Markov blanket

A Markov blanket of a variable is a set that makes it conditionally independent of every remaining variable. In a Directed acyclic graph, the parents, children, and other parents of the children form the standard Markov blanket.

Under faithfulness of a directed acyclic graph, every conditioning set screening a vertex from all remaining vertices contains its graphical Markov blanket. A missing adjacent vertex leaves a one-edge active path; a missing other parent of a child leaves the active path $k\to h\leftarrow u$, since the child $h$ is necessarily conditioned. Thus the graphical blanket is the unique smallest screening set.

In a Directed acyclic graph, let $B$ contain the parents, children, and other parents of children of vertex $k$. Then $k$ is D-separated from all remaining vertices by $B$. Every path out of $k$ meets a conditioned noncollider either at a parent, at a child, or at another parent just after a conditioned child-collider. The global Markov property for a directed acyclic graph gives $Z_k\perp Z_{(B\cup \{k\}{)}^c}\mid Z_B$.