Benutzer:Hourssales/Informationsmaße

${\displaystyle {\begin{aligned}{\text{Shannon: }}&I^{\text{S}}(p)=\sum _{i}p_{i}\ln p_{i}\\{\text{Renyi: }}&I_{q}^{\text{R}}(p)={\frac {1}{q-1}}\ln \left(\sum _{i}p_{i}^{q}\right)\\{\text{Tsallis: }}&I_{q}^{\text{T}}(p)={\frac {1}{1-q}}\left(1-\sum _{i}p_{i}^{q}\right)\end{aligned}}}${\displaystyle {\begin{aligned}{\text{Shannon: }}&I^{\text{S}}(p)=\sum _{i}p_{i}\ln p_{i}\\{\text{Renyi: }}&I_{q}^{\text{R}}(p)={\frac {1}{q-1}}\ln \left(\sum _{i}p_{i}^{q}\right)\\{\text{Tsallis: }}&I_{q}^{\text{T}}(p)={\frac {1}{1-q}}\left(1-\sum _{i}p_{i}^{q}\right)\end{aligned}}} Wir wissen: ${\displaystyle \lim _{x\rightarrow 0}\ln(x+1)=x}${\displaystyle \lim _{x\rightarrow 0}\ln(x+1)=x}

${\displaystyle {\begin{aligned}I_{1}^{\text{R}}(p)&:=\lim _{q\rightarrow 1}I_{q}^{\text{R}}(p)\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\ln \left(\sum _{i}p_{i}^{q}\right)\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\underbrace {\ln \left(\underbrace {\sum _{i}p_{i}^{q}} _{\rightarrow 1}-\underbrace {\sum _{i}p_{i}} _{=1}+1\right)} _{\rightarrow \sum _{i}p_{i}^{q}-\sum _{i}p_{i}}\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\left(\sum _{i}p_{i}^{q}-\sum _{i}p_{i}\right)\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\sum _{i}\left(p_{i}^{q}-p_{i}\right)\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\sum _{i}p_{i}\underbrace {\left(p_{i}^{q-1}-1\right)} _{\rightarrow \ln p_{i}^{q-1}}\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\sum _{i}p_{i}\ln p_{i}^{q-1}\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\sum _{i}p_{i}(q-1)\ln p_{i}\\&=\lim _{q\rightarrow 1}{\frac {q-1}{q-1}}\sum _{i}p_{i}\ln p_{i}\\&=\sum _{i}p_{i}\ln p_{i}\\&=I^{\text{S}}(p)\end{aligned}}}${\displaystyle {\begin{aligned}I_{1}^{\text{R}}(p)&:=\lim _{q\rightarrow 1}I_{q}^{\text{R}}(p)\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\ln \left(\sum _{i}p_{i}^{q}\right)\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\underbrace {\ln \left(\underbrace {\sum _{i}p_{i}^{q}} _{\rightarrow 1}-\underbrace {\sum _{i}p_{i}} _{=1}+1\right)} _{\rightarrow \sum _{i}p_{i}^{q}-\sum _{i}p_{i}}\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\left(\sum _{i}p_{i}^{q}-\sum _{i}p_{i}\right)\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\sum _{i}\left(p_{i}^{q}-p_{i}\right)\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\sum _{i}p_{i}\underbrace {\left(p_{i}^{q-1}-1\right)} _{\rightarrow \ln p_{i}^{q-1}}\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\sum _{i}p_{i}\ln p_{i}^{q-1}\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\sum _{i}p_{i}(q-1)\ln p_{i}\\&=\lim _{q\rightarrow 1}{\frac {q-1}{q-1}}\sum _{i}p_{i}\ln p_{i}\\&=\sum _{i}p_{i}\ln p_{i}\\&=I^{\text{S}}(p)\end{aligned}}}

${\displaystyle {\begin{aligned}I_{1}^{\text{T}}(p)&:=\lim _{q\rightarrow 1}I_{q}^{\text{T}}(p)\\&=\lim _{q\rightarrow 1}{\frac {1}{1-q}}\left(1-\sum _{i}p_{i}^{q}\right)\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\underbrace {\left(\underbrace {\sum _{i}p_{i}^{q}} _{\rightarrow 1}-1\right)} _{\rightarrow \ln \sum _{i}p_{i}^{q}-1+1}\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\ln \sum _{i}p_{i}^{q}\\&=\lim _{q\rightarrow 1}I_{q}^{\text{R}}(p)\\&=I^{\text{S}}(p)\end{aligned}}}${\displaystyle {\begin{aligned}I_{1}^{\text{T}}(p)&:=\lim _{q\rightarrow 1}I_{q}^{\text{T}}(p)\\&=\lim _{q\rightarrow 1}{\frac {1}{1-q}}\left(1-\sum _{i}p_{i}^{q}\right)\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\underbrace {\left(\underbrace {\sum _{i}p_{i}^{q}} _{\rightarrow 1}-1\right)} _{\rightarrow \ln \sum _{i}p_{i}^{q}-1+1}\\&=\lim _{q\rightarrow 1}{\frac {1}{q-1}}\ln \sum _{i}p_{i}^{q}\\&=\lim _{q\rightarrow 1}I_{q}^{\text{R}}(p)\\&=I^{\text{S}}(p)\end{aligned}}}