Markov blanket

A Markov blanket of a random variable $Y$ in a random variable set $\backslash mathcal\{S\}=\backslash \{X\_1,\backslash ldots,X\_n\backslash \}$ is any subset $\backslash mathcal\{S\}\_1$ of $\backslash mathcal\{S\}$, conditioned on which other variables are independent with $Y$:

$$Y\backslash perp\; \backslash !\backslash !\backslash !\; \backslash perp\backslash mathcal\{S\}\backslash backslash\backslash mathcal\{S\}\_1\; \backslash mid\; \backslash mathcal\{S\}\_1.$$

It means that $\backslash mathcal\{S\}\_1$ contains at least all the information one needs to infer $Y$, where the variables in $\backslash mathcal\{S\}\backslash backslash\backslash mathcal\{S\}\_1$ are redundant.

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

A Markov boundary of $Y$ in $\backslash mathcal\{S\}$ is a subset $\backslash mathcal\{S\}\_2$ of $\backslash mathcal\{S\}$, such that $\backslash mathcal\{S\}\_2$ itself is a Markov blanket of $Y$, but any proper subset of $\backslash mathcal\{S\}\_2$ 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.

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.{{cite journal |last1=Statnikov |first1=Alexander |last2=Lytkin |first2=Nikita I. |last3=Lemeire |first3=Jan |last4=Aliferis |first4=Constantin F. |title=Algorithms for discovery of multiple Markov boundaries |journal=Journal of Machine Learning Research |date=2013 |volume=14 |pages=499-566 |url=http://www.jmlr.org/papers/volume14/statnikov13a/statnikov13a.pdf}} When there are multiple Markov boundaries, quantities measuring causal effect could fail.{{cite journal |last1=Wang |first1=Yue |last2=Wang |first2=Linbo |title=Causal inference in degenerate systems: An impossibility result |journal=Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics |date=2020 |pages=3383-3392 |url=http://proceedings.mlr.press/v108/wang20i.html}}

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

Category:Bayesian networks

Category:Markov networks