Two generalizations of Markov blankets
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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 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 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.
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