Question 9 [3+3+3

Question 9 [3+3+3.5+3+5.5=18 points] Consider the linear regression model Y = MB + E, where Y = (Y], ..., Yn)T is the nx 1 response vector, M is the n xp model matrix with full column rank, ß is the p < 1 vector of unknown regression coefficients, and E is the nx1 vector of random errors. Assume E~Nn (Onx1,1--In), where 2 > 0 is an unknown parameter, Onxi is the n x 1 vector of 0's, and In is the n x n identity matrix. Write the vector of observed values of the Yi's as y = 1 (V1, ... , Yn)". Assume the prior for (B, 2) is given by g(6,2) 01-27(2 > 0). Note that information in this paragraph should be used to answer the sub-questions below unless stated otherwise. (a) Write down the posterior PDF of (2, 2), i.e., g(6,2]y), up to a proportionality constant. Please write it in a form that depends on y only through B = (M"M)-M"y and RSS = y? [In – M(MTM)-1M"]y. (b) Derive the marginal posterior PDF of 1, i.e., g(|y), up to a proportionality constant and identify the marginal posterior distribution of 1 (state the name of the distribution and its parameters). Note: You need to write the marginal posterior PDF of 1 into a form that allows you to identify the marginal posterior distribution of 1. (c) Derive the marginal posterior PDF of ß, i.e., g(ly), up to a proportionality constant and identify the marginal posterior distribution of ß (state the name of the distribution and its parameters). Note: You need to write the marginal posterior PDF of ß into a form that allows you to identify the marginal posterior distribution of ß. (d) Based on the result you obtained in part (a), write down: 1. The full conditional PDF of ß up to a proportionality constant. 2. The full conditional PDF of 2 up to a proportionality constant. Identify the full conditional distribution of ß and the full conditional distribution of 1 (state the name and parameters of each of the two full conditional distributions). Note: You need to write each of the two full conditional PDFs g(lla, y) and g(218,y) into a form that allows you to identify the distribution it represents. (e) Suppose Y(xº)|, 1, y = Y(xº) 8, ~N(m(xº)ß,1-1). Derive and identify the posterior predictive distribution y(x)]y (please state the name of the distribution and give explicit expressions for its parameters).