Hammersley–Clifford theorem

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 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.

[1] 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[2] and Frank Spitzer[3] in the context of statistical mechanics.

The theorem is named after John Hammersley and Peter Clifford, who proved the equivalence in an unpublished paper in 1971.

[4][5] Simpler proofs using the inclusion–exclusion principle were given independently by Geoffrey Grimmett,[6] Preston[7] and Sherman[8] in 1973, with a further proof by Julian Besag in 1974.

[9] It is a trivial matter to show that a Gibbs random field satisfies every Markov property.

As an example of this fact, see the following: In the image to the right, a Gibbs random field over the provided graph has the form

are fixed, then the global Markov property requires that:

(see conditional independence), since

forms a barrier between

To establish that every positive probability distribution that satisfies the local Markov property is also a Gibbs random field, the following lemma, which provides a means for combining different factorizations, needs to be proved: Lemma 1 Let

denote the set of all random variables under consideration, and let

denote arbitrary sets of variables.

(Here, given an arbitrary set of variables

will also denote an arbitrary assignment to the variables from

provides a template for further factorization of

as a template to further factorize

To this end, let

be an arbitrary fixed assignment to the variables from

For an arbitrary set of variables

denote the assignment

, excluding the variables from

need to be rendered moot for the variables from

have been fixed to the values prescribed by

do not conflict with the values prescribed by

Lemma 1 provides a means of combining two different factorizations of

The local Markov property implies that for any random variable

Applying Lemma 1 repeatedly eventually factors

into a product of clique potentials (see the image on the right).