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g_i(X) 是X 来自 omega_i 的 Posterior probability | g_i(X) 是X 来自 omega_i 的 Posterior probability | ||
Decision rule: 如果 个g_1(X) > g_2(X),就选 omega1, 不然选 omega 2 | Decision rule: 如果 个g_1(X) > g_2(X),就选 omega1, 不然选 omega 2 | ||
| − | 据Bayes theorem, decision rule 可以用likelihood ratio 表示 | + | 据Bayes theorem, decision rule 可以用likelihood ratio 表示 <math> l(X) </math> |
<math>\begin{align} | <math>\begin{align} | ||
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\Rightarrow & P(\omega_1|X) > P(\omega_2|X) \\ | \Rightarrow & P(\omega_1|X) > P(\omega_2|X) \\ | ||
\Rightarrow & \frac{P(X|\omega_1)P(\omega_1)}{P(X)} > \frac{P(X|\omega_2)P(\omega_2)}{P(X)} \\ | \Rightarrow & \frac{P(X|\omega_1)P(\omega_1)}{P(X)} > \frac{P(X|\omega_2)P(\omega_2)}{P(X)} \\ | ||
| − | \Rightarrow & P(X|\omega_1)P(\omega_1) > P(X|\omega_2)P(\omega_2) | + | \Rightarrow & P(X|\omega_1)P(\omega_1) > P(X|\omega_2)P(\omega_2) \\ |
| + | \Rightarrow & l(X)=\frac{P(X|\omega_1)}{P(X|\omega_2)} > \frac{P(\omega_2)}{P(\omega_1)} = k | ||
\end{align} | \end{align} | ||
| − | |||
</math> | </math> | ||
| + | |||
| + | |||
| + | |||
Neyman -- Pearson Test | Neyman -- Pearson Test | ||
Revision as of 14:14, 1 May 2014
Hypothesis Testing
PR 的目标是将新的sample进行分类。 分类的决定通过 假设sample是rv, 其conditional density来自其类别 如果知道conditional density， pr的问题就变成statistical hyp testing 的问题 如下假设sample属于两个class其中一个、知道conditional density 和 prior
Bayes Decision Rule for Minum Error 假设X是个observation vector. g_i(X) 是X 来自 omega_i 的 Posterior probability Decision rule: 如果 个g_1(X) > g_2(X),就选 omega1, 不然选 omega 2 据Bayes theorem, decision rule 可以用likelihood ratio 表示 $ l(X) $
$ \begin{align} & g_1(X) > g_2(X) \\ \Rightarrow & P(\omega_1|X) > P(\omega_2|X) \\ \Rightarrow & \frac{P(X|\omega_1)P(\omega_1)}{P(X)} > \frac{P(X|\omega_2)P(\omega_2)}{P(X)} \\ \Rightarrow & P(X|\omega_1)P(\omega_1) > P(X|\omega_2)P(\omega_2) \\ \Rightarrow & l(X)=\frac{P(X|\omega_1)}{P(X|\omega_2)} > \frac{P(\omega_2)}{P(\omega_1)} = k \end{align} $
Neyman -- Pearson Test
