Hammersley–Clifford theorem

E899013

result in mathematical statistics theorem in probability theory theorem in statistics

The Hammersley–Clifford theorem is a fundamental result in probability theory and statistics that links Markov random fields with Gibbs distributions by showing that, under positivity conditions, the Markov property is equivalent to factorization over cliques of an underlying graph.

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All labels observed (1)

Label	Occurrences
Hammersley–Clifford theorem canonical	1

Statements (48)

Predicate	Object
instanceOf	result in mathematical statistics ⓘ theorem in probability theory ⓘ theorem in statistics ⓘ
appliesTo	positive probability distributions ⓘ
assumes	finite set of random variables ⓘ strict positivity of the distribution ⓘ
category	theorem about Gibbs measures ⓘ theorem about Markov random fields ⓘ
concerns	cliques of a graph ⓘ undirected graphs ⓘ
equates	Gibbs random field NERFINISHED ⓘ Markov random field NERFINISHED ⓘ
equivalenceBetween	Gibbs factorization ⓘ Markov property ⓘ
field	Markov random fields NERFINISHED ⓘ graphical models ⓘ probability theory ⓘ statistical mechanics ⓘ statistics ⓘ
hasCondition	Markov property with respect to an undirected graph ⓘ positivity condition ⓘ
historicalContext	developed in the context of Gibbs fields and Markov random fields ⓘ
implies	local Markov property is equivalent to global Markov property under positivity ⓘ pairwise Markov property is equivalent to clique factorization under positivity ⓘ
importance	links conditional independence structure to factorization structure ⓘ provides theoretical foundation for undirected probabilistic graphical models ⓘ
namedAfter	John Michael Hammersley NERFINISHED ⓘ Peter Clifford NERFINISHED ⓘ
relatedTo	Dobrushin–Lanford–Ruelle equations NERFINISHED ⓘ Gibbs–Markov equivalence NERFINISHED ⓘ
relatesConcept	Gibbs distribution NERFINISHED ⓘ Gibbs measure NERFINISHED ⓘ Markov property ⓘ Markov random field NERFINISHED ⓘ clique factorization ⓘ clique potential ⓘ conditional independence ⓘ factorization of probability distributions ⓘ positivity condition ⓘ undirected graphical model ⓘ
states	a positive distribution that is Markov with respect to an undirected graph factorizes over the cliques of that graph ⓘ for positive distributions, the global Markov property is equivalent to factorization over cliques ⓘ
typicalFormulation	a strictly positive distribution on a finite set of variables is a Markov random field with respect to a graph if and only if it is a Gibbs distribution with respect to the cliques of that graph ⓘ
usedIn	Bayesian networks and graphical models theory NERFINISHED ⓘ Markov random field modeling ⓘ image analysis ⓘ spatial statistics ⓘ statistical physics ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Markov random fields → isCharacterizedBy → Hammersley–Clifford theorem ⓘ

subject surface form: Markov random field