Qns for the class/Blog (on Bayes nets)

Here are questions for which I will be looking for answers at the beginning of the next class.

1. You have been given the topology of a bayes network, but haven't yet gotten the conditional probability tables
    (to be concrete, you may think of the pearl alarm-earth quake scenario bayes net).
    Your friend shows up and says he has the joint distribution all ready for you. You don't quite trust your
    friend and think he is making these numbers up. Is there any way you can prove that your friends' joint
    distribution is not correct?


2. Continuing bad friends, in the question above, suppose a second friend comes along and says that he can give you
   the conditional probabilities that you want to complete the specification of your bayes net. You ask him a CPT entry,
   and pat comes a response--some number between 0 and 1. This friend is well meaning, but you are worried that the
   numbers he is giving may lead to some sort of inconsistent joint probability distribution. Is your worry justified ( i.e., can your
   friend give you numbers that can lead to an inconsistency?)

  (To understand "inconsistency", consider someone who insists on giving you P(A), P(B), P(A&B) as well as P(AVB)  and they
wind up not satisfying the P(AVB)= P(A)+P(B) -P(A&B)
[or alternately, they insist on giving you P(A|B), P(B|A), P(A) and P(B), and the four numbers dont satisfy the bayes rule]

3. Your friend heard your claims that Bayes Nets can represent any possible conditional independence assertions exactly. He comes to you
and says he has four random variables, X, Y, W and Z, and only TWO conditional independence assertions:

X .ind. Y |  {W,Z}
W .ind. X  |  {X, Y}

He dares you to give him a bayes network topology on these four nodes that exactly represents these and only these conditional independencies.
Can you? (Note that you only need to look at 4 vertex directed graphs).

=================

Rao