Class o [[Probabilistic Graphical Model|PGM]], not to be confused with [[Bayesian Neural Networks]]. --- Because of the directedness, we need additional results to connect graph structure and distribution, see [[Markov Equivalence Classes]]. >[!info] Definition - Bayesian Networks >A Bayesian Network is a pair $(\mathcal{G}, P)$ where $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ is a [[Graph|directed acyclic graph]] and each node $X_i$ carries a conditional probability distribution $P(X_i \mid \text{Pa}(X_i))$ suhc that the [[Probabilistic Graphical Model|Markov property]] is satisfied (is an I-map of the graph). The topology of the graph encodes how the joint factorizes via parents: $P(X_1, \dots, X_n) = \prod_{i=1}^{n} P(X_i \mid \text{Pa}(X_i))$