Updating Beliefs (3 points)P(A∣n)/P(B∣n) = P(A)/P(B) x P(n∣A)/P(n∣B)The above equation says that the posterior probability ratio of modelsAandBequals the ratio of prior probabilities associated with the models multiplied by the probability of receiving signalnconditional on each model being the 'truth'. Recall the model introduced in the recorded lectures. A professor is trying to decide if a student is good or bad. The prior belief that the student is good is1/2. Then the professor receives signals (tests), wheregrepresents a good test performance andbrepresents a bad test performance. Once a professor believes it more than50%likely that a student is good (bad), she will misinterpret a bad (good) test performance as good (bad) with probability q. Also good students do good on tests and bad students do bad on tests withP(g∣G)=P(b∣B)=p. A. Supposeq= 1/5andp=1/2Given a naive professor observes three good tests from a student (denotedgggwhat is her posterior probability that the student is good? B. Suppose the sophisticated professor's prior probability that the student is good is50/100What is her posterior given she observesgg(two good tests consecutively)? C. Suppose the sophisticated professor's prior probability that the student is good is51%. What is her posterior given she observes ggD. Compare your answers to (b) and (c). What is counter-intuitive here? Where is it coming from in the model? E. What is the professor's posterior if it is equally likely that good students and bad students do well on tests and she has no bias? Answer in words, appealing to the likelihood-ratio formulation of Bayes' Rule above.(
