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In statistics and machine learning, when one wants to infer a random variable with a set of variables, usually a subset is enough, and other variables are useless. Such a subset that contains all the useful information is called a Markov blanket. If a Markov blanket is minimal, meaning that it cannot drop any variable without losing information, it is called a Markov boundary. Identifying a Markov blanket or a Markov boundary helps to extract useful features. The terms of Markov blanket and Markov boundary were coined by Judea Pearl in 1988. A Markov blanket can be constituted by a set of Markov chains.

Markov blanket

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

$${\displaystyle Y\perp \!\!\!\perp {\mathcal {S}}\backslash {\mathcal {S}}_{1}\mid {\mathcal {S}}_{1}.}$$

It means that ${\displaystyle {\mathcal {S}}_{1}}$ contains at least all the information one needs to infer ${\displaystyle Y}$, where the variables in ${\displaystyle {\mathcal {S}}\backslash {\mathcal {S}}_{1}}$ are redundant.

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

Markov boundary

A Markov boundary of ${\displaystyle Y}$ in ${\displaystyle {\mathcal {S}}}$ is a subset ${\displaystyle {\mathcal {S}}_{2}}$ of ${\displaystyle {\mathcal {S}}}$, such that ${\displaystyle {\mathcal {S}}_{2}}$ itself is a Markov blanket of ${\displaystyle Y}$, but any proper subset of ${\displaystyle {\mathcal {S}}_{2}}$ is not a Markov blanket of ${\displaystyle Y}$. In other words, a Markov boundary is a minimal Markov blanket.

The Markov boundary of a node ${\displaystyle A}$ in a Bayesian network is the set of nodes composed of ${\displaystyle A}$'s parents, ${\displaystyle A}$'s children, and ${\displaystyle A}$'s children's other parents. In a Markov random field, the Markov boundary for a node is the set of its neighboring nodes. In a dependency network, the Markov boundary for a node is the set of its parents.

Uniqueness of Markov boundary

The Markov boundary always exists. Under some mild conditions, the Markov boundary is unique. However, for most practical and theoretical scenarios multiple Markov boundaries may provide alternative solutions. When there are multiple Markov boundaries, quantities measuring causal effect could fail.

See also

• Andrey Markov
• Free energy minimisation
• Moral graph
• Separation of concerns
• Causality
• Causal inference

Notes

1. Pearl, Judea (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Representation and Reasoning Series. San Mateo CA: Morgan Kaufmann. ISBN 0-934613-73-7.
2. Statnikov, Alexander; Lytkin, Nikita I.; Lemeire, Jan; Aliferis, Constantin F. (2013). "Algorithms for discovery of multiple Markov boundaries" (PDF). Journal of Machine Learning Research. 14: 499–566.
3. Wang, Yue; Wang, Linbo (2020). "Causal inference in degenerate systems: An impossibility result". Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics: 3383–3392.

• Bayesian networks
• Markov networks