# d-separation in Bayes Network vs separation in undirected graph

I've been teaching myself about d-separation and am trying to answer the following question. Given the graphs below, write down all conditional independence relationships involving the random variable C i.e. $$C \perp X|Y$$

Note: I've had to use $$\perp$$ for conditional independence as I don't know how to write the proper symbol in LaTeX without adding stuff to preamble (which stackexchange doesn't have).

Here are the graphs:

For image (i), we have a tail-to-tail relationship at B which means C and A are d-separated and so I have $$C \perp A|B$$. I cannot find any more conditional independence relationships here of the form $$C \perp X | Y$$ since both $$B$$ and $$D$$ are adjacent to $$C$$ and therefore not valid. Is this right?

For image (ii), we have a head-to-tail relationship at B and D. I think this means $$C \perp A|B$$ and $$C \perp A|D$$ and $$C \perp A| \{ B,D\}$$. I was tempted to write $$C \perp B|D$$ but I don't think this is valid since only one of the possible paths from C to B would be d-separated. So I find a total of three valid conditional indpendence relationships. Is that right?

For image (iii) and (iv), the problem is now that the graphs are undirected and I have not been able to find as much material online to help with this. I was wondering if someone could help with the following:

(a) explain the meaning of the black squares in (iv) (b) link to some notes/videos that might help with finding conditional independencies in undirected graphs

Thanks!

(a) The black squares represent factors in a factor graph. Wikipedia is the wiki page about factor graphs.

(b) A pdf from stat cmu is a reference about Undirected Graph / Markov Random Field.

• Thanks. Yeah I found some helpful stuff as well. Understand it all now :) – user11128 Sep 27 at 8:03