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, ..., Xn,
then the joint probability
- P(X1, ..., Xn),
is equal to the product of the conditional probabilities
- P(Xi | parents of Xi) 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, also called Markov networks, in which graph separation encodes conditional independencies (these are also known as graphical Gaussian models, or GGMs).
A recent application of graphical models is to describe gene regulatory networks.
See also belief propagation.