スコア凾数の期待値と分散

$$\frac{\partial \log f(x;\theta )}{\partial \theta }:\mathrm{ス}\mathrm{コ}\mathrm{ア}\mathrm{凾}\mathrm{数}$$

スコア凾数の期待値

$$\begin{array}{rcl}\mathrm{E}\left[\frac{\partial \log f(x;\theta )}{\partial \theta }\right]& =& \int \frac{\partial \log f(x;\theta )}{\partial \theta }f(x;\theta )\mathrm{d}x\\ & =& \int \frac{1}{f(x;\theta )}\frac{\partial f(x;\theta )}{\partial \theta }f(x;\theta )\mathrm{d}x\cdots \frac{\mathrm{d}}{\mathrm{d}x}\log f(x)=\frac{1}{f(x)}\frac{\mathrm{d}f(x)}{\mathrm{d}x}\\ & =& \int \frac{\partial f(x;\theta )}{\partial \theta }\mathrm{d}x\\ & =& \frac{\partial }{\partial \theta }\int f(x;\theta )\mathrm{d}x\\ & =& \frac{\partial }{\partial \theta }1\cdots \int f(x;\theta )\mathrm{d}x=1\\ & =& 0\cdots \mathrm{定}\mathrm{数}\mathrm{の}\mathrm{微}\mathrm{分}\mathrm{は}0\end{array}$$

スコア凾数の分散

$$\mathcal{I}=\mathrm{V}\left[\frac{\partial \log f(x;\theta )}{\partial \theta }\right]:\mathrm{フ}\mathrm{ィ}\mathrm{ッ}\mathrm{シ}\mathrm{ャ}\mathrm{ー}\mathrm{情}\mathrm{報}\mathrm{量}$$