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 [ 注釈 1]
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
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
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
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
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
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
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)[ 注釈 2]
p
1
,
…
,
p
k
{\displaystyle p_{1},\dots {},p_{k}}
∑
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}}}
∑
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)[ 注釈 2]
p
1
,
…
,
p
k
{\displaystyle p_{1},\dots {},p_{k}}
∑
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}}}
∑
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)[ 注釈 2]
p
1
,
…
,
p
k
{\displaystyle p_{1},\dots {},p_{k}}
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}}}
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)
n
{\displaystyle n}
p
1
,
…
,
p
k
{\displaystyle p_{1},\dots {},p_{k}}
∑
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}}}
∑
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)
n
{\displaystyle n}
p
1
,
…
,
p
k
{\displaystyle p_{1},\dots {},p_{k}}
∑
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)
n
{\displaystyle n}
p
1
,
…
,
p
k
{\displaystyle p_{1},\dots {},p_{k}}
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 [ 注釈 3]
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 }
![{\displaystyle f_{X}(x\mid \theta )=h(x)\exp {\left[\eta (\theta )\cdot T(x)-A(\theta )\right]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28915cbbd82ba210921c004db984fc61b9df0097)
![{\displaystyle f_{X}(x\mid \theta )=h(x)\,g(\theta )\,\exp {\left[\eta (\theta )\cdot T(x)\right]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/127a6ae82d0c516603c518b893b60b72e5c37300)
![{\displaystyle f_{X}(x\mid \theta )=\exp {\left[\eta (\theta )\cdot T(x)-A(\theta )+B(x)\right]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/080a2cc8ebf0cba5e8c0b0ab14a78d82085e4bcb)
![{\displaystyle {\begin{aligned}h_{\sigma }(x)&={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp \left(-{\frac {x^{2}}{2\sigma ^{2}}}\right)\\[4pt]T_{\sigma }(x)&={\frac {x}{\sigma }}\\[4pt]A_{\sigma }(\mu )&={\frac {\mu ^{2}}{2\sigma ^{2}}}\\[4pt]\eta _{\sigma }(\mu )&={\frac {\mu }{\sigma }}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/232dcd73674bd4ce165b3ab9a5a3507de825d6fc)
![{\displaystyle {\begin{bmatrix}{\dfrac {\mu }{\sigma ^{2}}}\\[10pt]-{\dfrac {1}{2\sigma ^{2}}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e15ab0bb474390f6abb3c5326fcaffe750a720d)
![{\displaystyle {\begin{bmatrix}-{\dfrac {\eta _{1}}{2\eta _{2}}}\\[15pt]-{\dfrac {1}{2\eta _{2}}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f2c34af3e73bf19b2a84778a3271eea2a66190c)
![{\displaystyle {\begin{bmatrix}-{\dfrac {\lambda }{2\mu ^{2}}}\\[15pt]-{\dfrac {\lambda }{2}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/801f80ae3e3f69c9f7c539c5c19172db4906b32e)
![{\displaystyle {\begin{bmatrix}{\sqrt {\dfrac {\eta _{2}}{\eta _{1}}}}\\[15pt]-2\eta _{2}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/233b92f05323f4fe678aa94ed71b074a385890ed)
![{\displaystyle {\begin{bmatrix}x\\[5pt]{\dfrac {1}{x}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f50097b01afa108c0ec9fccdcc20eecd6eda3c5)
![{\displaystyle {\begin{bmatrix}k-1\\[5pt]-{\dfrac {1}{\theta }}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4337f3072e1ca84b753404461afd7f999aac085)
![{\displaystyle {\begin{bmatrix}\eta _{1}+1\\[5pt]-{\dfrac {1}{\eta _{2}}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e266294768392795079ca2000d7e9823de86f7a)
![{\displaystyle {\begin{bmatrix}-{\dfrac {\nu }{2}}-1\\[10pt]-{\dfrac {\nu \sigma ^{2}}{2}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/632abc32dc2c7d336f377bd32b4fcf4575e4357f)
![{\displaystyle {\begin{bmatrix}-2(\eta _{1}+1)\\[10pt]{\dfrac {\eta _{2}}{\eta _{1}+1}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ec5dddecabff3cd5e6d5f59cb729c7de4862c12)
![{\displaystyle {\begin{bmatrix}{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}\\[5pt]-{\frac {1}{2}}{\boldsymbol {\Sigma }}^{-1}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74b0a40444a6664bc5713a3b8f54e7b01847e249)
![{\displaystyle {\begin{bmatrix}-{\frac {1}{2}}{\boldsymbol {\eta }}_{2}^{-1}{\boldsymbol {\eta }}_{1}\\[5pt]-{\frac {1}{2}}{\boldsymbol {\eta }}_{2}^{-1}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/346f39d6f70932a36d285283fcfb3b8224b41295)
![{\displaystyle {\begin{bmatrix}\mathbf {x} \\[5pt]\mathbf {x} \mathbf {x} ^{\mathrm {T} }\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e0ab71035a29cddb125f8a7dd172beb6bb21375)
![{\displaystyle {\begin{bmatrix}[x=1]\\\vdots \\{[x=k]}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fba20bad4ecfebabb4dc21e96182eb9a64ed7820)
![{\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/724572f1c127ce9b2c7eac1df03ece501981f647)
![{\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}\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffe48da05da1c76249b03c91aabbc2d1be6b33b)
![{\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)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57b5e213c7c56952e9d02e5776b86e0f6375d58c)
![{\displaystyle {\begin{bmatrix}-{\frac {1}{2}}\mathbf {V} ^{-1}\\[5pt]{\dfrac {n-p-1}{2}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dea6c3a692bd8beaf10ba5b29b9730b9ef9f80a)
![{\displaystyle {\begin{bmatrix}-{\frac {1}{2}}{{\boldsymbol {\eta }}_{1}}^{-1}\\[5pt]2\eta _{2}+p+1\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da3e3baebd5b1b79f6aa2a37ff93e3a96854c83a)
![{\displaystyle {\begin{bmatrix}-{\frac {1}{2}}{\boldsymbol {\Psi }}\\[5pt]-{\dfrac {m+p+1}{2}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89298742ba48b82f09239ae72925c7be14962cec)
![{\displaystyle {\begin{bmatrix}-2{\boldsymbol {\eta }}_{1}\\[5pt]-(2\eta _{2}+p+1)\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dde83f85b4fdefb13eb836efae8d668fdf3474b)
![{\displaystyle [x=i]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/747094cb4b84a3d9ede55bd7667e9da65c8cf6a8)
