The factor graph
The Markov network doesn't give a very clear picture of the Gibbs parameterization of the distribution because we can't conclude whether the factors in it involve the maximal cliques or subgraphs. To overcome this limitation of the Markov network, we require a representation that can show the parameterization explicitly. The factor graph is one such representation.
A factor graph is a bipartite graph, one disjoint set being variable nodes, representing the variables, and the other being factor nodes, representing factors. An edge between a variable node and a factor node denotes that the random variable belongs to the scope of the factor. Thus, a factor graph is parameterized by a set of factors, where each of them is associated with a factor node, whose scope is all sets of all the random variables that it is neighbor to.
Generally, all the variable nodes are represented by a circle and all the factor nodes are represented by a square. Here's an example:
Fig 2.3 Factor graph...
