p \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right), \text {w h e r e} t > t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j} (35) \\ \end{array}
$$
$$
p _ {t \in \pi_ {(k, s)} ^ {j}} \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) := p \left(x _ {t} ^ {j}, \operatorname {P a} \left(x _ {t} ^ {j}\right)\right), \text {w h e r e} t > t ^ {\prime}, t \in \pi_ {(k, s)} ^ {j} \tag {36}
$$
Based on the definition of homogenous time partition and time partition,  $p_{t \in \pi_{(k,s)}^j}(x_t^j | \mathrm{Pa}(x_t^j)) = p_{t \in \Pi_k^j}(x_t^j | \mathrm{Pa}(x_t^j))$ ,  $\forall s \in [\frac{\delta}{\omega_j}]$ .
Similar to Eq.(22), one estimator of  $p_{t\in \Pi_k^j}(x_t^j |\mathrm{Pa}(x_t^j))$ $\forall k = [\omega_j]$  is
$$
\begin{array}{l} \hat {p} _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} \mid \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) = \frac {\sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j} , \operatorname {P a} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(\operatorname {P a} \left(x _ {t} ^ {j}\right)\right)} (37) \\ = \frac {\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} \sum_ {t \in \pi_ {(k , s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j} , \operatorname {P a} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} \sum_ {t \in \pi_ {(k , s)} ^ {j}} \mathbb {1} \left(\operatorname {P a} \left(x _ {t} ^ {j}\right)\right)} (38) \\ = \frac {\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} \frac {T}{\delta} \sum_ {t \in \pi_ {(k , s)} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j} , \operatorname {P a} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} \frac {T}{\delta} \sum_ {t \in \pi_ {(k , s)} ^ {j}} \mathbb {1} \left(\operatorname {P a} \left(x _ {t} ^ {j}\right)\right)} (39) \\ \end{array}
$$
Take a limit of Eq.(37), we have:
$$
\begin{array}{l} \lim  _ {T \rightarrow \infty} \hat {p} _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} \mid \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) (40) \\ \underline {{\underline {{\operatorname {E q} \left[ 3 5 \right]}}}} \frac {\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k , s)} ^ {j}} \left(x _ {t} ^ {j} , \mathrm {P a} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k , s)} ^ {j}} \left(\mathrm {P a} \left(x _ {t} ^ {j}\right)\right)} (41) \\ = \frac {\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k , s)} ^ {j}} \left(x _ {t} ^ {j} \mid \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) p _ {t \in \pi_ {(k , s)} ^ {j}} \left(\operatorname {P a} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k , s)} ^ {j}} \left(\operatorname {P a} \left(x _ {t} ^ {j}\right)\right)} (42) \\ \underline {{\frac {p _ {t \in \pi_ {(k , s)} ^ {j}} \left(x _ {t} ^ {j} \mid \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) \text {a r e s a m e f o r a l l} s}{\sum_ {s = 1} ^ {\delta} p _ {t \in \pi_ {(k , s)} ^ {j}} \left(\operatorname {P a} \left(x _ {t} ^ {j}\right)\right)}} \frac {p _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} \mid \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k , s)} ^ {j}} \left(\operatorname {P a} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {s = 1} ^ {\delta} p _ {t \in \pi_ {(k , s)} ^ {j}} \left(\operatorname {P a} \left(x _ {t} ^ {j}\right)\right)}} (43) \\ = p _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} \mid \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) (44) \\ \end{array}
$$
Hence,  $\hat{p}_{t\in \Pi_k^j}(x_t^j |\mathrm{Pa}(x_t^j))$  is a consistent estimator of  $p_{t\in \Pi_k^j}(x_t^j |\mathrm{Pa}(x_t^j))$ .
Similarly, we construct an estimator of  $p(x_{t}^{j}|\cup_{h}\mathrm{Pa}_{h}(x_{t}^{j}))$  where  $t\in [T]$ :
$$
\begin{array}{l} \hat {p} \left(x _ {t} ^ {j} \mid \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right) = \sum_ {t} \mathbb {1} \left(x _ {t} ^ {j} \mid \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right) (45) \\ = \frac {\sum_ {t} \mathbb {1} \left(x _ {t} ^ {j} , \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {t} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)}. (46) \\ \end{array}
$$
We will prove that this estimator is converged as  $T$  goes to infinity in Lemma D.2. Hence, it is a consistent estimator.
