“条件熵”的版本间的差异

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此词条由Jie翻译。
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此词条由Jie翻译。由Lincent审校。
  
 
{{Short description|Measure of relative information in probability theory}}
 
{{Short description|Measure of relative information in probability theory}}
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In [[information theory]], the '''conditional entropy''' quantifies the amount of information needed to describe the outcome of a [[random variable]] <math>Y</math> given that the value of another random variable <math>X</math> is known. Here, information is measured in [[Shannon (unit)|shannon]]s, [[Nat (unit)|nat]]s, or [[Hartley (unit)|hartley]]s. The ''entropy of <math>Y</math> conditioned on <math>X</math>'' is written as H(X ǀ Y).
 
In [[information theory]], the '''conditional entropy''' quantifies the amount of information needed to describe the outcome of a [[random variable]] <math>Y</math> given that the value of another random variable <math>X</math> is known. Here, information is measured in [[Shannon (unit)|shannon]]s, [[Nat (unit)|nat]]s, or [[Hartley (unit)|hartley]]s. The ''entropy of <math>Y</math> conditioned on <math>X</math>'' is written as H(X ǀ Y).
  
在'''<font color="#ff8000"> 信息论Information theory</font>'''中,假设随机变量<math>X</math>的值已知,那么'''<font color="#ff8000"> 条件熵Conditional entropy</font>'''则用于去量化描述随机变量<math>Y</math>结果所需的信息量。此时,信息以'''<font color="#ff8000"> 香农Shannon </font>''','''<font color="#ff8000"> 奈特nat</font>'''或'''<font color="#ff8000"> 哈特莱hartley</font>'''来衡量。以<math>X</math>为条件的<math>Y</math>熵写为<math>H(X ǀ Y)</math>。
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在'''<font color="#ff8000"> 信息论Information theory</font>'''中,假设随机变量<math>X</math>的值已知,那么'''<font color="#ff8000"> 条件熵Conditional entropy</font>'''则用于去定量描述随机变量<math>Y</math>表示的信息量。此时,信息以'''<font color="#ff8000"> 香农Shannon </font>''','''<font color="#ff8000"> 奈特nat</font>'''或'''<font color="#ff8000"> 哈特莱hartley</font>'''来衡量。已知<math>X</math>的条件下<math>Y</math>的熵记为<math>H(X ǀ Y)</math>。
  
  
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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。
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其中<math>\mathcal X</math>和<math>\mathcal Y</math>表示<math>X</math>和<math>Y</math>的<font color="#32cd32">支撑集</font>。
  
  
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''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? -->
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注意: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? -->
  
  
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Intuitive explanation of the definition : According to the definition, <math>\displaystyle H( Y|X) =\mathbb{E}( \ f( X,Y) \ )</math> where <math>\displaystyle f:( x,y) \ \rightarrow -\log( \ p( y|x) \ ) .</math> <math>\displaystyle f</math> associates to  <math>\displaystyle ( x,y)</math> the information content of <math>\displaystyle ( Y=y)</math> given <math>\displaystyle (X=x)</math>, which is the amount of information needed to describe the event <math>\displaystyle (Y=y)</math> given <math>(X=x)</math>.  According to the law of large numbers, <math>\displaystyle H(Y|X)</math> is the arithmetic mean of a large number of independent realizations of <math>\displaystyle f(X,Y)</math>.
 
Intuitive explanation of the definition : According to the definition, <math>\displaystyle H( Y|X) =\mathbb{E}( \ f( X,Y) \ )</math> where <math>\displaystyle f:( x,y) \ \rightarrow -\log( \ p( y|x) \ ) .</math> <math>\displaystyle f</math> associates to  <math>\displaystyle ( x,y)</math> the information content of <math>\displaystyle ( Y=y)</math> given <math>\displaystyle (X=x)</math>, which is the amount of information needed to describe the event <math>\displaystyle (Y=y)</math> given <math>(X=x)</math>.  According to the law of large numbers, <math>\displaystyle H(Y|X)</math> is the arithmetic mean of a large number of independent realizations of <math>\displaystyle f(X,Y)</math>.
  
对该定义的直观解释是:根据定义<math>\displaystyle H( Y|X) =\mathbb{E}( \ f( X,Y) \ )</math>,其中<math>\displaystyle f:( x,y) \ \rightarrow -\log( \ p( y|x) \ ) </math>. <math>\displaystyle f</math>将给定<math>\displaystyle (X=x)</math><math>\displaystyle ( Y=y)</math>的信息内容与<math>\displaystyle ( x,y)</math>相关联,这是描述在给定<math>(X=x)</math>条件下的事件<math>\displaystyle (Y=y)</math>所需的信息量。根据大数定律,<math>H(Y ǀ X)</math><math>\displaystyle f(X,Y)</math>的大量独立实现的算术平均值。
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对该定义的直观解释是:根据定义<math>\displaystyle H( Y|X) =\mathbb{E}( \ f( X,Y) \ )</math>,其中<math>\displaystyle f:( x,y) \ \rightarrow -\log( \ p( y|x) \ ) </math>. <math>\displaystyle f</math>将给定<math>\displaystyle (X=x)</math>时的<math>\displaystyle ( Y=y)</math>的信息内容与<math>\displaystyle ( x,y)</math>相关联,这是描述在给定<math>(X=x)</math>条件下的事件<math>\displaystyle (Y=y)</math>所需的信息量。根据大数定律,<math>H(Y ǀ X)</math>是大量<math>\displaystyle f(X,Y)</math>独立实验结果的算术平均值。
  
