$$p(x | \eta)=h(x) \exp \left(\eta^{\top} \phi(x)-A(\eta)\right) \\
\eta: 参数\ parameter \\
\phi(x) 充分统计量\ sufficient\ statistics \\
A(\eta)对数配分函数\ log\ particion\ function$$
对数配分函数与充分统计量
$$\begin{aligned}
p(x | \eta) &=h(x) \exp \left(\eta^{T} \phi(x)\right) \cdot \exp (-A(n)) \\
&=\frac{1}{\exp (A(n))} h(x) \cdot \exp \left(\eta^{\top} \phi(x)\right)
\end{aligned}$$
\begin{aligned}
\exp (A(\eta))=\int & h(x) \cdot \exp \left(\eta^{\top} \phi(x)\right) d x \\
\exp (A(\eta)) \cdot A^{\prime}(\eta) &=\frac{\partial}{\partial \eta}\left(\int h(x) \exp \left(\eta^{\top} \phi(x)\right) d x\right) \\
&=\int h(x) \exp \left(\eta^{\top} \phi(x)\right) \cdot \phi(x) d x
\end{aligned}
\begin{aligned}
A^{\prime}(n) &=\frac{\int h(x) \exp \left(\eta^{\top} \phi(x)\right) \cdot \phi(x) d x}{\exp (A(\eta))} \\
&=\int {h(x) \exp \left(\eta^{\top} \phi(x)-A(\eta)\right) \cdot \phi(x) d x} \\
&=\int p(x | \eta) \phi(x) dx\\
&=E_{p(x | \eta)}[\phi(x)]
\end{aligned}
$$A^{\prime \prime}(\eta)=\operatorname{Var}[\phi(x)]$$
最大似然估计
数据$$D=\left\lbrace x_{1}, x_{2}, \cdots, x_{N}\right\rbrace$$
$$
\begin{aligned}
\eta_{MLE} &=\arg \max \log p(D | \eta) \\
&=\arg \max \log \prod_{i=1}^{N} p\left(x_{i} | \eta\right) \\
&=\arg \max \sum_{i=1}^{N}\left[ \log h\left(x_{i}\right) \cdot \exp \left(\eta^{T} \phi\left(x_{i}\right)-A(\eta)\right)\right] \\
&=\arg \max\sum {i=1}^N \left[\log h\left(x_i \right)+\eta^{\top} \phi\left(x_i \right)-A(\eta)\right]\\
&=\arg \max \sum{i=1}^N \left(\eta^{\top} \phi\left(x_i \right)-A(\eta)\right) \\
\end{aligned}
$$
$$
\begin{aligned}
& \frac{\partial}{\partial \eta} \left(\sum_{i=1}^N \left(\eta^{\top} \phi\left(x_{i}\right)-A(\eta)\right)\right)\\
=& \sum_{i=1}^{N} \frac{\partial}{\partial \eta}\left(\eta^{\top} \phi\left(x_{i}\right)-A(\eta)\right) \\
=& \sum_{i=1}^{N} \phi\left(x_{i}\right)-A^{\prime}(\eta) \\
=& \sum_{i=1}^{N} \phi\left(x_{i}\right)-N A^{\prime}(\eta) \\
=& 0
\end{aligned}
$$
$$
A^{\prime}\left(\eta_{MLE}\right)=\frac{1}{N} \sum_{i=1}^{N} \phi\left(x_{i}\right)
$$
$$\eta_{M L E}={A}’^{(-1)} (\eta)$$
