# 逻辑回归2

* 逻辑斯蒂分布
  * 分布函数：$$F(x)=P(X \leq x)=\frac{1}{1+e^{-(x-\mu)/\gamma}}$$
  * 密度函数：$$f(x)=F'(x)=\frac{e^{-(x-\mu)/\gamma}}{\gamma(1+e^{-(x-\mu)/\gamma})^2}$$
* 二项逻辑斯蒂回归模型概率分布：

  $$P(Y=1|x)=\frac{exp(w\.x+b)}{1+exp(w\.x+b)}$$

  $$P(Y=0|x)=\frac{1}{1+exp(w\.x+b)}$$
* 通常合并w、b：

  $$P(Y=1|x)=\frac{exp(w\.x)}{1+exp(w\.x)}$$

  $$P(Y=0|x)=\frac{1}{1+exp(w\.x)}$$
* 特点：对数几率
  * 正负概率只比$$\frac{p}{1-p}$$
  * 取对数$$log\frac{p}{1-p}=w\.b$$
  * 意义：$$w .b$$越大正类的概率比负类的比越大
* 模型参数估计：极大似然

  记：$$P(Y=1|x)=\pi(x),Y(Y=0|x)=1-\pi(x)$$

  似然估计函数：$$\prod\_{i=1}^N \[\pi(x\_i)]^{y\_i}\[1-\pi(x\_i)]^{1-y\_i}$$

  `对数似然`：

  $$
  \begin{aligned}
  L(w)=\sum\_{i=1}^{N}\[y\_ilog\pi(x\_i)+(1-y\_i)log(1-\pi(x\_i))] \\
  \= \sum\_{i=1}^{N}\[y\_ilog\frac{\pi(x\_i)}{1-\pi(x\_i)}+log(1-\pi(x\_i))] \\
  \= \sum\_{i=1}^{N}\[y\_i(w\.x\_i)-log(1+exp(w\.x\_i))] \\
  \end{aligned}
  $$
* 然后对w求偏导数，得到梯度下降等进行参数求解


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