  
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Let <math>H(Y ǀ X = x)</math> be the [[Shannon Entropy|entropy]] of the discrete random variable <math>Y</math> conditioned on the discrete random variable <math>X</math> taking a certain value <math>x</math>. Denote the support sets of <math>X</math> and <math>Y</math> by <math>\mathcal X</math> and <math>\mathcal Y</math>. Let <math>Y</math> have [[probability mass function]] <math>p_Y{(y)}</math>. The unconditional entropy of <math>Y</math> is calculated as <math>H(Y):=E[I(Y)</math>, i.e.
 
Let <math>H(Y ǀ X = x)</math> be the [[Shannon Entropy|entropy]] of the discrete random variable <math>Y</math> conditioned on the discrete random variable <math>X</math> taking a certain value <math>x</math>. Denote the support sets of <math>X</math> and <math>Y</math> by <math>\mathcal X</math> and <math>\mathcal Y</math>. Let <math>Y</math> have [[probability mass function]] <math>p_Y{(y)}</math>. The unconditional entropy of <math>Y</math> is calculated as <math>H(Y):=E[I(Y)</math>, i.e.
  
设<math>H(Y ǀ X = x)</math>为离散随机变量<math>Y</math>的熵,条件是离散随机变量<math>X</math>取一定值<math>x</math>。用<math>\mathcal X</math>和<math>\mathcal Y</math>表示<math>X</math>和<math>Y</math>的支撑集。令<math>Y</math>具有概率质量函数<math>p_Y{(y)}</math>。<math>Y</math>的无条件熵计算为<math>H(Y):=E[I(Y)</math>。
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设<math>H(Y ǀ X = x)</math>为离散随机变量<math>Y</math>在离散随机变量<math>X</math>取定值<math>x</math>时的熵。用<math>\mathcal X</math>和<math>\mathcal Y</math>表示<math>X</math>和<math>Y</math>的支撑集。令<math>Y</math>的概率密度函数为<math>p_Y{(y)}</math>。<math>Y</math>的无条件熵计算为<math>H(Y):=E[I(Y)</math>。
  
  
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where <math>\operatorname{I}(y_i)</math> is the [[information content]] of the [[Outcome (probability)|outcome]] of <math>Y</math> taking the value <math>y_i</math>. The entropy of <math>Y</math> conditioned on <math>X</math> taking the value <math>x</math> is defined analogously by [[conditional expectation]]:  
 
where <math>\operatorname{I}(y_i)</math> is the [[information content]] of the [[Outcome (probability)|outcome]] of <math>Y</math> taking the value <math>y_i</math>. The entropy of <math>Y</math> conditioned on <math>X</math> taking the value <math>x</math> is defined analogously by [[conditional expectation]]:  
  
这里当取值为<math>y_i</math>时,<math>\operatorname{I}(y_i)</math>是其结果<math>Y</math>的信息内容。类似地以<math>X</math>为条件的<math>Y</math>的熵,当值为<math>x</math>时,也可以通过条件期望来定义:
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当<math>Y</math>取值为<math>y_i</math>时,<math>\operatorname{I}(y_i)</math>是其结果<math>Y</math>的信息内容。类似地,当<math>X</math>值为<math>x</math>时以<math>X</math>为条件的<math>Y</math>的熵,也可以通过条件期望来定义:
  
  
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Note that<math> H(Y ǀ X)</math> is the result of averaging <math>H(Y ǀ X = x)</math> over all possible values <math>x</math> that <math>X</math> may take. Also, if the above sum is taken over a sample <math>y_1, \dots, y_n</math>, the expected value <math>E_X[ H(y_1, \dots, y_n \mid X = x)]</math> is known in some domains as '''equivocation'''.<ref>{{cite journal|author1=Hellman, M.|author2=Raviv, J.|year=1970|title=Probability of error, equivocation, and the Chernoff bound|journal=IEEE Transactions on Information Theory|volume=16|issue=4|pp=368-372}}</ref>
 
Note that<math> H(Y ǀ X)</math> is the result of averaging <math>H(Y ǀ X = x)</math> over all possible values <math>x</math> that <math>X</math> may take. Also, if the above sum is taken over a sample <math>y_1, \dots, y_n</math>, the expected value <math>E_X[ H(y_1, \dots, y_n \mid X = x)]</math> is known in some domains as '''equivocation'''.<ref>{{cite journal|author1=Hellman, M.|author2=Raviv, J.|year=1970|title=Probability of error, equivocation, and the Chernoff bound|journal=IEEE Transactions on Information Theory|volume=16|issue=4|pp=368-372}}</ref>
  
