An interesting feature of Bayesian updating is that it is pathindependent: 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 2 (,ul, p52) 2 (1/3, 2/3), where an is the prior pro|bability of state n. Consider the information structure s t wl 3/4 1/4 (.02 1/2 1/2 (a) Given prior ,u = (1/3, 2 / 3), compute the Bayesian posterior if s is observed. Denote this by us 2 (ref, reg). (Hint. Use Bayes' formula to compute psi. Since probabilities sum to 1, this immediately gives a; = 1 ,uf. I do not recommend using a calculator anywhere in this questionithe fractions all work out nicely). (b) Now suppose the agent observes t after having observed 3. This means we treat ,us from part (a) as the prior, and can use Bayes' rule to compute a posterior, denoted it\". Find st ,u . (C) Now we will apply the signals in the opposite order. First, suppose if is generated and compute the Bayesian posterior ,uIt using the original prior ,u. = (1/3, 2/3). Then, treating at as the new prior, suppose s is generated and use Bayes' rule to compute it\". Finally, verify that [its : y.\" (if your calculations don't give this result, you have made a mistake somewhere). Suppose the two states correspond to personal characteristics that the individual might care about. For example, state an could indicate high intelligence and 002 could indicate low intelligence. Might the agent respond in a nonBayesian fashion to some signals? Would such an agent still satisfy the pathindependence property? (Intuitive explana tions are ne, so don't worry about providing technical details)