信息熵和条件熵的计算

本是古典何须时尚 2021-09-29 12:24 712阅读 0赞

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### 目录 ###

*  1. 信息熵
 *  2. 条件熵
 *  3. 参考文章

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**表1. 目标值为PlayTennis的14个训练样例**

<table> 
 <thead> 
 <tr> 
 <th>Day</th> 
 <th>Outlook</th> 
 <th>Temperature</th> 
 <th>Humidity</th> 
 <th>Wind</th> 
 <th>PlayTennis</th> 
 </tr> 
 </thead> 
 <tbody> 
 <tr> 
 <td> D 1 D_1 D1</td> 
 <td>Sunny</td> 
 <td>Hot</td> 
 <td>High</td> 
 <td>Weak</td> 
 <td>No</td> 
 </tr> 
 <tr> 
 <td> D 2 D_2 D2</td> 
 <td>Sunny</td> 
 <td>Hot</td> 
 <td>High</td> 
 <td>Strong</td> 
 <td>No</td> 
 </tr> 
 <tr> 
 <td> D 3 D_3 D3</td> 
 <td>Overcast</td> 
 <td>Hot</td> 
 <td>High</td> 
 <td>Weak</td> 
 <td>Yes</td> 
 </tr> 
 <tr> 
 <td> D 4 D_4 D4</td> 
 <td>Rain</td> 
 <td>Mild</td> 
 <td>High</td> 
 <td>Weak</td> 
 <td>Yes</td> 
 </tr> 
 <tr> 
 <td> D 5 D_5 D5</td> 
 <td>Rain</td> 
 <td>Cool</td> 
 <td>Normal</td> 
 <td>Weak</td> 
 <td>Yes</td> 
 </tr> 
 <tr> 
 <td> D 6 D_6 D6</td> 
 <td>Rain</td> 
 <td>Cool</td> 
 <td>Normal</td> 
 <td>Strong</td> 
 <td>No</td> 
 </tr> 
 <tr> 
 <td> D 7 D_7 D7</td> 
 <td>Overcast</td> 
 <td>Cool</td> 
 <td>Normal</td> 
 <td>Strong</td> 
 <td>Yes</td> 
 </tr> 
 <tr> 
 <td> D 8 D_8 D8</td> 
 <td>Sunny</td> 
 <td>Mild</td> 
 <td>High</td> 
 <td>Weak</td> 
 <td>No</td> 
 </tr> 
 <tr> 
 <td> D 9 D_9 D9</td> 
 <td>Sunny</td> 
 <td>Cool</td> 
 <td>Normal</td> 
 <td>Weak</td> 
 <td>Yes</td> 
 </tr> 
 <tr> 
 <td> D 10 D_{10} D10</td> 
 <td>Rain</td> 
 <td>Mild</td> 
 <td>Normal</td> 
 <td>Weak</td> 
 <td>Yes</td> 
 </tr> 
 <tr> 
 <td> D 11 D_{11} D11</td> 
 <td>Sunny</td> 
 <td>Mild</td> 
 <td>Normal</td> 
 <td>Strong</td> 
 <td>Yes</td> 
 </tr> 
 <tr> 
 <td> D 12 D_{12} D12</td> 
 <td>Overcast</td> 
 <td>Mild</td> 
 <td>High</td> 
 <td>Strong</td> 
 <td>Yes</td> 
 </tr> 
 <tr> 
 <td> D 13 D_{13} D13</td> 
 <td>Overcast</td> 
 <td>Hot</td> 
 <td>Normal</td> 
 <td>Weak</td> 
 <td>Yes</td> 
 </tr> 
 <tr> 
 <td> D 14 D_{14} D14</td> 
 <td>Rain</td> 
 <td>Mild</td> 
 <td>High</td> 
 <td>Strong</td> 
 <td>No</td> 
 </tr> 
 </tbody> 
</table>

如表1所示，目标值是：**PlayTennis**，也就是是否打球。  
表1中有四个特征，分别是天气（Outlook）、温度（Temperature）、湿度（Humidity）以及风（Wind）。

