2. (10 points] In this exercise we derive the Bayesian connection to the Lasso. (a) Consider the following linear regression without an intercept: 3) ya = Zita/3;: + 5i: j=1 where 51, . . . i an are independent and identically distributed as NH), 02) and 0'2 is known. What is the likelihood for the data?I [Hint: the density of the NM, or?) distribution is pm at?) = (maa-op)\". (1)) Assume the following prior for ,8: 61, . . . ,p are independent and identically distributed as double exponential random variables with density function Using the Bayes\" Theorem, derive the posterior for ,6. (c) By choosing T appropriately, show that the mode of the posterior distribution 33(3 l y) is the same as the Lasso estimate. [Hint: the value of a that maximizes f(z) is the same as the value of 2 that minimizes f(z).]