Editing Markov random field (section)

== Definition ==
Given an undirected graph <math>G=(V,E)</math>, a set of random variables <math>X = (X_v)_{v\in V}</math> indexed by <math>V</math>&nbsp; form a Markov random field with respect to <math>G</math>&nbsp; if they satisfy the local Markov properties:

:Pairwise Markov property: Any two non-adjacent variables are [[conditional independence|conditionally independent]] given all other variables:

::<math>X_u \perp\!\!\!\perp X_v \mid X_{V \smallsetminus \{u,v\}} </math>

:Local Markov property: A variable is conditionally independent of all other variables given its neighbors:

::<math>X_v \perp\!\!\!\perp X_{V\smallsetminus \operatorname{N}[v]} \mid X_{\operatorname{N}(v)}</math>
:where <math display="inline">\operatorname{N}(v)</math> is the set of neighbors of <math>v</math>, and <math>\operatorname{N}[v] = v \cup \operatorname{N}(v)</math> is the [[Neighborhood (graph theory)|closed neighbourhood]] of <math>v</math>.

:Global Markov property: Any two subsets of variables are conditionally independent given a separating subset:

::<math>X_A \perp\!\!\!\perp X_B \mid X_S</math>
:where every path from a node in <math>A</math> to a node in <math>B</math> passes through <math>S</math>.

The Global Markov property is stronger than the Local Markov property, which in turn is stronger than the Pairwise one.<ref>{{cite book |last1=Lauritzen |first1=Steffen |title=Graphical models |date=1996 |publisher=Clarendon Press |location=Oxford |isbn=978-0198522195 |page=33}}</ref> However, the above three Markov properties are equivalent for positive distributions<ref>{{cite book|title=Probabilistic Graphical Models|last1=Koller|last2=Friedman|publisher=MIT Press|date=2009|isbn=9780262013192|first1=Daphne|first2=Nir|page=114-122}}</ref> (those that assign only nonzero probabilities to the associated variables).

The relation between the three Markov properties is particularly clear in the following formulation:

* Pairwise: For any <math>i, j \in V</math> not equal or adjacent, <math>X_i \perp\!\!\!\perp X_j | X_{V \smallsetminus \{i, j\}}</math>.
* Local: For any <math>i\in V</math> and <math>J\subset V</math> not containing or adjacent to <math>i</math>, <math>X_i \perp\!\!\!\perp X_J | X_{V \smallsetminus (\{i\}\cup J)}</math>.
* Global: For any <math>I, J\subset V</math> not intersecting or adjacent, <math>X_I \perp\!\!\!\perp X_J | X_{V \smallsetminus (I\cup J)}</math>.