# 1. 信息熵 #

信息熵的公式：  
 H ( X ) = − ∑ x ∈ X p ( x ) log ⁡ p ( x ) H(X) = - \\sum\_\{x \\in X\} p(x) \\log p(x) H(X)=−x∈X∑p(x)logp(x)  
顺带一提，  
 0 ≤ H ( X ) ≤ log ⁡ n 0 \\leq H(X) \\leq \\log n 0≤H(X)≤logn

以表1为例，设是否打球这一随机变量为 Y Y Y，则  
 p ( Y = Yes ) = 9 14 p(Y = \\text\{Yes\}) = \\frac\{9\}\{14\} p(Y=Yes)=149  
 p ( Y = No ) = 5 14 p(Y = \\text\{No\}) = \\frac\{5\}\{14\} p(Y=No)=145  
所以，  
 H ( Y ) = − ∑ y ∈ Y p ( y ) log ⁡ p ( y ) = − ( p ( Y = Yes ) ∗ log ⁡ p ( Y = Yes ) + p ( Y = No ) ∗ log ⁡ p ( Y = No ) ) = − ( 9 14 ∗ log ⁡ 2 9 14 + 5 14 ∗ log ⁡ 2 5 14 ) = 0.9403 \\begin\{aligned\} H(Y) &= - \\sum\_\{y \\in Y\} p(y) \\log p(y) \\\\ &= - ( p(Y=\\text\{Yes\}) \\ast \\log p(Y=\\text\{Yes\}) + p(Y=\\text\{No\}) \\ast \\log p(Y=\\text\{No\}) ) \\\\ &= - ( \\frac\{9\}\{14\} \\ast \\log\_2 \\frac\{9\}\{14\} + \\frac\{5\}\{14\} \\ast \\log\_2 \\frac\{5\}\{14\}) \\\\ &= 0.9403 \\end\{aligned\} H(Y)=−y∈Y∑p(y)logp(y)=−(p(Y=Yes)∗logp(Y=Yes)\+p(Y=No)∗logp(Y=No))=−(149∗log2149\+145∗log2145)=0.9403

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# 2. 条件熵 #

条件熵表示在条件 X X X下 Y Y Y的信息熵。  
公式如下：  
 H ( Y ∣ X ) = ∑ x ∈ X p ( x ) H ( Y ∣ X = x ) H(Y|X) = \\sum\_\{x \\in X\} p(x) H(Y|X=x) H(Y∣X)=x∈X∑p(x)H(Y∣X=x)

在表1的例子中，设湿度（Humidity）为随机变量 X X X，则：  
 p ( X = High ) = 7 14 = 1 2 p(X=\\text\{High\}) = \\frac\{7\}\{14\} = \\frac\{1\}\{2\} p(X=High)=147=21

p ( X = Normal ) = 7 14 = 1 2 p(X=\\text\{Normal\}) = \\frac\{7\}\{14\} = \\frac\{1\}\{2\} p(X=Normal)=147=21  
所以，  
 H ( Y ∣ X ) = ∑ x ∈ X p ( x ) H ( Y ∣ X = x ) = p ( X = High ) ∗ H ( Y ∣ X = High ) + p ( X = Normal ) ∗ H ( Y ∣ X = Normal ) \\begin\{aligned\} H(Y|X) &= \\sum\_\{x \\in X\} p(x) H(Y|X=x) \\\\ &= p(X=\\text\{High\}) \\ast H(Y|X=\\text\{High\}) + p(X=\\text\{Normal\}) \\ast H(Y|X=\\text\{Normal\}) \\end\{aligned\} H(Y∣X)=x∈X∑p(x)H(Y∣X=x)=p(X=High)∗H(Y∣X=High)\+p(X=Normal)∗H(Y∣X=Normal)

接下来计算 H ( Y ∣ X = High ) H(Y|X=\\text\{High\}) H(Y∣X=High)和 H ( Y ∣ X = Normal ) H(Y|X=\\text\{Normal\}) H(Y∣X=Normal)。

