A Bayesian network or Bayesian belief network is a directed acyclic graph of nodes representing variables and arcs representing dependence relations among the variables. If there is an arc from node A to another node B then we say that A is a parent of B. If a node has a known value, it is said to be an evidence node. A node can represent any kind of variable, be it an observed measurement, a parameter, a latent variable, or a hypothesis. Nodes are not restricted to representing random variables; this is what is "Bayesian" about a Bayesian network.
A Bayesian network is a representation of the joint distribution over all the variables represented by nodes in the graph. Let the variables be X(1), ..., X(n). Let parents(A) be the parents of the node A. Then the joint distribution for X(1) through X(n) is represented as the product of the probability distributions p(X(i) | parents(X(i))) for i from 1 to n. If X has no parents, its probability distribution is said to be unconditional, otherwise it is conditional.
Questions about dependence among variables can be answered by studying the graph alone. It can be shown that the graphical notion called d-separation corresponds to the notion of conditional independence: if nodes X and Y are d-separated (given specified evidence nodes), then variables X and Y are independent given the evidence variables.
In order to carry out numerical calculations, it is necessary to further specify for each node X the probability distribution for X conditional on its parents. The distribution of X given its parents may have any form. However, it is common to work with discrete or Gaussian distributions, since that simplifies calculations.
The goal of inference is typically to find the conditional distribution of a subset of the variables, conditional on known values for some other subset (the evidence), and integrating over any other variables. Thus a Bayesian network can be considered a mechanism for automatically constructing extensions of Bayes' theorem to more complex problems.
See also: Machine learning