分布表
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2021/09/17 08:32 UTC 版)
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： f X ( x ∣ θ ) = h ( x ) exp ( η ( θ ) T ( x ) − A ( η ) ) {\displaystyle f_{X}(x\mid \theta )=h(x)\,\exp {{\Big (}\eta (\theta )\,T(x)-A(\eta ){\Big )}}} スカラー変数とベクトルパラメータの場合： f X ( x ∣ θ ) = h ( x ) exp ( η ( θ ) ⊺ T ( x ) − A ( η ) ) {\displaystyle f_{X}(x\mid {\boldsymbol {\theta }})=h(x)\,\exp {{\Big (}{\boldsymbol {\eta }}({\boldsymbol {\theta }})^{\intercal }\,{\boldsymbol {T}}(x)-A({\boldsymbol {\eta }}){\Big )}}} f X ( x ∣ θ ) = h ( x ) g ( θ ) exp ( η ( θ ) ⊺ T ( x ) ) {\displaystyle f_{X}(x\mid {\boldsymbol {\theta }})=h(x)\,g({\boldsymbol {\theta }})\,\exp {{\Big (}{\boldsymbol {\eta }}({\boldsymbol {\theta }})^{\intercal }\,{\boldsymbol {T}}(x){\Big )}}} ベクトル変数とベクトルパラメータの場合： f X ( x ∣ θ ) = h ( x ) exp ( η ( θ ) ⊺ T ( x ) − A ( η ) ) {\displaystyle f_{X}(\mathbf {x} \mid {\boldsymbol {\theta }})=h(\mathbf {x} )\,\exp {{\Big (}{\boldsymbol {\eta }}({\boldsymbol {\theta }})^{\intercal }\,{\boldsymbol {T}}({\boldsymbol {x}})-A({\boldsymbol {\eta }}){\Big )}}} 確率分布 パラメータ θ {\displaystyle {\boldsymbol {\theta }}} 自然パラメータ η {\displaystyle {\boldsymbol {\eta }}} パラメータの逆写像Base measure h ( x ) {\displaystyle h(x)} 十分統計量 T ( x ) {\displaystyle T(x)} Log-partition A ( η ) {\displaystyle A({\boldsymbol {\eta }})} Log-partition A ( θ ) {\displaystyle A({\boldsymbol {\theta }})} ベルヌーイ分布 p {\displaystyle p} log p 1 − p {\displaystyle \log {\frac {p}{1-p}}} 1 1 + e − η = e η 1 + e η {\displaystyle {\frac {1}{1+e^{-\eta }}}={\frac {e^{\eta }}{1+e^{\eta }}}} 1 {\displaystyle 1} x {\displaystyle x} log ( 1 + e η ) {\displaystyle \log(1+e^{\eta })} − log ( 1 − p ) {\displaystyle -\log(1-p)} 二項分布既知の試行回数 n {\displaystyle n} p {\displaystyle p} log p 1 − p {\displaystyle \log {\frac {p}{1-p}}} 1 1 + e − η = e η 1 + e η {\displaystyle {\frac {1}{1+e^{-\eta }}}={\frac {e^{\eta }}{1+e^{\eta }}}} ( n x ) {\displaystyle {n \choose x}} x {\displaystyle x} n log ( 1 + e η ) {\displaystyle n\log(1+e^{\eta })} − n log ( 1 − p ) {\displaystyle -n\log(1-p)} ポアソン分布 λ {\displaystyle \lambda } log λ {\displaystyle \log \lambda } e η {\displaystyle e^{\eta }} 1 x ! {\displaystyle {\frac {1}{x!}}} x {\displaystyle x} e η {\displaystyle e^{\eta }} λ {\displaystyle \lambda } 負の二項分布with known number of failures r {\displaystyle r} p {\displaystyle p} log p {\displaystyle \log p} e η {\displaystyle e^{\eta }} ( x + r − 1 x ) {\displaystyle {x+r-1 \choose x}} x {\displaystyle x} − r log ( 1 − e η ) {\displaystyle -r\log(1-e^{\eta })} − r log ( 1 − p ) {\displaystyle -r\log(1-p)} 指数分布 λ {\displaystyle \lambda } − λ {\displaystyle -\lambda } − η {\displaystyle -\eta } 1 {\displaystyle 1} x {\displaystyle x} − log ( − η ) {\displaystyle -\log(-\eta )} − log λ {\displaystyle -\log \lambda } パレート分布with known minimum value x m {\displaystyle x_{m}} α {\displaystyle \alpha } − α − 1 {\displaystyle -\alpha -1} − 1 − η {\displaystyle -1-\eta } 1 {\displaystyle 1} log x {\displaystyle \log x} − log ( − 1 − η ) + ( 1 + η ) log x m {\displaystyle -\log(-1-\eta )+(1+\eta )\log x_{\mathrm {m} }} − log α − α log x m {\displaystyle -\log \alpha -\alpha \log x_{\mathrm {m} }} ワイブル分布with known shape k {\displaystyle k} λ {\displaystyle \lambda } − 1 λ k {\displaystyle -{\frac {1}{\lambda ^{k}}}} ( − η ) − 1 k {\displaystyle (-\eta )^{-{\frac {1}{k}}}} x k − 1 {\displaystyle x^{k-1}} x k {\displaystyle x^{k}} − log ( − η ) − log k {\displaystyle -\log(-\eta )-\log k} k log λ − log k {\displaystyle k\log \lambda -\log k} ラプラス分布既知の平均 μ {\displaystyle \mu } b {\displaystyle b} − 1 b {\displaystyle -{\frac {1}{b}}} − 1 η {\displaystyle -{\frac {1}{\eta }}} 1 {\displaystyle 1} | x − μ | {\displaystyle |x-\mu |} log ( − 2 η ) {\displaystyle \log \left(-{\frac {2}{\eta }}\right)} log 2 b {\displaystyle \log 2b} カイ二乗分布 ν {\displaystyle \nu } ν 2 − 1 {\displaystyle {\frac {\nu }{2}}-1} 2 ( η + 1 ) {\displaystyle 2(\eta +1)} e − x 2 {\displaystyle e^{-{\frac {x}{2}}}} log x {\displaystyle \log x} log Γ ( η + 1 ) + ( η + 1 ) log 2 {\displaystyle \log \Gamma (\eta +1)+(\eta +1)\log 2} log Γ ( ν 2 ) + ν 2 log 2 {\displaystyle \log \Gamma \left({\frac {\nu }{2}}\right)+{\frac {\nu }{2}}\log 2} 正規分布 既知の分散 σ 2 {\displaystyle \sigma ^{2}} μ {\displaystyle \mu } μ σ {\displaystyle {\frac {\mu }{\sigma }}} σ η {\displaystyle \sigma \eta } e − x 2 2 σ 2 2 π σ {\displaystyle {\frac {e^{-{\frac {x^{2}}{2\sigma ^{2}}}}}{{\sqrt {2\pi }}\sigma }}} x σ {\displaystyle {\frac {x}{\sigma }}} η 2 2 {\displaystyle {\frac {\eta ^{2}}{2}}} μ 2 2 σ 2 {\displaystyle {\frac {\mu ^{2}}{2\sigma ^{2}}}} 正規分布 μ {\displaystyle \mu } , σ 2 {\displaystyle \sigma ^{2}} [ μ σ 2 − 1 2 σ 2 ] {\displaystyle {\begin{bmatrix}{\dfrac {\mu }{\sigma ^{2}}}\\[10pt]-{\dfrac {1}{2\sigma ^{2}}}\end{bmatrix}}} [ − η 1 2 η 2 − 1 2 η 2 ] {\displaystyle {\begin{bmatrix}-{\dfrac {\eta _{1}}{2\eta _{2}}}\\[15pt]-{\dfrac {1}{2\eta _{2}}}\end{bmatrix}}} 1 2 π {\displaystyle {\frac {1}{\sqrt {2\pi }}}} [ x x 2 ] {\displaystyle {\begin{bmatrix}x\\x^{2}\end{bmatrix}}} − η 1 2 4 η 2 − 1 2 log ( − 2 η 2 ) {\displaystyle -{\frac {\eta _{1}^{2}}{4\eta _{2}}}-{\frac {1}{2}}\log(-2\eta _{2})} μ 2 2 σ 2 + log σ {\displaystyle {\frac {\mu ^{2}}{2\sigma ^{2}}}+\log \sigma } 対数正規分布 μ {\displaystyle \mu } , σ 