In this section, we have proved that  $\hat{p}(x_t^j, \mathrm{Pa}(x_t^j))$  in Eq.(22) is a consistent estimator of  $p(x_t^j, \mathrm{Pa}(x_t^j))$  using samples with  $t$  in the same homogenous time partition subset and  $\hat{p}(x_t^j | \mathrm{Pa}(x_t^j))$  in Eq.(37) is a consistent estimator of  $p(x_t^j | \mathrm{Pa}(x_t^j))$  using samples with  $t$  in the same time partition subset.
D Theorem
Theorem D.1. Let  $\widehat{\mathcal{G}}$  be the estimated graph using the Algorithm  $PCMCI_{\Omega}$ . Under assumptions A1-A7 and with an oracle (infinite sample size limit), we have that:
$$
\widehat {\mathcal {G}} = \mathcal {G} \tag {47}
$$
Lemma D.2. Denote that  $\{\mathrm{Pa}_k(X_t^j)\}_{k\in [\omega_j]}$  contain the true and illusory parent sets, where  $\omega_{j}$  is the true periodicity of  $\mathbf{X}^j$ . For any random variable  $X_{t}^{j}$  with large enough  $t$ , under assumptions A1-A7 and with an oracle (infinite sample size limit), we have:
$$
p \left(p \left(X _ {t} ^ {j} \mid \cup_ {k = 1} ^ {\omega_ {j}} \operatorname {P a} _ {k} \left(X _ {t} ^ {j}\right)\right) \neq p \left(X _ {t} ^ {j} \mid \cup_ {k = 1} ^ {\omega_ {j}} \operatorname {P a} _ {k} \left(X _ {t} ^ {j}\right) \backslash y\right)\right) = 1, \forall y \in \cup_ {k = 1} ^ {\omega_ {j}} \operatorname {P a} _ {k} \left(X _ {t} ^ {j}\right) \tag {48}
$$
Here,  $p(X_{t}^{j}|\cup_{k = 1}^{\omega_{j}}\mathrm{Pa}_{k}(X_{t}^{j})) = \lim_{T\to \infty}\hat{p} (X_{t}^{j}|\cup_{k = 1}^{\omega_{j}}\mathrm{Pa}_{k}(X_{t}^{j}))$
Proof. We first prove that there exist a sequence of coefficients  $\{\alpha_{k}\}_{k\in [\omega_{j}]}$  satisfying  $\sum_{k = 1}^{\omega_j}\alpha_k = 1$  so that:
$\forall$  configuration  $\cup_{h}\mathrm{Pa}_{h}(x_{t}^{j})$
$$
\hat {p} \left(x _ {t} ^ {j} \mid \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right) = \sum_ {k = 1} ^ {\omega_ {j}} \alpha_ {k} \hat {p} _ {k} \left(x _ {t} ^ {j} \mid \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) \tag {49}
$$
If this is correct, then  $\hat{p}(x_t^j | \cup_h \mathrm{Pa}_h(x_t^j))$  would be a consistent estimator of  $p(x_t^j | \cup_h \mathrm{Pa}_h(x_t^j))$ . Based on Eq.(46), we have:
$$
\text {w h e r e} \alpha_ {k} (T) = \frac {\sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {t} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)}. \tag {61}
$$
Using the same logic in Eq.(30)-(35), we can decompose the numerator and denominator of  $\alpha_{k}$  with homogenous time partition until each component converges to a stationary distribution.
