In probability theory, statistics, and machine learning, a graphical model (GM) is a graph that represents independencies among random variables by a graph in which each node is a random variable, and the missing edges between the nodes represent conditional independencies.
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. Any two nodes that are not connected by an arrow are conditionally independent given the values of their parents. In general, any two sets of nodes are conditionally independent, given a third set if a criterion called "d-separation" holds in the graph. (link d-separation to wiki entry).
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.
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).
A computational reasoning approach is provided in Pearl, Probaiblistic Reasoning in Intelligence Systems (1988)[1] were the relationships between graphs and probabilities were formally introduced.
See also
Reference
- ^ Pearl, J. (1988) Probabilistic Reasoning in Intelligent Systems, (Revised Second Printing) San Mateo, CA: Morgan Kaufmann.