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{{Short description|Measure of relative information in probability theory}}
 
{{Short description|Measure of relative information in probability theory}}
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{{Information theory}}
 
{{Information theory}}
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[[Image:Entropy-mutual-information-relative-entropy-relation-diagram.svg|thumb|256px|right|[[Venn diagram]] showing additive and subtractive relationships various [[Quantities of information|information measures]] associated with correlated variables <math>X</math> and <math>Y</math>. The area contained by both circles is the [[joint entropy]] <math>\Eta(X,Y)</math>. The circle on the left (red and violet) is the [[Entropy (information theory)|individual entropy]] <math>\Eta(X)</math>, with the red being the [[conditional entropy]] <math>\Eta(X|Y)</math>. The circle on the right (blue and violet) is <math>\Eta(Y)</math>, with the blue being <math>\Eta(Y|X)</math>. The violet is the [[mutual information]] <math>\operatorname{I}(X;Y)</math>.]]
 
[[Image:Entropy-mutual-information-relative-entropy-relation-diagram.svg|thumb|256px|right|[[Venn diagram]] showing additive and subtractive relationships various [[Quantities of information|information measures]] associated with correlated variables <math>X</math> and <math>Y</math>. The area contained by both circles is the [[joint entropy]] <math>\Eta(X,Y)</math>. The circle on the left (red and violet) is the [[Entropy (information theory)|individual entropy]] <math>\Eta(X)</math>, with the red being the [[conditional entropy]] <math>\Eta(X|Y)</math>. The circle on the right (blue and violet) is <math>\Eta(Y)</math>, with the blue being <math>\Eta(Y|X)</math>. The violet is the [[mutual information]] <math>\operatorname{I}(X;Y)</math>.]]
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[[Venn diagram showing additive and subtractive relationships various information measures associated with correlated variables <math>X</math> and <math>Y</math>. The area contained by both circles is the joint entropy <math>\Eta(X,Y)</math>. The circle on the left (red and violet) is the individual entropy <math>\Eta(X)</math>, with the red being the conditional entropy <math>\Eta(X|Y)</math>. The circle on the right (blue and violet) is <math>\Eta(Y)</math>, with the blue being <math>\Eta(Y|X)</math>. The violet is the mutual information <math>\operatorname{I}(X;Y)</math>.]]
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文恩图显示了相加和相减的关系，各种信息测量与相关变量相关。两个圆圈所包含的区域是联合熵。左边的圆圈(红色和紫色)代表个体熵。左边的圆圈代表个体熵。右边的圆圈代表个体熵。右边的圆圈代表个体熵。右边的圆圈代表个体熵。右边的圆圈代表个体熵。右边的圆圈代表个体熵。右边的圆圈代表个体熵。右边的圆圈代表个体熵。右边的圆圈代表个体熵。右边的圆圈代表个体。右边的圆圈(蓝色和紫色)是 < math > Eta (y) </math > ，蓝色的是 < math > Eta (y | x) </math > 。紫色是共同的信息[ math > 操作者名称{ i }(x; y) </math > ]
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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 <math>\Eta(Y|X)</math>.
 
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 <math>\Eta(Y|X)</math>.
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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 shannons, nats, or hartleys. The entropy of <math>Y</math> conditioned on <math>X</math> is written as <math>\Eta(Y|X)</math>.
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在信息论中，如果另一个随机变量的值是已知的，那么条件熵就会量化描述一个随机变量的结果所需的信息量。在这里，信息是用夏农、纳特斯或哈特利来衡量的。“数学”的熵取决于“数学” ，“ x”表示“数学” ，“埃塔”表示“数学”。
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== Definition ==
 
== Definition ==
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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
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The conditional entropy of <math>Y</math> given <math>X</math> is defined as
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给定的 x 条件熵被定义为
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{{Equation box 1
      