$$
\alpha_ {k} (T) = \frac {\sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {k} \sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)} \tag {62}
$$
$$
= \frac {\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} \frac {T}{\delta} \sum_ {t \in \pi_ {(k , s)} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {k} \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} \frac {T}{\delta} \sum_ {t \in \pi_ {(k , s)} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)} \tag {63}
$$
$$
\lim  _ {T \rightarrow \infty} \alpha_ {k} (T) = \frac {\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi^ {j} _ {(k , s)}} \left(\cup_ {h} \mathrm {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {k} \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi^ {j} _ {(k , s)}} \left(\cup_ {h} \mathrm {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)} \tag {64}
$$
Without loss of generality, assume  $y \in \mathrm{Pa}(x_t^j)$ , where  $t \in \Pi_j^1$  and  $y \notin \mathrm{Pa}(x_t^j)$ , where  $t \notin \Pi_j^1$ . Then we have
$$
\hat {p} \left(x _ {t} ^ {j} \mid \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right) = \frac {\sum_ {t} \mathbb {1} \left(x _ {t} ^ {j} , \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right)}{\sum_ {t} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right)} \tag {65}
$$
$$
= \sum_ {k = 2} ^ {\omega_ {j}} \left(\hat {p} _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} \mid \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right) \frac {\sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right)}{\sum_ {t} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right)}\right) \tag {66}
$$
$$
+ \frac {\sum_ {t \in \Pi_ {1} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j} , \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \setminus y\right)}{\sum_ {t \in \Pi_ {1} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \setminus y\right)} \frac {\sum_ {t \in \Pi_ {1} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \setminus y\right)}{\sum_ {t} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \setminus y\right)}
$$
$$
= \sum_ {k = 2} ^ {\omega_ {j}} \beta_ {k} (T) \hat {p} _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} | \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) + \beta_ {1} (T) \hat {p} _ {t \in \Pi_ {1} ^ {j}} \left(x _ {t} ^ {j} | \operatorname {P a} \left(x _ {t} ^ {j}\right) \backslash y\right) \tag {67}
$$
$$
\text {w h e r e} \beta_ {k} (T) = \frac {\sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right)}{\sum_ {t} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right)} \tag {68}
$$
Similarly, we have:
$$
\beta_ {k} (T) = \frac {\sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right)}{\sum_ {k} \sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right)} \tag {69}
$$
$$
= \frac {\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} \frac {T}{\delta} \sum_ {t \in \pi_ {(k , s)} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right)}{\sum_ {k} \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} \frac {T}{\delta} \sum_ {t \in \pi_ {(k , s)} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right)} \tag {70}
$$
$$
\lim  _ {T \rightarrow \infty} \beta_ {k} (T) = \frac {\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k , s)} ^ {j}} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right)}{\sum_ {k} \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k , s)} ^ {j}} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right)} \tag {71}
$$
Proving  $p(x_{t}^{j}| \cup_{h} \mathrm{Pa}_{h}(x_{t}^{j})) \neq p(x_{t}^{j}| \cup_{h} \mathrm{Pa}_{h}(x_{t}^{j}) \setminus y)$  is equal to proving:
$$
p \left(x _ {t} ^ {j} \mid \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right) - p \left(x _ {t} ^ {j} \mid \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right) \neq 0 \tag {72}
$$
Substitutes Eq. (60) and Eq. (67) in Eq. (72), we have the following derivation:
$$
\begin{array}{l} p \left(x _ {t} ^ {j} \mid \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right) - p \left(x _ {t} ^ {j} \mid \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right) (73) \\ = \lim  _ {T \rightarrow \infty} \left(\sum_ {k = 1} ^ {\omega_ {j}} \alpha_ {k} (T) \hat {p} _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} \mid \operatorname {P a} \left(x _ {t} ^ {j}\right)\right)\right) - (74) \\ \end{array}
$$
$$
