4) Bayesian linear models, MAP estimator, choosing priors (20 pts.) Suppose we are given a linear relationship between a random vector X E RM and a random parameter vector O E R" for some given A E Rman with measurements x = A0 + w, where W ~ N(0, 62 D). a) (9 pts.) Derive the MAP for 0 if its prior distribution is a normal distribution, O ~ N(0, T2 1). b) (9 pts.) Assuming A = I, derive the MAP for 0 if its prior distribution is a Laplace Distribution, O ~ Laplace(0, , I), p(0) = 2 exp(->||0| |2). c) (2 pts.) Compare the qualitative effects of each prior