注意,<math> H(Y ǀ X)</math>是在<math>X</math>可能取的所有可能值<math>x</math>上对<math>H(Y ǀ X = x)</math>求平均值的结果。同样,如果将上述总和接管到样本<math>y_1, \dots, y_n</math>上,则预期值<math>E_X[ H(y_1, \dots, y_n \mid X = x)]</math>在某些领域中会变得模糊。<ref>{{cite journal|author1=Hellman, M.|author2=Raviv, J.|year=1970|title=Probability of error, equivocation, and the Chernoff bound|journal=IEEE Transactions on Information Theory|volume=16|issue=4|pp=368-372}}</ref>
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注意,<math> H(Y ǀ X)</math>是在<math>X</math>可能取的所有可能值<math>x</math>时对<math>H(Y ǀ X = x)</math>求平均值的结果。同样,如果上述和取自样本<math>y_1, \dots, y_n</math>上,则期望值<math>E_X[ H(y_1, \dots, y_n \mid X = x)]</math><font color="#32cd32"> 在某些领域中认为是模糊值</font>。<ref>{{cite journal|author1=Hellman, M.|author2=Raviv, J.|year=1970|title=Probability of error, equivocation, and the Chernoff bound|journal=IEEE Transactions on Information Theory|volume=16|issue=4|pp=368-372}}</ref>
  
  
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Given [[Discrete random variable|discrete random variables]] <math>X</math> with image <math>\mathcal X</math> and <math>Y</math> with image <math>\mathcal Y</math>, the conditional entropy of <math>Y</math> given <math>X</math> is defined as the weighted sum of <math>H(Y|X=x)</math> for each possible value of <math>x</math>, using  <math>p(x)</math> as the weights:<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}
 
Given [[Discrete random variable|discrete random variables]] <math>X</math> with image <math>\mathcal X</math> and <math>Y</math> with image <math>\mathcal Y</math>, the conditional entropy of <math>Y</math> given <math>X</math> is defined as the weighted sum of <math>H(Y|X=x)</math> for each possible value of <math>x</math>, using  <math>p(x)</math> as the weights:<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}
  
给定具有像<math>\mathcal X</math>的离散随机变量<math>X</math>和具有像<math>\mathcal Y</math>的离散随机变量<math>Y</math>,将给定<math>X</math>的<math>Y</math>的条件熵定义为<math>H(Y|X=x)</math>的权重之和,以<math>x</math>的每个可能值为准,并使用<math>p(x)</math>作为权重,其表达式如下:<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}
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给定具有像<math>\mathcal X</math>的离散随机变量<math>X</math>和具有像<math>\mathcal Y</math>的离散随机变量<math>Y</math>,将给定<math>X</math>的<math>Y</math>的条件熵定义为以<math>p(x)</math>作为权重,对<math>x</math>的每个可能取值得到的<math>H(Y|X=x)</math>的加权和。其表达式如下:<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}
  
  
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<math>H(Y|X)=0</math> if and only if the value of <math>Y</math> is completely determined by the value of <math>X</math>.
 
<math>H(Y|X)=0</math> if and only if the value of <math>Y</math> is completely determined by the value of <math>X</math>.
  
当且仅当<math>Y</math>的值完全由<math>X</math>的值确定时,才为<math>H(Y|X)=0</math>。
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当且仅当<math>Y</math>的值完全由<math>X</math>的值确定时,<math>H(Y|X)=0</math>。
  
  
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Conversely, <math>H(Y|X) = H(Y)</math> if and only if <math>Y</math> and <math>X</math> are [[independent random variables]].
 
Conversely, <math>H(Y|X) = H(Y)</math> if and only if <math>Y</math> and <math>X</math> are [[independent random variables]].
  
相反,当且仅当<math>Y</math>和<math>X</math>是独立随机变量时,则为<math>H(Y|X) =H(Y)</math>。
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相反,当且仅当<math>Y</math>和<math>X</math>是互相独立的随机变量时,则<math>H(Y|X) =H(Y)</math>。
  
  
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Assume that the combined system determined by two random variables <math>X</math> and <math>Y</math> has [[joint entropy]] <math>H(X,Y)</math>, that is, we need <math>H(X,Y)</math> bits of information on average to describe its exact state. Now if we first learn the value of <math>X</math>, we have gained <math>H(X)</math> bits of information. Once <math>X</math> is known, we only need <math>H(X,Y)-H(X)</math> bits to describe the state of the whole system. This quantity is exactly <math>H(Y|X)</math>, which gives the ''chain rule'' of conditional entropy:
 
Assume that the combined system determined by two random variables <math>X</math> and <math>Y</math> has [[joint entropy]] <math>H(X,Y)</math>, that is, we need <math>H(X,Y)</math> bits of information on average to describe its exact state. Now if we first learn the value of <math>X</math>, we have gained <math>H(X)</math> bits of information. Once <math>X</math> is known, we only need <math>H(X,Y)-H(X)</math> bits to describe the state of the whole system. This quantity is exactly <math>H(Y|X)</math>, which gives the ''chain rule'' of conditional entropy:
  
假设由两个随机变量<math>X</math>和<math>Y</math>确定的组合系统具有联合熵<math>H(X,Y)</math>,也就是说,我们通常需要<math>H(X,Y)</math>位信息来描述其确切状态。现在,如果我们首先获得<math>X</math>的值,我们将知晓<math>H(X)</math>位信息。一旦知道了<math>X</math>的值,我们就可以通过<math>H(X,Y)</math>-<math>H(X)</math>位来描述整个系统的状态。这个数量恰好是<math>H(Y|X)</math>,它给出了条件熵的链式法则:
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假设由两个随机变量<math>X</math>和<math>Y</math>确定的组合系统具有联合熵<math>H(X,Y)</math>,也就是说,我们通常需要<math>H(X,Y)</math>位信息来描述其确切状态。现在,如果我们首先尝试获得<math>X</math>的值,我们将知晓<math>H(X)</math>位信息。一旦<math>X</math>的值确定,我们就可以通过<math>H(X,Y)</math>-<math>H(X)</math>位来描述整个系统的状态。这个数量恰好是<math>H(Y|X)</math>,它给出了条件熵的链式法则:
  
  
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where <math>\operatorname{I}(X;Y)</math> is the [[mutual information]] between <math>X</math> and <math>Y</math>.
 
where <math>\operatorname{I}(X;Y)</math> is the [[mutual information]] between <math>X</math> and <math>Y</math>.
  
