Editing Bayesian network (section)

===Local Markov property===
''X'' is a Bayesian network with respect to ''G'' if it satisfies the ''local Markov property'': each variable is [[Conditional independence|conditionally independent]] of its non-descendants given its parent variables:{{sfn|Russell|Norvig|2003|p=499}}

:<math> X_v \perp\!\!\!\perp X_{V \,\smallsetminus\, \operatorname{de}(v)} \mid X_{\operatorname{pa}(v)} \quad\text{for all }v \in V</math>

where de(''v'') is the set of descendants and ''V''&nbsp;\&nbsp;de(''v'') is the set of non-descendants of ''v''.

This can be expressed in terms similar to the first definition, as

:<math>
\begin{align}
& \operatorname P(X_v=x_v \mid  X_i=x_i \text{ for each } X_i \text{ that is not a descendant of } X_v\, ) \\[6pt]
= {} & P(X_v=x_v \mid X_j=x_j \text{ for each } X_j \text{ that is a parent of } X_v\, )
\end{align}
</math>

The set of parents is a subset of the set of non-descendants because the graph is [[Cycle (graph theory)|acyclic]].