Consider five BOOLEAN random variables representing the following facts:

G = true indicates that you will go to graduate school

S = true indicates that you will study

W = true indicates that you will work part-time.

D = true indicates that you will be able to successful in getting your advanced degree

M = true indicates that you will have money to make your ends meet

Given the above representations, now you need to do the following inference:

a. The probability that you will go to a graduate school and that you will study and be successful getting your advanced degree, but will not work part-time or make your ends meet. i.e., P(G=true, S=true, W=false, D=true, M=false)

b. The probability that you will have success in getting your advanced degree if you can make your ends meet and study, but do not work part-time. i.e., P(D = true | M = true, S = true, G = true, W = false)

c. The probability that you will go to a graduate school and will get your advanced degree and that you will make your ends meet. i.e., P(M=true, G=true, D = true).

d. The probability that you will get your degree and also makes your ends meet given that you go to a graduate school, study hard, and work part time. i.e., P(D = true, M = true |G = true, S = true, W = true).

Given above representations from problem:

Give a Bayesian Network representing the following dependency relationships:

if you go to graduate school, this will affect the likelihood that you will study and the likelihood that you will work part-time. Studying and working affect your chances of getting the advanced degree, and working affects your chances of making your ends meet.

Use the following additional information gathered from observing a large set of data to compute ALL the probability values for your Bayesian network:

a. There is a 20% chance that you will go to a graduate school.

b. If you go to a graduate school, there is a 80% of chance that you will study. However, even if you don't go to a graduate school, there is still a 20% of chance that you will study.

c. If you go to a graduate school, there is a 60% of chance that you will work part-time to make your living. However, even if you don't go to a graduate school, there is a 50% of chance that you might still work part-time.

d. If you study and work part-time, there is a 60% of chance that you will successfully get an advanced degree. If you study and do not work part time, the change to get an advance degree is much higher, about 90%. However, if you do not study, the chance for you to get an advanced degree is 10% if you work part-time and 20% if you do not work at all.

e. If you work part-time, the chance to make your ends meet is 90%. However, if you do not work, then the chance to make your ends meet is only 20%

ART.- r 3|] 1 a5 (b) Figure 1: More left to right, we will refer to the above Bayesian networks as the original network, network (a), and network (b). 1. 5 points. In lecture we mentioned that the directionality of edges in Bayesian networks do not necessarily reect causality, but having causal edges often lead to simpler network structures {and hence greater inference eiciency). We will explore this concept further using the burglar alarm example, shown above. {a} Write out all the independence / conditional independence assumptions implied by the graphical structure of the original network. {Use B, E, A, J, M as the random variables.) Hint: There should be 21 in total. Recall that AlLBlC' is equivalent to BlLAlC' (so they only count as one conditional independence], but A_|_|_B|C is dierent from AJLB|C,D. {b} Now consider networks (a) and (b) that involve the same variables. In each of these cases, we consider variables in a different order, and when introducing the variable, we introduced all the edges necessary to ensure the original probability distribution can still be represented. For example, in network (a), when A was introduced, both arrows leading from M and J were necessary, otherwise conditional independence assumptions not present in the original network would have been introduced (e.g., if M } A was not present, then M LAlJ would be introduced). Write out all the independence ,1 conditional independence assumptions implied by the graphical structure of network [a]. The geometric distribution is a probability distribution of the number X of Bernoulli trials needed to get one success. For example, how many attempts does a basketball player need to get a goal. Given the probability of success in a single trial is p, the probability that the xth trial is the first success is: Pr(X = xlp) = (1 -p)*-1p for x=1,2,3,.. Suppose, you observe n basketball players trying to score and record the number of attempts required to get the first goal: X1, X2..... X for each of them. (a) Write down your likelihood and find the MLE of p. (1pt) (b) Using Bayesian approach, assign a beta prior to p and obtain the posterior distribution for p. (1pt) (c) Compare the posterior mean of p to the MLE. What can you say about sensitivity of posterior inference to prior assumptions? What happens to the difference between the posterior mean and the MLE of p as the sample size n increases? (1pt)Problem #1 (25 points): A Bayesian model and Gibbs sampling can be applied to estimate the change point in the annual number of coal mining accidents. Let y1,...In be a sample of the number of accidents occurred in a year, which is a random variable from a Poisson distribution for which there is a suspicion of a change point m along the observation process where the means change, m = 1, ..., n. Given m, the observation distributions are: Vi| A - Poi (1), i = 1,...,m vil - Poi(). i = m+1,...,n The model is completed with independent prior distributions A - Gamma(c, B) - Gamma(x, 5). m - Uniform Discrete (1, n). Then, the posterior joint density is: a( ), p, my...))