其中<math>\operatorname{I}(X;Y)</math>是<math>X</math>和<math>Y</math>之间的相互信息。
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其中<math>\operatorname{I}(X;Y)</math>是<math>X</math>和<math>Y</math>之间的<font color="#ff8000"> 互信息</font>。
  
  
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Although the specific-conditional entropy <math>H(X|Y=y)</math> can be either less or greater than <math>H(X)</math> for a given [[random variate]] <math>y</math> of <math>Y</math>, <math>H(X|Y)</math> can never exceed <math>H(X)</math>.
 
Although the specific-conditional entropy <math>H(X|Y=y)</math> can be either less or greater than <math>H(X)</math> for a given [[random variate]] <math>y</math> of <math>Y</math>, <math>H(X|Y)</math> can never exceed <math>H(X)</math>.
  
尽管对于给定的<math>Y</math>随机变量<math>y</math>,特定条件熵<math>H(X|Y=y)</math>可以小于或大于<math>H(X)</math>,但<math>H(X|Y)</math>永远不会超过<math>H(X)</math>。
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对于给定随机变量<math>Y</math>的值<math>y</math>,尽管特定条件熵<math>H(X|Y=y)</math>可以小于或大于<math>H(X)</math>,但<math>H(X|Y)</math>永远不会超过<math>H(X)</math>。
  
  
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Notice however that this rule may not be true if the involved differential entropies do not exist or are infinite.
 
Notice however that this rule may not be true if the involved differential entropies do not exist or are infinite.
  
但是请注意,如果所涉及的微分熵不存在或无限,则此规则可能不成立。
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但是请注意,如果所涉及的微分熵不存在或无限,则此法则可能不成立。
  
  
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Joint differential entropy is also used in the definition of the [[mutual information]] between continuous random variables:
 
Joint differential entropy is also used in the definition of the [[mutual information]] between continuous random variables:
  
联合微分熵也用于定义连续随机变量之间的交互信息:
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联合微分熵也用于定义连续随机变量之间的互信息:
  
  
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<math>h(X|Y) \le h(X)</math> with equality if and only if <math>X</math> and <math>Y</math> are independent.<ref name=cover1991 />{{rp|253}}
 
<math>h(X|Y) \le h(X)</math> with equality if and only if <math>X</math> and <math>Y</math> are independent.<ref name=cover1991 />{{rp|253}}
  
当且仅当X和Y是独立的时,<math>h(X|Y) \le h(X)</math>才相等。
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当且仅当X和Y是独立的,<math>h(X|Y) \le h(X)</math>等号成立。
  
  
  
=== Relation to estimator error 与预估误差的关系 ===
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=== Relation to estimator error 与估计量误差的关系 ===
 
The conditional differential entropy yields a lower bound on the expected squared error of an [[estimator]]. For any random variable <math>X</math>, observation <math>Y</math> and estimator <math>\widehat{X}</math> the following holds:<ref name=cover1991 />{{rp|255}}
 
The conditional differential entropy yields a lower bound on the expected squared error of an [[estimator]]. For any random variable <math>X</math>, observation <math>Y</math> and estimator <math>\widehat{X}</math> the following holds:<ref name=cover1991 />{{rp|255}}
  
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* '''<font color="#ff8000"> 熵(信息论)Entropy (information theory)</font>'''
 
* '''<font color="#ff8000"> 熵(信息论)Entropy (information theory)</font>'''
* '''<font color="#ff8000"> 交互信息Mutual information</font>'''
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* '''<font color="#ff8000"> 互信息Mutual information</font>'''
 
* '''<font color="#ff8000"> 条件量子熵Conditional quantum entropy</font>'''
 
* '''<font color="#ff8000"> 条件量子熵Conditional quantum entropy</font>'''
* '''<font color="#ff8000"> 信息变差Variation of information</font>'''
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* '''<font color="#ff8000"> 信息差异Variation of information</font>'''
 
* '''<font color="#ff8000"> 熵幂不等式Entropy power inequality</font>'''
 
* '''<font color="#ff8000"> 熵幂不等式Entropy power inequality</font>'''
 
* '''<font color="#ff8000"> 似然函数Likelihood function</font>'''
 
* '''<font color="#ff8000"> 似然函数Likelihood function</font>'''

2020年12月3日 (四) 19:10的版本

此词条由Jie翻译。由Lincent审校。

模板:Information theory

该图表示在变量X、Y相关联的各种信息量之间,进行加减关系的维恩图。两个圆所包含的区域是联合熵H(X,Y)。左侧的圆圈(红色和紫色)是单个熵H(X),红色是条件熵H(X ǀ Y)。右侧的圆圈(蓝色和紫色)为H(Y),蓝色为H(Y ǀ X)。中间紫色的是相互信息i(X; Y)。

In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable [math]\displaystyle{ Y }[/math] given that the value of another random variable [math]\displaystyle{ X }[/math] is known. Here, information is measured in shannons, nats, or hartleys. The entropy of [math]\displaystyle{ Y }[/math] conditioned on [math]\displaystyle{ X }[/math] is written as H(X ǀ Y).