{{Equation box 1
 
{{Equation box 1
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{方程式方框1
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|indent =
 
|indent =
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2012年10月22日
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|title=
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|title=
 
|title=
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2012年10月11日
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|equation = {{NumBlk||<math>\Eta(Y|X)\ = -\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x,y)} {p(x)}</math>|{{EquationRef|Eq.1}}}}
 
|equation = {{NumBlk||<math>\Eta(Y|X)\ = -\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x,y)} {p(x)}</math>|{{EquationRef|Eq.1}}}}
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|equation = }}
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| equation = }
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|cellpadding= 6
 
|cellpadding= 6
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|cellpadding= 6
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6
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|border
 
|border
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|border
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边界
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|border colour = #0073CF
 
|border colour = #0073CF
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|border colour = #0073CF
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0073CF
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|background colour=#F5FFFA}}
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|background colour=#F5FFFA}}
 
|background colour=#F5FFFA}}
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5/fffa }}
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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>.
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where <math>\mathcal X</math> and <math>\mathcal Y</math> denote the support sets of <math>X</math> and <math>Y</math>.
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这里 < math > 数学 x </math > 和 < math > > 数学 y </math > 表示 < math > x </math > 和 < math > y </math > 的支持集。
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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? -->
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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>
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注意: 常规的表达式 < math > 0 log 0 </math > 和 < math > 0 log c/0 </math > 对于 fixed < math > c > 0 </math > 应该被视为等于零。这是因为 < math > lim { theta to0 ^ + } theta，log，c/theta = 0 </math > 和 < math > lim { theta to0 ^ + } theta，log theta = 0 </math >
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Intuitive explanation of the definition :  
 
Intuitive explanation of the definition :  
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The chain rule follows from the above definition of conditional entropy:
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链式规则遵循了上述条件熵的定义:
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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>.
 
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>.
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<math>\begin{align}
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1.1.1.2.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.4.3.3.3.3.3.3.3.3.3.3.3.4.3.3.3.3.3.3.3.3.3
      
== Motivation ==
 
== Motivation ==
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\Eta(Y|X) &= \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \left(\frac{p(x)}{p(x,y)} \right) \\[4pt]
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Eta (y | x) & = sum _ { x in mathcal x，y in mathcal y } p (x，y) log left (frac { p (x)}{ p (x，y)} right)[4 pt ]
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Let <math>\Eta(Y|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>\Eta(Y) := \mathbb{E}[\operatorname{I}(Y)]</math>, i.e.
 
Let <math>\Eta(Y|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>\Eta(Y) := \mathbb{E}[\operatorname{I}(Y)]</math>, i.e.
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&= \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)(\log (p(x))-\log (p(x,y))) \\[4pt]
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& = sum _ { x in mathcal x，y in mathcal y } p (x，y)(log (p (x))-log (p (x，y)))[4 pt ]
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&= -\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]
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& =-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))}[4 pt ]
      
:<math>\Eta(Y) = \sum_{y\in\mathcal Y} {\mathrm{Pr}(Y=y)\,\mathrm{I}(y)}  
 
:<math>\Eta(Y) = \sum_{y\in\mathcal Y} {\mathrm{Pr}(Y=y)\,\mathrm{I}(y)}  
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& = \Eta(X,Y) + \sum_{x \in \mathcal X} p(x)\log (p(x) ) \\[4pt]
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& = Eta (x，y) + sum _ { x in mathcal x } p (x) log (p (x))[4 pt ]
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= -\sum_{y\in\mathcal Y} {p_Y(y) \log_2{p_Y(y)}},</math>
 
= -\sum_{y\in\mathcal Y} {p_Y(y) \log_2{p_Y(y)}},</math>
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& = \Eta(X,Y) - \Eta(X).
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& = Eta (x，y)-Eta (x).
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\end{align}</math>
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结束{ align } </math >
      
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]]:  
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In general, a chain rule for multiple random variables holds:
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一般来说，多个随机变量的链式规则适用于:
      
:<math>\Eta(Y|X=x)
 
:<math>\Eta(Y|X=x)
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= -\sum_{y\in\mathcal Y} {\Pr(Y = y|X=x) \log_2{\Pr(Y = y|X=x)}}.</math>
 
