指数型分布族
分布表

次の表は、多くの一般的な分布を、正準型の指数型分布族として書き換える方法を示している。
スカラー変数とスカラーパラメータの場合：
スカラー変数とベクトルパラメータの場合：
ベクトル変数とベクトルパラメータの場合：
| 確率分布 |  パラメータ  {\boldsymbol {\theta }} | 自然パラメータ  {\boldsymbol {\eta }} | パラメータの逆写像 | Base measure  h(x) | 十分統計量  T(x) | Log-partition  A({\boldsymbol {\eta }}) | Log-partition  A({\boldsymbol {\theta }}) |
| --- | --- | --- | --- | --- | --- | --- | --- |
| ベルヌーイ分布 |  p |  \log {\frac {p}{1-p}} |  {\frac {1}{1+e^{-\eta }}}={\frac {e^{\eta }}{1+e^{\eta }}} |  1 |  x |  \log(1+e^{\eta }) |  -\log(1-p) |
| 二項分布 既知の試行回数  n |  p |  \log {\frac {p}{1-p}} |  {\frac {1}{1+e^{-\eta }}}={\frac {e^{\eta }}{1+e^{\eta }}} |  {n \choose x} |  x |  n\log(1+e^{\eta }) |  -n\log(1-p) |
| ポアソン分布 |  \lambda  |  \log \lambda  |  e^{\eta } |  {\frac {1}{x!}} |  x |  e^{\eta } |  \lambda  |
| 負の二項分布 with known number of failures  r |  p |  \log p |  e^{\eta } |  {x+r-1 \choose x} |  x |  -r\log(1-e^{\eta }) |  -r\log(1-p) |
| 指数分布 |  \lambda  |  -\lambda  |  -\eta  |  1 |  x |  -\log(-\eta ) |  -\log \lambda  |
| パレート分布 with known minimum value  x_{m} |  \alpha  |  -\alpha -1 |  -1-\eta  |  1 |  \log x |  -\log(-1-\eta )+(1+\eta )\log x_{\mathrm {m} } |  -\log \alpha -\alpha \log x_{\mathrm {m} } |
| ワイブル分布 with known shape  k |  \lambda  |  -{\frac {1}{\lambda ^{k}}} |  (-\eta )^{-{\frac {1}{k}}} |  x^{k-1} |  x^{k} |  -\log(-\eta )-\log k |  k\log \lambda -\log k |
| ラプラス分布 既知の平均  \mu  |  b |  -{\frac {1}{b}} |  -{\frac {1}{\eta }} |  1 |  |x-\mu | |  \log \left(-{\frac {2}{\eta }}\right) |  \log 2b |
| カイ二乗分布 |  \nu  |  {\frac {\nu }{2}}-1 |  2(\eta +1) |  e^{-{\frac {x}{2}}} |  \log x |  \log \Gamma (\eta +1)+(\eta +1)\log 2 |  \log \Gamma \left({\frac {\nu }{2}}\right)+{\frac {\nu }{2}}\log 2 |
| 正規分布  既知の分散  \sigma ^{2} |  \mu  |  {\frac {\mu }{\sigma }} |  \sigma \eta  |  {\frac {e^{-{\frac {x^{2}}{2\sigma ^{2}}}}}{{\sqrt {2\pi }}\sigma }} |  {\frac {x}{\sigma }} |  {\frac {\eta ^{2}}{2}} |  {\frac {\mu ^{2}}{2\sigma ^{2}}} |
| 正規分布 |  \mu ,  \sigma ^{2} |  {\begin{bmatrix}{\dfrac {\mu }{\sigma ^{2}}}\\[10pt]-{\dfrac {1}{2\sigma ^{2}}}\end{bmatrix}} |  {\begin{bmatrix}-{\dfrac {\eta _{1}}{2\eta _{2}}}\\[15pt]-{\dfrac {1}{2\eta _{2}}}\end{bmatrix}} |  {\frac {1}{\sqrt {2\pi }}} |  {\begin{bmatrix}x\\x^{2}\end{bmatrix}} |  -{\frac {\eta _{1}^{2}}{4\eta _{2}}}-{\frac {1}{2}}\log(-2\eta _{2}) |  {\frac {\mu ^{2}}{2\sigma ^{2}}}+\log \sigma  |
