Exercise 15.7.11. Consider the polio vaccine example from Exercise 6.8.9, where Xy is the number of subjects in the vaccine group that contracted polio, and Xc is the number in the control group. We model Xy and Xe as independent, where Xv ~ Poisson(cyev) and Xc ~ Poisson(ccec). (15.72) Here ey and de are the population rates of polio cases per 100,000 subjects for the two groups. The sample sizes for the two groups are ny = 200, 745 and nc = 201, 229, so that cy = 2.00745 and cc = 2.01229. We wish to test Ho: dy = ec versus HA : By # ec. (15.73) Here we will perform a Bayesian test. (a) Let X|O = 0 ~ Poisson(ce) and ( ~ Gamma(a, A) for given & > 0, A > 0. Show that the marginal density of X is f(x]a,A) = r(x+ a) xll(a) (ctAjita' (15.74) (If a is a positive integer, then this f is the negative binomial pmf. The density is in fact a generalization of the negative binomial to real-valued a > 0.) (b) Take Xy and Xc in (15.72) and the hypotheses in (15.73). Suppose the prior given the alternative hypothesis has Oc and Ov independent, both Gamma(a, A). Show that the marginal for (Xc. Xy) under the alternative is f(xy | a, A)f(xca. A). (c) Under the null, let Ov = Oc = 0, their common value, and set the prior O ~ Gamma(a, A). Show that the marginal joint pmf of (Xy, Xc) is f(xXCHA) = [(avtxcta) CVCC xylxc!( a) (Atcytec)intacta (15.75) [This distribution is the negative multinomial] (d) For the vaccine group, there were xy = 57 cases of polio, and for the control group, there were xc = 142 cases. Find (numerically) the Bayes factor BAo(xy, xc), the ratio of integrated likelihoods as in (15.40), when & = 1, A = 1/25. [Hint: You may wish to find the logs of the various quantities first.] (e) What is the posterior probability of the null? What do you conclude about the effectiveness of the polio vaccine based on this analysis? (f) [Extra