Thinking Cap qns on Bayes Networks...

[Subbarao Kambhampati <rao@asu.edu>]

0. In class, we seemed to convince ourselves that the CPT entries don't have to add up to 1. Suppose you have a boolean node with m boolean parents. What is the maximum value of the sum of CPT entries? When does it happen?

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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. Afterall, your friend is a bayesian and is making up is *personal* probabilities that may not have any interpretation from a frequency point of view. 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 other friend (okay--your social life is full of geeks ever since you started taking this course) heard your claims that Bayes Nets can represent any possible conditional independence assertions exactly. She 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}

She 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).

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4. If your? answer to 3 above is going to be "No", how serious an issue do you think this is? In particular, suppose your domain has exactly set A of conditional independencies. You have two bayes network configurations B1 and B2. The CIA(B1) is?a?superset of

A and CIA(B1) is a subset of A.? ?Clearly, neither B1 nor B2 exactly represent what you know about the domain. If you have to choose one to model the domain, what are the tradeoffs in choosing B1 vs. B2?

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Rao

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