\begin{array}{l} \lim  _ {T \rightarrow \infty} \left(\sum_ {k = 2} ^ {\omega_ {j}} \beta_ {k} (T) \hat {p} _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} | \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) + \beta_ {1} (T) \hat {p} _ {t \in \Pi_ {1} ^ {j}} \left(x _ {t} ^ {j} | \operatorname {P a} \left(x _ {t} ^ {j}\right) \setminus y\right)\right) \\ = \sum_ {k = 1} ^ {\omega_ {j}} \frac {\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k , s)} ^ {j}} \left(\cup_ {h} \mathrm {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {k} \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k , s)} ^ {j}} \left(\cup_ {h} \mathrm {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)} p _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} | \mathrm {P a} \left(x _ {t} ^ {j}\right)\right) \tag {75} \\ - \Bigg(\sum_{k = 2}^{\omega_{j}}\frac{\sum_{s = 1}^{\frac{\delta}{\omega_{j}}}p_{t\in\pi^{j}_{(k,s)}}(\cup_{h}\mathrm{Pa}_{h}(x^{j}_{t})\setminus y)}{\sum_{k}\sum_{s = 1}^{\frac{\delta}{\omega_{j}}}p_{t\in\pi^{j}_{(k,s)}}(\cup_{h}\mathrm{Pa}_{h}(x^{j}_{t})\setminus y)} p_{t\in\Pi^{j}_{k}}(x^{j}_{t}|\mathrm{Pa}(x^{j}_{t})) \\ + \frac {\sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \Pi_ {(1 , s)} ^ {j}} \left(\cup_ {h} \mathrm {P a} _ {h} \left(x _ {t} ^ {j}\right) \setminus y\right)}{\sum_ {k} \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k , s)} ^ {j}} \left(\cup_ {h} \mathrm {P a} _ {h} \left(x _ {t} ^ {j}\right) \setminus y\right)} p _ {t \in \Pi_ {1} ^ {j}} \left(x _ {t} ^ {j} | \mathrm {P a} \left(x _ {t} ^ {j}\right) \setminus y\right)) \\ \end{array}
$$
After equating the denominators, the numerator is:
$$
\begin{array}{l} \left(\sum_ {k = 1} ^ {\omega_ {j}} \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k, s)} ^ {j}} \left(\cup_ {h} \mathrm {P a} _ {h} \left(x _ {t} ^ {j}\right)\right) p _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} | \mathrm {P a} \left(x _ {t} ^ {j}\right)\right)\right) \left(\sum_ {k = 1} ^ {\omega_ {j}} \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k, s)} ^ {j}} \left(\cup_ {h} \mathrm {P a} _ {h} \left(x _ {t} ^ {j}\right) \setminus y\right)\right) \\ - \left(\sum_ {k = 2} ^ {\omega_ {j}} \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k, s)} ^ {j}} \left(\cup_ {h} \mathrm {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right) p _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} | \mathrm {P a} \left(x _ {t} ^ {j}\right)\right) \right. \\ \left. + \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \Pi_ {(1, s)} ^ {j}} \left(\cup_ {h} \mathrm {P a} _ {h} \left(x _ {t} ^ {j}\right) \setminus y\right) p _ {t \in \Pi_ {1} ^ {j}} \left(x _ {t} ^ {j} | \mathrm {P a} \left(x _ {t} ^ {j}\right) \setminus y\right)\right) \\ \times \left(\sum_ {k = 1} ^ {\omega_ {j}} \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k, s)} ^ {j}} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)\right) \tag {76} \\ \end{array}
$$
For the sake of simplicity, denote
$$
a _ {k} := \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k, s)} ^ {j}} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right) \tag {77}
$$
$$
b _ {k} := \sum_ {s = 1} ^ {\frac {\delta}{\omega_ {j}}} p _ {t \in \pi_ {(k, s)} ^ {j}} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right) \tag {78}
$$
$$
c _ {k} := p _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} \mid \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) \tag {79}
$$
$$
c _ {1} ^ {\prime} := p _ {t \in \Pi_ {1} ^ {j}} \left(x _ {t} ^ {j} \mid \operatorname {P a} \left(x _ {t} ^ {j}\right) \backslash y\right) \tag {80}
$$
After substituting the simple notations in Eq.(76):
$$
\begin{array}{l} \left(\sum_ {k = 1} ^ {\omega_ {j}} a _ {k} c _ {k}\right) \left(\sum_ {k = 1} ^ {\omega_ {j}} b _ {k}\right) - \left(\sum_ {k = 2} ^ {\omega_ {j}} b _ {k} c _ {k} + b _ {1} c _ {1} ^ {\prime}\right) \left(\sum_ {k = 1} ^ {\omega_ {j}} a _ {k}\right) (81) \\ = \sum_ {k = 1} ^ {\omega_ {j}} \left(c _ {k} - c _ {1 ^ {\prime}}\right) a _ {k} b _ {1} + \sum_ {k = 1} ^ {\omega_ {j}} \sum_ {i > 1, i \neq k} ^ {\omega_ {j}} \left(c _ {k} - c _ {i}\right) a _ {k} b _ {i} (82) \\ = b _ {1} \sum_ {k = 1} ^ {\omega_ {j}} c _ {k} a _ {k} - c _ {1 ^ {\prime}} b _ {1} \sum_ {k = 1} ^ {\omega_ {j}} a _ {k} + \sum_ {k = 1} ^ {\omega_ {j}} c _ {k} a _ {k} \sum_ {i > 1, i \neq k} ^ {\omega_ {j}} b _ {i} - \sum_ {k = 1} ^ {\omega_ {j}} a _ {k} \sum_ {i > 1, i \neq k} ^ {\omega_ {j}} c _ {i} b _ {i} (83) \\ \end{array}
$$
$$
V _ {t} = \left\{\mathbf {X} _ {t ^ {\prime}} \mid 0 <   t ^ {\prime} <   t \right\} \tag {84}
$$
That is,  $V_{t}$  contains all the nodes before time point  $t$ .
