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# 证明：对偶函数的极大化=模型的极大似然估计

## 模型的极大似然估计

条件概率分布P(Y|X)的对数似然函数表示为：

```
\begin{aligned}
L_{\tilde P}(P_w) = \log \prod_{x,y}P(y|x)^{\tilde P(x, y)} \\
= \sum_{x,y}\tilde P(x, y)\log P(y|x) && {1}
\end{aligned}
```

当条件概率分布P(Y|X)是最大熵模型时，即

$$
\begin{aligned}
P(y|x) = \frac{exp(\sum\_{i=0}^n)w\_{i+1}f\_i(x, y)}{exp(1-w+0)} && {2}
\end{aligned}
$$

将公式（2）代入公式（1）得：

$$
L\_{\tilde P}(P\_w) = \sum\_{x,y} \tilde P(x, y) \sum\_{i=0}^n w\_{i+1}f\_i(x, y) - \sum\_x \tilde P(x) \log Z\_w(x)
$$

## 对偶函数的极大化

对偶函数定义如下：

$$
\begin{aligned}
\Psi (w) =  L(P\_w, w) \\
\= \sum\_{x,y}\tilde P(x) P\_w(y|x)\log P\_w(y|x) \\
\= \sum\_{x,y} \tilde P(x, y) \sum\_{i=0}^n w\_{i+1}f\_i(x, y) - \sum\_x \tilde P(x) \log Z\_w(x)
\end{aligned}
$$

## 结论

$$
L\_{\tilde P}(P\_w) = \Psi (w)
$$

即：对偶函数的极大化=模型的极大似然估计