  Markov random fields provide a compact representation of joint probability
distributions by representing its independence properties in an undirected
graph. The well-known Hammersley-Clifford theorem uses these conditional
independences to factorize a Gibbs distribution into a set of factors. However,
an important issue of using a graph to represent independences is that it
cannot encode some types of independence relations, such as the
context-specific independences (CSIs). They are a particular case of
conditional independences that is true only for a certain assignment of its
conditioning set; in contrast to conditional independences that must hold for
all its assignments. This work presents a method for factorizing a Markov
random field according to CSIs present in a distribution, and formally
guarantees that this factorization is correct. This is presented in our main
contribution, the context-specific Hammersley-Clifford theorem, a
generalization to CSIs of the Hammersley-Clifford theorem that applies for
conditional independences.
