توزیع |
![{\displaystyle {\boldsymbol {\theta }}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Na2eQaDrFzAdAztvBoNmOnto1oNG3aqvDa2s1yjC3oNCOatwPyqzA) |
![{\displaystyle {\boldsymbol {\eta }}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80yti4otG3oDJDajm1aDK1aDnBaAa3ajs1ajCPajlAzqa0yto4nDC3) |
تابع پارامتر معکوس |
![{\displaystyle h(x)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QaAaQzNGOzqvFnte4zNm1ngoQa2wPzqvFnqe4yjw0njw5oNvCoAw0) |
![{\displaystyle T(x)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PatKPoNe5oDvDaAe1ztK1nDsQotvBotCNatzAytm0zjFFaAzEntm1) |
![{\displaystyle A({\boldsymbol {\eta }})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OztePnghAajJFaDnAnja0oDG4ajvEatw4zgsPnDiQa2w0oDC5aDCQ) |
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Bernoulli distribution |
p
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binomial distribution with known number of trials n |
p
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Poisson distribution |
λ
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negative binomial distribution with known number of failures r |
p
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exponential distribution |
λ
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Pareto distribution with known minimum value xm |
α
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Weibull distribution with known shape k |
λ
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Laplace distribution with known mean μ |
b
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chi-squared distribution |
ν
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normal distribution known variance |
μ
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normal distribution |
μ,σ2
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lognormal distribution |
μ,σ2
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inverse Gaussian distribution |
μ,λ
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gamma distribution |
α,β
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k, θ
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inverse gamma distribution |
α,β
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scaled inverse chi-squared distribution |
ν,σ2
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beta distribution |
α,β
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multivariate normal distribution |
μ,Σ
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categorical distribution |
p1,...,pk
where
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![{\displaystyle {\begin{bmatrix}e^{\eta _{1}}\\\vdots \\e^{\eta _{k}}\end{bmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DytlFoNJAoqvAajsPzjo0nqo1nDmQaDJCntvFzjvCzqaPzDFEotaP)
where
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is the Iverson bracket (1 if , 0 otherwise).
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categorical distribution |
p1,...,pk
where
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![{\displaystyle {\begin{bmatrix}{\dfrac {1}{C}}e^{\eta _{1}}\\\vdots \\{\dfrac {1}{C}}e^{\eta _{k}}\end{bmatrix}}=}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Pyqi4o2rEaDm0ztrFzjm5zDCOotzFa2wPzAo5zNnBotKOoti0otnC)
where
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is the Iverson bracket (1 if , 0 otherwise).
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categorical distribution |
p1,...,pk
where
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![{\displaystyle {\begin{bmatrix}\ln {\dfrac {p_{1}}{p_{k}}}\\[10pt]\vdots \\[5pt]\ln {\dfrac {p_{k-1}}{p_{k}}}\\[15pt]0\end{bmatrix}}=}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DatdFo2vDnts4o2rCaqnCajzEz2eOoDnBzNlBatBAnDvDytG0z2zC)
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![{\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://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QzNK5zAo1oNmNoAzAytFCntlFztK3zNzDzDiPzNhAyteNyqiQnAe0)
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is the Iverson bracket (1 if , 0 otherwise).
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multinomial distribution with known number of trials n |
p1,...,pk
where
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![{\displaystyle {\begin{bmatrix}e^{\eta _{1}}\\\vdots \\e^{\eta _{k}}\end{bmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DytlFoNJAoqvAajsPzjo0nqo1nDmQaDJCntvFzjvCzqaPzDFEotaP)
where
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multinomial distribution with known number of trials n |
p1,...,pk
where
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![{\displaystyle {\begin{bmatrix}{\dfrac {1}{C}}e^{\eta _{1}}\\\vdots \\{\dfrac {1}{C}}e^{\eta _{k}}\end{bmatrix}}=}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Pyqi4o2rEaDm0ztrFzjm5zDCOotzFa2wPzAo5zNnBotKOoti0otnC)
where
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multinomial distribution with known number of trials n
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p1,...,pk
where
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![{\displaystyle {\begin{bmatrix}\ln {\dfrac {p_{1}}{p_{k}}}\\[10pt]\vdots \\[5pt]\ln {\dfrac {p_{k-1}}{p_{k}}}\\[15pt]0\end{bmatrix}}=}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DatdFo2vDnts4o2rCaqnCajzEz2eOoDnBzNlBatBAnDvDytG0z2zC)
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![{\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://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QzNK5zAo1oNmNoAzAytFCntlFztK3zNzDzDiPzNhAyteNyqiQnAe0)
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Dirichlet distribution |
α1,...,αk
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Wishart distribution |
V,n
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![{\displaystyle -\left(\eta _{2}+{\frac {p+1}{2}}\right)\ln |-{\boldsymbol {\eta }}_{1}|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PyjdAnqsPajhBatC1zNBEaji0nAi2njKPzAvFzjsQzAs2ajoQaNFD)
![{\displaystyle +\ln \Gamma _{p}\left(\eta _{2}+{\frac {p+1}{2}}\right)=}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85a2eQzNwOaqe1ngdCoAdAaNm2zqrDotKNagi3yjs2aqnCzjs5njs1)
![{\displaystyle -{\frac {n}{2}}\ln |-{\boldsymbol {\eta }}_{1}|+\ln \Gamma _{p}\left({\frac {n}{2}}\right)=}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PnjC3ytwOatFAnqnBoArDytnCzgsOnDa4aNm5zNwPaNsOzjrEo2rC)
![{\displaystyle \left(\eta _{2}+{\frac {p+1}{2}}\right)(p\ln 2+\ln |\mathbf {V} |)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83nAiNzAnBzjdCytGQzqvEzgnDnAwNagaNythDyjhEagdDaAhDotm3)
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inverse Wishart distribution |
Ψ,m
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normal-gamma distribution |
α,β,μ,λ
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![{\displaystyle \ln \Gamma \left(\eta _{1}+{\frac {1}{2}}\right)-{\frac {1}{2}}\ln \left(-2\eta _{4}\right)-}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Da2w2oNmPzqa2agrDyjJEoNm0ajGOotrDzgi4oqePzAs0ntmPatoP)
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