2
$\begingroup$

I would like to know whether $B \perp\kern-5pt\perp C | A $ in the following two graphical models and would like to know if my reasoning is correct:

enter image description here

For the left graphical model, which is a Belief Network, here's how I deduce it:

$P(B,C|A) \propto \sum_{D,E} P(B)P(A|B)P(C|A)P(D|B,C)P(E|D) = \underbrace{P(B)P(A|B)}_{f(B)}\underbrace{P(C|A)}_{f(C)}$

Since the probability factors into a product of functions of $B$ and $C$, we can say that they are independent given $A$.

For the right graphical model, which is a Markov Network, I use the following method:

  1. Remove all edges from $A$
  2. Check if there is a path leading from $B$ to $C$

So we remove all edges from $A$ and we can see that we have a path, $B-D-C$ and hence $B$ and $C$ are not independent given $A$.

Is my reasoning correct? Also, for the Belief network deduction, is there a faster way to see this? My reasoning, without writing anything, would be to just say that since $A$ is not a collider for $B$ and $C$, then they are independent. Would this be correct reasoning?

$\endgroup$
1

1 Answer 1

2
$\begingroup$

Your arguments both for the BN and the MN are correct, provided those graphs are perfect maps for your two distributions (which are then necessarily different).

The formula-free explanation for the BN is correct, too, although I would add that the path $B\to D\leftarrow C$ is blocked because of $D$ being a collider.

$\endgroup$
3
  • $\begingroup$ Thank you. I just don't understand how is $A\rightarrow D \rightarrow C$ a path? It seems like in the BN it doesn't exist $\endgroup$
    – user
    Jul 27, 2022 at 11:01
  • 1
    $\begingroup$ Sorry, it is the path $B\to D \leftarrow C$ , I fixed it. $\endgroup$
    – frank
    Jul 27, 2022 at 11:03
  • $\begingroup$ Hi Frank, do you mind taking a look at this and this Q's. $\endgroup$
    – user
    Aug 3, 2022 at 10:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.