| 対数正規分布 |  \mu ,  \sigma ^{2} |  {\begin{bmatrix}{\dfrac {\mu }{\sigma ^{2}}}\\[10pt]-{\dfrac {1}{2\sigma ^{2}}}\end{bmatrix}} |  {\begin{bmatrix}-{\dfrac {\eta _{1}}{2\eta _{2}}}\\[15pt]-{\dfrac {1}{2\eta _{2}}}\end{bmatrix}} |  {\frac {1}{{\sqrt {2\pi }}x}} |  {\begin{bmatrix}\log x\\(\log x)^{2}\end{bmatrix}} |  -{\frac {\eta _{1}^{2}}{4\eta _{2}}}-{\frac {1}{2}}\log(-2\eta _{2}) |  {\frac {\mu ^{2}}{2\sigma ^{2}}}+\log \sigma  |
| 逆ガウス分布 |  \mu ,  \lambda  |  {\begin{bmatrix}-{\dfrac {\lambda }{2\mu ^{2}}}\\[15pt]-{\dfrac {\lambda }{2}}\end{bmatrix}} |  {\begin{bmatrix}{\sqrt {\dfrac {\eta _{2}}{\eta _{1}}}}\\[15pt]-2\eta _{2}\end{bmatrix}} |  {\frac {1}{{\sqrt {2\pi }}x^{\frac {3}{2}}}} |  {\begin{bmatrix}x\\[5pt]{\dfrac {1}{x}}\end{bmatrix}} |  2{\sqrt {\eta _{1}\eta _{2}}}-{\frac {1}{2}}\log(-2\eta _{2}) |  -{\frac {\lambda }{\mu }}-{\frac {1}{2}}\log \lambda  |
| ガンマ分布 |  \alpha ,  \beta  |  {\begin{bmatrix}\alpha -1\\-\beta \end{bmatrix}} |  {\begin{bmatrix}\eta _{1}+1\\-\eta _{2}\end{bmatrix}} |  1 |  {\begin{bmatrix}\log x\\x\end{bmatrix}} |  \log \Gamma (\eta _{1}+1)-(\eta _{1}+1)\log(-\eta _{2}) |  \log \Gamma (\alpha )-\alpha \log \beta  |
| ガンマ分布 |  k,  \theta  |  {\begin{bmatrix}k-1\\[5pt]-{\dfrac {1}{\theta }}\end{bmatrix}} |  {\begin{bmatrix}\eta _{1}+1\\[5pt]-{\dfrac {1}{\eta _{2}}}\end{bmatrix}} |  1 |  {\begin{bmatrix}\log x\\x\end{bmatrix}} |  \log \Gamma (\eta _{1}+1)-(\eta _{1}+1)\log(-\eta _{2}) |  \log \Gamma (k)+k\log \theta  |
| 逆ガンマ分布 |  \alpha ,  \beta  |  {\begin{bmatrix}-\alpha -1\\-\beta \end{bmatrix}} |  {\begin{bmatrix}-\eta _{1}-1\\-\eta _{2}\end{bmatrix}} |  1 |  {\begin{bmatrix}\log x\\{\frac {1}{x}}\end{bmatrix}} |  \log \Gamma (-\eta _{1}-1)-(-\eta _{1}-1)\log(-\eta _{2}) |  \log \Gamma (\alpha )-\alpha \log \beta  |
| 一般化逆ガウス分布 |  p,  a,  b |  {\begin{bmatrix}p-1\\-a/2\\-b/2\end{bmatrix}} |  {\begin{bmatrix}\eta _{1}+1\\-2\eta _{2}\\-2\eta _{3}\end{bmatrix}} |  1 |  {\begin{bmatrix}\log x\\x\\{\frac {1}{x}}\end{bmatrix}} |  \log 2K_{\eta _{1}+1}({\sqrt {4\eta _{2}\eta _{3}}})-{\frac {\eta _{1}+1}{2}}\log {\frac {\eta _{2}}{\eta _{3}}} |  \log 2K_{p}({\sqrt {ab}})-{\frac {p}{2}}\log {\frac {a}{b}} |
