There has been an outbreak of mumps in your college. You feel fine, but you’re worried that you might already be infected. You decide to use Bayes nets to analyze the probability that you’ve contracted the mumps.

You first think about the following two factors:

• You think you have immunity from the mumps (+i) due to being vaccinated recently, but the vaccine is not completely effective, so you might not be immune (−i). 

• Your roommate didn’t feel well yesterday, and though you aren’t sure yet, you suspect they might have the mumps (+r).

Denote these random variables by I and R. Let the random variable M take the value +m if you have the mumps, and −m if you do not. You write down the Bayes net in Figure S13.20 to describe your chances of being sick: 

a. Fill in the following table with the joint distribution over I, M, and R, P(I, M, R).

b. What is the marginal probability P(+m) that you have the mumps? 

c. Assuming you do have the mumps, you’re concerned that your roommate may have the disease as well. What is the probability P(+r | + m) that your roommate has the mumps given that you have the mumps? Note that you still don’t know whether or not you have immunity.

You’re still not sure if you have enough information about your chances of having the mumps, so you decide to include two new variables in the Bayes net. Your roommate went to a party over the weekend, and there’s some chance another person at the party had the mumps (+f). Furthermore, both you and your roommate were vaccinated at a clinic that reported a vaccine mix-up. Whether or not you got the right vaccine (+v or −v) has ramifications for both your immunity (I) and the probability that your roommate has since contracted the disease (R). Accounting for these, you draw the modified Bayes net shown in Figure S13.21: 

d. Which of the following statements are guaranteed to be true for this Bayes net? 

(i) V ⊥⊥ M | I, R 

(ii) V ⊥⊥ M | R 

(iii) M ⊥⊥ F | R 

(iv) V ⊥⊥ F 

(v) V ⊥⊥ F | M 

(vi) V ⊥⊥ F | I

Figure S13.20

Figure S13.21

Transcribed Image Text:

IRM +i+r+m +i+r-m +ir+m +i-r-m -i+r+m -i+r-m -i-r+m -i-r-m P(I, R, M) 0 0 0.056 0.024 0.024 0.096