part a - f please

2. Consider a linear regression setting where the response variable is y = (11, .... .) and there is one feature, or else predictor, I = (21, ..., 2.). We are interested in fitting the following model Vi = Bri +6, i=1, ....n, where the error terms e,'s are independent and distributed according to the Normal distribution with mean 0 and known variance o'. Equivalently, we can write that given a each y, is independent and distributed according to the Normal distribution with mean Br, and known variance g?. (a) Derive the likelihood function for the unknown parameter B. [2 marks] (b) Derive the Jeffreys prior for B. Use it to obtain the corresponding posterior distribution. [5 marks] (c) Consider the Normal distribution prior for S with zero mean and variance w. Use it to obtain the corresponding posterior distribution. [5 marks] (d) Consider the least squares criterion (1) and show that the estimator of B that minimises equation (1), also maximises the likelihood function derived in part (a). Derive this estimator and, in addition, consider the following penalised least squares criterion (2) given a > > 0. Derive the estimator of & that minimises equation (2) and compare with the one that minimises equation (1). [4 marks] (e) Provide a Bayes estimator for each of the posteriors in parts (b) and (c) and compare them with the estimators of part (d). [5 marks] (f) Let In+1 represent a future observation from the same model given the corresponding value of the predictor I,41. Find the posterior predictive distribution of Un+1 for one of the posteriors in parts (b) or (c). [4 marks]