Editing Bayesian network (section)

===={{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.)