Second Hand Car Purchase
You want to buy a used car. You build a decision netwrok given below. In your decision network, the variable A is the car’s age (old A = +o, not old A = −o), D is the current owner’s driving style (aggressive D = a, neutral D = n, defensive D = d) and S is the shape of the car (good shape S = +s, bad shape S = −s). You further have the option of testing the car, which is represented by the variable T (success T = +t, fail T = −t). You decision is whether to buy the car or not. The price that the owner is asking is 5000 below the market value. However, if it is not in good shape, you will need to spend an additional 11000 on it. This is given in the utility node (not buying has no utility).
Q1 (5 Points) What is the joint distribution of this Bayesian Network?
Q2 (25 Points) Calculate the probability of the car being in good shape if you get a positive test result (i.e. calculate P(+s| + t)), using variable elimination. Make sure to highlight all your factors at each time step. If you do not use variable elimination, there is a chance that you will not receive any points.
Q3 (10 points) Calculate P(+s) however you want. You may use your result from the previous question.
Q4 (10 points) What is the expected utility of the buy (+b) action without any evidence? You may use your results to the previous questions.
Q5 (15 points) What is the maximum amount of money you would be willing to pay to get the car tested? (You need to do calculations, this is not an essay question). You may use your results to the previous questions.
Q6 (10 points) You want to perform rejection sampling to find P(d|+o,+t). Calculate the weights of the samples given below.
A
D
S
T
Weight
+o
d
+s
+t
+o
d
−s
+t
Q7 (25 points) You want to perform Gibbs sampling. Given {A = −o,D = n,S = +s,T = −t}, calculate the probability distribution which will be used to sample S.
