In probability theory, statistics, and machine learning, a graphical model (GM) is a graph that represents dependencies among random variables by a graph in which each node is a random variable, and the edges between the nodes represent conditional dependencies.
Two common types of GMs correspond to graphs with directed and undirected edges. If the network structure of the model is a directed acyclic graph (DAG), 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. Classic machine learning models like hidden Markov models, neural networks and newer models such as variable-order Markov models can be considered as special cases of Bayesian networks.
Graphical models with undirected edges are generally called Markov random fields or Markov networks. It can be shown that they have the same representational capacity as directed graphical models. However, while directed models are better at explicitly representing the joint probability, undirected models are better for representing conditional independences.
Applications of graphical models include modeling of gene regulatory networks, speech recognition, gene finding, computer vision and diagnosis of diseases.
A good reference for learning the basics of graphical models is written by Neapolitan, Learning Bayesian networks (2004). A more advanced and statistically oriented book is by Cowell, Dawid, Lauritzen and Spiegelhalter, Probabilistic networks and expert systems (1999).