Throughout the representation section of the Bayesian network, we have encountered two representations: one being a graph, and the other being a probability distribution. We asked ourselves questions such as are they equivalent and when we switch from one view to the other, do we lose or gain information? We will now examine these questions in the context of a Markov network.
What is the equivalency of a distribution D and graph G? When does D factorize over G? In other words, when can D be represented using G? One way to understand factorization is to think of it as a decomposition problem. We have a problem (a huge joint distribution, for example), and we want to decompose it into smaller pieces (such as conditional probability distributions in the case of the Bayesian network).
We can state that the distribution D factorizes over G if we have a set of factors 
(which is a product of its individual factors), and that G is the induced graph for the set of factors .
Unlike in...