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 of a random variable in a random variable set is any subset of , conditioned on which other variables are independent with :
It means that contains all the information one needs to infer , and the variables in are redundant.
In general, Markov blanket is not unique. Any set in that contains a Markov blanket is also a Markov blanket itself. Specifically, is a Markov blanket of in .
A Markov boundary of in is a subset of , that itself is a Markov blanket of , but any proper subset of is not a Markov blanket of . In other words, a Markov boundary is a minimal Markov blanket.
The Markov boundary of a node in a Bayesian network is the set of nodes composed of 's parents, 's children, and '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. When there are multiple Markov boundaries, quantities measuring causal effect could fail.
- Andrey Markov
- Free energy minimisation
- Moral graph
- Separation of concerns
- Causal inference
- 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.