Editing Bayesian network (section)

===Markov blanket===
The [[Markov blanket]] of a node is the set of nodes consisting of its parents, its children, and any other parents of its children. The Markov blanket renders the node independent of the rest of the network; the joint distribution of the variables in the Markov blanket of a node is sufficient knowledge for calculating the distribution of the node. ''X'' is a Bayesian network with respect to ''G'' if every node is conditionally independent of all other nodes in the network, given its [[Markov blanket]].{{sfn|Russell|Norvig|2003|p=499}}

===={{anchor|d-separation}}''d''-separation====

This definition can be made more general by defining the "d"-separation of two nodes, where d stands for directional.<ref name=pearl2000/> We first define the "d"-separation of a trail and then we will define the "d"-separation of two nodes in terms of that.

Let ''P'' be a trail from node ''u'' to ''v''. A trail is a loop-free, undirected (i.e. all edge directions are ignored) path between two nodes. Then ''P'' is said to be ''d''-separated by a set of nodes ''Z'' if any of the following conditions holds:

*''P'' contains (but does not need to be entirely) a directed chain, <math> u \cdots \leftarrow m \leftarrow \cdots v</math> or <math> u \cdots \rightarrow m \rightarrow \cdots v</math>, such that the middle node ''m'' is in ''Z'',
*''P'' contains a fork, <math> u \cdots \leftarrow m \rightarrow \cdots v</math>, such that the middle node ''m'' is in ''Z'', or
*''P'' contains an inverted fork (or collider), <math> u \cdots \rightarrow m \leftarrow \cdots v</math>, such that the middle node ''m'' is not in ''Z'' and no descendant of ''m'' is in ''Z''.

The nodes ''u'' and ''v'' are ''d''-separated by ''Z'' if all trails between them are ''d''-separated. If ''u'' and ''v'' are not d-separated, they are d-connected.

''X'' is a Bayesian network with respect to ''G'' if, for any two nodes ''u'', ''v'':

: <math>X_u \perp\!\!\!\perp X_v \mid X_Z</math>

where ''Z'' is a set which ''d''-separates ''u'' and ''v''. (The [[Markov blanket]] is the minimal set of nodes which ''d''-separates node ''v'' from all other nodes.)