Editing Markov blanket

{{Short description|Subset of variables that contains all the useful information}}
[[Image:Diagram of a Markov blanket.svg|frame|In a [[Bayesian network]], the Markov boundary of node ''A'' includes its parents, children and the other parents of all of its children.]]

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.<ref>{{cite book |last=Pearl |first=Judea |authorlink=Judea Pearl |title=Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference |publisher=Morgan Kaufmann |location=San Mateo CA |year=1988 |isbn=0-934613-73-7 |series=Representation and Reasoning Series |url-access=registration |url=https://archive.org/details/probabilisticrea00pear }}</ref> A Markov blanket can be constituted by a set of [[Markov chain]]s.<!--[[Markov chain#Testing]]-->

== Markov blanket ==

A Markov blanket of a random variable <math>Y</math> in a random variable set <math>\mathcal{S}=\{X_1,\ldots,X_n\}</math> is any subset <math>\mathcal{S}_1</math> of <math>\mathcal{S}</math>, conditioned on which other variables are independent with <math>Y</math>:

<math display="block">Y\perp \!\!\! \perp\mathcal{S}\backslash\mathcal{S}_1 \mid \mathcal{S}_1.</math>

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

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

== Markov boundary ==
A '''Markov boundary''' of <math>Y</math> in <math>\mathcal{S}</math> is a subset <math>\mathcal{S}_2</math> of <math>\mathcal{S}</math>, such that <math>\mathcal{S}_2</math> itself is a Markov blanket of <math>Y</math>, but any proper subset of <math>\mathcal{S}_2</math> is not a Markov blanket of <math>Y</math>. In other words, a Markov boundary is a minimal Markov blanket.

The Markov boundary of a [[Vertex (graph theory)|node]] <math>A</math> in a [[Bayesian network]] is the set of nodes composed of <math>A</math>'s parents, <math>A</math>'s children, and <math>A</math>'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 (graphical model)|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.<ref>{{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}}</ref> When there are multiple Markov boundaries, quantities measuring causal effect could fail.<ref>{{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}}</ref>

== See also ==
* [[Andrey Markov]]
* [[Free_energy_principle#Free energy minimisation|Free energy minimisation]]
* [[Moral graph]]
* [[Separation of concerns]]
* [[Causality]]
* [[Causal inference]]

==Notes==
<references/>

[[Category:Bayesian networks]]
[[Category:Markov networks]]