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→Definition
== Definition ==
== Definition ==
The conditional entropy of <math>Y</math> given <math>X</math> is defined as
The conditional entropy of <math>Y</math> given <math>X</math> is defined as
在给定<math>X</math>的情况下，<math>Y</math>的条件熵定义为：
{{Equation box 1
{{Equation box 1
|border colour = #0073CF
|border colour = #0073CF
|background colour=#F5FFFA}}
|background colour=#F5FFFA}}
where <math>\mathcal X</math> and <math>\mathcal Y</math> denote the [[Support (mathematics)|support sets]] of <math>X</math> and <math>Y</math>.
where <math>\mathcal X</math> and <math>\mathcal Y</math> denote the [[Support (mathematics)|support sets]] of <math>X</math> and <math>Y</math>.
其中<math>\mathcal X</math>和<math>\mathcal Y</math>表示<math>X</math>和<math>Y</math>的支撑集。
''Note:'' It is conventioned that the expressions <math>0 \log 0</math> and <math>0 \log c/0</math> for fixed <math>c > 0</math> should be treated as being equal to zero. This is because <math>\lim_{\theta\to0^+} \theta\, \log \,c/\theta = 0</math> and <math>\lim_{\theta\to0^+} \theta\, \log \theta = 0</math><ref>{{Cite web|url=http://www.inference.org.uk/mackay/itprnn/book.html|title=David MacKay: Information Theory, Pattern Recognition and Neural Networks: The Book|website=www.inference.org.uk|access-date=2019-10-25}}</ref> <!-- because p(x,y) could still equal 0 even if p(x) != 0 and p(y) != 0. What about p(x,y)=p(x)=0? -->
''Note:'' It is conventioned that the expressions <math>0 \log 0</math> and <math>0 \log c/0</math> for fixed <math>c > 0</math> should be treated as being equal to zero. This is because <math>\lim_{\theta\to0^+} \theta\, \log \,c/\theta = 0</math> and <math>\lim_{\theta\to0^+} \theta\, \log \theta = 0</math><ref>{{Cite web|url=http://www.inference.org.uk/mackay/itprnn/book.html|title=David MacKay: Information Theory, Pattern Recognition and Neural Networks: The Book|website=www.inference.org.uk|access-date=2019-10-25}}</ref> <!-- because p(x,y) could still equal 0 even if p(x) != 0 and p(y) != 0. What about p(x,y)=p(x)=0? -->
注意：在约定<math>c > 0</math>始终成立时，表达式<math>0 \log 0</math>和<math>0 \log c/0</math>视为等于零。这是因为<math>\lim_{\theta\to0^+} \theta\, \log \,c/\theta = 0</math>，而且<math>\lim_{\theta\to0^+} \theta\, \log \theta = 0</math>><ref>{{Cite web|url=http://www.inference.org.uk/mackay/itprnn/book.html|title=David MacKay: Information Theory, Pattern Recognition and Neural Networks: The Book|website=www.inference.org.uk|access-date=2019-10-25}}</ref> <!-- because p(x,y) could still equal 0 even if p(x) != 0 and p(y) != 0. What about p(x,y)=p(x)=0? -->
Intuitive explanation of the definition :
Intuitive explanation of the definition :