Markov field models

Markov fields make it possible to build data models with many variables and a reduced number of parameters, by imposing that the variables have only local interactions. These are defined on a non-directional graph, such as an image grid. A Markov field captures local interactions through conditional dependencies. The Markov property imposes localization of interactions through conditional independence of a variable with all other variables outside a neighborhood. A Markov field is defined by a probability density that factorizes into a product of terms, which depend only on values in neighborhoods. An application is studied for Gaussian processes and energies in physics defined by a scalar potential. The Hammersley-Clifford theorem demonstrates an equivalence between the factorization of Markov fields and the Markov property for positive measures. Markov fields can also be applied to trees to define hierarchical models.