= -\sum_{y\in\mathcal Y} {\Pr(Y = y|X=x) \log_2{\Pr(Y = y|X=x)}}.</math>
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<math> \Eta(X_1,X_2,\ldots,X_n) =
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< math > Eta (x1，x2，ldots，xn) =
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Note that <math>\Eta(Y|X)</math> is the result of averaging <math>\Eta(Y|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[ \Eta(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>\Eta(Y|X)</math> is the result of averaging <math>\Eta(Y|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[ \Eta(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>
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\sum_{i=1}^n \Eta(X_i | X_1, \ldots, X_{i-1}) </math>
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Sum { i = 1} ^ n Eta (x _ i | x _ 1，ldots，x _ { i-1}) </math >
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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>\Eta(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>\Eta(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}}
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It has a similar form to chain rule in probability theory, except that addition instead of multiplication is used.
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除了用加法代替乘法之外，它的形式与概率论的链式法则相似。
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:<math>
 
:<math>
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\begin{align}
 
\begin{align}
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Bayes' rule for conditional entropy states
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条件熵的贝叶斯规则
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\Eta(Y|X)\ &\equiv \sum_{x\in\mathcal X}\,p(x)\,\Eta(Y|X=x)\\
 
\Eta(Y|X)\ &\equiv \sum_{x\in\mathcal X}\,p(x)\,\Eta(Y|X=x)\\
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<math>\Eta(Y|X) \,=\, \Eta(X|Y) - \Eta(X) + \Eta(Y).</math>
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[数学] Eta (y | x) ，= ，Eta (x | y)-Eta (x) + Eta (y)
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& =-\sum_{x\in\mathcal X} p(x)\sum_{y\in\mathcal Y}\,p(y|x)\,\log\, p(y|x)\\
 
& =-\sum_{x\in\mathcal X} p(x)\sum_{y\in\mathcal Y}\,p(y|x)\,\log\, p(y|x)\\
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& =-\sum_{x\in\mathcal X}\sum_{y\in\mathcal Y}\,p(x,y)\,\log\,p(y|x)\\
 
& =-\sum_{x\in\mathcal X}\sum_{y\in\mathcal Y}\,p(x,y)\,\log\,p(y|x)\\
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Proof. <math>\Eta(Y|X) = \Eta(X,Y) - \Eta(X)</math> and <math>\Eta(X|Y) = \Eta(Y,X) - \Eta(Y)</math>. Symmetry entails <math>\Eta(X,Y) = \Eta(Y,X)</math>. Subtracting the two equations implies Bayes' rule.
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证据。Eta (y | x) = Eta (x，y)-Eta (x) | math > Eta (x | y) = Eta (y，x)-Eta (y).对称意味着 Eta (x，y) = Eta (y，x)。减去这两个方程就得到了贝叶斯定律。
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& =-\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\,p(y|x)\\
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& =-\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,y)} {p(x)}. \\
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If <math>Y</math> is conditionally independent of <math>Z</math> given <math>X</math> we have:
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如果[数学] y </math > 是条件独立于[数学] z </math > 给定 < 数学 > x </math > 我们有:
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& = \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x)} {p(x,y)}. \\
 
& = \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x)} {p(x,y)}. \\
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\end{align}
 
\end{align}
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<math>\Eta(Y|X,Z) \,=\, \Eta(Y|X).</math>
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[ math ] Eta (y | x，z) ，= ，Eta (y | x)
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</math>
 
</math>
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<!-- This paragraph is incorrect; the last line is not the KL divergence between any two distributions, since p(x) is [in general] not a valid distribution over the domains of X and Y. The last formula above is the [[Kullback-Leibler divergence]], also known as relative entropy. Relative entropy is always positive, and vanishes if and only if <math>p(x,y) = p(x)</math>. This is when knowing <math>x</math> tells us everything about <math>y</math>.  ADDED: Could this comment be out of date since the KL divergence is not mentioned above? November 2014 -->
 