信息论Information theory中,假设随机变量[math]\displaystyle{ X }[/math]的值已知,那么 条件熵Conditional entropy则用于去定量描述随机变量[math]\displaystyle{ Y }[/math]表示的信息量。此时,信息以 香农Shannon 奈特nat 哈特莱hartley来衡量。已知[math]\displaystyle{ X }[/math]的条件下[math]\displaystyle{ Y }[/math]的熵记为[math]\displaystyle{ H(X ǀ Y) }[/math]


Definition 定义

The conditional entropy of [math]\displaystyle{ Y }[/math] given [math]\displaystyle{ X }[/math] is defined as

在给定[math]\displaystyle{ X }[/math]的情况下,[math]\displaystyle{ Y }[/math]的条件熵定义为:


[math]\displaystyle{ \Eta(Y|X)\ = -\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x,y)} {p(x)} }[/math]

 

 

 

 

(Eq.1)


where [math]\displaystyle{ \mathcal X }[/math] and [math]\displaystyle{ \mathcal Y }[/math] denote the support sets of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math].

其中[math]\displaystyle{ \mathcal X }[/math][math]\displaystyle{ \mathcal Y }[/math]表示[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]支撑集


Note: It is conventioned that the expressions [math]\displaystyle{ 0 \log 0 }[/math] and [math]\displaystyle{ 0 \log c/0 }[/math] for fixed [math]\displaystyle{ c \gt 0 }[/math] should be treated as being equal to zero. This is because [math]\displaystyle{ \lim_{\theta\to0^+} \theta\, \log \,c/\theta = 0 }[/math] and [math]\displaystyle{ \lim_{\theta\to0^+} \theta\, \log \theta = 0 }[/math][1]

注意:约定[math]\displaystyle{ c \gt 0 }[/math]始终成立时,表达式[math]\displaystyle{ 0 \log 0 }[/math][math]\displaystyle{ 0 \log c/0 }[/math]视为等于零。这是因为[math]\displaystyle{ \lim_{\theta\to0^+} \theta\, \log \,c/\theta = 0 }[/math],而且[math]\displaystyle{ \lim_{\theta\to0^+} \theta\, \log \theta = 0 }[/math]>[2]


Intuitive explanation of the definition : According to the definition, [math]\displaystyle{ \displaystyle H( Y|X) =\mathbb{E}( \ f( X,Y) \ ) }[/math] where [math]\displaystyle{ \displaystyle f:( x,y) \ \rightarrow -\log( \ p( y|x) \ ) . }[/math] [math]\displaystyle{ \displaystyle f }[/math] associates to [math]\displaystyle{ \displaystyle ( x,y) }[/math] the information content of [math]\displaystyle{ \displaystyle ( Y=y) }[/math] given [math]\displaystyle{ \displaystyle (X=x) }[/math], which is the amount of information needed to describe the event [math]\displaystyle{ \displaystyle (Y=y) }[/math] given [math]\displaystyle{ (X=x) }[/math]. According to the law of large numbers, [math]\displaystyle{ \displaystyle H(Y|X) }[/math] is the arithmetic mean of a large number of independent realizations of [math]\displaystyle{ \displaystyle f(X,Y) }[/math].

对该定义的直观解释是:根据定义[math]\displaystyle{ \displaystyle H( Y|X) =\mathbb{E}( \ f( X,Y) \ ) }[/math],其中[math]\displaystyle{ \displaystyle f:( x,y) \ \rightarrow -\log( \ p( y|x) \ ) }[/math]. [math]\displaystyle{ \displaystyle f }[/math]将给定[math]\displaystyle{ \displaystyle (X=x) }[/math]时的[math]\displaystyle{ \displaystyle ( Y=y) }[/math]的信息内容与[math]\displaystyle{ \displaystyle ( x,y) }[/math]相关联,这是描述在给定[math]\displaystyle{ (X=x) }[/math]条件下的事件[math]\displaystyle{ \displaystyle (Y=y) }[/math]所需的信息量。根据大数定律,[math]\displaystyle{ H(Y ǀ X) }[/math]是大量[math]\displaystyle{ \displaystyle f(X,Y) }[/math]独立实验结果的算术平均值。


Motivation 动机

Let [math]\displaystyle{ H(Y ǀ X = x) }[/math] be the entropy of the discrete random variable [math]\displaystyle{ Y }[/math] conditioned on the discrete random variable [math]\displaystyle{ X }[/math] taking a certain value [math]\displaystyle{ x }[/math]. Denote the support sets of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] by [math]\displaystyle{ \mathcal X }[/math] and [math]\displaystyle{ \mathcal Y }[/math]. Let [math]\displaystyle{ Y }[/math] have probability mass function [math]\displaystyle{ p_Y{(y)} }[/math]. The unconditional entropy of [math]\displaystyle{ Y }[/math] is calculated as [math]\displaystyle{ H(Y):=E[I(Y) }[/math], i.e.

