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The belief network for the following questions is given below:

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Q1: Which variables must be independent of A given C and S according to the belief network above?

My ans:

Since C is given, there is no active trail through A-C-D, hence D is independent of A.

Since C is given, there is an active trail through A-C-B, hence B may be dependent on A.

Since B may be dependent on A and E is dependent on B, E may be dependent on A.

Therefore, the only variable independent of A is D.

Q2: Which variables must be independent of A given D and S according to the belief network above?

My ans:

C is dependent on A.

Since D is given, there is no active trail through C-D-E.

There is no active trail through A-C-B.

Therefore, both B and E are independent of A.

Would really appreciate it if someone can verify whether my reasoning and answers are correct. Thank you for your help in advance.

Alex Williams
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  • In your proposed solution for $Q1$, don't you conclude that the only variable independent of $A$ is $D$? – lulu Aug 07 '22 at 22:22
  • No problem. For $Q2$, perhaps I've forgotten what the notation means, but is it not the case that knowing $A,D$ occurred gives me some information regarding $C$, and information regarding $C$ gives me information regarding $B$, no? – lulu Aug 07 '22 at 22:31
  • As I am testing whether or not $A,B$ are independent, I can assume that $A$ occurred. that is to say, if $A$ occurs then we have learned something about $C$, and knowing something about $C$ surely tells me something about $B$. – lulu Aug 07 '22 at 22:46
  • Suppose, for instance, that, given $A$, $C$ occurs with probability $.01$ and that given $B$, $C$ occurs with probability $.99$ In that case, given $A$, I really don't think $C$ is terribly likely, so I therefore don't think $B$ is terribly likely. So information about $A$ can tell me something about $B$. – lulu Aug 07 '22 at 22:50
  • Does the notation require that $A,B$ are independent? Otherwise, I think my example works. But, as I say, I am not really clear on what the notation implies. – lulu Aug 07 '22 at 23:04

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