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**Exam Feb. 19, 2020: Finding stochastically independent variables in a Bayesian Network**

Hello,

in the above mentioned exam in subtask 1.2 there is a BN given and the task is to state which variables are a) conditionally independent given which other variables, and b) stochastically independent.

Subtask a) is not too difficult: A node X is conditionally independent of its non-descendants given its parents.

However, I’m having troubles finding out which variables are stochastically independent.

How can we obtain the information from a BN, that two variables are stochastically independent, if the BN models only conditional dependencies?

Thanks for the help in advance and have a nice weekend.

Hi,

I think that’s the same question that was covered in this post:

https://fsi.cs.fau.de/forum/thread/18556-Exam-SS19-Question-1-2-Soundness-and-Completeness

I suggest you read it. Short summary: it’s not easy in this case.

I think the question is slightly different. The problem addressed in the thread regards finding variables that are guaranteed to be stochastically dependent. This is indeed difficult to do for reasons explained in the thread. However the question…

How can we obtain the information from a BN, that two variables are stochastically independent, if the BN models only conditional dependencies?

can be answered in this way:

Two nodes A, B in a Bayesian network are conditionally independent given a set of nodes Z if they are d-separated by Z. If A, B are d-separated by the empty set then they are stochastically independent.

D-separation was not covered in the lecture but it is explained here: https://fsi.cs.fau.de/forum/post/164641 or on Wikipedia: https://de.wikipedia.org/wiki/D-Separation

In the lecture only an intuitive, simplified version of d-separation was used: “Each node X in a BN is conditionally independent of its non-descendants given its parents Parents(X).” (observation 20.3.1, slide 703). From this you can immediately infer that two nodes that are non-descendents of each other and have no parents must be stochastically independent. This is a sufficient condition for stochastic independence and suffices to solve any of the past exam questions on stochastic independence in BN. Given that d-separation was not covered in the lecture, you can assume that it will also suffice this year.

Thanks for the replies!

One short question remaining:

That’s the case because conditional independence of A and B given C for an empty C is just the same as stochastic independence of A and B, correct?

It would be more accurate to say for C with P(C)=1, but essentially yes.

Hi,

unfortunately there’s still something that’s pretty unclear to me:

The solution of this task also states that for example X1 and X2 are stochastically independent. This isn’t the same situation as with A1 and A2 since neither A1 nor A2 have parents. Same with A1 and X3.

Could you maybe explain again, why for example X1 and X2 are stochastically independent? I get it now for A1 and A2, but not for those two (as well as A1 and X3).

Also, a completely different thing that’s unclear to me: Why are X3 and A1 not conditionally independent given A2? In my opinion they fit the description “Each node X is conditionally independent of its non descendents given its parents Parents(X)” with X = X3 and Parents(X) = Parents(X3) = A2.

The solution doesn’t say that X1 and X2 are stochastically independent (they are conditionally independent given A1), if I’m thinking of the same problem (retake 2017, problem 1.2).

Regarding X3 and A1, you can either start with A1, say that it is conditionally independent of X3 given its parents (none), and thus the variables are stochastically independent. Or you could start with X3, and say that it is conditionally independent of A1 given A2, this would also be correct. I think the solution was not meant to be exhaustive

Does this help?

Thank you, your explanation is very helpful. The fact that the solutions aren’t exhaustive made it clearer to me, I thought it was a complete list of independencies.

Oh sorry, I mixed something up! The solution says that X1 and X3 (accidentally wrote X2 instead of X3) are stochastically independent. That’s something I still don’t get.

They are indeed stochastically independent because they are d-separated. However, we did not cover d-separation in the lecture. So you would not lose points if you did not spot that independency. The reference solution goes somewhat beyond the lecture material here.