machine learning,

i dont understand part b and some parts of d

answer is provided, please explain the answer in detail to me

9 (a) A linear fit to the discriminant function g(x) = w x for a two-class Bayes classifier leads to a logistic sigmoid function for the posterior p(C1 | 2). Prove this statement starting from the following expression for the discrim- inant function 9(x) = In PC / x) =W wx p(C2x) (b) Show that the negative log-likelihood for logistic regression training is E(w) =-CIn o(w"x;) + (1 - C:) ln(1 0(w"x;)) where C; {0,1} is the class of training feature x; and w the set of weights. (e) Show for the logistic sigmoid function o(2) that do(2) = o(2) (1 - 0(z)) dz (d) Hence show that the gradient of E in part (b) is n aw C; -o)x; we have: w . slf - W x ) - (+) T 1 , ) w . P (4:01 W) 1 Y Taking independent ply I w ) o (w*x)"(1-'o (WT observations (w "x)) (-) . o the likelihood of - logh (w) given y 1(w) log(a (wa)*( 0(ww.) )*'* *:)) : (8: logowa) = (1 - 7:) log (1-0(wa))) (Q9cd) dz (2) (1e7 c) (1-2) : : (-) -. ( ) -)*() ( : 2 %) * { ) : ( ) 4 e ( from 14 where does this cane (Q9cd) (2-I : : (1-) (14) 11e-1 : () * *(( ) -)*( : ( :12 * ( :( |1=G) (: - . ( () pleare derive T -2 {G 4 2 { (( -1 ) 4 ( - 1) + (x & c } - ) 1 1- ... { (-1) - (:) ( - ) ( 3 (x ; } : ( G - sx ( -) - 3 ( -1) } 2- (8) dox Hence please explain these equations M. 5 Feb 07 6) Odborne have 2013 we slf -() wT*) P (1 T X . olu Tx) (1-'(w+x)) (-4) Y . ply: 11w) ) P (4:0W) 1 PL w o(wx) Taking independent observations To(w'*:)*(1- o (Wx;)) this i the likelihood of w 6(w) - logh (w) . Plog (a (wx)i (k- o(wx:))."'-:)) yiloyo (wx) * ( I -y:) log (1-0 (w7x:))) (w?r:)) (vi) given y 13 11