Introduction

Let $U=\{x_1,\ldots,x_n\}$, $n\ge 1$ be a set of variables. A Bayesian network $B$ over a set of variables $U$ is a network structure $B_S$, which is a directed acyclic graph (DAG) over $U$ and a set of probability tables $B_P = \{p(u\vert pa(u)) \vert u\in U\}$ where $pa(u)$ is the set of parents of $u$ in $B_S$. A Bayesian network represents a probability distributions $P(U) = \prod_{u\in U}p(u\vert pa(u))$.

Below, a Bayesian network is shown for the variables in the iris data set. Note that the links between the nodes class, petallength and petalwidth do not form a directed cycle, so the graph is a proper DAG.

This picture just shows the network structure of the Bayes net, but for each of the nodes a probability distribution for the node given its parents are specified as well. For example, in the Bayes net above there is a conditional distribution for petallength given the value of class. Since class has no parents, there is an unconditional distribution for sepalwidth.