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Consider the problem of comparing proportions from two binomial distributions, 61 and 92. We observe yl distributed as Binomial(n1, 6'1) and y2 distributed as Binomial(n2,62). We want to derive the posterior distributions of 61 and 62. Let's consider the case of dependent priors for 61 and 62. That is, knowledge of the value of 61 may inuence the prior belief about the location of the second proportion 62. For example, the Australian Technical Advisory Group on Immunisation (ATAGI) has recommended that Pzer is the preferred vaccine for people aged 60 and under. So what proportion of people under age 60 are willing jump the Pzer queue and get the alternative AstraZeneca vaccine which is more readily available? Let 61 denote the proportion of people aged 30-39 who are willing to get the AstraZeneca vaccine. Let 62 denote the proportion of people aged 40-49 who are willing to get the AstraZeneca vaccine. Because we are considering adjacent age groups, the vaccine preferences of people in the rst age group may affect the vaccine preferences of people in the second age group and vice versa. That is, the belief that 61 is close to say 7% might lead us to believe that the value of 62 is also close to 7%. This belief implies the use of dependent priors for 91 and 92. What are the options for a dependent prior? Howard (1998) proposed a special form of dependent prior between 61 and 62 expressed as follows. First, consider a legit transformation of the parameters 61 and 62. That is, dene 61 and lo .92 161 72 gel62- To model the dependency, let 72m ~ Normal(mean = 71, stdev = J). Howard (1998) proposed the following general form of the dependent prior 71 = loge M9132) oc 31/2\"\"aslu a-lsg-lu 6'2)\" where u = iloge (3:2)). (a) [3 marks] Explain the role of each of the hyperparameters (a, ,6, 19,5, 0) (b) [2 marks] Is the joint prior on 330%, 62) dened above a conjugate prior? Explain why or why HOW