could you help me on this problem ? Thank you .

In this exercise, we solve a simple optimization problem related to the multinomial naive Bayes classifier. Fix a finite set % = (x1,...,XM} and real numbers c1, . .., CM 2 0. Consider the following optimiza tion problem W = argmax L(w) := _ clog wi (1) W=[WI....>WM]TERM 1= 1 subject to: wis a PMF on . We will show that the solution is given by C1 CM W = , . . (2) E Ci (i)) The constraint set of (1) is the 'simplex' 6 = (we RM|0 S w1, . .., WM S 1, EM, w; = 1). Denote the larger constraint set 6 = (we RM | EM, wi = 1). Let A ER be a Lagrange multiplier for 6. Then the Lagrangian is M g (1, w) = L(w) -_ _ wi -1. (3) Show that 1 needs to satisfy ag(1, w) C1 CM - A [1,. .., 1] = 0. (4) Ow W1 WM Conclude that (2) is the global maximum L(w) over 6. (ii) From (i), conclude that (2) is the global maximum L(w) over 6