<!-- This paragraph is incorrect; the last line is not the KL divergence between any two distributions, since p(x) is [in general] not a valid distribution over the domains of X and Y. The last formula above is the [[Kullback-Leibler divergence]], also known as relative entropy. Relative entropy is always positive, and vanishes if and only if <math>p(x,y) = p(x)</math>. This is when knowing <math>x</math> tells us everything about <math>y</math>.  ADDED: Could this comment be out of date since the KL divergence is not mentioned above? November 2014 -->
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For any <math>X</math> and <math>Y</math>:
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对于任意的 < math > x </math > 和 < math > > y </math > :
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<math display="block">\begin{align}
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(数学显示 = “ block” > begin { align })
      
==Properties==
 
==Properties==
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  \Eta(Y|X) &\le \Eta(Y) \, \\
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埃塔(y | x)及埃塔(y) ,
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===Conditional entropy equals zero===
 
===Conditional entropy equals zero===
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  \Eta(X,Y) &= \Eta(X|Y) + \Eta(Y|X) + \operatorname{I}(X;Y),\qquad \\
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eta (x，y) & = Eta (x | y) + Eta (y | x) + 操作数名{ i }(x; y) ，qquad
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<math>\Eta(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>\Eta(Y|X)=0</math> if and only if the value of <math>Y</math> is completely determined by the value of <math>X</math>.
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  \Eta(X,Y) &= \Eta(X) + \Eta(Y) - \operatorname{I}(X;Y),\, \\
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Eta (x，y) & = Eta (x) + Eta (y)-操作员名称{ i }(x; y) ，,
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  −
  \operatorname{I}(X;Y) &\le \Eta(X),\,
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  −
操作者名{ i }(x; y) & le Eta (x) ，,
      
===Conditional entropy of independent random variables===
 
===Conditional entropy of independent random variables===
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  −
\end{align}</math>
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  −
结束{ align } </math >
  −
   
Conversely, <math>\Eta(Y|X) = \Eta(Y)</math> if and only if <math>Y</math> and <math>X</math> are [[independent random variables]].
 
Conversely, <math>\Eta(Y|X) = \Eta(Y)</math> if and only if <math>Y</math> and <math>X</math> are [[independent random variables]].
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  −
  −
  −
where <math>\operatorname{I}(X;Y)</math> is the mutual information between <math>X</math> and <math>Y</math>.
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  −
其中，“数学”和“数学”之间的相互信息。
      
===Chain rule===
 
===Chain rule===
−
   
Assume that the combined system determined by two random variables <math>X</math> and <math>Y</math> has [[joint entropy]] <math>\Eta(X,Y)</math>, that is, we need <math>\Eta(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>\Eta(X)</math> bits of information. Once <math>X</math> is known, we only need <math>\Eta(X,Y)-\Eta(X)</math> bits to describe the state of the whole system. This quantity is exactly <math>\Eta(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>\Eta(X,Y)</math>, that is, we need <math>\Eta(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>\Eta(X)</math> bits of information. Once <math>X</math> is known, we only need <math>\Eta(X,Y)-\Eta(X)</math> bits to describe the state of the whole system. This quantity is exactly <math>\Eta(Y|X)</math>, which gives the ''chain rule'' of conditional entropy:
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For independent <math>X</math> and <math>Y</math>:
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对于独立的《数学》和《数学》 :
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:<math>\Eta(Y|X)\, = \, \Eta(X,Y)- \Eta(X).</math><ref name=cover1991 />{{rp|17}}
 
:<math>\Eta(Y|X)\, = \, \Eta(X,Y)- \Eta(X).</math><ref name=cover1991 />{{rp|17}}
−
  −
<math>\Eta(Y|X) = \Eta(Y) </math> and <math>\Eta(X|Y) = \Eta(X) \, </math>
  −
  −
Eta (y | x) = Eta (y) </math > and < math > Eta (x | y) = Eta (x) ，</math >
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  −
      
The chain rule follows from the above definition of conditional entropy:
 