| スケールされた逆カイ二乗分布 |  \nu ,  \sigma ^{2} |  {\begin{bmatrix}-{\dfrac {\nu }{2}}-1\\[10pt]-{\dfrac {\nu \sigma ^{2}}{2}}\end{bmatrix}} |  {\begin{bmatrix}-2(\eta _{1}+1)\\[10pt]{\dfrac {\eta _{2}}{\eta _{1}+1}}\end{bmatrix}} |  1 |  {\begin{bmatrix}\log x\\{\frac {1}{x}}\end{bmatrix}} |  \log \Gamma (-\eta _{1}-1)-(-\eta _{1}-1)\log(-\eta _{2}) |  \log \Gamma \left({\frac {\nu }{2}}\right)-{\frac {\nu }{2}}\log {\frac {\nu \sigma ^{2}}{2}} |
| ベータ分布 （variant 1) |  \alpha ,  \beta  |  {\begin{bmatrix}\alpha \\\beta \end{bmatrix}} |  {\begin{bmatrix}\eta _{1}\\\eta _{2}\end{bmatrix}} |  {\frac {1}{x(1-x)}} |  {\begin{bmatrix}\log x\\\log(1-x)\end{bmatrix}} |  \log \Gamma (\eta _{1})+\log \Gamma (\eta _{2})-\log \Gamma (\eta _{1}+\eta _{2}) |  \log \Gamma (\alpha )+\log \Gamma (\beta )-\log \Gamma (\alpha +\beta ) |
| ベータ分布 (variant 2) |  \alpha ,  \beta  |  {\begin{bmatrix}\alpha -1\\\beta -1\end{bmatrix}} |  {\begin{bmatrix}\eta _{1}+1\\\eta _{2}+1\end{bmatrix}} |  1 |  {\begin{bmatrix}\log x\\\log(1-x)\end{bmatrix}} |  \log \Gamma (\eta _{1}+1)+\log \Gamma (\eta _{2}+1)-\log \Gamma (\eta _{1}+\eta _{2}+2) |  \log \Gamma (\alpha )+\log \Gamma (\beta )-\log \Gamma (\alpha +\beta ) |
| 多変量正規分布 |  {\boldsymbol {\mu }},  {\boldsymbol {\sigma }} |  {\begin{bmatrix}{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}\\[5pt]-{\frac {1}{2}}{\boldsymbol {\Sigma }}^{-1}\end{bmatrix}} |  {\begin{bmatrix}-{\frac {1}{2}}{\boldsymbol {\eta }}_{2}^{-1}{\boldsymbol {\eta }}_{1}\\[5pt]-{\frac {1}{2}}{\boldsymbol {\eta }}_{2}^{-1}\end{bmatrix}} |  (2\pi )^{-{\frac {k}{2}}} |  {\begin{bmatrix}\mathbf {x} \\[5pt]\mathbf {x} \mathbf {x} ^{\mathrm {T} }\end{bmatrix}} |  -{\frac {1}{4}}{\boldsymbol {\eta }}_{1}^{\rm {T}}{\boldsymbol {\eta }}_{2}^{-1}{\boldsymbol {\eta }}_{1}-{\frac {1}{2}}\log \left|-2{\boldsymbol {\eta }}_{2}\right| |  {\frac {1}{2}}{\boldsymbol {\mu }}^{\rm {T}}{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}+{\frac {1}{2}}\log |{\boldsymbol {\Sigma }}| |
