信息论 二元离散熵的导数

\[(p\log_{2}p)' = \log_{2}p + p \frac{1}{\ln 2}\frac{1}{p} = \frac{\ln p+1}{\ln 2} \]

\[\begin{align*} H'(p) &= -\frac{1}{\ln 2}(\ln p + 1 -\ln(1-p) -1) \\&=-\frac{1}{\ln 2}(\ln \frac{p}{1-p}) \\&=-\log_{2}\frac{p}{1-p} \\&=\log_{2}\frac{1-p}{p} \end{align*}\]

\[H'(f(p)) = \boldsymbol{f'(p)}\log_{2}\frac{1-f(p)}{f(p)} \]