About: Hammersley–Clifford theorem

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The Hammersley–Clifford theorem is a result in probability theory, mathematical statistics and statistical mechanics that gives necessary and sufficient conditions under which a strictly positive probability distribution (of events in a probability space) can be represented as events generated by a Markov network (also known as a Markov random field). It is the fundamental theorem of random fields. It states that a probability distribution that has a strictly positive mass or density satisfies one of the Markov properties with respect to an undirected graph G if and only if it is a Gibbs random field, that is, its density can be factorized over the cliques (or complete subgraphs) of the graph.

Attributes	Values
rdf:type	yago:WikicatMarkovNetworks yago:WikicatStatisticalTheorems yago:Abstraction100002137 yago:Communication100033020 yago:Group100031264 yago:Message106598915 yago:Network108434259 yago:Proposition106750804 yago:Statement106722453 yago:System108435388 yago:Theorem106752293 yago:WikicatProbabilityTheorems
rdfs:label	Teoremo de Hammersley–Clifford (eo) Hammersley–Clifford theorem (en)
rdfs:comment	La Teoremo de Hammersley–Clifford temas pri probabloteorio, statistiko kaj statistika mekaniko. Ĝi difinas la necesan kaj sufiĉan kondiĉojn tial, kial probabla distribuo povas reprezentiĝi kiel Markova reto. Ĝi estas la fundamenta teoremo pri hazarda kampo. La teoremo asertas, ke probablo distribuo, kiu havas strikte pozitivan mason aŭ denson, estas Markova al sendirekta grafeo G, se kaj nur se ĝi estas Gibsa. Tio estas, ĝia denso eblas faktoriĝi laŭ la klikoj de la grafeo. (eo) The Hammersley–Clifford theorem is a result in probability theory, mathematical statistics and statistical mechanics that gives necessary and sufficient conditions under which a strictly positive probability distribution (of events in a probability space) can be represented as events generated by a Markov network (also known as a Markov random field). It is the fundamental theorem of random fields. It states that a probability distribution that has a strictly positive mass or density satisfies one of the Markov properties with respect to an undirected graph G if and only if it is a Gibbs random field, that is, its density can be factorized over the cliques (or complete subgraphs) of the graph. (en)
foaf:depiction
dcterms:subject	Probability theorems Markov networks Theorems in statistics
Wikipage page ID	21963702 (xsd:integer)
Wikipage revision ID	1086768670 (xsd:integer)
Link from a Wikipage to another Wikipage	Probability distribution Probability space Roland Dobrushin Probability theorems Julian Besag University of Washington Complete graph Conditional independence Conditional random field Frank Spitzer Geoffrey Grimmett Markov networks Mathematical statistics Probability density function Probability theory Theorems in statistics Statistical mechanics John Hammersley Inclusion–exclusion principle Markov random field Probability mass function Gibbs random field Markov network dbr:Peter_Clifford_(statistician)
Link from a Wikipage to an external page	http://www.statslab.cam.ac.uk/~grg/books/pgs.html http://ssli.ee.washington.edu/courses/ee512/handout2.pdf%7Ctitle=Handout http://www.idi.ntnu.no/~helgel/thesis/forelesning.pdf
sameAs	Hammersley–Clifford theorem Hammersley–Clifford theorem Hammersley–Clifford theorem Hammersley–Clifford theorem
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Clarify dbt:Reflist dbt:Collapse_bottom dbt:Collapse_top
thumbnail	wiki-commons:Special:FilePath/A_simple_Markov_network.png?width=300
date	April 2016 (en)
reason	This is completely underspecified!! A probability distribution on what? Without specifying the graph, lattice, state, process, or other underlying structure where the "randomness" is occuring, the definition makes little sense. (en)
has abstract	La Teoremo de Hammersley–Clifford temas pri probabloteorio, statistiko kaj statistika mekaniko. Ĝi difinas la necesan kaj sufiĉan kondiĉojn tial, kial probabla distribuo povas reprezentiĝi kiel Markova reto. Ĝi estas la fundamenta teoremo pri hazarda kampo. La teoremo asertas, ke probablo distribuo, kiu havas strikte pozitivan mason aŭ denson, estas Markova al sendirekta grafeo G, se kaj nur se ĝi estas Gibsa. Tio estas, ĝia denso eblas faktoriĝi laŭ la klikoj de la grafeo. La rilaton inter Markova kaj Gibsa reto unue studis Roland Dobrushin kaj Frank Spitzer sub kunteksto de statistika mekaniko. La teoremo nomiĝas memore al John Hammersley kaj Peter Clifford, kiuj pruvis la ekvivalenton en nepublikigita raporto en 1971. Pli simpla pruvo per inkluziveco-ekskluda principo donis sendepende , Preston kaj Sherman en 1973, kun posta pruvo fare de en 1974. (eo) The Hammersley–Clifford theorem is a result in probability theory, mathematical statistics and statistical mechanics that gives necessary and sufficient conditions under which a strictly positive probability distribution (of events in a probability space) can be represented as events generated by a Markov network (also known as a Markov random field). It is the fundamental theorem of random fields. It states that a probability distribution that has a strictly positive mass or density satisfies one of the Markov properties with respect to an undirected graph G if and only if it is a Gibbs random field, that is, its density can be factorized over the cliques (or complete subgraphs) of the graph. The relationship between Markov and Gibbs random fields was initiated by Roland Dobrushin and Frank Spitzer in the context of statistical mechanics. The theorem is named after John Hammersley and , who proved the equivalence in an unpublished paper in 1971. Simpler proofs using the inclusion–exclusion principle were given independently by Geoffrey Grimmett, Preston and Sherman in 1973, with a further proof by Julian Besag in 1974. (en)
gold:hypernym	Result
prov:wasDerivedFrom	wikipedia-en:Hammersley–Clifford_theorem?oldid=1086768670&ns=0
page length (characters) of wiki page	11974 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Hammersley–Clifford_theorem
is Link from a Wikipage to another Wikipage of	Conditional random field Gibbs measure Clifford's theorem Hammersley-Clifford theorem Catalog of articles in probability theory Markov random field Factor graph List of statistics articles List of theorems Outline of machine learning Fundamental theorem of random fields
is Wikipage redirect of	Hammersley-Clifford theorem Fundamental theorem of random fields
is foaf:primaryTopic of	wikipedia-en:Hammersley–Clifford_theorem

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