The Jensen Inequality

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  

                        （1）

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

              （2）

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

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  that are normalized（标准化） (). The general result（通式） is  

  　　　　(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）.