多分类逻辑回归模型

假设Y的取值集合是${1, 2, \cdots, K}$，则

$$\begin{aligned} P(Y=k|x) = \frac{\exp (w_k \cdot x)}{1+\sum_{k=1}^{K-1}exp(w_k \cdot x)} \\ P(Y=K) = \frac{1}{1+\sum_{k=1}^{K-1}exp(w_k \cdot x)} \end{aligned}$$

其中$x \in R^{n+1}$，$w_k \in R^{n+1}$