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Consider a dataset with n data points (@', y' ), a' ER", following from the following linear model: y' = Brite, i=1,...,m, where el are i.i.d. Gaussian noise with zero mean and variance o', and * is the true parameter. Consider the ridge regression as follows: B(A) = arg min B m (1) i=1 where A 2 0 is the regularized parameter. (a) (5 points) Find the closed form solution for B(A) and its distribution conditioning on {r } (i.e., treat them as fixed). (b) (5 points) Calculate the bias Er?B(X)] - "8* as a function of A and some fixed test point . (c) (5 points) Calculate the variance term E (&B(X) - EB(X)]) as a function of 1. (d) (5 points) Use the results from parts (b) and (c) and the bias-variance decomposition to analyze the impact of A in the mean squared error. Specifically, which term dominates when A is small, and large, respectively? (e) (5 points) Now suppose we have m = 100 samples. Write a pseudo-code to explain how to use cross validation to find the optimal 1. (f) (5 points) Explain if we would like to perform variable selection, how should we change the regularization term in Equation (1) to achieve this goal