[math]\displaystyle{ H(Y ǀ X = x) }[/math]为离散随机变量[math]\displaystyle{ Y }[/math]在离散随机变量[math]\displaystyle{ X }[/math]取定值[math]\displaystyle{ x }[/math]时的熵。用[math]\displaystyle{ \mathcal X }[/math][math]\displaystyle{ \mathcal Y }[/math]表示[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]的支撑集。令[math]\displaystyle{ Y }[/math]的概率密度函数为[math]\displaystyle{ p_Y{(y)} }[/math][math]\displaystyle{ Y }[/math]的无条件熵计算为[math]\displaystyle{ H(Y):=E[I(Y) }[/math]


[math]\displaystyle{ H(Y) = \sum_{y\in\mathcal Y} {\mathrm{Pr}(Y=y)\,\mathrm{I}(y)} = -\sum_{y\in\mathcal Y} {p_Y(y) \log_2{p_Y(y)}}, }[/math]


where [math]\displaystyle{ \operatorname{I}(y_i) }[/math] is the information content of the outcome of [math]\displaystyle{ Y }[/math] taking the value [math]\displaystyle{ y_i }[/math]. The entropy of [math]\displaystyle{ Y }[/math] conditioned on [math]\displaystyle{ X }[/math] taking the value [math]\displaystyle{ x }[/math] is defined analogously by conditional expectation:

[math]\displaystyle{ Y }[/math]取值为[math]\displaystyle{ y_i }[/math]时,[math]\displaystyle{ \operatorname{I}(y_i) }[/math]是其结果[math]\displaystyle{ Y }[/math]的信息内容。类似地,当[math]\displaystyle{ X }[/math]值为[math]\displaystyle{ x }[/math]时以[math]\displaystyle{ X }[/math]为条件的[math]\displaystyle{ Y }[/math]的熵,也可以通过条件期望来定义:


[math]\displaystyle{ H(Y|X=x) = -\sum_{y\in\mathcal Y} {\Pr(Y = y|X=x) \log_2{\Pr(Y = y|X=x)}}. }[/math]


Note that[math]\displaystyle{ H(Y ǀ X) }[/math] is the result of averaging [math]\displaystyle{ H(Y ǀ X = x) }[/math] over all possible values [math]\displaystyle{ x }[/math] that [math]\displaystyle{ X }[/math] may take. Also, if the above sum is taken over a sample [math]\displaystyle{ y_1, \dots, y_n }[/math], the expected value [math]\displaystyle{ E_X[ H(y_1, \dots, y_n \mid X = x)] }[/math] is known in some domains as equivocation.[3]

注意,[math]\displaystyle{ H(Y ǀ X) }[/math]是在[math]\displaystyle{ X }[/math]可能取的所有可能值[math]\displaystyle{ x }[/math]时对[math]\displaystyle{ H(Y ǀ X = x) }[/math]求平均值的结果。同样,如果上述和取自样本[math]\displaystyle{ y_1, \dots, y_n }[/math]上,则期望值[math]\displaystyle{ E_X[ H(y_1, \dots, y_n \mid X = x)] }[/math] 在某些领域中认为是模糊值[4]


Given discrete random variables [math]\displaystyle{ X }[/math] with image [math]\displaystyle{ \mathcal X }[/math] and [math]\displaystyle{ Y }[/math] with image [math]\displaystyle{ \mathcal Y }[/math], the conditional entropy of [math]\displaystyle{ Y }[/math] given [math]\displaystyle{ X }[/math] is defined as the weighted sum of [math]\displaystyle{ H(Y|X=x) }[/math] for each possible value of [math]\displaystyle{ x }[/math], using [math]\displaystyle{ p(x) }[/math] as the weights:[5]:15

给定具有像[math]\displaystyle{ \mathcal X }[/math]的离散随机变量[math]\displaystyle{ X }[/math]和具有像[math]\displaystyle{ \mathcal Y }[/math]的离散随机变量[math]\displaystyle{ Y }[/math],将给定[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]的条件熵定义为以[math]\displaystyle{ p(x) }[/math]作为权重,对[math]\displaystyle{ x }[/math]的每个可能取值得到的[math]\displaystyle{ H(Y|X=x) }[/math]的加权和。其表达式如下:[5]:15


[math]\displaystyle{ \begin{align} H(Y|X)\ &\equiv \sum_{x\in\mathcal X}\,p(x)\,H(Y|X=x)\\ & =-\sum_{x\in\mathcal X} p(x)\sum_{y\in\mathcal Y}\,p(y|x)\,\log\, p(y|x)\\ & =-\sum_{x\in\mathcal X}\sum_{y\in\mathcal Y}\,p(x,y)\,\log\,p(y|x)\\ & =-\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log\,p(y|x)\\ & =-\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x,y)} {p(x)}. \\ & = \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x)} {p(x,y)}. \\ \end{align} }[/math]



Properties 属性

Conditional entropy equals zero 条件熵等于零

[math]\displaystyle{ H(Y|X)=0 }[/math] if and only if the value of [math]\displaystyle{ Y }[/math] is completely determined by the value of [math]\displaystyle{ X }[/math].

