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KL散度
KL 散度是一种衡量两个概率分布的匹配程度的指标，两个分布差异越大，KL散度越大。
定义：
其中 p(x) 是真实分布，q(x)是目标分布（建模目标分布），如果两个分布完全匹配，那么
第一组：数据集为采集100个人的年龄，为以下表格所示，我们使用KL散度去研究最符合的分布类型。
age012345678910总数
count3671113181511754100
尝试1：使用均匀分布建立模型
可视化为：黄色的为建立的目标均匀分布模型，与蓝色真实分布的对比。
尝试2：使用高斯分布建立模型
可视化为：（红色虚线是相同与拟合的一条正态分布曲线，蓝色条形图为概率密度）：
计算分析 ：
如何判断真实分布更接近均匀分布还是高斯分布，用肉眼看是很不准确的，用KL散度来衡量真实分布去匹配目标分布所损失的信息量。所以能够将模型量化去比较从而判断出接近哪种分布。
1、计算与均匀分布的KL散度：
import numpy as np import math count=np.array([3,6,7,11,13,18,15,11,7,5,4])
count_rate=count/100 balance_rate=1/11 sum=0 for i in range(11):
sum+=count_rate[i]*math.log(count_rate[i]/balance_rate) print(sum)
计算结果为：0.12899493763053263
2、计算与高斯分布的KL散度：
def gaosi(x): mu=5.03 sigma=2.4349743325135895
t1=1/(sigma*math.sqrt(2*math.pi)) t2=((x-mu)**2)/(2*sigma*sigma) return
math.exp(-1*t2)*t1 count=np.array([3,6,7,11,13,18,15,11,7,5,4])
count_rate=count/100 sum=0 for i in range(11):
sum+=count_rate[i]*math.log(count_rate[i]/gaosi(i)) print(sum)
计算结果为：0.03997441345364968
结论：
在只考虑均匀分布模型与高斯分布模型的情况下，用本身去拟合目标模型时，匹配高斯分布所损失的信息量最少，计算发现该数据集的分布更符合高斯分布。
