توزیع
θ {\displaystyle {\boldsymbol {\theta }}}
η {\displaystyle {\boldsymbol {\eta }}}
تابع پارامتر معکوس
h ( x ) {\displaystyle h(x)}
T ( x ) {\displaystyle T(x)}
A ( η ) {\displaystyle A({\boldsymbol {\eta }})}
A ( θ ) {\displaystyle A({\boldsymbol {\theta }})}
Bernoulli distribution
p
ln p 1 − p {\displaystyle \ln {\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}
ln ( 1 + e η ) {\displaystyle \ln(1+e^{\eta })}
− ln ( 1 − p ) {\displaystyle -\ln(1-p)}
binomial distribution with known number of trials n
p
ln p 1 − p {\displaystyle \ln {\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 ln ( 1 + e η ) {\displaystyle n\ln(1+e^{\eta })}
− n ln ( 1 − p ) {\displaystyle -n\ln(1-p)}
Poisson distribution
λ
ln λ {\displaystyle \ln \lambda }
e η {\displaystyle e^{\eta }}
1 x ! {\displaystyle {\frac {1}{x!}}}
x {\displaystyle x}
e η {\displaystyle e^{\eta }}
λ {\displaystyle \lambda }
negative binomial distribution with known number of failures r
p
ln p {\displaystyle \ln p}
e η {\displaystyle e^{\eta }}
( x + r − 1 x ) {\displaystyle {x+r-1 \choose x}}
x {\displaystyle x}
− r ln ( 1 − e η ) {\displaystyle -r\ln(1-e^{\eta })}
− r ln ( 1 − p ) {\displaystyle -r\ln(1-p)}
exponential distribution
λ
− λ {\displaystyle -\lambda }
− η {\displaystyle -\eta }
1 {\displaystyle 1}
x {\displaystyle x}
− ln ( − η ) {\displaystyle -\ln(-\eta )}
− ln λ {\displaystyle -\ln \lambda }
Pareto distribution with known minimum value x m
α
− α − 1 {\displaystyle -\alpha -1}
− 1 − η {\displaystyle -1-\eta }
1 {\displaystyle 1}
ln x {\displaystyle \ln x}
− ln ( − 1 − η ) + ( 1 + η ) ln x m {\displaystyle -\ln(-1-\eta )+(1+\eta )\ln x_{\mathrm {m} }}
− ln α − α ln x m {\displaystyle -\ln \alpha -\alpha \ln x_{\mathrm {m} }}
Weibull distribution with known shape k
λ
− 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}}
− ln ( − η ) − ln k {\displaystyle -\ln(-\eta )-\ln k}
k ln λ − ln k {\displaystyle k\ln \lambda -\ln k}
Laplace distribution with known mean μ
b
− 1 b {\displaystyle -{\frac {1}{b}}}
− 1 η {\displaystyle -{\frac {1}{\eta }}}
1 {\displaystyle 1}
| x − μ | {\displaystyle |x-\mu |}
ln ( − 2 η ) {\displaystyle \ln \left(-{\frac {2}{\eta }}\right)}
ln 2 b {\displaystyle \ln 2b}
chi-squared distribution
ν
ν 2 − 1 {\displaystyle {\frac {\nu }{2}}-1}
2 ( η + 1 ) {\displaystyle 2(\eta +1)}
e − x 2 {\displaystyle e^{-{\frac {x}{2}}}}
ln x {\displaystyle \ln x}
ln Γ ( η + 1 ) + ( η + 1 ) ln 2 {\displaystyle \ln \Gamma (\eta +1)+(\eta +1)\ln 2}
ln Γ ( ν 2 ) + ν 2 ln 2 {\displaystyle \ln \Gamma \left({\frac {\nu }{2}}\right)+{\frac {\nu }{2}}\ln 2}
normal distribution known variance
μ
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}}}}
normal distribution
μ,σ2
[ 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 ln ( − 2 η 2 ) {\displaystyle -{\frac {\eta _{1}^{2}}{4\eta _{2}}}-{\frac {1}{2}}\ln(-2\eta _{2})}
μ 2 2 σ 2 + ln σ {\displaystyle {\frac {\mu ^{2}}{2\sigma ^{2}}}+\ln \sigma }
lognormal distribution
μ,σ2
[ 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}}}
