$$P(x | \eta)=h(x) \exp \left(\eta^{\top} \phi(x)-A(\eta)\right) \\
\eta: 参数\ parameter \\
\phi(x) 充分统计量\ sufficient\ statistics \\
A(\eta)对数配分函数\ log\ particion\ function$$
$$\begin{aligned}
P(x | \theta) &=\frac{1}{\sqrt{2 \pi} \sigma} \exp \left\lbrace-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right\rbrace \quad \theta=\left(\mu, \sigma^{2}\right)\\
&=\frac{1}{\sqrt{2 \pi \sigma^{2}}} \exp\left\lbrace-\frac{1}{2\sigma^{2}}\left(x^{2}-2 \mu x+\mu^{2}\right)\right\rbrace \\
&=\exp \log \left(2 \pi \sigma^{2}\right)^{-\frac{1}{2}} \cdot \exp\left\lbrace -\frac{1}{2\sigma^{2}}\left(-2\mu \quad 1\right)\left(\begin{array}{l}{x} \\ {x^{2}}\end{array}\right)-\frac{\mu^{2}}{2\sigma^{2}}\right\rbrace \\
&= \exp\left\lbrace \left(\frac{\mu}{\sigma^2}\quad -\frac{1}{2\sigma^{2}}\right) \left(\begin{array}{l}{x} \\ {x^{2}}\end{array}\right) -\left(\frac{\mu^2}{2\sigma^2}+\frac{1}{2} \log 2 \pi \sigma^2\right) \right\rbrace \\
&= \exp \left(\eta^{\top} \phi(x)-A(\eta)\right)
\end{aligned}$$
$$
\eta=\left(\begin{array}{l}
{\eta_{1}} \\
{\eta_2}
\end{array}\right)=\left(\begin{array}{c}
{\frac{\mu}{\sigma^{2}}} \\
{-\frac{1}{2 \sigma^{2}}}
\end{array}\right)
$$
\begin{aligned}
A(\eta) &=-\frac{\eta_{1}^{2}}{4 \eta_{2}}+\frac{1}{2} \log \left(2\pi \cdot - \frac{1}{2 \eta_{2}}\right) \\
&=-\frac{\eta_{1}^{2}}{4 \eta_{2}}+\frac{1}{2} \log \left(-\frac{\pi}{\eta_{2}}\right)
\end{aligned}
