In the previous chapter, we understood that the two concepts of a graph and a distribution are both encoded in a graphical model. We now turn to the equivalence of factorization and independence, and we would like to know whether they are both respected in both the views, in the context of Markov nets.
The following are the questions that we wish to address:
The first one is if the two nodes are conditionally independent in the graph, does the distribution respect that independence?
The second one is that is the factorization (or decomposition) of a distribution into a graph a valid decomposition?
This is a theorem that has parallels from Bayesian networks. If a distribution P factorizes over a graph G, and suppose that two random variables X and Y are separated (in the graph G) given 
, then the distribution P satisfies the independence statement, and X is conditionally independent of Y given Z.
In other words, the independence defined by the graph H by the...