| カテゴリカル分布 (variant 1) |  p_{1},\dots {},p_{k}  where  \textstyle \sum _{i=1}^{k}p_{i}=1 |  {\begin{bmatrix}\log p_{1}\\\vdots \\\log p_{k}\end{bmatrix}} |  {\begin{bmatrix}e^{\eta _{1}}\\\vdots \\e^{\eta _{k}}\end{bmatrix}}  where  \textstyle \sum _{i=1}^{k}e^{\eta _{i}}=1 |  1 |  {\begin{bmatrix}[x=1]\\\vdots \\{[x=k]}\end{bmatrix}} |  0 |  0 |
| カテゴリカル分布 (variant 2) |  p_{1},\dots {},p_{k}  where  \textstyle \sum _{i=1}^{k}p_{i}=1 |  {\begin{bmatrix}\log p_{1}+C\\\vdots \\\log p_{k}+C\end{bmatrix}} |  {\begin{bmatrix}{\dfrac {1}{C}}e^{\eta _{1}}\\\vdots \\{\dfrac {1}{C}}e^{\eta _{k}}\end{bmatrix}}=   {\begin{bmatrix}{\dfrac {e^{\eta _{1}}}{\sum _{i=1}^{k}e^{\eta _{i}}}}\\[10pt]\vdots \\[5pt]{\dfrac {e^{\eta _{k}}}{\sum _{i=1}^{k}e^{\eta _{i}}}}\end{bmatrix}} where  \textstyle \sum _{i=1}^{k}e^{\eta _{i}}=C |  1 |  {\begin{bmatrix}[x=1]\\\vdots \\{[x=k]}\end{bmatrix}} |  0 |  0 |
| カテゴリカル分布 (variant 3) |  p_{1},\dots {},p_{k}  where  p_{k}=1-\textstyle \sum _{i=1}^{k-1}p_{i} |  {\begin{bmatrix}\log {\dfrac {p_{1}}{p_{k}}}\\[10pt]\vdots \\[5pt]\log {\dfrac {p_{k-1}}{p_{k}}}\\[15pt]0\end{bmatrix}}=   {\begin{bmatrix}\log {\dfrac {p_{1}}{1-\sum _{i=1}^{k-1}p_{i}}}\\[10pt]\vdots \\[5pt]\log {\dfrac {p_{k-1}}{1-\sum _{i=1}^{k-1}p_{i}}}\\[15pt]0\end{bmatrix}} This is the inverse softmax function, a generalization of the logit function. |  {\begin{bmatrix}{\dfrac {e^{\eta _{1}}}{\sum _{i=1}^{k}e^{\eta _{i}}}}\\[10pt]\vdots \\[5pt]{\dfrac {e^{\eta _{k}}}{\sum _{i=1}^{k}e^{\eta _{i}}}}\end{bmatrix}}=    {\begin{bmatrix}{\dfrac {e^{\eta _{1}}}{1+\sum _{i=1}^{k-1}e^{\eta _{i}}}}\\[10pt]\vdots \\[5pt]{\dfrac {e^{\eta _{k-1}}}{1+\sum _{i=1}^{k-1}e^{\eta _{i}}}}\\[15pt]{\dfrac {1}{1+\sum _{i=1}^{k-1}e^{\eta _{i}}}}\end{bmatrix}} This is the softmax function, a generalization of the logistic function. |  1 |  {\begin{bmatrix}[x=1]\\\vdots \\{[x=k]}\end{bmatrix}} |  \log \left(\sum _{i=1}^{k}e^{\eta _{i}}\right)=\log \left(1+\sum _{i=1}^{k-1}e^{\eta _{i}}\right) |  -\log p_{k}=-\log \left(1-\sum _{i=1}^{k-1}p_{i}\right) |
| 多項分布 (variant 1) 既知の試行回数  n |  p_{1},\dots {},p_{k}  where  \textstyle \sum _{i=1}^{k}p_{i}=1 |  {\begin{bmatrix}\log p_{1}\\\vdots \\\log p_{k}\end{bmatrix}} |  {\begin{bmatrix}e^{\eta _{1}}\\\vdots \\e^{\eta _{k}}\end{bmatrix}}  where  \textstyle \sum _{i=1}^{k}e^{\eta _{i}}=1 |  {\frac {n!}{\prod _{i=1}^{k}x_{i}!}} |  {\begin{bmatrix}x_{1}\\\vdots \\x_{k}\end{bmatrix}} |  0 |  0 |
