User:Nippashish/sandbox

In the following table the support of the density is denoted $x$. The symbol $\theta$ stands for the distribution parameters as they appear in their own wiki page, for example in the entry for the Bernoulli, $\theta =p$ whereas in the entry for the Normal distribution, we use $\theta =\{\mu ,\sigma \}$.

Distribution	$h(x)$	$T(x)$	$\eta (\theta )$	$A(\theta )$	$A(\eta )$
Bernoulli	$1$	$x$	$\log \left({\frac {p}{1-p}}\right)$	$-\log(1-p)$	$\log(1+\exp(\eta ))$
Binomial (known n)	${n \choose x}$	$x$	$\log \left({\frac {p}{1-p}}\right)$	$-n\log(1-p)$	$n\log(1+\exp(\eta ))$
Multinomial (known n)	$n!\left(\prod _{i=1}^{K}x_{i}!\right)^{-1}$	${\begin{bmatrix}x_{1}\\\vdots \\x_{K}\end{bmatrix}}$	${\begin{bmatrix}\log(p_{1})\\\vdots \\\log(p_{K})\end{bmatrix}}$	$1$	$1$
Poisson	${\frac {1}{x!}}$	$x$	$\log(\lambda )$	$\lambda$	$\exp(\eta )$
Normal (known variance)	${\frac {1}{\sqrt {2\pi }}}\exp \left(-{\frac {x^{2}}{2\sigma ^{2}}}\right)$	${\frac {x}{\sigma }}$	${\frac {\mu }{\sigma }}$	${\frac {\mu ^{2}}{2\sigma ^{2}}}$	${\frac {\eta ^{2}}{2}}$
Normal	${\frac {1}{\sqrt {2\pi }}}$	${\begin{bmatrix}x\\x^{2}\end{bmatrix}}$	${\begin{bmatrix}{\frac {\mu }{\sigma ^{2}}}\\-{\frac {1}{2\sigma ^{2}}}\end{bmatrix}}$	${\frac {\mu ^{2}}{2\sigma ^{2}}}+\log(\sigma )$	$-{\frac {1\eta _{1}^{2}}{4\eta _{2}}}-{\frac {1}{2}}\log(-2\eta _{2})$
Multivariate Normal (d dimensions)	$(2\pi )^{-d/2}$	${\begin{bmatrix}x\\xx^{\mathrm {T} }\end{bmatrix}}$	${\begin{bmatrix}\Sigma ^{-1}\mu \\-{\frac {1}{2}}\Sigma ^{-1}\end{bmatrix}}$	$1$	$1$
Exponential (rate)	$1$	$x$	$-\lambda$	$-\log(\lambda )$	$-\log(-\eta )$
Gamma (rate)	$1$	${\begin{bmatrix}x\\\log(x)\end{bmatrix}}$	${\begin{bmatrix}-\lambda \\k-1\end{bmatrix}}$	$\log(\Gamma (k))-k\log(\lambda )$	$\log(\Gamma (\eta _{2}+1))-(\eta _{2}+1)\log(-\eta _{1})$
Chi-squared	$\exp \left(-{\frac {x}{2}}\right)$	$\log(x)$	${\frac {k}{2}}-1$	${\frac {k}{2}}\log(2)+\log(\Gamma ({\frac {k}{2}}))$	$(\eta +1)\log(2)+\log(\Gamma (\eta +1))$
Beta	$1$	${\begin{bmatrix}\log(x)\\\log(1-x)\end{bmatrix}}$	${\begin{bmatrix}\alpha -1\\\beta -1\end{bmatrix}}$	$\log(B(\alpha ,\beta ))$	$\log(B(\eta _{1}+1,\eta _{2}+1))$
Dirichlet	$1$	${\begin{bmatrix}x_{1}\\\vdots \\x_{K}\end{bmatrix}}$	${\begin{bmatrix}\alpha _{1}-1\\\vdots \\\alpha _{K}-1\end{bmatrix}}$	$\sum _{i=1}^{K}\log(\Gamma (\alpha _{i}))-\log(\Gamma (\sum _{i=1}^{K}\alpha _{i}))$	$\sum _{i=1}^{K}\log(\Gamma (\eta _{i}+1))-\log(\Gamma (\sum _{i=1}^{K}\eta _{i}-K))$