Gibbs Random Field
A Gibbs Random Field is a Markov random field with joint probability densityes (of the random variables) that is strictly positive.
- See: Gibbs Measure, Phase Transition, Probability Measure, Partition Function (Mathematics), Intensive Property.
References
2016
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/markov_random_field Retrieved:2016-8-13.
- … When the joint probability density of the random variables is strictly positive, it is also referred to as a Gibbs random field, because, according to the Hammersley–Clifford theorem, it can then be represented by a Gibbs measure for an appropriate (locally defined) energy function. The prototypical Markov random field is the Ising model; indeed, the Markov random field was introduced as the general setting for the Ising model. In the domain of artificial intelligence, a Markov random field is used to model various low- to mid-level tasks in image processing and computer vision.
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Gibbs_measure Retrieved:2016-8-13.
- … A measure is a Gibbs measure if the conditional probabilities it induces on each finite subsystem satisfy a consistency condition: if all degrees of freedom outside the finite subsystem are frozen, the canonical ensemble for the subsystem subject to these boundary conditions matches the probabilities in the Gibbs measure conditional on the frozen degrees of freedom.
The Hammersley–Clifford theorem implies that any probability measure that satisfies a Markov property is a Gibbs measure for an appropriate choice of (locally defined) energy function. Therefore, the Gibbs measure applies to widespread problems outside of physics, such as Hopfield networks, Markov networks, and Markov logic networks.
A Gibbs measure in a system with local (finite-range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density.
The Gibbs measure of an infinite system is not necessarily unique, in contrast to the canonical ensemble of a finite system, which is unique. The existence of more than one Gibbs measures is associated with statistical phenomena such as symmetry breaking and phase coexistence.
- … A measure is a Gibbs measure if the conditional probabilities it induces on each finite subsystem satisfy a consistency condition: if all degrees of freedom outside the finite subsystem are frozen, the canonical ensemble for the subsystem subject to these boundary conditions matches the probabilities in the Gibbs measure conditional on the frozen degrees of freedom.