| 多項分布 (variant 2) 既知の試行回数  n |  p_{1},\dots {},p_{k}  where  \textstyle \sum _{i=1}^{k}p_{i}=1 |  {\begin{bmatrix}\log p_{1}+C\\\vdots \\\log p_{k}+C\end{bmatrix}} |  {\begin{bmatrix}{\dfrac {1}{C}}e^{\eta _{1}}\\\vdots \\{\dfrac {1}{C}}e^{\eta _{k}}\end{bmatrix}}=   {\begin{bmatrix}{\dfrac {e^{\eta _{1}}}{\sum _{i=1}^{k}e^{\eta _{i}}}}\\[10pt]\vdots \\[5pt]{\dfrac {e^{\eta _{k}}}{\sum _{i=1}^{k}e^{\eta _{i}}}}\end{bmatrix}} where  \textstyle \sum _{i=1}^{k}e^{\eta _{i}}=C |  {\frac {n!}{\prod _{i=1}^{k}x_{i}!}} |  {\begin{bmatrix}x_{1}\\\vdots \\x_{k}\end{bmatrix}} |  0 |  0 |
| 多項分布 (variant 3) 既知の試行回数  n |  p_{1},\dots {},p_{k}  where  p_{k}=1-\textstyle \sum _{i=1}^{k-1}p_{i} |  {\begin{bmatrix}\log {\dfrac {p_{1}}{p_{k}}}\\[10pt]\vdots \\[5pt]\log {\dfrac {p_{k-1}}{p_{k}}}\\[15pt]0\end{bmatrix}}=   {\begin{bmatrix}\log {\dfrac {p_{1}}{1-\sum _{i=1}^{k-1}p_{i}}}\\[10pt]\vdots \\[5pt]\log {\dfrac {p_{k-1}}{1-\sum _{i=1}^{k-1}p_{i}}}\\[15pt]0\end{bmatrix}} |  {\begin{bmatrix}{\dfrac {e^{\eta _{1}}}{\sum _{i=1}^{k}e^{\eta _{i}}}}\\[10pt]\vdots \\[5pt]{\dfrac {e^{\eta _{k}}}{\sum _{i=1}^{k}e^{\eta _{i}}}}\end{bmatrix}}=    {\begin{bmatrix}{\dfrac {e^{\eta _{1}}}{1+\sum _{i=1}^{k-1}e^{\eta _{i}}}}\\[10pt]\vdots \\[5pt]{\dfrac {e^{\eta _{k-1}}}{1+\sum _{i=1}^{k-1}e^{\eta _{i}}}}\\[15pt]{\dfrac {1}{1+\sum _{i=1}^{k-1}e^{\eta _{i}}}}\end{bmatrix}} |  {\frac {n!}{\prod _{i=1}^{k}x_{i}!}} |  {\begin{bmatrix}x_{1}\\\vdots \\x_{k}\end{bmatrix}} |  n\log \left(\sum _{i=1}^{k}e^{\eta _{i}}\right)=n\log \left(1+\sum _{i=1}^{k-1}e^{\eta _{i}}\right) |  -n\log p_{k}=-n\log \left(1-\sum _{i=1}^{k-1}p_{i}\right) |
| ディリクレ分布 (variant 1) |  \alpha _{1},\dots {},\alpha _{k} |  {\begin{bmatrix}\alpha _{1}\\\vdots \\\alpha _{k}\end{bmatrix}} |  {\begin{bmatrix}\eta _{1}\\\vdots \\\eta _{k}\end{bmatrix}} |  {\frac {1}{\prod _{i=1}^{k}x_{i}}} |  {\begin{bmatrix}\log x_{1}\\\vdots \\\log x_{k}\end{bmatrix}} |  \sum _{i=1}^{k}\log \Gamma (\eta _{i})-\log \Gamma \left(\sum _{i=1}^{k}\eta _{i}\right) |  \sum _{i=1}^{k}\log \Gamma (\alpha _{i})-\log \Gamma \left(\sum _{i=1}^{k}\alpha _{i}\right) |
