I am a bit puzzled by the use of polytree to infer a posterior in a Bayesian Network (BN).
BN are defined as directed acyclic graphs. A polytree is DAG whose underlying undirected graph is a tree. It is known that if a BN is a polytree, exact inference algorithms such a belief propagation can be efficiently applied. If a BN is not a polytree, one may have to use approximate inference algorithms to avoid the potential non-convergence of a belief propagation algorithm.
First, I don't understand why BNs should be treated as polytrees. A BN is a DAG by definition, so why one should use polytree to describe BNs?
Second, using polytrees seems to degrade the original topology of the BN. If a BN (which is a DAG) is not a polytree, then it is considered to have cycles. It is highly confusing (many research papers on the topic often refer to cyclic BN, where it should be, I believe, a non-polytree BN). Why isn't it possible to use the structure of a BN (i.e., knowledge about the directed edges) to create exact inference algorithms for non-polytree BNs? It seems that the common message-passing algorithms do not use these topological information, which may explain the poor performance/in-feasibility of exact inference on non-polytree BN.
If someone knows enough to answer these questions, I would be glad to have some hints. I really do not understand why the DAG structure of BNs seems to be ignored.
