memo

Poisson回帰メモ

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p(y|\lambda) = \frac{\lambda^y \exp(-\lambda)}{y!}
L = \log\left(\prod_i p(y_i|\lambda_i)\right)
L = \sum_i \log( p(y_i|\lambda_i) )
L = \sum_i y_i \log(\lambda_i) - \lambda_i + \mbox{cont}
\log(\lambda) = {\bf w} \cdot {\bf x} + b
L = \sum_i y_i ({\bf w} \cdot {\bf x}_i + b) - \exp({\bf w} \cdot {\bf x}_i + b) + \mbox{cont}

Gaussian prior

wにgaussian priorを入れる

L = \sum_i y_i ({\bf w} \cdot {\bf x}_i + b) - \exp({\bf w} \cdot {\bf x}_i + b) 
 - c \ |{\bf w}|^2

オフセット項を入れる

L = \sum_i y_i ({\bf w} \cdot {\bf x}_i + b + \log A_i) - \exp({\bf w} \cdot {\bf x}_i + b + \log A_i) 
 - c \ |{\bf w}|^2


Lを最大化