更改

添加112字节 、 2020年12月3日 (四) 19:10
无编辑摘要
第1行: 第1行: −
此词条由Jie翻译。
+
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}}
第8行: 第8行:  
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>。
+
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>。
      第32行: 第32行:  
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。
+
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>。
      第38行: 第38行:  
''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? -->
+
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? -->
      第44行: 第44行:  
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。
+
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。
      第51行: 第51行:  
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>。
+
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>。
      第60行: 第60行:  
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:
+
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:
      第69行: 第69行:  
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>
+
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>
      第75行: 第75行:  
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}}
+
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}}
      第97行: 第97行:  
<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>。
+
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>。
      第104行: 第104行:  
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>。
+
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>。
      第111行: 第111行:  
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:
+
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:
      第185行: 第185行:  
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。
+
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>。
      第199行: 第199行:  
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>。
+
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>。
      第237行: 第237行:  
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。
+
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。
      第243行: 第243行:  
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:
+
联合微分熵也用于定义连续随机变量之间的互信息:
      第251行: 第251行:  
<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。
+
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 ===
+
=== 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}}
   第287行: 第287行:     
* '''<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>'''
+
* '''<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>'''
+
* '''<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>'''
19

个编辑