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.[1]
Markov blanket
A Markov blanket of a random variable Y in a random variable set \( {\displaystyle {\mathcal {S}}=\{X_{1},\ldots ,X_{n}\}} \) is any subset \( {\displaystyle {\mathcal {S}}_{1}} \) of \( {\mathcal {S}} \) , conditioned on which other variables are independent with Y:
\( {\displaystyle Y\perp \!\!\!\perp {\mathcal {S}}\backslash {\mathcal {S}}_{1}\mid {\mathcal {S}}_{1}.} \)
It means that \( {\displaystyle {\mathcal {S}}_{1}} \) contains all the information one needs to infer Y, and the variables in \( {\displaystyle {\mathcal {S}}\backslash {\mathcal {S}}_{1}} \) are redundant.
In general, Markov blanket is not unique. Any set in \( {\mathcal {S}} \) that contains a Markov blanket is also a Markov blanket itself. Specifically, \( {\mathcal {S}} \) is a Markov blanket of Y in \( {\mathcal {S}} \) .
Markov boundary
A Markov boundary of Y in \( {\mathcal {S}} \) is a subset \( {\displaystyle {\mathcal {S}}_{2}} \) of \( {\mathcal {S}} \) , that \( {\displaystyle {\mathcal {S}}_{2}} \) itself is a Markov blanket of Y, but any proper subset of \( {\displaystyle {\mathcal {S}}_{2}} \) is not a Markov blanket of Y. In other words, a Markov boundary is a minimal Markov blanket.
The Markov boundary of a node A in a Bayesian network is the set of nodes composed of A's parents, A's children, and 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, there are some theoretical and practical cases with multiple Markov boundaries[2]. When there are multiple Markov boundaries, quantities measuring causal effect could fail[3].
See also
Andrey Markov
Free energy minimisation
Moral graph
Separation of concerns
Causality
Causal inference
Notes
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.
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.
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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