Markov random fields factorization with context-specific independences
This work addresses a limitation in probabilistic graphical models for researchers in machine learning and statistics, but it is incremental as it extends an existing theorem.
The paper tackles the problem that Markov random fields cannot encode context-specific independences (CSIs), which are conditional independences true only for certain assignments, by presenting a method to factorize a Markov random field according to CSIs and formally guaranteeing its correctness with a generalization of the Hammersley-Clifford theorem.
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.