Consider the Bayesian linear model where i=1nxi=0,i=1nxi2=n and i=1nxiyi=. The prior for is given by ()=0.51(=0)+0.5N(0,2), where 1(=0) is the probability mass function which puts all probability on =0. Suppose and are known. (1) ( 8 pts) Write down the posterior density for and note that this is always of the form prior times likelihood. (2) (12 pts). By integrating out , find the marginal likelihood m(y1,,yn)=i=1nN(yixi,2)N(0,2)d. (3) (12 pts) Hence, show that (y1,,yn){i=1nN(yi0,2)}1(=0)+m(y1,,yn)N(,2) and find and . (4) (8 pts) Hence, write down (=0y1,,yn). (5) (10 pts) What is the significance of the term i=1nN(yi0,2)m(y1,,yn) and show that this converges to infinity as y2 grows