beta函数与置信度估计

可信度的估计

• 二项分布中的\(p\) 服从Beta分布 $ {\rm beta}(\alpha, \beta)$, 密度函数 \(\frac1{B(\alpha, \beta)} x^{\alpha-1} (1-x)^{\beta -1}\)
• 均值 \(\frac \alpha {\alpha + \beta}\)
• 方差 \(\frac {\alpha \beta} {(\alpha+\beta)^2 (\alpha+ \beta + 1) } ?\)

from scipy.stats import beta

def confidence(n_bad, n_good, tol=2):
    ''' 返回估计的坏率p, 以及在tol倍标准差下的可信度'''
    a, b = n_bad+1, n_good+1
    p = a / (a+b)
    v = beta.std(a, b)
    up, low =  p + v*tol, p - v*tol
    up, low = min(up,1), max(0, low)
    d = beta.cdf(up, a,b) -  beta.cdf(low, a,b)
    return p, v, d



test_set = [
    (500, 20000, 2), 
    (1000, 200000, 2), 
    (2000, 200000, 2), 
    (5000, 200000, 2),
    (500,  100000, 2), 
    (1000, 100000, 2), 
    (2000, 100000, 2), 
    (5000, 100000, 2), 
    (2000, 10000, 2), 
]
for (n_bad, n_good, tol)  in  test_set:

    p,v,d = confidence(n_bad, n_good, tol)
    ss = ('bad = {}; total = {}; 均值p = {p:0.4f}; 标准差v = {v:0.6f}; 变异系数c = {c:0.3f}; 均值的相对误差为 {e:0.3f};'  
         + '均值落在区间[p - {t}v, p + {t}v]下的概率为{d:2.2f}%'
         ).format(n_bad, n_bad+n_good, p=p,v=v, c=v/p, d =d*100,t=tol, e=tol*v/p)
    print(ss)

bad = 500; total = 20500; 均值p = 0.0244; 标准差v = 0.001078; 变异系数c = 0.044; 均值的相对误差为 0.088;均值落在区间[p - 2v, p + 2v]下的概率为95.46%
bad = 1000; total = 201000; 均值p = 0.0050; 标准差v = 0.000157; 变异系数c = 0.032; 均值的相对误差为 0.063;均值落在区间[p - 2v, p + 2v]下的概率为95.46%
bad = 2000; total = 202000; 均值p = 0.0099; 标准差v = 0.000220; 变异系数c = 0.022; 均值的相对误差为 0.044;均值落在区间[p - 2v, p + 2v]下的概率为95.45%
bad = 5000; total = 205000; 均值p = 0.0244; 标准差v = 0.000341; 变异系数c = 0.014; 均值的相对误差为 0.028;均值落在区间[p - 2v, p + 2v]下的概率为95.45%
bad = 500; total = 100500; 均值p = 0.0050; 标准差v = 0.000222; 变异系数c = 0.045; 均值的相对误差为 0.089;均值落在区间[p - 2v, p + 2v]下的概率为95.46%
bad = 1000; total = 11000; 均值p = 0.0910; 标准差v = 0.002742; 变异系数c = 0.030; 均值的相对误差为 0.060;均值落在区间[p - 2v, p + 2v]下的概率为95.45%
bad = 2000; total = 102000; 均值p = 0.0196; 标准差v = 0.000434; 变异系数c = 0.022; 均值的相对误差为 0.044;均值落在区间[p - 2v, p + 2v]下的概率为95.45%
bad = 5000; total = 105000; 均值p = 0.0476; 标准差v = 0.000657; 变异系数c = 0.014; 均值的相对误差为 0.028;均值落在区间[p - 2v, p + 2v]下的概率为95.45%
bad = 2000; total = 12000; 均值p = 0.1667; 标准差v = 0.003402; 变异系数c = 0.020; 均值的相对误差为 0.041;均值落在区间[p - 2v, p + 2v]下的概率为95.45%

beta函数与置信度估计

原文：https://www.cnblogs.com/bregman/p/10510308.html