当且仅当[math]\displaystyle{ Y }[/math]的值完全由[math]\displaystyle{ X }[/math]的值确定时,[math]\displaystyle{ H(Y|X)=0 }[/math]


Conditional entropy of independent random variables 独立随机变量的条件熵

Conversely, [math]\displaystyle{ H(Y|X) = H(Y) }[/math] if and only if [math]\displaystyle{ Y }[/math] and [math]\displaystyle{ X }[/math] are independent random variables.

相反,当且仅当[math]\displaystyle{ Y }[/math][math]\displaystyle{ X }[/math]是互相独立的随机变量时,则[math]\displaystyle{ H(Y|X) =H(Y) }[/math]


Chain rule 链式法则

Assume that the combined system determined by two random variables [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] has joint entropy [math]\displaystyle{ H(X,Y) }[/math], that is, we need [math]\displaystyle{ H(X,Y) }[/math] bits of information on average to describe its exact state. Now if we first learn the value of [math]\displaystyle{ X }[/math], we have gained [math]\displaystyle{ H(X) }[/math] bits of information. Once [math]\displaystyle{ X }[/math] is known, we only need [math]\displaystyle{ H(X,Y)-H(X) }[/math] bits to describe the state of the whole system. This quantity is exactly [math]\displaystyle{ H(Y|X) }[/math], which gives the chain rule of conditional entropy:

假设由两个随机变量[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]确定的组合系统具有联合熵[math]\displaystyle{ H(X,Y) }[/math],也就是说,我们通常需要[math]\displaystyle{ H(X,Y) }[/math]位信息来描述其确切状态。现在,如果我们首先尝试获得[math]\displaystyle{ X }[/math]的值,我们将知晓[math]\displaystyle{ H(X) }[/math]位信息。一旦[math]\displaystyle{ X }[/math]的值确定,我们就可以通过[math]\displaystyle{ H(X,Y) }[/math]-[math]\displaystyle{ H(X) }[/math]位来描述整个系统的状态。这个数量恰好是[math]\displaystyle{ H(Y|X) }[/math],它给出了条件熵的链式法则:


[math]\displaystyle{ H(Y|X)\, = \, H(X,Y)- H(X). }[/math][5]:17


The chain rule follows from the above definition of conditional entropy:

链式法则遵循以上条件熵的定义:


[math]\displaystyle{ \begin{align} H(Y|X) &= \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \left(\frac{p(x)}{p(x,y)} \right) \\[4pt] &= \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)(\log (p(x))-\log (p(x,y))) \\[4pt] &= -\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log (p(x,y)) + \sum_{x\in\mathcal X, y\in\mathcal Y}{p(x,y)\log(p(x))} \\[4pt] & = H(X,Y) + \sum_{x \in \mathcal X} p(x)\log (p(x) ) \\[4pt] & = H(X,Y) - H(X). \end{align} }[/math]


In general, a chain rule for multiple random variables holds:

通常情况下,多个随机变量的链式法则表示为:


[math]\displaystyle{ H(X_1,X_2,\ldots,X_n) = \sum_{i=1}^n H(X_i | X_1, \ldots, X_{i-1}) }[/math][5]:22


It has a similar form to chain rule in probability theory, except that addition instead of multiplication is used.

除了使用加法而不是乘法之外,它具有与概率论中的链式法则类似的形式。


Bayes' rule 贝叶斯法则

Bayes' rule for conditional entropy states

条件熵状态的贝叶斯法则


[math]\displaystyle{ H(Y|X) \,=\, H(X|Y) - H(X) + H(Y). }[/math]


Proof. [math]\displaystyle{ H(Y|X) = H(X,Y) - H(X) }[/math] and [math]\displaystyle{ H(X|Y) = H(Y,X) - H(Y) }[/math]. Symmetry entails [math]\displaystyle{ H(X,Y) = H(Y,X) }[/math]. Subtracting the two equations implies Bayes' rule.

证明,[math]\displaystyle{ H(Y|X) = H(X,Y) - H(X) }[/math][math]\displaystyle{ H(X|Y) = H(Y,X) - H(Y) }[/math]。对称性要求[math]\displaystyle{ H(X,Y) = H(Y,X) }[/math]。将两个方程式相减就意味着贝叶斯定律。


If [math]\displaystyle{ Y }[/math] is conditionally independent of [math]\displaystyle{ Z }[/math] given [math]\displaystyle{ X }[/math] we have:

如果给定[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]有条件地独立于[math]\displaystyle{ Z }[/math],则我们有:


[math]\displaystyle{ H(Y|X,Z) \,=\, H(Y|X). }[/math]


Other properties 其他性质

For any [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math]:

对于任何[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]


[math]\displaystyle{ \begin{align} H(Y|X) &\le H(Y) \, \\ H(X,Y) &= H(X|Y) + H(Y|X) + \operatorname{I}(X;Y),\qquad \\ H(X,Y) &= H(X) + H(Y) - \operatorname{I}(X;Y),\, \\ \operatorname{I}(X;Y) &\le H(X),\, \end{align} }[/math]

where [math]\displaystyle{ \operatorname{I}(X;Y) }[/math] is the mutual information between [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math].

