1 导论
1.1
[
abla_mathbf{w}E(mathbf{w})=sum_{n=1}^N{y(x_n,mathbf{w})-t_n}phi(x_n)=0
]
[sum_{n=1}^Nphi^ ext{T}(x_n)mathbf{w}phi(x_n)=sum_{n=1}^Nt_nphi(x_n)
]
[mathbf{w}sum_{n=1}^Nphi(x_n)phi^ ext{T}(x_n) = sum_{n=1}^Nt_nphi(x_n)
]
1.2
1.29
由于(ln(cdot))是凹函数，利用琴生不等式，
[egin{aligned} ext{H}[x]&=-sum_{i=1}^Mp_iln p_i\&=sum_{i=1}^Mp_iln frac{1}{p_i}\&leqlnleft(sum_{i=1}^Mp_icdot frac{1}{p_i}
ight)\&=ln M.end{aligned}
]
1.30
[egin{aligned} ext{KL}(pVert q)&=-int p(x)lnfrac{q(x)}{p(x)}{
m{d}}x\&=-int p(x)lnleft(frac{sigma}{s}expleft(-frac{1}{2}left[frac{(x-m)^2}{s^2}-frac{(x-mu)^2}{sigma^2}
ight]
ight)
ight){
m{d}}x\&=-int p(x)left[lnfrac{sigma}{s}-frac{1}{2}left(frac{(x-m)^2}{s^2}-frac{(x-mu)^2}{sigma^2}
ight)
ight]{
m{d}}x
\&=lnfrac{s}{sigma}-frac{1}{2}intleft[left(frac{1}{sigma^2}-frac{1}{s^2}
ight)x^2+2left(frac{m}{s^2}-frac{mu}{sigma^2}
ight)x+left(frac{mu^2}{sigma^2}-frac{m^2}{s^2}
ight)
ight]p(x){
m{d}}x
\&=lnfrac{s}{sigma}+frac{1}{2}+frac{sigma^2+(mu-m)^2}{2s^2}.
end{aligned}]