reply to http://disq.us/p/254p777

docs/02-Overview-of-Supervised-Learning/2.4-Statistical-Decision-Theory.md

@@ -108,16 +108,29 @@ $$
 \hat{G}(x) = \argmin_{g\in \cal{G}}\sum\limits_{k=1}^KL({\cal{G}}_k,g)\Pr({\cal G}_k\mid X = x)\tag{2.21}
 $$
 
-结合 $0-1$ 损失函数上式简化为
+结合 0-1 损失函数上式简化为
 
 $$
-\hat{G}(x) = \argmin_{g\in \cal{G}}[1 − \Pr(g\mid X = x)]\tag{2.22}
+\hat{G}(x) = \argmin_{g\in \cal{G}}[1 − \Pr(g\mid X = x)]\tag{2.22}\label{2.22}\,.
 $$
 
+!!! note "weiya 注：推导 \eqref{2.22}"
+    注意到，对于 0-1 损失，
+    $$
+    L(\cG_k, g) = \begin{cases}
+        0 & \text{if } g=\cG_k\\
+        1 & \text{if } g\neq \cG_k
+    \end{cases}\,,
+    $$
+    则我们有
+    $$
+    \sum_{k=1}^KL({\cal{G}}_k,g)\Pr({\cal G}_k\mid X = x) = \sum_{k=1}^K\Pr(\cG_k\neq g\mid X=x)=1-\Pr(\cG_k=g\mid X=x)\,.
+    $$
+
 或者简单地
 
 $$
-\hat{G}(X) = {\cal{G}}_k \text{ if } \Pr({\cal{G}}_k\mid X = x) = \underset{g\in{\cal{G}}}{max } \Pr(g\mid X = x)\tag{2.23}
+\hat{G}(X) = {\cal{G}}_k \text{ if } \Pr({\cal{G}}_k\mid X = x) = \underset{g\in{\cal{G}}}{\max } \Pr(g\mid X = x)\tag{2.23}
 $$
 
 合理的解决方法被称作 **贝叶斯分类 (Bayes classifier)**，利用条件（离散）分布 $\Pr(G\mid X)$ 分到最合适的类别．对于我们模拟的例子图 2.5 显示了最优的贝叶斯判别边界．贝叶斯分类的误差阶被称作 **贝叶斯阶 (Bayes rate)**．

docs/js/mathjax.js

@@ -38,6 +38,7 @@ window.MathJax = {
       calC: "{{\\cal{C}}}",
       calS: "{{\\cal{S}}}",
       calI: "{{\\cal{I}}}",
+      cG: "{{\\cal{G}}}",
       cH: "{{\\cal{H}}}",
       cM: "{{\\cal{M}}}",
       cP: "{{\\cal{P}}}",