For this question, assume you observe n (data point, label) pairs (xi, yi) n i=1, with xi 2 Rd and yi 2 R for all i = 1,..., n. We denote X as the data matrix containing all the data points and y as the label vector containing all the labels: X = 2 6666666666664 x> 1 . . . x> n 3 7777777777775 2 Rnd , y = 2 6666666666664 y1 . . . yn 3 7777777777775 2 Rn . (a) (4 points) Ignoring y for now, suppose we model the data points as coming from a d-dimensional Gaussian with diagonal covariance: 8i = 1,..., n, xi i.i.d. N(, ); = 2 6666666666664 2 1 ... 0 . . . ... . . . 0 ... 2 d 3 7777777777775 . If we consider 2 Rd and (2 1,...,2 d), where each 2 i > 0, to be unknown, the parameter space here is 2d-dimensional. When we refer to as a parameter, we are referring to the d-tuple (2 1,...,2 d), but inside a linear algebraic expression, denotes the diagonal matrix diag(2 1,...,2 d). Compute the log-likelihood `(, ) = log p(X |