A Bayesian network is a carrier of the conditional independencies of a set of variables, not of their causal connections.

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Once fully specified, a Bayesian network compactly represents the joint probability distribution (JPD) and, thus, can be used for computing the posterior probabilities of any subset of variables given evidence about any other subset.

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The minimal set of nodes which d-separates node X from all other nodes is given by Xs Markov blanket.

A path p (allowing paths that are not directed) is said to be d-separated (or blocked) by a set of nodes Z if and only if one of the following holds: A set Z is said to d-separate x from y in a directed acyclic graph G if all paths from x to y in G are d-separated by Z.

Probabilistic models based on directed acyclic graphs (DAG) have a long and rich tradition, beginning with the work of geneticist Sewall Wright in the 1920s. Within statistics, such models are known as directed graphical models; within cognitive science and artificial intelligence, such models are known as Bayesian networks. Thomas Bayes (1702-1761), whose rule for updating probabilities in the light of new evidence is the foundation of the approach. Bayes addressed both the case of discrete probability distributions of data and the more complicated case of continuous probability distributions.

In the discrete case, Bayesâ€™ theorem relates the conditional and marginal probabilities of events.Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called influence diagrams. A directed acyclic graph is a Bayesian Network relative to a set of variables if the joint distribution of the node values can be written as the product of the local distributions of each node and its parents: has no parents, its local probability distribution is said to be unconditional, otherwise it is conditional.If the value of a node is observed, then the node is said to be an evidence node.Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.The term "Bayesian networks" was coined by Pearl (1985) to emphasize three aspects: Formally, Bayesian networks are directed acyclic graphs whose nodes represent variables, and whose arcs encode conditional independencies between the variables.Efficient algorithms exist that perform inference and learning in Bayesian networks.