確率分布 | パラメータ | 自然パラメータ | パラメータの逆写像 | Base measure | 十分統計量 | Log-partition | Log-partition |
ベルヌーイ分布[注釈 1] | | | | | | | |
二項分布 既知の試行回数 ![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b) | | | | | | | |
ポアソン分布 | | | | | | | |
負の二項分布 with known number of failures ![{\displaystyle r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538) | | | | | | | |
指数分布 | | | | | | | |
パレート分布 with known minimum value ![{\displaystyle x_{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f12100e1dc5769ced8c9806b219abc06ab321d3) | | | | | | | |
ワイブル分布 with known shape ![{\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) | | | | | | | |
ラプラス分布 既知の平均 ![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) | | | | | | | |
カイ二乗分布 | | | | | | | |
正規分布 既知の分散 ![{\displaystyle \sigma ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5) | | | | | | | |
正規分布 | , | | | | | | |
対数正規分布 | , | | | | | | |
逆ガウス分布 | , | | | | | | |
ガンマ分布 | , | | | | | | |
, | | | |
逆ガンマ分布 | , | | | | | | |
一般化逆ガウス分布 | , , | | | | | | |
スケールされた逆カイ二乗分布 | , | | | | | | |
ベータ分布 (variant 1) | , | | | | | | |
ベータ分布 (variant 2) | , | | | | | | |
多変量正規分布 | , | | | | | | |
カテゴリカル分布 (variant 1)[注釈 2] | ![{\displaystyle p_{1},\dots {},p_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a2eeaa41628dabc1467d4ea30b6a8c7b941e24)
where | | ![{\displaystyle {\begin{bmatrix}e^{\eta _{1}}\\\vdots \\e^{\eta _{k}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c90ac7fadf051464ef5f0027de4a44d5c168ba31)
where | | | | |
カテゴリカル分布 (variant 2)[注釈 2] | ![{\displaystyle p_{1},\dots {},p_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a2eeaa41628dabc1467d4ea30b6a8c7b941e24)
where | | ![{\displaystyle {\begin{bmatrix}{\dfrac {1}{C}}e^{\eta _{1}}\\\vdots \\{\dfrac {1}{C}}e^{\eta _{k}}\end{bmatrix}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19a8ceb20455a409692a3a3d16f976ea72a14a6d) where | | | | |
カテゴリカル分布 (variant 3)[注釈 2] | ![{\displaystyle p_{1},\dots {},p_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a2eeaa41628dabc1467d4ea30b6a8c7b941e24)
where | ![{\displaystyle {\begin{bmatrix}\log {\dfrac {p_{1}}{p_{k}}}\\[10pt]\vdots \\[5pt]\log {\dfrac {p_{k-1}}{p_{k}}}\\[15pt]0\end{bmatrix}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac88e81cef128af36a3101295d1fec7d9e37c385)
- This is the inverse softmax function, a generalization of the logit function.
| ![{\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}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07796f5c03bcf98de0a57773c61171f9239a0fb4)
- This is the softmax function, a generalization of the logistic function.
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多項分布 (variant 1) 既知の試行回数 ![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b) | ![{\displaystyle p_{1},\dots {},p_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a2eeaa41628dabc1467d4ea30b6a8c7b941e24)
where | | ![{\displaystyle {\begin{bmatrix}e^{\eta _{1}}\\\vdots \\e^{\eta _{k}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c90ac7fadf051464ef5f0027de4a44d5c168ba31)
where | | | | |
多項分布 (variant 2) 既知の試行回数 ![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b) | ![{\displaystyle p_{1},\dots {},p_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a2eeaa41628dabc1467d4ea30b6a8c7b941e24)
where | | ![{\displaystyle {\begin{bmatrix}{\dfrac {1}{C}}e^{\eta _{1}}\\\vdots \\{\dfrac {1}{C}}e^{\eta _{k}}\end{bmatrix}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19a8ceb20455a409692a3a3d16f976ea72a14a6d)
where | | | | |
多項分布 (variant 3) 既知の試行回数 | ![{\displaystyle p_{1},\dots {},p_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a2eeaa41628dabc1467d4ea30b6a8c7b941e24)
where | ![{\displaystyle {\begin{bmatrix}\log {\dfrac {p_{1}}{p_{k}}}\\[10pt]\vdots \\[5pt]\log {\dfrac {p_{k-1}}{p_{k}}}\\[15pt]0\end{bmatrix}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac88e81cef128af36a3101295d1fec7d9e37c385)
| ![{\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}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07796f5c03bcf98de0a57773c61171f9239a0fb4)
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ディリクレ分布 (variant 1) | | | | | | | |
ディリクレ分布 (variant 2) | | | | | | | |
ウィッシャート分布[注釈 3] | , | | | | | ![{\displaystyle -\left(\eta _{2}+{\frac {p+1}{2}}\right)\log |-{\boldsymbol {\eta }}_{1}|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bf5fd0ab2368a5f2f09df909654f23d737231c5) ![{\displaystyle +\log \Gamma _{p}\left(\eta _{2}+{\frac {p+1}{2}}\right)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0740dd46ae4f2ef5e660de27e8a5d0897524f242)
![{\displaystyle -{\frac {n}{2}}\log |-{\boldsymbol {\eta }}_{1}|+\log \Gamma _{p}\left({\frac {n}{2}}\right)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90976f6cee86b06595a134e91874ff03e97a2525)
![{\displaystyle \left(\eta _{2}+{\frac {p+1}{2}}\right)(p\log 2+\log |\mathbf {V} |)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4663021fdcd2bf2f69b2d2e615c22dfbbef13ed4)
- Three variants with different parameterizations are given, to facilitate computing moments of the sufficient statistics.
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逆ウィッシャート分布 | , | | | | | | |
ガウス・ガンマ分布 | , , , | | | | | ![{\displaystyle \log \Gamma \left(\eta _{1}+{\frac {1}{2}}\right)-{\frac {1}{2}}\log \left(-2\eta _{4}\right)-}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4c11ea053d86cb0b80b636baa2bfcdbfd223360) | |