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== Motivation ==

== Motivation ==

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 H（Y）:=E[I（Y）, 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H（Y）:=E[I（Y）。

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>。

Note that<math> H（Y ǀ X）</math> is the result of averaging H（Y ǀ X = x） 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H（Y ǀ X = x）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>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>