SIT743 - Bayesian Learning & Graphical Models - Statistics Assignment Help

Assignment Task

Q1)
Melbourne City council conducted a survey to study the relationship between the type of dwelling and the income profile of the people living in Melbourne city. A list of factors that influence the type of dwelling, along with their possible values, and a Bayesian network that represents the relationship between these factors (variables) are given below.
E (Education) ∈ {Graduate, Non-graduate}
G (Gender) ∈ { Male, Female}
A (Age) ∈ {35 or less, 35+ }
S (Salary) ∈ {$50k or less per annum, $50k+ per annum}
J (Job type) ∈ {Professional, Laborer, Student, Unemployed}

1.1) Write down the joint distribution for the above network.
1.2) Find the minimum number of parameters required to fully specify the distribution according to the above network.
1.3)
a) Write down a joint probability density function if there are no independence among the variables is assumed.
b) How many parameters are required, at a minimum, if there are no independencies among the variables is assumed?
c) Compare with the result of the above question (Q1.2) and comment.

1.4) The Melbourne city council, from a previous study, found out that the Marriage is conditionally independent of job type, given the salary and Age. The Melbourne city council wants to modify the Bayesian network given in Figure 1 by incorporating this new information. Assume now that the Marriage is conditionally independent of job type, given the salary and Age, perform the following.
a) What change will happen to the Bayesian network (shown in Figure 1) when the above assumption is considered. Draw the new Bayesian network considering the above assumption (you may draw this by hand).
b) Compute the change in the minimum number of parameters required for this new Bayesian network, compared to the minimum number of parameters required for the Bayesian network shown in Figure 1. Comment on the results.

1.5) d-separation method can be used to find two sets of independent or conditionally independent variables in a Bayesian network. Use the Bayesian network given in Figure 1 to answer the following:
For each of the statements/questions given below from (a) to (b), perform the following:
• List all the possible paths from the first (set of) node/s to the second (set of) node/s considered for the independence check.
• State if each of those paths is blocking or non-blocking with reasons.
• Hence, answer the question about independence.
a) Is dwelling (D) conditionally independent of gender (G) given salary (S) and job type (J)?
b) Is {, } ⊥  | {, } ?

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