Graphical Models:  Assignment 2
Due February 5, 2019

  1. Answer the following true/false questions about the Bayesian network shown below.  Explain your reasoning.
    1. K is independent of A given F.
    2. F d-separates A from B and D.
    3. E is independent of H given B.
    4. B is independent of C given G and H.
    5. J is independent of D given E, H and C.
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  1. Find the Markov blanket of F.  Find the Markov blanket of E.  Explain your reasoning.
  2. Assume each of these random variables has 5 possible states. How many probabilities are needed to specify this Bayesian network?  How many probabilities are needed to specify a fully general probability distribution on these random variables (i.e., 10 random variables, 5 states for each random variable)?
  3. Repeat Problem 3 if there are 10 states per random variable.
  4. Find a general formula for the number of probabilities needed to define a fully specified joint distribution on n random variables with k states per random variable?  How many probabilities are needed for a Bayesian network with n random variables with k states per random variable if there is one root node, two nodes with one parent, and the rest of the nodes with two parents? Compare the number of probabilities for a full distribution and Bayesian network for n=10 and k=5. Repeat for n=100 and k=10. Comment on your results.