| ディリクレ分布 (variant 2) |  \alpha _{1},\dots {},\alpha _{k} |  {\begin{bmatrix}\alpha _{1}-1\\\vdots \\\alpha _{k}-1\end{bmatrix}} |  {\begin{bmatrix}\eta _{1}+1\\\vdots \\\eta _{k}+1\end{bmatrix}} |  1 |  {\begin{bmatrix}\log x_{1}\\\vdots \\\log x_{k}\end{bmatrix}} |  \sum _{i=1}^{k}\log \Gamma (\eta _{i}+1)-\log \Gamma \left(\sum _{i=1}^{k}(\eta _{i}+1)\right) |  \sum _{i=1}^{k}\log \Gamma (\alpha _{i})-\log \Gamma \left(\sum _{i=1}^{k}\alpha _{i}\right) |
| ウィッシャート分布 |  \mathbf {V} ,  n |  {\begin{bmatrix}-{\frac {1}{2}}\mathbf {V} ^{-1}\\[5pt]{\dfrac {n-p-1}{2}}\end{bmatrix}} |  {\begin{bmatrix}-{\frac {1}{2}}{{\boldsymbol {\eta }}_{1}}^{-1}\\[5pt]2\eta _{2}+p+1\end{bmatrix}} |  1 |  {\begin{bmatrix}\mathbf {X} \\\log |\mathbf {X} |\end{bmatrix}} |  -\left(\eta _{2}+{\frac {p+1}{2}}\right)\log |-{\boldsymbol {\eta }}_{1}|         +\log \Gamma _{p}\left(\eta _{2}+{\frac {p+1}{2}}\right)=   -{\frac {n}{2}}\log |-{\boldsymbol {\eta }}_{1}|+\log \Gamma _{p}\left({\frac {n}{2}}\right)=   \left(\eta _{2}+{\frac {p+1}{2}}\right)(p\log 2+\log |\mathbf {V} |)         +\log \Gamma _{p}\left(\eta _{2}+{\frac {p+1}{2}}\right) Three variants with different parameterizations are given, to facilitate computing moments of the sufficient statistics. |  {\frac {n}{2}}(p\log 2+\log |\mathbf {V} |)+\log \Gamma _{p}\left({\frac {n}{2}}\right) |
| 逆ウィッシャート分布 |  {\boldsymbol {\Psi }},  m |  {\begin{bmatrix}-{\frac {1}{2}}{\boldsymbol {\Psi }}\\[5pt]-{\dfrac {m+p+1}{2}}\end{bmatrix}} |  {\begin{bmatrix}-2{\boldsymbol {\eta }}_{1}\\[5pt]-(2\eta _{2}+p+1)\end{bmatrix}} |  1 |  {\begin{bmatrix}\mathbf {X} ^{-1}\\\log |\mathbf {X} |\end{bmatrix}} |  {\begin{aligned}&\left(\eta _{2}+{\frac {p+1}{2}}\right)\log |-{\boldsymbol {\eta }}_{1}|+\log \Gamma _{p}\left(-{\Big (}\eta _{2}+{\frac {p+1}{2}}{\Big )}\right)\\&=-{\frac {m}{2}}\log |-{\boldsymbol {\eta }}_{1}|+\log \Gamma _{p}\left({\frac {m}{2}}\right)\\&=-\left(\eta _{2}+{\frac {p+1}{2}}\right)(p\log 2-\log |{\boldsymbol {\Psi }}|)+\log \Gamma _{p}\left(-{\Big (}\eta _{2}+{\frac {p+1}{2}}{\Big )}\right)\end{aligned}} |  {\frac {m}{2}}(p\log 2-\log |{\boldsymbol {\Psi }}|)+\log \Gamma _{p}\left({\frac {m}{2}}\right) |