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第54行： − {{Equation box 1 − |indent = − |title= − |equation = − <math> − I(X;Y) = D_{\mathrm{KL}}( P_{(X,Y)} \| P_{X} \otimes P_{Y} ) − </math> − |cellpadding= 1 − |border − |border colour = #0073CF − |background colour=#F5FFFA}} − Notice, as per property of the [[Kullback–Leibler divergence]], that <math>I(X;Y)</math> is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when <math>X</math> and <math>Y</math> are independent (and hence observing <math>Y</math> tells you nothing about <math>X</math>). In general <math>I(X;Y)</math> is non-negative, it is a measure of the price for encoding <math>(X,Y)</math> as a pair of independent random variables, when in reality they are not. + − {{Equation box 1 − {{Equation box 1 − {方程式方框1+ − − |indent = − − |indent = − − 不会有事的 − − |title= − − |title= − − 标题 − − |equation = − − |equation = − − 方程式 − − <math> − − <math> − − 数学 − − I(X;Y) = D_{\mathrm{KL}}( P_{(X,Y)} \| P_{X} \otimes P_{Y} ) − − I(X;Y) = D_{\mathrm{KL}}( P_{(X,Y)} \| P_{X} \otimes P_{Y} ) − − I (x; y) d { mathrm { KL }(p {(x，y)} | p { x }乘 p { y }) − − </math> − − </math> − − 数学 − − |cellpadding= 1 − − |cellpadding= 1 − − 1号牢房 − − |border − − |border − − 边界 − − |border colour = #0073CF − − |border colour = #0073CF − − 0073CF − − |background colour=#F5FFFA}} − − |background colour=#F5FFFA}} − − 5 / fffa } − − where <math>D_{\mathrm{KL}}</math> is the [[Kullback–Leibler divergence]]. − − where <math>D_{\mathrm{KL}}</math> is the Kullback–Leibler divergence. − − 其中数学 d (mathrm { KL } / math)是 Kullback-Leibler 分歧。 − − − − Notice, as per property of the Kullback–Leibler divergence, that <math>I(X;Y)</math> is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when <math>X</math> and <math>Y</math> are independent (and hence observing <math>Y</math> tells your nothing about <math>X</math>). In general <math>I(X;Y)</math> is non-negative, it is a measure of the price for encoding <math>(X,Y)</math> as a pair of independent random variables, when in reality they are not. − +
→定义 Definition
where <math>D_{\mathrm{KL}}</math> is the [[Kullback–Leibler divergence]].
where <math>D_{\mathrm{KL}}</math> is the [[Kullback–Leibler divergence]].
其中<math>D_{\mathrm{KL}}</math>表示Kullback-Leibler散度。
Notice, as per property of the [[Kullback–Leibler divergence]], that <math>I(X;Y)</math> is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when <math>X</math> and <math>Y</math> are independent (and hence observing <math>Y</math> tells you nothing about <math>X</math>). In general <math>I(X;Y)</math> is non-negative, it is a measure of the price for encoding <math>(X,Y)</math> as a pair of independent random variables, when in reality they are not.
Notice, as per property of the [[Kullback–Leibler divergence]], that <math>I(X;Y)</math> is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when <math>X</math> and <math>Y</math> are independent (and hence observing <math>Y</math> tells your nothing about <math>X</math>). In general <math>I(X;Y)</math> is non-negative, it is a measure of the price for encoding <math>(X,Y)</math> as a pair of independent random variables, when in reality they are not.
注意，根据 Kullback-Leibler 散度的性质，当联合分布与边际乘积重合时，math i (x; y) / math 恰好等于零，即。当数学 x / 数学和数学 y / 数学是独立的(因此观察数学 y / 数学并不能告诉你数学 x / 数学)。在一般的数学 i (x; y) / math 是非负的，它是一种测量方法，用于将 math (x，y) / math 作为一对独立的随机变量进行编码，而实际上它们并不是。
注意，根据 Kullback-Leibler 散度的性质，当联合分布与边际乘积重合时，<math>I(X;Y)</math> 恰好等于零，即。当数学 x / 数学和数学 y / 数学是独立的(因此观察数学 y / 数学并不能告诉你数学 x / 数学)。在一般的数学 i (x; y) / math 是非负的，它是一种测量方法，用于将 math (x，y) / math 作为一对独立的随机变量进行编码，而实际上它们并不是。
== 关于离散分布的PMF In terms of PMFs for discrete distributions ==
== 关于离散分布的PMF In terms of PMFs for discrete distributions ==