The chain rule follows from the above definition of conditional entropy:
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  −
Although the specific-conditional entropy <math>\Eta(X|Y=y)</math> can be either less or greater than <math>\Eta(X)</math> for a given random variate <math>y</math> of <math>Y</math>, <math>\Eta(X|Y)</math> can never exceed <math>\Eta(X)</math>.
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虽然对于给定的随机变量来说，特定条件熵的 Eta (x | y = y) </math > </math > 可能比 </math > Eta (x) </math > </math > ，< math > Eta (x | y) </math > 不能超过 math > Eta (x) </math > 。
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:<math>\begin{align}  
 
:<math>\begin{align}  
−
   
\Eta(Y|X) &= \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \left(\frac{p(x)}{p(x,y)} \right) \\[4pt]
 
\Eta(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))-\log (p(x,y))) \\[4pt]
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  −
The above definition is for discrete random variables. The continuous version of discrete conditional entropy is called conditional differential (or continuous) entropy. Let <math>X</math> and <math>Y</math> be a continuous random variables with a joint probability density function <math>f(x,y)</math>. The differential conditional entropy <math>h(X|Y)</math> is defined as
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  −
上面的定义适用于离散随机变量。离散条件熵的连续形式称为条件微分(或连续)熵。设 x 是连续随机变量，f (x，y)是连续随机概率密度函数。微分条件熵 < math > h (x | y) </math > 被定义为
  −
   
  &= -\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]
 
  &= -\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]
−
   
  & = \Eta(X,Y) + \sum_{x \in \mathcal X} p(x)\log (p(x) ) \\[4pt]
 
  & = \Eta(X,Y) + \sum_{x \in \mathcal X} p(x)\log (p(x) ) \\[4pt]
−
  −
{{Equation box 1
  −
  −
{方程式方框1
  −
   
  & = \Eta(X,Y) - \Eta(X).  
 
  & = \Eta(X,Y) - \Eta(X).  
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  −
|indent =
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2012年10月22日
  −
   
\end{align}</math>
 
\end{align}</math>
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|title=
  −
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2012年10月11日
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  −
|equation = }}
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| equation = }
      
In general, a chain rule for multiple random variables holds:
 
In general, a chain rule for multiple random variables holds:
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|cellpadding= 6
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6
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|border
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边界
      
:<math> \Eta(X_1,X_2,\ldots,X_n) =
 
:<math> \Eta(X_1,X_2,\ldots,X_n) =
−
  −
|border colour = #0073CF
  −
  −
0073CF
  −
   
  \sum_{i=1}^n \Eta(X_i | X_1, \ldots, X_{i-1}) </math><ref name=cover1991 />{{rp|22}}
 
  \sum_{i=1}^n \Eta(X_i | X_1, \ldots, X_{i-1}) </math><ref name=cover1991 />{{rp|22}}
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  −
|background colour=#F5FFFA}}
  −
  −
5/fffa }}
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  −
      
It has a similar form to [[Chain rule (probability)|chain rule]] in probability theory, except that addition instead of multiplication is used.
 
It has a similar form to [[Chain rule (probability)|chain rule]] in probability theory, except that addition instead of multiplication is used.
−
  −
  −
  −
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反，条件微分熵可能是负的。
      
===Bayes' rule===
 
===Bayes' rule===
−
   
[[Bayes' rule]] for conditional entropy states
 
[[Bayes' rule]] for conditional entropy states
−
  −
As in the discrete case there is a chain rule for differential entropy:
  −
  −
在离散情况下，微分熵有一个链式规则:
  −
   
:<math>\Eta(Y|X) \,=\, \Eta(X|Y) - \Eta(X) + \Eta(Y).</math>
 
:<math>\Eta(Y|X) \,=\, \Eta(X|Y) - \Eta(X) + \Eta(Y).</math>
−
  −
<math>h(Y|X)\,=\,h(X,Y)-h(X)</math>
  −
  −
H (y | x) ，= ，h (x，y)-h (x)
  −
  −
  −
  −
Notice however that this rule may not be true if the involved differential entropies do not exist or are infinite.
  −
  −
然而，请注意，如果所涉及的微分熵不存在或者是无限的，那么这个规则可能不成立。
      
''Proof.'' <math>\Eta(Y|X) = \Eta(X,Y) - \Eta(X)</math> and <math>\Eta(X|Y) = \Eta(Y,X) - \Eta(Y)</math>. Symmetry entails <math>\Eta(X,Y) = \Eta(Y,X)</math>. Subtracting the two equations implies Bayes' rule.
 
