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2. Gibbs Sampler and Lifetimes with Multiplicative frailty. Exponentially distributed lifetimes have constant hazard rate equal to the rate parameter When A is a constant hazard rate, a simple way to model heterogeneity of hazards is to introduce a multiplicative frailty parameter so that lifetimes Ti have distribution The prior on (A, is that is, and are apriori independent with distributions ga(c, o) and ga(d, ), respectively. The hyperparameters c, d, and are known (elicited) and positive. Assume that lifetimes tl, t2, , are observed. (a) Show that full conditionals for and are gamma, and by symmetry, R+d,AYti+ (b) Using the result from (a) develop Gibbs Sampler algorithm that will sample 51000 pairs (A, g) from the pFterior and burn-in the first 1000 simulations. Assume that n = 20 and that the sum of observed lifetimes is ti = 512. Assume further that the priors are specified by hyperparameters c = g, d = l, = 100, and = 5. Start the chain with = 0.1. (c) From the produced chain, plot the scatterplot of (A, as well as histograms of individual components, and Estimate posterior means and variances for and g. Find 95% equitailed credible sets for and (d) A frequentist statistician estimates the product as 20/512 = 0.0391. What is the Bayes estimator of this product? (Hint: It is not the product of averages, it is the average of products, so you will need to save products in the MCMC loop).