Editing Markov random field (section)

== Clique factorization ==
As the Markov property of an arbitrary probability distribution can be difficult to establish, a commonly used class of Markov random fields are those that can be factorized according to the [[Clique (graph theory)|clique]]s of the graph.

Given a set of random variables <math>X = (X_v)_{v\in V}</math>, let <math>P(X=x)</math> be the [[Probability density function|probability]] of a particular field configuration <math>x</math> in&nbsp;<math>X</math>—that is, <math>P(X=x)</math> is the probability of finding that the random variables <math>X</math> take on the particular value <math>x</math>.  Because <math>X</math> is a set, the probability of <math>x</math> should be understood to be taken with respect to a ''joint distribution'' of the <math>X_v</math>.

If this joint density can be factorized over the cliques of <math>G</math> as

:<math>P(X=x) = \prod_{C \in \operatorname{cl}(G)} \varphi_C (x_C) </math>

then <math>X</math> forms a Markov random field with respect to <math>G</math>.  Here, <math>\operatorname{cl}(G)</math> is the set of cliques of <math>G</math>.   The definition is equivalent if only maximal cliques are used. The functions <math>\varphi_C</math> are sometimes referred to as ''factor potentials'' or ''clique potentials''. Note, however, conflicting terminology is in use: the word ''potential'' is often applied to the logarithm of <math>\varphi_C</math>.  This is because, in [[statistical mechanics]], <math>\log(\varphi_C)</math> has a direct interpretation as the [[potential energy]] of a [[Configuration space (physics)|configuration]]&nbsp;<math>x_C</math>.

Some MRF's do not factorize: a simple example can be constructed on a cycle of 4 nodes with some infinite energies, i.e. configurations of zero probabilities,<ref>{{cite journal
 |first=John |last=Moussouris
 |title=Gibbs and Markov random systems with constraints 
 |journal=Journal of Statistical Physics
 |volume=10 |issue=1 |pages=11–33 |year=1974
 |doi=10.1007/BF01011714 |mr=0432132
|hdl=10338.dmlcz/135184
 |bibcode=1974JSP....10...11M
 |s2cid=121299906
 |hdl-access=free}}</ref> even if one, more appropriately, allows the infinite energies to act on the complete graph on <math>V</math>.<ref>{{cite journal
 | last1 = Gandolfi | first1 = Alberto
 | last2 = Lenarda | first2 = Pietro
 |title= A note on Gibbs and Markov Random Fields with constraints and their moments
 |journal=Mathematics and Mechanics of Complex Systems
 |volume=4 |issue=3–4 |pages=407–422 |year=2016 | doi=10.2140/memocs.2016.4.407
 |doi-access=free }}</ref>

MRF's factorize if at least one of the following conditions is fulfilled:
* the density is strictly positive (by the [[Hammersley–Clifford theorem]])
* the graph is [[Chordal graph|chordal]] (by equivalence to a [[Bayesian network]])

When such a factorization does exist, it is possible to construct a [[factor graph]] for the network.