[ ln x ( ln x ) 2 ] {\displaystyle {\begin{bmatrix}\ln x\\(\ln x)^{2}\end{bmatrix}}}
− η 1 2 4 η 2 − 1 2 ln ( − 2 η 2 ) {\displaystyle -{\frac {\eta _{1}^{2}}{4\eta _{2}}}-{\frac {1}{2}}\ln(-2\eta _{2})}
μ 2 2 σ 2 + ln σ {\displaystyle {\frac {\mu ^{2}}{2\sigma ^{2}}}+\ln \sigma }
inverse Gaussian distribution
μ,λ
[ − 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 ln ( − 2 η 2 ) {\displaystyle -2{\sqrt {\eta _{1}\eta _{2}}}-{\frac {1}{2}}\ln(-2\eta _{2})}
− λ μ − 1 2 ln λ {\displaystyle -{\frac {\lambda }{\mu }}-{\frac {1}{2}}\ln \lambda }
gamma distribution
α,β
[ 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}
[ ln x x ] {\displaystyle {\begin{bmatrix}\ln x\\x\end{bmatrix}}}
ln Γ ( η 1 + 1 ) − ( η 1 + 1 ) ln ( − η 2 ) {\displaystyle \ln \Gamma (\eta _{1}+1)-(\eta _{1}+1)\ln(-\eta _{2})}
ln Γ ( α ) − α ln β {\displaystyle \ln \Gamma (\alpha )-\alpha \ln \beta }
k , θ
[ 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}}}
ln Γ ( k ) + k ln θ {\displaystyle \ln \Gamma (k)+k\ln \theta }
inverse gamma distribution
α,β
[ − 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}
[ ln x 1 x ] {\displaystyle {\begin{bmatrix}\ln x\\{\frac {1}{x}}\end{bmatrix}}}
ln Γ ( − η 1 − 1 ) − ( − η 1 − 1 ) ln ( − η 2 ) {\displaystyle \ln \Gamma (-\eta _{1}-1)-(-\eta _{1}-1)\ln(-\eta _{2})}
ln Γ ( α ) − α ln β {\displaystyle \ln \Gamma (\alpha )-\alpha \ln \beta }
scaled inverse chi-squared distribution
ν,σ2
[ − 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}
[ ln x 1 x ] {\displaystyle {\begin{bmatrix}\ln x\\{\frac {1}{x}}\end{bmatrix}}}
ln Γ ( − η 1 − 1 ) − ( − η 1 − 1 ) ln ( − η 2 ) {\displaystyle \ln \Gamma (-\eta _{1}-1)-(-\eta _{1}-1)\ln(-\eta _{2})}
ln Γ ( ν 2 ) − ν 2 ln ν σ 2 2 {\displaystyle \ln \Gamma \left({\frac {\nu }{2}}\right)-{\frac {\nu }{2}}\ln {\frac {\nu \sigma ^{2}}{2}}}
beta distribution
α,β
[ 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)}}}
[ ln x ln ( 1 − x ) ] {\displaystyle {\begin{bmatrix}\ln x\\\ln(1-x)\end{bmatrix}}}
ln Γ ( η 1 ) + ln Γ ( η 2 ) − ln Γ ( η 1 + η 2 ) {\displaystyle \ln \Gamma (\eta _{1})+\ln \Gamma (\eta _{2})-\ln \Gamma (\eta _{1}+\eta _{2})}
ln Γ ( α ) + ln Γ ( β ) − ln Γ ( α + β ) {\displaystyle \ln \Gamma (\alpha )+\ln \Gamma (\beta )-\ln \Gamma (\alpha +\beta )}
multivariate normal distribution
μ ,Σ
[ 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 ln | − 2 η 2 | {\displaystyle -{\frac {1}{4}}{\boldsymbol {\eta }}_{1}^{\rm {T}}{\boldsymbol {\eta }}_{2}^{-1}{\boldsymbol {\eta }}_{1}-{\frac {1}{2}}\ln \left|-2{\boldsymbol {\eta }}_{2}\right|}
1 2 μ T Σ − 1 μ + 1 2 ln | Σ | {\displaystyle {\frac {1}{2}}{\boldsymbol {\mu }}^{\rm {T}}{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}+{\frac {1}{2}}\ln |{\boldsymbol {\Sigma }}|}
categorical distribution
p1 ,...,pk where ∑ i = 1 k p i = 1 {\displaystyle \textstyle \sum _{i=1}^{k}p_{i}=1}
[ ln p 1 ⋮ ln p k ] {\displaystyle {\begin{bmatrix}\ln p_{1}\\\vdots \\\ln 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}}}
[ x = i ] {\displaystyle [x=i]} is the Iverson bracket (1 if x = i {\displaystyle x=i} , 0 otherwise).
