The Jensen Inequality

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原文链接：http://www.cnblogs.com/kevinGaoblog/archive/2012/03/30/2425986.html

本文探讨了具有正二阶导数的凸函数，并通过不等式解释了函数值与其加权平均之间的关系。文中利用具体例子x²进行了图解说明，并推广到更多数值的加权平均情况。

Let（设） f be a function with a positive second derivative（二阶导数）. Such a function is called “convex"（凸的，注意是向下凸，向上凹，如 x²） and satisfies the inequality

$\begin{displaymath} {f(a)\ +\ f(b)\over 2}\ -\ f\left( {a + b\over 2}\right)\quad \geq \quad 0\end{displaymath}$ （1）

inequation (1) relates a function of an average to an average of the function. The average can be weighted, for example,

$\begin{displaymath} { \frac{1}{3} \, f(a)\ +\ \frac{2}{3} \, f(b)}\ -\ f\left( { \frac{1}{3} a + \frac{2}{3} b}\right) \quad \geq \quad 0\end{displaymath}$ （2）

Figure 1 is a graphical interpretation of inequation (2) for the function f=x².

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There is nothing special about f=x², except that it is convex. Given three numbers a, b, and c, the inequality (2) can first be applied to a and b, and then to c and the average of a and b. Thus, recursively, an inequality like (2) can be built for a weighted average（加权平均数） of three or more numbers. Define weights $w_j \geq 0$ that are normalized（标准化） ( $\textstyle {\sum_j} w_j = 1$ ). The general result（通式） is

$\begin{displaymath} S(p_j) \eq \sum_{j=1}^N w_j f(p_j)\ -\ f\left( \sum_{j=1}^N w_j p_j \right) \quad \geq \quad 0\end{displaymath}$ 　　　　(3)

If all the p_j are the same, then both of the two terms in S are the same, and S vanishes. Hence, minimizing S is like urging all the p_j to be identical（完全一样）. Equilibrium is when S is reduced to the smallest possible value which satisfies any constraints that may be applicable. The function S defined by (3) is like the entropy（熵） defined in thermodynamics（热力学）.

转载于:https://www.cnblogs.com/kevinGaoblog/archive/2012/03/30/2425986.html