Distribution

q_i

b(q_i)

f

m_i

Bernoulli

$\ln \left(\frac{{\pi }_i}{1-{\pi }_i}\right)$

$\ln \left(1+\exp \left({\theta }_i\right)\right)$

1

${\pi }_i=\frac{\exp \left({\theta }_i\right)}{1+\exp \left({\theta }_i\right)}$

Poisson

$\ln \left({\lambda }_i\right)$

$\exp \left({\theta }_i\right)$

1

$\exp \left({\theta }_i\right)={\lambda }_i$

Normal

${\mu }_i$

$\frac{{\theta }_i^2}{2}$

${\sigma }^2$

${\theta }_i={\mu }_i$

Gamma

$-\frac{1}{{\mu }_i}$

$-\frac{1}{{\mu }_i}$

$\frac{1}{\upsilon }$

$-\frac{1}{{\theta }_i}={\mu }_i$

Inverse Gaussian

$-\frac{1}{2{\mu }_i^2}$

$-{\left(-2{\theta }_i\right)}^{1/2}$

${\sigma }^2$

$\frac{1}{{\left(-2{\theta }_i\right)}^{1/2}}={\mu }_i$