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Question 4 An interesting feature of Bayesian updating is that it is path-independent: given a set of observed signals, a Bayesian ends up with the same nal beliefs no matter what order the signals are processed in. This question explores this property. Suppose there are two states of the world and that an individual has a prior ,u = (#1, p2) = (1 / 3, 2/ 3), where p\" is the prior probability of state n. Consider the information structure .9 t .1 .. (a) Given prior ,u = (1/3, 2/3), compute the Bayesian posterior if s is observed. Denote this by ,u.\"3 = (p.51, p3). (Hint. Use Bayes' formula to compute of. Since probabilities sum to 1, this immediately give; is; = l ,uf. I do not recommend using a calculator anywhere in this questionthe fractions all work out nicely). (b) Now suppose the agent observes t after having observed .5. This means we treat p5 from part (a) as the prior, and can use Bayes' rule to compute a posterior, denoted p\". Find 81. n . (c) Now we will apply the signals in the opposite order. First, suppose it is generated and compute the Bayesian posterior pt using the original prior ,u = (1 f 3, 2/ 3). Then, treating n' as the new prior, suppose s is generated and use Bayes' rule to compute p\". Finally, verify that ,u\" = n\" (if your calculations don't give this result, you have made a mistake somewhere). (d) Suppose the two states correspond to personal characteristics that the individual might care about. For example, state ail could indicate high intelligence and {412 could indicate low intelligence. Might the agent respond in a non-Bayesian fashion to some signals? Would such an agent still satisfy the path-independence property? (Intuitive explana- tions are ne, so don't worry about providing technical details)