Target is a top US retailer. By analyzing consumer's purchase history, Target data scientist could figure out if she is pregnant. Following is a true story cited from the New York Time article http://www.nytimes.com/2012/02/19/magazine/shopping-habits.html?_r=2&hp=&pagewanted=all A man walked into a Target outside Minneapolis and demanded to see the manager. He was clutching coupons that had been sent to his daughter, and he was angry, according to an employee who participated in the conversation. "My daughter got this in the mail!" he said. "She's still in high school, and you're sending her coupons for baby clothes and cribs? Are you trying to encourage her to get pregnant?" The manager didn't have any idea what the man was talking about. He looked at the mailer. Sure enough, it was addressed to the man's daughter and contained advertisements for maternity clothing, nursery furniture and pictures of smiling infants. The manager apologized and then called a few days later to apologize again. On the phone, though, the father was somewhat abashed. "I had a talk with my daughter," he said. "It turns out there's been some activities in my house I haven't been completely aware of. She's due in August. I owe you an apology."In 2014, there were 3,988,076 new births in the US. In that year, the US female population is 125.9 million. Assume all pregnancy leads to birth and assume all female population is in reproductive age. Please calculate the prior p(a woman is pregnant). Target identified that purchasing 25 specific products together can indicate pregnancy. Suppose Target data scientist finds that among pregnant female customers, 95% of them purchase these 25 products all together and among non-pregnant female customers, 0.5% of them purchase these 25 products together. Using Bayesian inference to calculate the probability that a woman is pregnant if Target observes that she purchases all these 25 products.

Figure 3: Probabilities for the second car model (Part 2) Part 2: Bayesian Networks In a different model of the car, the alternator (A) can stop working due to an electric fault (E) or due to the breaking of the drive belt (D). The failure of the alternator causes complete discharge of the battery (B) that supplies current to the radio (R) and lights (L). The battery, the lights and the radio may also stop working for internal reasons. 1. Draw the Bayesian network that represents the model of the car, show- ing the variables and the dependence/ independence relationships between them. 2. Use the obtained network and the probabilities listed in Figure 3 to com- pute the probability of: 0 Phil, 6, a, b, -'r, -l) o P(-ud, s, -u&, b, r, I) Part 3: Exact Inference in Bayesian Networks To make a probability inference query means to compute the posterior prob- ability distribution for a set of query variables given some observed event. X denotes the query wariable, E denotes the set of evidence variables E1, ..., E\A company is interested in implementing some A / B testing on its website in order to improve sales. In order to do this they rst need to look at what their current website usage is. Let X.- represent the number of customers who visit the webpage in a given hour. Assume X1,X2, ...,Xn W PoissonOl). Let's use a Bayesian approach to make some inference about A. Use A ~ Exponentialm) as a prior distri bution. Where is the mean parameter. So f (A) = [lacM3 for /\\ 2 0. Derive the posterior distribution of A. Identify what wellknown distribution the posterior follows, and be sure to identify it's parameters. Note: the parameters of the posterior should be expressed as a function of the sample mean, sample size, ,6 and numbers (exclusively). Problem 2. Below is the Bayesian network for the WetGrass problem. Some prior probabilities and conditional probability tables are given. All variables are Boolean variables that can take values true (t) or false (t). 2.a Calculate the value for the joint probability (show your work): P(C=f, R=f, S=t, W=t) 2.!) You observe that W=t and S=f. Perform inference to obtain the posteriori probability that the weather is cloudy, that is: PIEC = t|W = LS = 1"). Show your work