Two generalizations of Markov blankets

TL;DR

This paper introduces two new generalizations of Markov blankets to analyze dependencies between variable sets in graphical models, with algorithms applicable to both directed and undirected models.

Contribution

It defines the Markov blanket of a set in a subset and in a direction, providing algorithms for their computation in graphical models.

Findings

Two new types of Markov blankets are defined and characterized.

Algorithms for computing these generalized blankets are developed.

Applications include feature selection and causality analysis.

Abstract

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.