其中[math]\displaystyle{ \operatorname{I}(X;Y) }[/math][math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]之间的 互信息


For independent [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math]:

对于独立的X和Y:


[math]\displaystyle{ H(Y|X) = H(Y) }[/math] and [math]\displaystyle{ H(X|Y) = H(X) \, }[/math]


Although the specific-conditional entropy [math]\displaystyle{ H(X|Y=y) }[/math] can be either less or greater than [math]\displaystyle{ H(X) }[/math] for a given random variate [math]\displaystyle{ y }[/math] of [math]\displaystyle{ Y }[/math], [math]\displaystyle{ H(X|Y) }[/math] can never exceed [math]\displaystyle{ H(X) }[/math].

对于给定随机变量[math]\displaystyle{ Y }[/math]的值[math]\displaystyle{ y }[/math],尽管特定条件熵[math]\displaystyle{ H(X|Y=y) }[/math]可以小于或大于[math]\displaystyle{ H(X) }[/math],但[math]\displaystyle{ H(X|Y) }[/math]永远不会超过[math]\displaystyle{ H(X) }[/math]


Conditional differential entropy 条件微分熵

Definition 定义

The above definition is for discrete random variables. The continuous version of discrete conditional entropy is called conditional differential (or continuous) entropy. Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be a continuous random variables with a joint probability density function [math]\displaystyle{ f(x,y) }[/math]. The differential conditional entropy [math]\displaystyle{ h(X|Y) }[/math] is defined as[5]:249

上面的定义是针对离散随机变量的。离散条件熵的连续形式称为 条件微分(或连续)熵Conditional differential (or continuous) entropy 。 令[math]\displaystyle{ X }[/math][math]\displaystyle{ Y }[/math]为具有联合概率密度函数[math]\displaystyle{ f(x,y) }[/math]的连续随机变量。则微分条件熵[math]\displaystyle{ h(X|Y) }[/math]定义为:[5]:249

[math]\displaystyle{ h(X|Y) = -\int_{\mathcal X, \mathcal Y} f(x,y)\log f(x|y)\,dx dy }[/math]

 

 

 

 

(Eq.2)


Properties 属性

In contrast to the conditional entropy for discrete random variables, the conditional differential entropy may be negative.

与离散随机变量的条件熵相比,条件微分熵可能为负。


As in the discrete case there is a chain rule for differential entropy:

与离散情况一样,微分熵也有链式法则:


[math]\displaystyle{ h(Y|X)\,=\,h(X,Y)-h(X) }[/math][5]:253


Notice however that this rule may not be true if the involved differential entropies do not exist or are infinite.

但是请注意,如果所涉及的微分熵不存在或无限,则此法则可能不成立。


Joint differential entropy is also used in the definition of the mutual information between continuous random variables:

联合微分熵也用于定义连续随机变量之间的互信息:


[math]\displaystyle{ \operatorname{I}(X,Y)=h(X)-h(X|Y)=h(Y)-h(Y|X) }[/math]


[math]\displaystyle{ h(X|Y) \le h(X) }[/math] with equality if and only if [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are independent.[5]:253

当且仅当X和Y是独立的,[math]\displaystyle{ h(X|Y) \le h(X) }[/math]等号成立。


Relation to estimator error 与估计量误差的关系

The conditional differential entropy yields a lower bound on the expected squared error of an estimator. For any random variable [math]\displaystyle{ X }[/math], observation [math]\displaystyle{ Y }[/math] and estimator [math]\displaystyle{ \widehat{X} }[/math] the following holds:[5]:255

条件微分熵在估计量的期望平方误差上有一个下限。对于任何随机变量[math]\displaystyle{ X }[/math],观察值[math]\displaystyle{ Y }[/math]和估计量[math]\displaystyle{ \widehat{X} }[/math],以下条件成立:


[math]\displaystyle{ \mathbb{E}\left[\bigl(X - \widehat{X}{(Y)}\bigr)^2\right] \ge \frac{1}{2\pi e}e^{2h(X|Y)} }[/math]


This is related to the uncertainty principle from quantum mechanics.

这与来自量子力学的不确定性原理有关。


Generalization to quantum theory 量子理论泛化

In quantum information theory, the conditional entropy is generalized to the conditional quantum entropy. The latter can take negative values, unlike its classical counterpart.

在量子信息论中,条件熵被广义化为条件量子熵。后者可以采用负值,这与经典方法不同。


See also 其他参考资料

  • 熵(信息论)Entropy (information theory)
  • 互信息Mutual information
  • 条件量子熵Conditional quantum entropy
  • 信息差异Variation of information
  • 熵幂不等式Entropy power inequality
  • 似然函数Likelihood function


References 参考文献

  1. "David MacKay: Information Theory, Pattern Recognition and Neural Networks: The Book". www.inference.org.uk. Retrieved 2019-10-25.
  2. "David MacKay: Information Theory, Pattern Recognition and Neural Networks: The Book". www.inference.org.uk. Retrieved 2019-10-25.
  3. Hellman, M.; Raviv, J. (1970). "Probability of error, equivocation, and the Chernoff bound". IEEE Transactions on Information Theory. 16 (4): 368–372.
  4. Hellman, M.; Raviv, J. (1970). "Probability of error, equivocation, and the Chernoff bound". IEEE Transactions on Information Theory. 16 (4): 368–372.
  5. 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 T. Cover; J. Thomas (1991). Elements of Information Theory. ISBN 0-471-06259-6. https://archive.org/details/elementsofinform0000cove.