2 {\displaystyle \sigma ^{2}} [ μ σ 2 − 1 2 σ 2 ] {\displaystyle {\begin{bmatrix}{\dfrac {\mu }{\sigma ^{2}}}\\[10pt]-{\dfrac {1}{2\sigma ^{2}}}\end{bmatrix}}} [ − η 1 2 η 2 − 1 2 η 2 ] {\displaystyle {\begin{bmatrix}-{\dfrac {\eta _{1}}{2\eta _{2}}}\\[15pt]-{\dfrac {1}{2\eta _{2}}}\end{bmatrix}}} 1 2 π x {\displaystyle {\frac {1}{{\sqrt {2\pi }}x}}} [ log x ( log x ) 2 ] {\displaystyle {\begin{bmatrix}\log x\\(\log x)^{2}\end{bmatrix}}} − η 1 2 4 η 2 − 1 2 log ( − 2 η 2 ) {\displaystyle -{\frac {\eta _{1}^{2}}{4\eta _{2}}}-{\frac {1}{2}}\log(-2\eta _{2})} μ 2 2 σ 2 + log σ {\displaystyle {\frac {\mu ^{2}}{2\sigma ^{2}}}+\log \sigma } 逆ガウス分布 μ {\displaystyle \mu } , λ {\displaystyle \lambda } [ − λ 2 μ 2 − λ 2 ] {\displaystyle {\begin{bmatrix}-{\dfrac {\lambda }{2\mu ^{2}}}\\[15pt]-{\dfrac {\lambda }{2}}\end{bmatrix}}} [ η 2 η 1 − 2 η 2 ] {\displaystyle {\begin{bmatrix}{\sqrt {\dfrac {\eta _{2}}{\eta _{1}}}}\\[15pt]-2\eta _{2}\end{bmatrix}}} 1 2 π x 3 2 {\displaystyle {\frac {1}{{\sqrt {2\pi }}x^{\frac {3}{2}}}}} [ x 1 x ] {\displaystyle {\begin{bmatrix}x\\[5pt]{\dfrac {1}{x}}\end{bmatrix}}} 2 η 1 η 2 − 1 2 log ( − 2 η 2 ) {\displaystyle 2{\sqrt {\eta _{1}\eta _{2}}}-{\frac {1}{2}}\log(-2\eta _{2})} − λ μ − 1 2 log λ {\displaystyle -{\frac {\lambda }{\mu }}-{\frac {1}{2}}\log \lambda } ガンマ分布 α {\displaystyle \alpha } , β {\displaystyle \beta } [ α − 1 − β ] {\displaystyle {\begin{bmatrix}\alpha -1\\-\beta \end{bmatrix}}} [ η 1 + 1 − η 2 ] {\displaystyle {\begin{bmatrix}\eta _{1}+1\\-\eta _{2}\end{bmatrix}}} 1 {\displaystyle 1} [ log x x ] {\displaystyle {\begin{bmatrix}\log x\\x\end{bmatrix}}} log Γ ( η 1 + 1 ) − ( η 1 + 1 ) log ( − η 2 ) {\displaystyle \log \Gamma (\eta _{1}+1)-(\eta _{1}+1)\log(-\eta _{2})} log Γ ( α ) − α log β {\displaystyle \log \Gamma (\alpha )-\alpha \log \beta } k {\displaystyle k} , θ {\displaystyle \theta } [ k − 1 − 1 θ ] {\displaystyle {\begin{bmatrix}k-1\\[5pt]-{\dfrac {1}{\theta }}\end{bmatrix}}} [ η 1 + 1 − 1 η 2 ] {\displaystyle {\begin{bmatrix}\eta _{1}+1\\[5pt]-{\dfrac {1}{\eta _{2}}}\end{bmatrix}}} log Γ ( k ) + k log θ {\displaystyle \log \Gamma (k)+k\log \theta } 逆ガンマ分布 α {\displaystyle \alpha } , β {\displaystyle \beta } [ − α − 1 − β ] {\displaystyle {\begin{bmatrix}-\alpha -1\\-\beta \end{bmatrix}}} [ − η 1 − 1 − η 2 ] {\displaystyle {\begin{bmatrix}-\eta _{1}-1\\-\eta _{2}\end{bmatrix}}} 1 {\displaystyle 1} [ log x 1 x ] {\displaystyle {\begin{bmatrix}\log x\\{\frac {1}{x}}\end{bmatrix}}} log Γ ( − η 1 − 1 ) − ( − η 1 − 1 ) log ( − η 2 ) {\displaystyle \log \Gamma (-\eta _{1}-1)-(-\eta _{1}-1)\log(-\eta _{2})} log Γ ( α ) − α log β {\displaystyle \log \Gamma (\alpha )-\alpha \log \beta } 一般化逆ガウス分布 p {\displaystyle p} , a {\displaystyle a} , b {\displaystyle b} [ p − 1 − a / 2 − b / 2 ] {\displaystyle {\begin{bmatrix}p-1\\-a/2\\-b/2\end{bmatrix}}} [ η 1 + 1 − 2 η 2 − 2 η 3 ] {\displaystyle {\begin{bmatrix}\eta _{1}+1\\-2\eta _{2}\\-2\eta _{3}\end{bmatrix}}} 1 {\displaystyle 1} [ log x x 1 x ] {\displaystyle {\begin{bmatrix}\log x\\x\\{\frac {1}{x}}\end{bmatrix}}} log 2 K η 1 + 1 ( 4 η 2 η 3 ) − η 1 + 1 2 log η 2 η 3 {\displaystyle \log 2K_{\eta _{1}+1}({\sqrt {4\eta _{2}\eta _{3}}})-{\frac {\eta _{1}+1}{2}}\log {\frac {\eta _{2}}{\eta _{3}}}} log 2 K p ( a b ) − p 2 log a b {\displaystyle \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 ν {\displaystyle \nu } , σ 2 {\displaystyle \sigma ^{2}} [ − ν 2 − 1 − ν σ 2 2 ] {\displaystyle {\begin{bmatrix}-{\dfrac {\nu }{2}}-1\\[10pt]-{\dfrac {\nu \sigma ^{2}}{2}}\end{bmatrix}}} [ − 2 ( η 1 + 1 ) η 2 η 1 + 1 ] {\displaystyle {\begin{bmatrix}-2(\eta _{1}+1)\\[10pt]{\dfrac {\eta _{2}}{\eta _{1}+1}}\end{bmatrix}}} 1 {\displaystyle 1} [ log x 1 x ] {\displaystyle {\begin{bmatrix}\log x\\{\frac {1}{x}}\end{bmatrix}}} log Γ ( − η 1 − 1 ) − ( − η 1 − 1 ) log ( − η 2 ) {\displaystyle \log \Gamma (-\eta _{1}-1)-(-\eta _{1}-1)\log(-\eta _{2})} log Γ ( ν 2 ) − ν 2 log ν σ 2 2 {\displaystyle \log \Gamma \left({\frac {\nu }{2}}\right)-{\frac {\nu }{2}}\log {\frac {\nu \sigma ^{2}}{2}}} ベータ分布 （variant 1) α {\displaystyle \alpha } , β {\displaystyle \beta } [ α β ] {\displaystyle {\begin{bmatrix}\alpha \\\beta \end{bmatrix}}} [ η 1 η 2 ] {\displaystyle {\begin{bmatrix}\eta _{1}\\\eta _{2}\end{bmatrix}}} 1 x ( 1 − x ) {\displaystyle {\frac {1}{x(1-x)}}} [ log x log ( 1 − x ) ] {\displaystyle {\begin{bmatrix}\log x\\\log(1-x)\end{bmatrix}}} log Γ ( η 1 ) + log Γ ( η 2 ) − log Γ ( η 1 + η 2 ) {\displaystyle \log \Gamma (\eta _{1})+\log \Gamma (\eta _{2})-\log \Gamma (\eta _{1}+\eta _{2})} log Γ ( α ) + log Γ ( β ) − log Γ ( α + β ) {\displaystyle \log \Gamma (\alpha )+\log \Gamma (\beta )-\log \Gamma (\alpha +\beta )} ベータ分布 (variant 2) α {\displaystyle \alpha } , β {\displaystyle \beta } [ α − 1 β − 1 ] {\displaystyle {\begin{bmatrix}\alpha -1\\\beta -1\end{bmatrix}}} [ η 1 + 1 η 2 + 1 ] {\displaystyle {\begin{bmatrix}\eta _{1}+1\\\eta _{2}+1\end{bmatrix}}} 1 {\displaystyle 1} [ log x log ( 1 − x ) ] {\displaystyle {\begin{bmatrix}\log x\\\log(1-x)\end{bmatrix}}} log Γ ( η 1 + 1 ) + log Γ ( η 2 + 1 ) − log Γ ( η 1 + η 2 + 2 ) {\displaystyle \log \Gamma (\eta _{1}+1)+\log \Gamma (\eta _{2}+1)-\log \Gamma (\eta _{1}+\eta _{2}+2)} log Γ ( α ) + log Γ ( β ) − log Γ ( α + β ) {\displaystyle \log \Gamma (\alpha )+\log \Gamma (\beta )-\log \Gamma (\alpha +\beta )} 多変量正規分布 μ {\displaystyle {\boldsymbol {\mu }}} , σ {\displaystyle {\boldsymbol {\sigma }}} [ Σ − 1 μ − 1 2 Σ − 1 ] {\displaystyle {\begin{bmatrix}{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}\\[5pt]-{\frac {1}{2}}{\boldsymbol {\Sigma }}^{-1}\end{bmatrix}}} [ − 1 2 η 2 − 1 η 1 − 1 2 η 2 − 1 ] {\displaystyle {\begin{bmatrix}-{\frac {1}{2}}{\boldsymbol {\eta }}_{2}^{-1}{\boldsymbol {\eta }}_{1}\\[5pt]-{\frac {1}{2}}{\boldsymbol {\eta }}_{2}^{-1}\end{bmatrix}}} ( 2 π ) − k 2 {\displaystyle (2\pi )^{-{\frac {k}{2}}}} [ x x x T ] {\displaystyle {\begin{bmatrix}\mathbf {x} \\[5pt]\mathbf {x} \mathbf {x} ^{\mathrm {T} }\end{bmatrix}}} − 1 4 η 1 T η 2 − 1 η 1 − 1 2 log | − 2 η 2 | {\displaystyle -{\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|} 1 2 μ T Σ − 1 μ + 1 2 log | Σ | {\displaystyle {\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 , … , p k {\displaystyle p_{1},\dots {},p_{k}} where ∑ i = 1 k p i = 1 {\displaystyle \textstyle \sum _{i=1}^{k}p_{i}=1} [ log p 1 ⋮ log p k ] {\displaystyle {\begin{bmatrix}\log p_{1}\\\vdots \\\log p_{k}\end{bmatrix}}} [ e η 1 ⋮ e η k ] {\displaystyle {\begin{bmatrix}e^{\eta _{1}}\\\vdots \\e^{\eta _{k}}\end{bmatrix}}} where ∑ i = 1 k e η i = 1 {\displaystyle \textstyle \sum _{i=1}^{k}e^{\eta _{i}}=1} 1 {\displaystyle 1} [ [ x = 1 ] ⋮ [ x = k ] ] {\displaystyle {\begin{bmatrix}[x=1]\\\vdots \\{[x=k]}\end{bmatrix}}} 0 {\displaystyle 0} 0 {\displaystyle 0} カテゴリカル分布 (variant 2) p 1 , … , p k {\displaystyle p_{1},\dots {},p_{k}} where ∑ i = 1 k p i = 1 {\displaystyle \textstyle \sum _{i=1}^{k}p_{i}=1} [ log p 1 + C ⋮ log p k + C ] {\displaystyle {\begin{bmatrix}\log p_{1}+C\\\vdots \\\log p_{k}+C\end{bmatrix}}} [ 1 C e η 1 ⋮ 1 C e η k ] = {\displaystyle {\begin{bmatrix}{\dfrac {1}{C}}e^{\eta _{1}}\\\vdots \\{\dfrac {1}{C}}e^{\eta _{k}}\end{bmatrix}}=} [ e η 1 ∑ i = 1 k e η i ⋮ e η k ∑ i = 1 k e η i ] {\displaystyle {\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 ∑ i = 1 k e η i = C {\displaystyle \textstyle \sum _{i=1}^{k}e^{\eta _{i}}=C} 1 {\displaystyle 1} [ [ x = 1 ] ⋮ [ x = k ] ] {\displaystyle {\begin{bmatrix}[x=1]\\\vdots \\{[x=k]}\end{bmatrix}}} 0 {\displaystyle 0} 0 {\displaystyle 0} カテゴリカル分布 (variant 3) p 1 , … , p k {\displaystyle p_{1},\dots {},p_{k}} where p k = 1 − ∑ i = 1 k − 1 p i {\displaystyle p_{k}=1-\textstyle \sum _{i=1}^{k-1}p_{i}} [ log p 1 p k ⋮ log p k − 1 p k 0 ] = {\displaystyle {\begin{bmatrix}\log {\dfrac {p_{1}}{p_{k}}}\\[10pt]\vdots \\[5pt]\log {\dfrac {p_{k-1}}{p_{k}}}\\[15pt]0\end{bmatrix}}=} [ log p 1 1 − ∑ i = 1 k − 1 p i ⋮ log p k − 1 1 − ∑ i = 1 k − 1 p i 0 ] {\displaystyle {\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. [ e η 1 ∑ i = 1 k e η i ⋮ e η k ∑ i = 1 k e η i ] = {\displaystyle {\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}}=} [ e η 1 1 + ∑ i = 1 k − 1 e η i ⋮ e η k − 1 1 + ∑ i = 1 k − 1 e η i 1 1 + ∑ i = 1 k − 1 e η i ] {\displaystyle {\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 {\displaystyle 1} [ [ x = 1 ] ⋮ [ x = k ] ] {\displaystyle {\begin{bmatrix}[x=1]\\\vdots \\{[x=k]}\end{bmatrix}}} log ( ∑ i = 1 k e η i ) = log ( 1 + ∑ i = 1 k − 1 e η i ) {\displaystyle \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 ( 1 − ∑ i = 1 k − 1 p i ) {\displaystyle -\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 {\displaystyle n} p 1 , … , p k {\displaystyle p_{1},\dots {},p_{k}} where ∑ i = 1 k p i = 1 {\displaystyle \textstyle \sum _{i=1}^{k}p_{i}=1} [ log p 1 ⋮ log p k ] {\displaystyle {\begin{bmatrix}\log p_{1}\\\vdots \\\log p_{k}\end{bmatrix}}} [ e η 1 ⋮ e η k ] {\displaystyle {\begin{bmatrix}e^{\eta _{1}}\\\vdots \\e^{\eta _{k}}\end{bmatrix}}} where ∑ i = 1 k e η i = 1 {\displaystyle \textstyle \sum _{i=1}^{k}e^{\eta _{i}}=1} n ! ∏ i = 1 k x i ! {\displaystyle {\frac {n!}{\prod _{i=1}^{k}x_{i}!}}} [ x 1 ⋮ x k ] {\displaystyle {\begin{bmatrix}x_{1}\\\vdots \\x_{k}\end{bmatrix}}} 0 {\displaystyle 0} 0 {\displaystyle 0} 多項分布 (variant 2)既知の試行回数 n {\displaystyle n} p 1 , … , p k {\displaystyle p_{1},\dots {},p_{k}} where ∑ i = 1 k p i = 1 {\displaystyle \textstyle \sum _{i=1}^{k}p_{i}=1} [ log p 1 + C ⋮ log p k + C ] {\displaystyle {\begin{bmatrix}\log p_{1}+C\\\vdots \\\log p_{k}+C\end{bmatrix}}} [ 1 C e η 1 ⋮ 1 C e η k ] = {\displaystyle {\begin{bmatrix}{\dfrac {1}{C}}e^{\eta _{1}}\\\vdots \\{\dfrac {1}{C}}e^{\eta _{k}}\end{bmatrix}}=} [ e η 1 ∑ i = 1 k e η i ⋮ e η k ∑ i = 1 k e η i ] {\displaystyle {\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 ∑ i = 1 k e η i = C {\displaystyle \textstyle \sum _{i=1}^{k}e^{\eta _{i}}=C} n ! ∏ i = 1 k x i ! {\displaystyle {\frac {n!}{\prod _{i=1}^{k}x_{i}!}}} [ x 1 ⋮ x k ] {\displaystyle {\begin{bmatrix}x_{1}\\\vdots \\x_{k}\end{bmatrix}}} 0 {\displaystyle 0} 0 {\displaystyle 0} 多項分布 (variant 3)既知の試行回数 n {\displaystyle n} p 1 , … , p k {\displaystyle p_{1},\dots {},p_{k}} where p k = 1 − ∑ i = 1 k − 1 p i {\displaystyle p_{k}=1-\textstyle \sum _{i=1}^{k-1}p_{i}} [ log p 1 p k ⋮ log p k − 1 p k 0 ] = {\displaystyle {\begin{bmatrix}\log {\dfrac {p_{1}}{p_{k}}}\\[10pt]\vdots \\[5pt]\log {\dfrac {p_{k-1}}{p_{k}}}\\[15pt]0\end{bmatrix}}=} [ log p 1 1 − ∑ i = 1 k − 1 p i ⋮ log p k − 1 1 − ∑ i = 1 k − 1 p i 0 ] {\displaystyle {\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}}} [ e η 1 ∑ i = 1 k e η i ⋮ e η k ∑ i = 1 k e η i ] = {\displaystyle {\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}}=} [ e η 1 1 + ∑ i = 1 k − 1 e η i ⋮ e η k − 1 1 + ∑ i = 1 k − 1 e η i 1 1 + ∑ i = 1 k − 1 e η i ] {\displaystyle {\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}}} n ! ∏ i = 1 k x i ! {\displaystyle {\frac {n!