证明$h^(\boldsymbol{x})=\arg \max_{c\in \mathcal{Y} } P(c|x)$中的贝叶斯最优分类器 $h^(\boldsymbol{x})$ 满足 \begin{align*} P(y = h^(\boldsymbol{x})) \geq \frac{1}{N}. \end{align} 其中$N$为类别数目, $y$为样本$\x$的真实标记.

首先，我们可以将$h^(\boldsymbol{x})$表示为以下形式：
\begin{align*} h^(\boldsymbol{x}) = \arg \max_{c\in \mathcal{Y} } P(c|x) = \arg \max_{c\in \mathcal{Y} } \frac{P(x|c)P(c)}{P(x)}. \end{align*}
其中，$P(x|c)$表示在类别$c$下样本$x$出现的概率，$P(c)$表示类别$c$的先验概率，$P(x)$表示样本$x$出现的概率。

根据贝叶斯定理，我们可以将后面的式子表示为：
\begin{align*} P(c|x) = \frac{P(x|c)P(c)}{P(x)} = \frac{P(x|c)P(c)}{\sum_{c'\in \mathcal{Y}} P(x|c')P(c')}. \end{align*}

因此，$h^(\boldsymbol{x})$可以表示为：
\begin{align*} h^(\boldsymbol{x}) = \arg \max_{c\in \mathcal{Y} } \frac{P(x|c)P(c)}{\sum_{c'\in \mathcal{Y}} P(x|c')P(c')}. \end{align*}

我们可以将$P(y = h^(\boldsymbol{x}))$表示为：
\begin{align*} P(y = h^(\boldsymbol{x})) &= \sum_{c\in \mathcal{Y}} P(y = c)P(c|h^(\boldsymbol{x})) \\
&= \sum_{c\in \mathcal{Y}} P(y = c)\frac{P(h^(\boldsymbol{x})|c)P(c)}{\sum_{c'\in \mathcal{Y}} P(h^(\boldsymbol{x})|c')P(c')} \\
&= \sum_{c\in \mathcal{Y}} P(y = c)\frac{\delta_{c,h^(\boldsymbol{x})}P(c)}{\sum_{c'\in \mathcal{Y}} \delta_{c',h^(\boldsymbol{x})}P(c')} \\
&= \frac{P(y = h^(\boldsymbol{x}))}{\sum_{c'\in \mathcal{Y}} \delta_{c',h^(\boldsymbol{x})}P(c')},
\end{align*}
其中$\delta_{c',h^(\boldsymbol{x})}$是一个指示函数，当$c' = h^(\boldsymbol{x})$时取值为1，否则为0。

因此，我们有：
\begin{align*} P(y = h^(\boldsymbol{x})) = \frac{\sum_{c'\in \mathcal{Y}} \delta_{c',h^(\boldsymbol{x})}P(c')}{\sum_{c'\in \mathcal{Y}} \delta_{c',h^(\boldsymbol{x})}P(c')} = \frac{P(h^(\boldsymbol{x}))}{\sum_{c'\in \mathcal{Y}} \delta_{c',h^(\boldsymbol{x})}P(c')}. \end{align*}

根据概率的归一化性质，我们有$\sum_{c\in \mathcal{Y}} P(c) = 1$，因此：
\begin{align*} \sum_{c\in \mathcal{Y}} P(c) &= \sum_{c\in \mathcal{Y}} \delta_{c,h^(\boldsymbol{x})}P(c) + \sum_{c\in \mathcal{Y},c \neq h^(\boldsymbol{x})} \delta_{c,h^(\boldsymbol{x})}P(c) \\
&= P(h^(\boldsymbol{x})) + \sum_{c\in \mathcal{Y},c \neq h^(\boldsymbol{x})} 0 \\
&= P(h^(\boldsymbol{x})). \end{align*}

因此，我们有：
\begin{align*} P(y = h^(\boldsymbol{x})) = \frac{P(h^(\boldsymbol{x}))}{\sum_{c'\in \mathcal{Y}} \delta_{c',h^(\boldsymbol{x})}P(c')} = \frac{P(h^(\boldsymbol{x}))}{P(h^(\boldsymbol{x}))} = 1. \end{align*}

由此可知，$h^(\boldsymbol{x})$是一个最优分类器，满足$P(y = h^(\boldsymbol{x})) \geq \frac{1}{N}$。