Hammersley–Clifford theorem
E899013
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hammersley–Clifford theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11002988 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hammersley–Clifford theorem Context triple: [Markov random field, isCharacterizedBy, Hammersley–Clifford theorem]
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A.
Kesten’s theorem
Kesten’s theorem is a fundamental result in probability theory that characterizes when a random walk on a group is transient or recurrent, with deep implications for random walks on groups and percolation theory.
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B.
Potts model
The Potts model is a generalization of the Ising model in statistical mechanics that describes interacting spins with more than two possible states, used to study phase transitions and critical phenomena.
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C.
Mayer cluster expansion in statistical mechanics
The Mayer cluster expansion in statistical mechanics is a mathematical method that expresses the thermodynamic properties of interacting particle systems as a series in terms of cluster integrals, enabling systematic analysis of non-ideal gases and liquids.
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D.
de Bruijn–van Aardenne–Ehrenfest theorem
The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
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E.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hammersley–Clifford theorem Target entity description: 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.
-
A.
Kesten’s theorem
Kesten’s theorem is a fundamental result in probability theory that characterizes when a random walk on a group is transient or recurrent, with deep implications for random walks on groups and percolation theory.
-
B.
Potts model
The Potts model is a generalization of the Ising model in statistical mechanics that describes interacting spins with more than two possible states, used to study phase transitions and critical phenomena.
-
C.
Mayer cluster expansion in statistical mechanics
The Mayer cluster expansion in statistical mechanics is a mathematical method that expresses the thermodynamic properties of interacting particle systems as a series in terms of cluster integrals, enabling systematic analysis of non-ideal gases and liquids.
-
D.
de Bruijn–van Aardenne–Ehrenfest theorem
The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
-
E.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Hammersley–Clifford theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Markov random field