Mutually compatible Gibbs random fields

A unified theory for the mathematical description of Gibbs random fields that answers some important theoretical and practical questions about their statistical behavior is presented. The local transfer function is introduced, and the joint probability measure of the general Gibbs random field is derived in terms of this function. The resulting probability structure is required to satisfy the property of mutual compatibility. A necessary and sufficient condition for a Gibbs random field to be mutually compatible is developed and used to prove that a mutually compatible Gibbs random field is a unilateral Markov random field. The existence of some special nontrivial cases of Gibbs random fields that are mutually compatible is demonstrated. Conditions on the translation invariance and isotropy of the general Gibbs random field with a free boundary are studied. The class of Gibbs random fields with a homogeneous local transfer function and the class of horizontally and vertically translation-invariant Gibbs random fields are introduced and treated. The concept of a translation-invariant Gibbs random field is also explored. The problem of the statistical inference of mutually compatible Gibbs random fields is discussed.>