0 {\displaystyle 0}
0 {\displaystyle 0}
categorical distribution
p1 ,...,pk where ∑ i = 1 k p i = 1 {\displaystyle \textstyle \sum _{i=1}^{k}p_{i}=1}
[ ln p 1 + C ⋮ ln p k + C ] {\displaystyle {\begin{bmatrix}\ln p_{1}+C\\\vdots \\\ln 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}}}
[ x = i ] {\displaystyle [x=i]} is the Iverson bracket (1 if x = i {\displaystyle x=i} , 0 otherwise).
0 {\displaystyle 0}
0 {\displaystyle 0}
categorical distribution
p1 ,...,pk where p k = 1 − ∑ i = 1 k − 1 p i {\displaystyle p_{k}=1-\textstyle \sum _{i=1}^{k-1}p_{i}}
[ ln p 1 p k ⋮ ln p k − 1 p k 0 ] = {\displaystyle {\begin{bmatrix}\ln {\dfrac {p_{1}}{p_{k}}}\\[10pt]\vdots \\[5pt]\ln {\dfrac {p_{k-1}}{p_{k}}}\\[15pt]0\end{bmatrix}}=} [ ln p 1 1 − ∑ i = 1 k − 1 p i ⋮ ln p k − 1 1 − ∑ i = 1 k − 1 p i 0 ] {\displaystyle {\begin{bmatrix}\ln {\dfrac {p_{1}}{1-\sum _{i=1}^{k-1}p_{i}}}\\[10pt]\vdots \\[5pt]\ln {\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}}}
[ x = i ] {\displaystyle [x=i]} is the Iverson bracket (1 if x = i {\displaystyle x=i} , 0 otherwise).
ln ( ∑ i = 1 k e η i ) = ln ( 1 + ∑ i = 1 k − 1 e η i ) {\displaystyle \ln \left(\sum _{i=1}^{k}e^{\eta _{i}}\right)=\ln \left(1+\sum _{i=1}^{k-1}e^{\eta _{i}}\right)}
− ln p k = − ln ( 1 − ∑ i = 1 k − 1 p i ) {\displaystyle -\ln p_{k}=-\ln \left(1-\sum _{i=1}^{k-1}p_{i}\right)}
multinomial distribution with known number of trials n
p1 ,...,pk where ∑ i = 1 k p i = 1 {\displaystyle \textstyle \sum _{i=1}^{k}p_{i}=1}
[ ln p 1 ⋮ ln p k ] {\displaystyle {\begin{bmatrix}\ln p_{1}\\\vdots \\\ln 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}
multinomial distribution with known number of trials n
p1 ,...,pk where ∑ i = 1 k p i = 1 {\displaystyle \textstyle \sum _{i=1}^{k}p_{i}=1}
[ ln p 1 + C ⋮ ln p k + C ] {\displaystyle {\begin{bmatrix}\ln p_{1}+C\\\vdots \\\ln 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}
multinomial distribution with known number of trials n
p1 ,...,pk where p k = 1 − ∑ i = 1 k − 1 p i {\displaystyle p_{k}=1-\textstyle \sum _{i=1}^{k-1}p_{i}}
[ ln p 1 p k ⋮ ln p k − 1 p k 0 ] = {\displaystyle {\begin{bmatrix}\ln {\dfrac {p_{1}}{p_{k}}}\\[10pt]\vdots \\[5pt]\ln {\dfrac {p_{k-1}}{p_{k}}}\\[15pt]0\end{bmatrix}}=} [ ln p 1 1 − ∑ i = 1 k − 1 p i ⋮ ln p k − 1 1 − ∑ i = 1 k − 1 p i 0 ] {\displaystyle {\begin{bmatrix}\ln {\dfrac {p_{1}}{1-\sum _{i=1}^{k-1}p_{i}}}\\[10pt]\vdots \\[5pt]\ln {\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 ln ( ∑ i = 1 k e η i ) = n ln ( 1 + ∑ i = 1 k − 1 e η i ) {\displaystyle n\ln \left(\sum _{i=1}^{k}e^{\eta _{i}}\right)=n\ln \left(1+\sum _{i=1}^{k-1}e^{\eta _{i}}\right)}
