Markov blanket

In a nutshell

In statistics and machine learning, a Markov blanket of a random variable is a set of variables that renders the variable conditionally independent of all other variables in the system. This concept is central in probabilistic graphical models and feature selection. If a Markov blanket is minimal—meaning that no variable in it can be removed without losing this conditional independence—it is called a Markov boundary. Identifying a Markov blanket or boundary allows for efficient inference and helps isolate relevant variables for prediction or causal reasoning. The terms Markov blanket and Markov boundary were coined by Judea Pearl in 1988. A Markov blanket may be derived from the structure of a probabilistic graphical model such as a Bayesian network or Markov random field.

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Definition

A Markov blanket of a random variable $Y$ in a random variable set $S = {X_{1}, \dots, X_{n}}$ is any subset $S_{1}$ of $S$ , conditioned on which other variables are independent with $Y$ :

$Y ⊥ ⊥ S ∖ S_{1} ∣ S_{1}$

It means that $S_{1}$ contains at least all the information one needs to infer $Y$ , where the variables in $S ∖ S_{1}$ are redundant.

In general, a given Markov blanket is not unique. Any set in $S$ that contains a Markov blanket is also a Markov blanket itself. Specifically, $S$ is a Markov blanket of $Y$ in $S$ .

Example

In a Bayesian network, the Markov blanket of a node consists of its parents, its children, and its children's other parents (i.e., co-parents). Knowing the values of these nodes makes the target node conditionally independent of the rest of the network. In a Markov random field, the Markov blanket of a node is simply its immediate neighbors.

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