$3-$ Suppose a multi$-$class logistic regression has been used as a model for digit recognition as

shown below

$P(Y=k|x=x)=\frac{\text{e}\text{x}\text{p}(w_k^{T}x)}{1+{\displaystyle \underset{l=1}{\overset{K-1}{}}}\text{e}\text{x}\text{p}(w_l^{T}x)}$ for $k=1,$dots,$K-1$

$P(Y=K|x=x)=\frac{1}{1+{\displaystyle \underset{l=1}{\overset{K-1}{}}}\text{e}\text{x}\text{p}(w_l^{T}x)}$

a$)$ Which parameters need to be estimated?

b$)$ How many parameters in total need to be estimated?

c$)$ With $n$ training samples as:

$\{(x_1,y_1),(x_2,y_2),$dots,$(x_n,y_n)\}$

d$)$ write down explicitly the log$-$likelihood function for multi$-$class logistic regression $($as

shown below$)$ and simplify it$.$

$L(w_1,$dots,$w_{K-1})={\displaystyle \underset{i=1}{\overset{n}{}}}lnP(Y=y_i|x=x_i)$

e$)$ Take the gradient with respect to w and simplify.