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