CS188 Fall 2011 Section 7: Probability and Bayes Nets1 Green Party PresidentIt’s election season, and the chosen president may or may not be the Green Party candidate. Pundits believe thatGreen Party presidents are more likely to legalize marijuana than candidates from other parties, but legalizationcould occur under any administration. Armed with the power of probability, the analysts model the situation withthe Bayes Net below.(G)reen Party President Elected(M)arijuana Legalized+g -gP(G) 0.1 0.9P(+m | G) P(-m | G)+g 0.667 0.333-g 0.25 0.751. Fill in the joint probability table over G and M.G M P(G, M)+g +m+g -m-g +m-g -m2. What is P(+m), the marginal probability that marijuana is legalized?3. News agencies air 24/7 coverage of the recent legalization of marijuana (+m), but you can’t seem to find outwho won the election. What is the conditional probability P(+g | +m) that a Green Party president waselected?We can make better inferences if we observe more evidence. On the next page, we will expand on the model(Bayes net) by introducing two new random variables: whether the budget is balanced (B), and whether classattendance increases (C). The expanded Bayes net and conditional distributions are shown below.1(G)reen Party President Elected(M)arijuana Legalized(B)alanced Budget (C)lass Attendance Increases+g -gP(G) 0.1 0.9P(+m | G) P(-m | G)+g 0.667 0.333-g 0.25 0.75P(+b | M) P(-b | M)+m 0.4 0.6-m 0.2 0.8P(+c | M) P(-c | M)+m 0.25 0.75-m 0.5 0.54. The full joint distribution is given below. Fill in the missing values.G M B C P (G, M, B, C)+ + + + 1/150+ + + -+ + - + 1/100+ + - - 3/100+ - + + 1/300+ - + - 1/300+ - - ++ - - - 1/75G M B C P (G, M, B, C)- + + + 9/400- + + - 27/400- + - + 27/800- + - - 81/800- - + + 27/400- - + - 27/400- - - +- - - - 27/1005. Compute the following quantities. You may use either the full joint distribution or the conditional tables,whichever is more convenient.(a) P(+b | +m) =(b) P(+b | +m, +g) =(c) P(+b) =(d) P(+c | +b) =26. Now, add a node S to the Bayes net that reflects the possibility that a new scientific study could influence theprobability that marijuana is legalized. Assume that the study does not directly influence B or C. Draw thenew Bayes net below. Which CPT or CPT’s need to be modified?7. Consider your augmented model. Just based on the structure, which of the following are guaranteed to be true,and which are guaranteed to be false?(i) B ⊥⊥ G(ii) C ⊥⊥ G|M(iii) G ⊥⊥ S(iv) G ⊥⊥ S|M(v) G ⊥⊥ S|B(vi) B ⊥⊥ C(vii) B ⊥⊥
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