− n ln p k = − n ln ( 1 − ∑ i = 1 k − 1 p i ) {\displaystyle -n\ln p_{k}=-n\ln \left(1-\sum _{i=1}^{k-1}p_{i}\right)}
Dirichlet distribution
α1 ,...,αk
[ 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}}}}
[ ln x 1 ⋮ ln x k ] {\displaystyle {\begin{bmatrix}\ln x_{1}\\\vdots \\\ln x_{k}\end{bmatrix}}}
∑ i = 1 k ln Γ ( η i ) − ln Γ ( ∑ i = 1 k η i ) {\displaystyle \sum _{i=1}^{k}\ln \Gamma (\eta _{i})-\ln \Gamma \left(\sum _{i=1}^{k}\eta _{i}\right)}
∑ i = 1 k ln Γ ( α i ) − ln Γ ( ∑ i = 1 k α i ) {\displaystyle \sum _{i=1}^{k}\ln \Gamma (\alpha _{i})-\ln \Gamma \left(\sum _{i=1}^{k}\alpha _{i}\right)}
Wishart distribution
V ,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 ln | X | ] {\displaystyle {\begin{bmatrix}\mathbf {X} \\\ln |\mathbf {X} |\end{bmatrix}}}
− ( η 2 + p + 1 2 ) ln | − η 1 | {\displaystyle -\left(\eta _{2}+{\frac {p+1}{2}}\right)\ln |-{\boldsymbol {\eta }}_{1}|} + ln Γ p ( η 2 + p + 1 2 ) = {\displaystyle +\ln \Gamma _{p}\left(\eta _{2}+{\frac {p+1}{2}}\right)=} − n 2 ln | − η 1 | + ln Γ p ( n 2 ) = {\displaystyle -{\frac {n}{2}}\ln |-{\boldsymbol {\eta }}_{1}|+\ln \Gamma _{p}\left({\frac {n}{2}}\right)=} ( η 2 + p + 1 2 ) ( p ln 2 + ln | V | ) {\displaystyle \left(\eta _{2}+{\frac {p+1}{2}}\right)(p\ln 2+\ln |\mathbf {V} |)}
+ ln Γ p ( η 2 + p + 1 2 ) {\displaystyle +\ln \Gamma _{p}\left(\eta _{2}+{\frac {p+1}{2}}\right)}
n 2 ( p ln 2 + ln | V | ) + ln Γ p ( n 2 ) {\displaystyle {\frac {n}{2}}(p\ln 2+\ln |\mathbf {V} |)+\ln \Gamma _{p}\left({\frac {n}{2}}\right)}
inverse Wishart distribution
Ψ ,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}
normal-gamma distribution
α,β,μ,λ
[ 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 }}}}
[ ln τ τ τ x τ x 2 ] {\displaystyle {\begin{bmatrix}\ln \tau \\\tau \\\tau x\\\tau x^{2}\end{bmatrix}}}
ln Γ ( η 1 + 1 2 ) − 1 2 ln ( − 2 η 4 ) − {\displaystyle \ln \Gamma \left(\eta _{1}+{\frac {1}{2}}\right)-{\frac {1}{2}}\ln \left(-2\eta _{4}\right)-} − ( η 1 + 1 2 ) ln ( − η 2 + η 3 2 4 η 4 ) {\displaystyle -\left(\eta _{1}+{\frac {1}{2}}\right)\ln \left(-\eta _{2}+{\dfrac {\eta _{3}^{2}}{4\eta _{4}}}\right)}
ln Γ ( α ) − α ln β − 1 2 ln λ {\displaystyle \ln \Gamma \left(\alpha \right)-\alpha \ln \beta -{\frac {1}{2}}\ln \lambda }