I know Bayesian or Directed Graphical models are good for fast inference using message passing techniques. But how does it work with undirected models? With directed models, you moralize the graph making it an undirected and triangular graph whose cliques are connected in a junction tree thus allowing the message passing.
If you start out with a triangulated undirected model, you just go right to the junction tree and skip the moralization. But what if it is not triangulated? If you triangulate this, it becomes a different model. Can you do fast inference with undirected models that are not triangulated?