Two generalizations of Markov blankets

In a probabilistic graphical model on a set of variables $V$, the Markov blanket of a random vector $B$ is the minimal set of variables conditioned to which $B$ is independent from the remaining of the variables $V \backslash B$. We generalize Markov blankets to study how a set $C$ of variables of interest depends on~$B$. Doing that, we must choose if we authorize vertices of $C$ or vertices of $V \backslash C$ in the blanket. We therefore introduce two generalizations. The Markov blanket of $B$ in $C$ is the minimal subset of $C$ conditionally to which $B$ and $C$ are independent. It is naturally interpreted as the inner boundary through which $C$ depends on $B$, and finds applications in feature selection. The Markov blanket of $B$ in the direction of $C$ is the nearest set to $B$ among the minimal sets conditionally to which ones $B$ and $C$ are independent, and finds applications in causality. It is the outer boundary of $B$ in the direction of $C$. We provide algorithms to compute them that are not slower than the usual algorithms for finding a d-separator in a directed graphical model. All our definitions and algorithms are provided for directed and undirected graphical models.