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In probability theory, statistics, and machine learning, a graphical model (GM) is a graph that represents independences among random variables. There are three main varieties. In the first two, each node is a random variable, and the missing edges between the nodes represent conditional independences; in the third, nodes can be either random variables or factors.
If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are
then the joint probability satisfies
![P[X_1,\ldots,X_n]=\prod_{i=1}^nP[X_i|pa_i]](http://upload.wikimedia.org/math/c/d/e/cdee2242c1e84db8c1cc05ee3f95fec7.png)
where pai is the set of parents of node Xi. In other words, the joint distribution factors into a product of conditional distributions. Any two nodes 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. Local independences and global independences are equivalent in Bayesian networks.
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 special cases of Bayesian networks.
Graphical models with undirected edges are generally called Markov random fields or Markov networks. A graphical model with many repeated subunits can be represented with plate notation.
A factor graph is an undirected bipartite graph connecting variables and factors. Each factor represents a probability distribution over the variables it is connected to. In contrast to a Bayesian network, a factor may be connected to more than two nodes.
Applications of graphical models include speech recognition, computer vision, decoding of low-density parity-check codes, modeling of gene regulatory networks, gene finding and diagnosis of diseases.
A good reference for learning the basics of graphical models is written by Neapolitan, Learning Bayesian networks (2004) and another is Finn Verner Jensen's An Introduction to Bayesian Networks from 1996.1 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 Judea Pearl's Probabilistic Reasoning in Intelligent Systems from 19882 where the relationships between graphs and probabilities were formally introduced.
See also
References
- ^ Finn Verner Jensen (1996). An Introduction to Bayesian Networks. New York: Springer Verlag. ISBN 0387915028.
- ^ Judea Pearl (1988). Probabilistic Reasoning in Intelligent Systems (Revised Second Printing ed.). San Mateo, CA: Morgan Kaufmann.
Others
- Graphical models, Chapter 8 of Pattern Recognition and Machine Learning by Christopher M. Bishop
- A Brief Introduction to Graphical Models and Bayesian Networks
- Heckerman's Bayes Net Learning Tutorial
- Edoardo M. Airoldi (2007). "Getting Started in Probabilistic Graphical Models". PLoS Computational Biology 3 (12): e252. doi:. http://compbiol.plosjournals.org/perlserv/?request=get-document&doi=10.1371/journal.pcbi.0030252&ct=1.
Wikipedia content modification information:
- This page was last modified on 1 January 2009, at 13:50.
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