Denote  $\{b_{t_i}\}_{i\in [n]} = \cup_h\mathrm{Pa}_h(x_t^j)$  and assume  $\{b_{t_i}\}_{1\leq i\leq n_1 <   n} = \mathrm{Pa}_1(x_t^j)$ , where  $t\in \Pi^{j}_{(k,s)}$
We express  $p_{t \in \Pi_{(k,s)}^j}(\cup_h \mathrm{Pa}_h(x_t^j))$  by marginalizing all other random variables occurring before the latest variables in  $\cup_h \mathrm{Pa}_h(x_t^j)$  and utilizing the Causal Markov assumption (A2):
$$
\begin{array}{l} p _ {t \in \Pi_ {(k, s)} ^ {j}} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right) (85) \\ = p \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \mid t \in \Pi_ {(k, s)} ^ {j}\right) (86) \\ = \sum_ {V _ {h} \backslash \left\{b _ {t _ {i}} \right\} _ {i \in [ n ]}} p \left(b _ {t _ {1}}, b _ {t _ {2}}, \dots b _ {t _ {n}}, V _ {h} \backslash \left\{b _ {t _ {i}} \right\} _ {i \in [ n ]} \mid h = \max  \left\{t _ {i}, 1 \leq i \leq n \right\}, t \in \Pi_ {(k, s)} ^ {j}\right) (87) \\ = \sum_ {\left\{\mathrm {P a} \left(b _ {t _ {i}}\right) \right\} _ {i \in [ n ]}} p \left(b _ {t _ {i}} \mid \mathrm {P a} \left(b _ {t _ {i}}\right)\right) \sum_ {V _ {\tau_ {\max }} \sum_ {\tau_ {\max } <   t ^ {\prime} \leq h} \sum_ {j \in [ n ]} \sum_ {x _ {t ^ {\prime}} ^ {j}, \mathrm {P a} \left(x _ {t ^ {\prime}} ^ {j}\right)}} p \left(x _ {t ^ {\prime}} ^ {j} \mid \mathrm {P a} \left(x _ {t ^ {\prime}} ^ {j}\right)\right) p \left(V _ {\tau_ {\max }}\right) (88) \\ \end{array}
$$
Note that  $x_{t'}^j \in V_h \setminus \{b_{t_i}\}_{i \in [n]}$ .
This joint distribution is now represented by conditional distributions of one related variable given its parents.
Similarly, assume  $y = b_{t_1}$ , we have
$$
\begin{array}{l} p _ {t \in \Pi_ {(k, s)} ^ {j}} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash y\right) (89) \\ = p _ {t \in \Pi_ {(k, s)} ^ {j}} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash b _ {t _ {1}}\right) (90) \\ = p \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j} \backslash y\right) \mid t \in \Pi_ {(k, s)} ^ {j}\right) (91) \\ = \sum_ {V _ {t _ {n}} \backslash \left\{b _ {t _ {i}} \right\} _ {i \neq 1}} p \left(b _ {t _ {2}}, \dots b _ {t _ {n - 1}}, b _ {t _ {n}}, V _ {t _ {n}} \backslash \left\{b _ {t _ {i}} \right\} _ {i \neq 1} \mid h = \max  \left\{t _ {i}, 2 \leq i \leq n \right\}\right) (92) \\ = \sum_ {\left\{\mathrm {P a} \left(b _ {t _ {i}}\right) \right\} _ {i \neq 1}} p \left(b _ {t _ {i}} \mid \mathrm {P a} \left(b _ {t _ {i}}\right)\right) \sum_ {V _ {\tau_ {\max }}} \sum_ {\tau_ {\max } <   t ^ {\prime} \leq h} \sum_ {j \in [ n ]} \sum_ {x _ {t ^ {\prime}} ^ {j}, \mathrm {P a} \left(x _ {t ^ {\prime}} ^ {j}\right)} p \left(x _ {t ^ {\prime}} ^ {j} \mid \mathrm {P a} \left(x _ {t ^ {\prime}} ^ {j}\right)\right) p \left(V _ {\tau_ {\max }}\right) (93) \\ \end{array}
$$
Note that  $x_{t'}^j \in V_{t_n} \setminus \{b_{t_i}\}_{i \neq 1}$ .
$p_{t\in \Pi_1^j}(x_t^j |\mathrm{Pa}(x_t^j)\setminus y)$  can also be represented by those conditional distributions based on Bayes rule.