根据信息熵的计算方法可得：  
 H ( Y ∣ X = High ) = − ∑ y ∈ Y p ( y ) log ⁡ p ( y ) = − ( p ( Y = Yes ∣ X = High ) ∗ log ⁡ p ( Y = Yes ∣ X = High ) + p ( Y = No ∣ X = High ) ∗ log ⁡ p ( Y = No ∣ X = High ) = − ( 3 7 ∗ log ⁡ 2 3 7 + 4 7 ∗ log ⁡ 2 4 7 ) = 0.9852 \\begin\{aligned\} H(Y|X=\\text\{High\}) &= - \\sum\_\{y \\in Y\} p(y) \\log p(y) \\\\ &= - ( p(Y=\\text\{Yes\} | X=\\text\{High\}) \\ast \\log p(Y=\\text\{Yes\} | X=\\text\{High\} ) \\\\ &+ p(Y=\\text\{No\} | X=\\text\{High\}) \\ast \\log p(Y=\\text\{No\} | X=\\text\{High\} ) \\\\ &= - ( \\frac\{3\}\{7\} \\ast \\log\_2 \\frac\{3\}\{7\} + \\frac\{4\}\{7\} \\ast \\log\_2 \\frac\{4\}\{7\} ) \\\\ &= 0.9852 \\end\{aligned\} H(Y∣X=High)=−y∈Y∑p(y)logp(y)=−(p(Y=Yes∣X=High)∗logp(Y=Yes∣X=High)\+p(Y=No∣X=High)∗logp(Y=No∣X=High)=−(73∗log273\+74∗log274)=0.9852

H ( Y ∣ X = Normal ) = − ∑ y ∈ Y p ( y ) log ⁡ p ( y ) = − ( p ( Y = Yes ∣ X = Normal ) ∗ log ⁡ p ( Y = Yes ∣ X = Normal ) + p ( Y = No ∣ X = Normal ) ∗ log ⁡ p ( Y = No ∣ X = Normal ) = − ( 6 7 ∗ log ⁡ 2 6 7 + 1 7 ∗ log ⁡ 2 1 7 ) = 0.5917 \\begin\{aligned\} H(Y|X=\\text\{Normal\}) &= - \\sum\_\{y \\in Y\} p(y) \\log p(y) \\\\ &= - ( p(Y=\\text\{Yes\} | X=\\text\{Normal\}) \\ast \\log p(Y=\\text\{Yes\} | X=\\text\{Normal\}) \\\\ &+ p(Y=\\text\{No\} | X=\\text\{Normal\}) \\ast \\log p(Y=\\text\{No\} | X=\\text\{Normal\}) \\\\ &= - ( \\frac\{6\}\{7\} \\ast \\log\_2 \\frac\{6\}\{7\} + \\frac\{1\}\{7\} \\ast \\log\_2 \\frac\{1\}\{7\} ) \\\\ &= 0.5917 \\end\{aligned\} H(Y∣X=Normal)=−y∈Y∑p(y)logp(y)=−(p(Y=Yes∣X=Normal)∗logp(Y=Yes∣X=Normal)\+p(Y=No∣X=Normal)∗logp(Y=No∣X=Normal)=−(76∗log276\+71∗log271)=0.5917

因此，  
 H ( Y ∣ X ) = ∑ x ∈ X p ( x ) H ( Y ∣ X = x ) = p ( X = High ) ∗ H ( Y ∣ X = High ) + p ( X = Normal ) ∗ H ( Y ∣ X = Normal ) = 1 2 ∗ 0.9852 + 1 2 ∗ 0.5917 = 0.7884 \\begin\{aligned\} H(Y|X) &= \\sum\_\{x \\in X\} p(x) H(Y|X=x) \\\\ &= p(X=\\text\{High\}) \\ast H(Y|X=\\text\{High\}) + p(X=\\text\{Normal\}) \\ast H(Y|X=\\text\{Normal\}) \\\\ &= \\frac\{1\}\{2\} \\ast 0.9852 + \\frac\{1\}\{2\} \\ast 0.5917 \\\\ &= 0.7884 \\end\{aligned\} H(Y∣X)=x∈X∑p(x)H(Y∣X=x)=p(X=High)∗H(Y∣X=High)\+p(X=Normal)∗H(Y∣X=Normal)=21∗0.9852\+21∗0.5917=0.7884

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# 3. 参考文章 #

1.  [什么是信息熵、条件熵和信息增益][Link 1]

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[Link 1]: https://www.jianshu.com/p/466ea79fbcde