# Dependency Trees (Bayesian Network)

A Bayesian network $(G, X)$ is a dependency tree if $G$ is a directed tree . That is, every node has at most one parent. Therefore, there is a function $\pi: V \rightarrow V \cup \{\emptyset\}$ which defines every vertex’s parent or lack thereof, which gives

$$\mathbb{P}[X] = \left ( \prod_{v \in V, \pi(v) = \emptyset} \mathbb{P}[X_v] \right ) \prod_{v \in V, \pi(v) \in V} \mathbb{P}[X_v \vert X_{\pi(v)}].$$