''Proof.'' <math>\Eta(Y|X) = \Eta(X,Y) - \Eta(X)</math> and <math>\Eta(X|Y) = \Eta(Y,X) - \Eta(Y)</math>. Symmetry entails <math>\Eta(X,Y) = \Eta(Y,X)</math>. Subtracting the two equations implies Bayes' rule.
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  −
  −
  −
Joint differential entropy is also used in the definition of the mutual information between continuous random variables:
  −
  −
联合微分熵也用于连续随机变量之间互信息的定义:
      
If <math>Y</math> is [[Conditional independence|conditionally independent]] of <math>Z</math> given <math>X</math> we have:
 
If <math>Y</math> is [[Conditional independence|conditionally independent]] of <math>Z</math> given <math>X</math> we have:
−
  −
<math>\operatorname{I}(X,Y)=h(X)-h(X|Y)=h(Y)-h(Y|X)</math>
  −
  −
(x，y) = h (x)-h (x | y) = h (y)-h (y | x) </math >
  −
  −
      
:<math>\Eta(Y|X,Z) \,=\, \Eta(Y|X).</math>
 
:<math>\Eta(Y|X,Z) \,=\, \Eta(Y|X).</math>
−
  −
<math>h(X|Y) \le h(X)</math> with equality if and only if <math>X</math> and <math>Y</math> are independent.
  −
  −
当且仅当 < math > x </math > 和 < math > y </math > 是独立的。
  −
  −
      
===Other properties===
 
===Other properties===
−
   
For any <math>X</math> and <math>Y</math>:
 
For any <math>X</math> and <math>Y</math>:
−
  −
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:
  −
  −
条件微分熵对估计量的期望平方误差产生一个下限。对于任何一个随机变量，观察值 < math > y </math > 和估计值 < math > widedhat { x } </math > ，下面是:
  −
   
:<math display="block">\begin{align}
 
:<math display="block">\begin{align}
−
  −
<math display="block">\mathbb{E}\left[\bigl(X - \widehat{X}{(Y)}\bigr)^2\right]
  −
  −
< math display = " block" > mathbb { e } left [ bigl (x-widehat { x }{(y)} bigr) ^ 2 right ]
  −
   
   \Eta(Y|X) &\le \Eta(Y) \, \\
 
   \Eta(Y|X) &\le \Eta(Y) \, \\
−
  −
\ge \frac{1}{2\pi e}e^{2h(X|Y)}</math>
  −
  −
1}{2 pi e } e ^ {2 h (x | y)} </math >
  −
   
   \Eta(X,Y) &= \Eta(X|Y) + \Eta(Y|X) + \operatorname{I}(X;Y),\qquad \\
 
   \Eta(X,Y) &= \Eta(X|Y) + \Eta(Y|X) + \operatorname{I}(X;Y),\qquad \\
−
   
   \Eta(X,Y) &= \Eta(X) + \Eta(Y) - \operatorname{I}(X;Y),\, \\
 
   \Eta(X,Y) &= \Eta(X) + \Eta(Y) - \operatorname{I}(X;Y),\, \\
−
  −
This is related to the uncertainty principle from quantum mechanics.
  −
  −
这与量子力学的不确定性原理有关。
  −
   
   \operatorname{I}(X;Y) &\le \Eta(X),\,
 
   \operatorname{I}(X;Y) &\le \Eta(X),\,
−
   
\end{align}</math>
 
\end{align}</math>
−
  −
  −
  −
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，条件熵被推广为条件量子熵。后者可以采取负值，不像它的古典对应物。
      
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>.
−
  −
      
For independent <math>X</math> and <math>Y</math>:
 
For independent <math>X</math> and <math>Y</math>:
−
  −
      
:<math>\Eta(Y|X) = \Eta(Y) </math> and <math>\Eta(X|Y) = \Eta(X) \, </math>
 
:<math>\Eta(Y|X) = \Eta(Y) </math> and <math>\Eta(X|Y) = \Eta(X) \, </math>
−
  −
      
Although the specific-conditional entropy <math>\Eta(X|Y=y)</math> can be either less or greater than <math>\Eta(X)</math> for a given [[random variate]] <math>y</math> of <math>Y</math>, <math>\Eta(X|Y)</math> can never exceed <math>\Eta(X)</math>.
 
