2.6. Statistical Models, supervised learning and Function approximation

Source: Internet
Author: User
Statical model

  • Regression$y_i=f_{\theta} (x_i) +\epsilon_i,e (\epsilon) =0$
    1.$\epsilon\sim N (0,\sigma^2) $2. Using maximum likelihood estimation $\rightarrow$ least squares
    $y \sim N (F_{\theta} (x), \sigma^2) $
    $L (\theta) =-\frac{n}{2}log (2\PI)-nlog\sigma-\frac{1}{2\sigma^2}\sum_i\left (Y_i-f_{\theta} (x_i) \right) ^2$
  • Classification $p _{\theta} (g_i=k| x=x_i), K=1\cdots k$
    Using the maximum likelihood estimate here is equivalent to cross entropy and KL divergence
    For a single data point $ (x,g=k) $, its owning category $g=k$ is 1 and the remaining category is 0
    • $L (\theta) =logp (g=k|x) $ needs to be maximized
    • $CE (P,Q) =-\sum_x p (x) logq (x) $
      Corresponds to this example $ce=-\sum_i P (g=i) Logp (g=i|x_i) =-logp (g=k|x) $ needs to be minimized
    • $KL (P,Q) =\sum_x p (x) log\frac{p (x)}{q (x)}$
      Corresponds to this example $kl=\sum_i P (g=i) log\frac{p (g=i)}{p (g=i|x)}=log\frac{1}{p (g=k|x)}=-logp (g=k|x) $ need to be minimized

2.6. Statistical Models, supervised learning and Function approximation

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