Thursday, March 19, 2009

Thinking Cap qns on Bayes Networks...

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?
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]

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).
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?


  1. I'll take a break from finishing project 2 (late) to give these a try.

    1. You can check if the JPD from your friend entails the same conditional probabilities from your BN structure. In other words, if the friend's JPD can't be expressed as the product of local distributions than he or she is wrong.

    2. Not quite understanding this one...sorry.

    3 and 4 - I've spent too long thinking. I'll come back to these after I finish the project and my other homework.

  2. 0. The sum should be 2^m, which corresponds to the case that this variable is always true regardless of the values of its parents.

    1. Michael already answered this question.

    2. The CPT entries should NOT lead to inconsistency, as long as they are specified according to the network topology. The Bayesian network is based on Bayesian perspective of probability, instead of the frequentist point of view. So the subjectiveness is justified in this context.

    3. No

    4. I assume the problem statement is “The CIA(B1) is a superset of A and CIA(B2) is a subset of A”. This should NOT be a huge problem, as we can always be more conservative and assume less CIA’s, at the price of assessing more CPT entries. In choosing B1, the resulting network encodes more CIA’s than the domain actually has, while choosing B2 results in network encodes less CIA’s.

  3. 0) 2^m seems correct. But if the variable is always true no matter what you put it then isnt it really independent... in which case all the connections to the parents are spurious? Which really begs the question that what ways can we think of to test for conditional independence given a network and cpt's(Say the network was badly designed)?
    One would be just check brute force enumeration.


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