Graphical Models:  Assignment 7
Due April 24, 2017

1. This problem uses the data set from Problem 6. Agent Hydziz I. Dennity collected data random sample of Depravians on PersonType, Gender, and HairLength. The data he collected is provided here.  Agent Dennity is considering these models for the relationship between PersonType, Gender and HairLength:
1. A fully connected Baysian network, as shown below, with no restrictions on the local distributions.
2. A Bayesian network in which HairLength depends on PersonType and Gender, but Gender does not depend on PersonType.
Assume uniform prior distributions on all local distributions, as in the K2 algorithm.  What is the probability of the data given each of these structures?
1. Find the probability of data given each of the following three structures:
1. Fully connected Bayesian network in which the gender distribution for government agents is the same as the gender distribution for dissidents;
2. Fully connected Bayesian network in which the hair length distribution is the same for all women and the hair length distributions are the same for male government agents, male government supporters, and apolitical males.
3. Fully connected Bayesian network in which the restrictions for both part a and part b hold.
2. If all five structures -- two from Problem 1 and three from Problem 2 -- have equal prior probability, what is the posterior probability of each of the structures?
3. This problem concerns the example network from Unit 6, shown below.  We are interested in using likelihood weighting to estimate the probability distribution of E given B=b2.  A sample of 250 cases was generated as follows:
• A was drawn from P(A)
• B was set to B=b2
• C was drawn from P(C|A), with A set to its sampled value.
• D was drawn from P(D|B,C), with B=b2 and C set to its sampled value.
• E was drawn from P(E|C), with C set to its sampled value.
The sampled values are given here.

Use likelihood weighting to estimate the probability distribution of E given B=b2.  Compare with the exact probability distribution. Comment.