}{\prod _{i=1}^{k}x_{i}!}}} [ x 1 ⋮ x k ] {\displaystyle {\begin{bmatrix}x_{1}\\\vdots \\x_{k}\end{bmatrix}}} n log ( ∑ i = 1 k e η i ) = n log ( 1 + ∑ i = 1 k − 1 e η i ) {\displaystyle 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 ( 1 − ∑ i = 1 k − 1 p i ) {\displaystyle -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) α 1 , … , α k {\displaystyle \alpha _{1},\dots {},\alpha _{k}} [ α 1 ⋮ α k ] {\displaystyle {\begin{bmatrix}\alpha _{1}\\\vdots \\\alpha _{k}\end{bmatrix}}} [ η 1 ⋮ η k ] {\displaystyle {\begin{bmatrix}\eta _{1}\\\vdots \\\eta _{k}\end{bmatrix}}} 1 ∏ i = 1 k x i {\displaystyle {\frac {1}{\prod _{i=1}^{k}x_{i}}}} [ log x 1 ⋮ log x k ] {\displaystyle {\begin{bmatrix}\log x_{1}\\\vdots \\\log x_{k}\end{bmatrix}}} ∑ i = 1 k log Γ ( η i ) − log Γ ( ∑ i = 1 k η i ) {\displaystyle \sum _{i=1}^{k}\log \Gamma (\eta _{i})-\log \Gamma \left(\sum _{i=1}^{k}\eta _{i}\right)} ∑ i = 1 k log Γ ( α i ) − log Γ ( ∑ i = 1 k α i ) {\displaystyle \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) α 1 , … , α k {\displaystyle \alpha _{1},\dots {},\alpha _{k}} [ α 1 − 1 ⋮ α k − 1 ] {\displaystyle {\begin{bmatrix}\alpha _{1}-1\\\vdots \\\alpha _{k}-1\end{bmatrix}}} [ η 1 + 1 ⋮ η k + 1 ] {\displaystyle {\begin{bmatrix}\eta _{1}+1\\\vdots \\\eta _{k}+1\end{bmatrix}}} 1 {\displaystyle 1} [ log x 1 ⋮ log x k ] {\displaystyle {\begin{bmatrix}\log x_{1}\\\vdots \\\log x_{k}\end{bmatrix}}} ∑ i = 1 k log Γ ( η i + 1 ) − log Γ ( ∑ i = 1 k ( η i + 1 ) ) {\displaystyle \sum _{i=1}^{k}\log \Gamma (\eta _{i}+1)-\log \Gamma \left(\sum _{i=1}^{k}(\eta _{i}+1)\right)} ∑ i = 1 k log Γ ( α i ) − log Γ ( ∑ i = 1 k α i ) {\displaystyle \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 V {\displaystyle \mathbf {V} } , n {\displaystyle n} [ − 1 2 V − 1 n − p − 1 2 ] {\displaystyle {\begin{bmatrix}-{\frac {1}{2}}\mathbf {V} ^{-1}\\[5pt]{\dfrac {n-p-1}{2}}\end{bmatrix}}} [ − 1 2 η 1 − 1 2 η 2 + p + 1 ] {\displaystyle {\begin{bmatrix}-{\frac {1}{2}}{{\boldsymbol {\eta }}_{1}}^{-1}\\[5pt]2\eta _{2}+p+1\end{bmatrix}}} 1 {\displaystyle 1} [ X log | X | ] {\displaystyle {\begin{bmatrix}\mathbf {X} \\\log |\mathbf {X} |\end{bmatrix}}} − ( η 2 + p + 1 2 ) log | − η 1 | {\displaystyle -\left(\eta _{2}+{\frac {p+1}{2}}\right)\log |-{\boldsymbol {\eta }}_{1}|} + log Γ p ( η 2 + p + 1 2 ) = {\displaystyle +\log \Gamma _{p}\left(\eta _{2}+{\frac {p+1}{2}}\right)=} − n 2 log | − η 1 | + log Γ p ( n 2 ) = {\displaystyle -{\frac {n}{2}}\log |-{\boldsymbol {\eta }}_{1}|+\log \Gamma _{p}\left({\frac {n}{2}}\right)=} ( η 2 + p + 1 2 ) ( p log 2 + log | V | ) {\displaystyle \left(\eta _{2}+{\frac {p+1}{2}}\right)(p\log 2+\log |\mathbf {V} |)} + log Γ p ( η 2 + p + 1 2 ) {\displaystyle +\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. n 2 ( p log 2 + log | V | ) + log Γ p ( n 2 ) {\displaystyle {\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 Ψ {\displaystyle {\boldsymbol {\Psi }}} , m {\displaystyle m} [ − 1 2 Ψ − m + p + 1 2 ] {\displaystyle {\begin{bmatrix}-{\frac {1}{2}}{\boldsymbol {\Psi }}\\[5pt]-{\dfrac {m+p+1}{2}}\end{bmatrix}}} [ − 2 η 1 − ( 2 η 2 + p + 1 ) ] {\displaystyle {\begin{bmatrix}-2{\boldsymbol {\eta }}_{1}\\[5pt]-(2\eta _{2}+p+1)\end{bmatrix}}} 1 {\displaystyle 1} [ X − 1 log | X | ] {\displaystyle {\begin{bmatrix}\mathbf {X} ^{-1}\\\log |\mathbf {X} |\end{bmatrix}}} ( η 2 + p + 1 2 ) log | − η 1 | + log Γ p ( − ( η 2 + p + 1 2 ) ) = − m 2 log | − η 1 | + log Γ p ( m 2 ) = − ( η 2 + p + 1 2 ) ( p log 2 − log | Ψ | ) + log Γ p ( − ( η 2 + p + 1 2 ) ) {\displaystyle {\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}}} m 2 ( p log 2 − log | Ψ | ) + log Γ p ( m 2 ) {\displaystyle {\frac {m}{2}}(p\log 2-\log |{\boldsymbol {\Psi }}|)+\log \Gamma _{p}\left({\frac {m}{2}}\right)} ガウス・ガンマ分布 α {\displaystyle \alpha } , β {\displaystyle \beta } , μ {\displaystyle \mu } , λ {\displaystyle \lambda } [ α − 1 2 − β − λ μ 2 2 λ μ − λ 2 ] {\displaystyle {\begin{bmatrix}\alpha -{\frac {1}{2}}\\-\beta -{\dfrac {\lambda \mu ^{2}}{2}}\\\lambda \mu \\-{\dfrac {\lambda }{2}}\end{bmatrix}}} [ η 1 + 1 2 − η 2 + η 3 2 4 η 4 − η 3 2 η 4 − 2 η 4 ] {\displaystyle {\begin{bmatrix}\eta _{1}+{\frac {1}{2}}\\-\eta _{2}+{\dfrac {\eta _{3}^{2}}{4\eta _{4}}}\\-{\dfrac {\eta _{3}}{2\eta _{4}}}\\-2\eta _{4}\end{bmatrix}}} 1 2 π {\displaystyle {\dfrac {1}{\sqrt {2\pi }}}} [ log τ τ τ x τ x 2 ] {\displaystyle {\begin{bmatrix}\log \tau \\\tau \\\tau x\\\tau x^{2}\end{bmatrix}}} log Γ ( η 1 + 1 2 ) − 1 2 log ( − 2 η 4 ) − {\displaystyle \log \Gamma \left(\eta _{1}+{\frac {1}{2}}\right)-{\frac {1}{2}}\log \left(-2\eta _{4}\right)-} − ( η 1 + 1 2 ) log ( − η 2 + η 3 2 4 η 4 ) {\displaystyle -\left(\eta _{1}+{\frac {1}{2}}\right)\log \left(-\eta _{2}+{\dfrac {\eta _{3}^{2}}{4\eta _{4}}}\right)} log Γ ( α ) − α log β − 1 2 log λ {\displaystyle \log \Gamma \left(\alpha \right)-\alpha \log \beta -{\frac {1}{2}}\log \lambda }
※この「分布表」の解説は、「指数型分布族」の解説の一部です。
「分布表」を含む「指数型分布族」の記事については、「指数型分布族」の概要を参照ください。
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