Graphical model

In probability theory and statistics, a graphical model (GM) represents dependencies among random variables by a graph in which each random variable is a node.

In the simplest case, the network structure of the model is a directed acyclic graph (DAG). Then the GM represents a factorization of the joint probability of all random variables. More precisely, if the events are

X₁, ..., X_n,

then the joint probability

P(X₁, ..., X_n),

is equal to the product of the conditional probabilities

P(X_i | parents of X_i) for i = 1,...,n.

In other words, the joint distribution factors into a product of conditional distributions. The graph structure indicates direct dependencies among random variables. Any two nodes that are not in a descendant/ancestor relationship are conditionally independent given the values of their parents. This type of graphical model is known as a directed graphical model, Bayesian network, or belief network. There are also undirected graphical models, a.k.a. Markov networks, in which graph separation encodes conditional independencies.