$$
\begin{array}{l} \hat {p} \left(x _ {t} ^ {j} \mid \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right) (50) \\ = \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)} (51) \\ = \frac {\sum_ {t} \sum_ {k} \mathbb {1} \left(x _ {t} ^ {j} , \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right) \mathbb {1} \left(t \in \Pi_ {k} ^ {j}\right)}{\sum_ {t} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)} (52) \\ = \sum_ {k} \frac {\sum_ {t} \mathbb {1} \left(x _ {t} ^ {j} , \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right) \mathbb {1} \left(t \in \Pi_ {k} ^ {j}\right)}{\sum_ {t} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)} (53) \\ = \sum_ {k} \frac {\sum_ {t \in \Pi_ {k} ^ {j}} \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)} (54) \\ = \sum_ {k} \left(\frac {\sum_ {t \in \Pi_ {k} ^ {j}} \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} \overline {{\operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)}}\right)} \frac {\sum_ {t} \mathbb {1} \left(\cup_ {h} \overline {{\operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)}}\right)}{\sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)} \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)}\right) (55) \\ = \sum_ {k} \left(\frac {\sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(x _ {t} ^ {j} , \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)} \frac {\sum_ {t \in \Pi_ {k} ^ {j}} \mathbb {1} \left(\cup_ {k} \operatorname {P a} _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j}\right)\right)}{\sum_ {t} \mathbb {1} \left(\cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right)}\right) (56) \\ = \sum_ {k} \left(\hat {p} _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} \mid \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right)\right) \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)}\right) (57) \\ = \sum_ {k} \left(\hat {p} _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} \mid \operatorname {P a} \left(x _ {t} ^ {j}\right), \cup_ {h} \operatorname {P a} _ {h} \left(x _ {t} ^ {j}\right) \backslash \operatorname {P a} \left(x _ {t} ^ {j}\right)\right) \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)}\right) (58) \\ = \sum_ {k} \left(\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(\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)}\right) (59) \\ = \sum_ {k} \alpha_ {k} (T) \hat {p} _ {t \in \Pi_ {k} ^ {j}} \left(x _ {t} ^ {j} \mid \operatorname {P a} \left(x _ {t} ^ {j}\right)\right), (60) \\ \end{array}
$$