We consider extending the binary logistic regression to a multiclass classification. Consider a dataset, (Xi, Vi)ty generated iid according to some unknown distribution with x; e Rand y; e [1; K']. For ke [1; K], we model the posterior probabilities n(x) - P [y = k(x] as Note that we have used our usual convention of appending a "1" to the vectors x. Our objective is to find the maximum likelihood estimator of the parameters |ke1;k] 1. Show that Onk (x) - m(x)(1 {k - j} - n;(x)) 2. Show that the log-likelihood function for the parameters @ = 6 kenapa- is given by NK ((0) - 1(y -blogn(x;) 3. Using (a), show that N Vo ((0) = [(1( = ] ) - nix; ) )x;. 4. The Hessian here a dk x dk matrix (since we have k parameters 0 of dimension d). Show that the Hessian is composed of d x d blocks given by Vex (Vof(9) ) - 1- 1 5. Can you find the maximum likelihood estimator explicitly? If yes, give an exact expression. If not, explain how you would find the answer numerically and justify why the algorithm would work (you can assume that the Hessian is semi definite positive, it's true but a bit hard to show )