The joint factorizes into unaries and pair-wise potentials. If that is the case, then why do we represent it as a fully connected graph? It is misleading and gives the impression that the joint cannot be factorized. Isn't it better to represent it using factor graphs rather than MRFs ?
Look at that! the maximal clique, is basically the entire graph !!
By definition, in an undirected graphical model, if two nodes are not directly connected by an edge, then they are conditionally independent (conditioned on all the neighbors of one node).
It's clear from inspection of the Boltzmann energy term that this is not the case for any two nodes... so it follows that the graph must be fully connected.
Isn't it better to represent it using factor graphs rather than MRFs ?
Well, that is subjective. Since pairwise potentials are a very common class of MRFs, if the reader is aware of the context, the meaning is clear, and there's no need to clutter the diagram with factor nodes.
