For the quasar data in Table 5.1, apply a logarithm transformation on the response Y so that Y ← log(Y ). Consider the semi-conjugate Bayesian linear model that includes all five predictors.

(a) With specification β0 = 0, V ≈ O, a = 0.0001, and b = 0.0001, obtain the Bayesian estimator of β and σ2, as well as their corresponding 95% confidence interval. Compare them to the results from ordinary least square estimation.

(b) With the same prior specification in part (a), obtain the posterior densities of each β and ν using two methods, the analytic method and Gibbs sampler. Plot them on the same figure and comment on how the MCMC method approximates the true posterior densities.

(c) Using the Gibbs sampler, construct a 95% Bayesian interval for predicting the response of a newcomer who has x = (3, −14, 45, 20, −27)t .

(d) Redo the above analysis with a different prior specification β0 = 0, V ≈ 0.04 · I, a = 0.1, and b = 0.1, and compare to see how an informative prior affects the results.