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:

enter image description here

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



1 Answer 1


(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.

  • $\begingroup$ Thanks. Yeah I found some helpful stuff as well. Understand it all now :) $\endgroup$
    – user11128
    Sep 27, 2019 at 8:03

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