D-separation alternative for Markov Random Fields¶
We cannot convert DGM to an UGM just by dropping the orientation of the edges. (To give an example we cannot convert a v-structure \(A \rightarrow B \leftarrow C\) into \(A-B-C\) which states that \(A \perp C | B\) but it muss not hold for the first). To avoid inccorect CI statements, we have to add edges between the “unmarried” parents A and C, and then drop the arrows from the edges, forming a fully connected undirected graph. This process is called normalization:
Example of normalization:
We need to interconnect 2,3 since they have an common child 5, and we interconnect 4,5,6 since they have a common child 7.
Downside of normalization is that we loose some CI information, and therefore we cannot use the normalized UGM to determine the Ci properties of DGM.