Although the specific-conditional entropy <math>\Eta(X|Y=y)</math> can be either less or greater than <math>\Eta(X)</math> for a given [[random variate]] <math>y</math> of <math>Y</math>, <math>\Eta(X|Y)</math> can never exceed <math>\Eta(X)</math>.
−
  −
      
== Conditional differential entropy ==
 
== Conditional differential entropy ==
−
   
=== Definition ===
 
=== Definition ===
−
   
The above definition is for discrete random variables. The continuous version of discrete conditional entropy is called ''conditional differential (or continuous) entropy''. Let <math>X</math> and <math>Y</math> be a continuous random variables with a [[joint probability density function]] <math>f(x,y)</math>. The differential conditional entropy <math>h(X|Y)</math> is defined as<ref name=cover1991 />{{rp|249}}
 
The above definition is for discrete random variables. The continuous version of discrete conditional entropy is called ''conditional differential (or continuous) entropy''. Let <math>X</math> and <math>Y</math> be a continuous random variables with a [[joint probability density function]] <math>f(x,y)</math>. The differential conditional entropy <math>h(X|Y)</math> is defined as<ref name=cover1991 />{{rp|249}}
    +
{{Equation box 1
 +
|indent =
 +
|title=
 +
|equation = {{NumBlk||<math>h(X|Y) = -\int_{\mathcal X, \mathcal Y} f(x,y)\log f(x|y)\,dx dy</math>|{{EquationRef|Eq.2}}}}
 +
|cellpadding= 6
 +
|border
 +
|border colour = #0073CF
 +
|background colour=#F5FFFA}}
    +
=== Properties ===
 +
In contrast to the conditional entropy for discrete random variables, the conditional differential entropy may be negative.
   −
{{Equation box 1
+
As in the discrete case there is a chain rule for differential entropy:
 +
:<math>h(Y|X)\,=\,h(X,Y)-h(X)</math><ref name=cover1991 />{{rp|253}}
 +
Notice however that this rule may not be true if the involved differential entropies do not exist or are infinite.
   −
Category:Entropy and information
+
Joint differential entropy is also used in the definition of the [[mutual information]] between continuous random variables:
 +
:<math>\operatorname{I}(X,Y)=h(X)-h(X|Y)=h(Y)-h(Y|X)</math>
   −
类别: 熵和信息
+
<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}}
   −
|indent =
+
===Relation to estimator error===
 +
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}}
 +
:<math display="block">\mathbb{E}\left[\bigl(X - \widehat{X}{(Y)}\bigr)^2\right]
 +
\ge \frac{1}{2\pi e}e^{2h(X|Y)}</math>
   −
Category:Information theory
+
This is related to the [[uncertainty principle]] from [[quantum mechanics]].
   −
范畴: 信息论
+
==Generalization to quantum theory==
 +
In [[quantum information theory]], the conditional entropy is generalized to the [[conditional quantum entropy]]. The latter can take negative values, unlike its classical counterpart.
   −
<noinclude>
+
== See also ==
 +
* [[Entropy (information theory)]]
 +
* [[Mutual information]]
 +
* [[Conditional quantum entropy]]
 +
* [[Variation of information]]
 +
* [[Entropy power inequality]]
 +
* [[Likelihood function]]
   −
<small>This page was moved from [[wikipedia:en:Conditional entropy]]. Its edit history can be viewed at [[条件熵/edithistory]]</small></noinclude>
+
==References==
 +
{{Reflist}}
   −
[[Category:待整理页面]]
+
[[Category:Entropy and information]]
 +
[[Category:Information theory]]
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