差分隐私入门——拉普拉斯分布

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$Laplace分布的概率密度函数:$
$p(x)=\frac{1}{2\lambda}e^{-\frac{|x-\mu|}{\lambda}}，一般取\mu=0，函数形式如：$
$p(x)=\frac{1}{2\lambda}e^{-\frac{|x|}{\lambda}}，又称为双指数函数分布。$
$标准Laplace分布的均值为0，方差为{2\lambda^2}，几种\lambda下其概率分布图如下：$

import matplotlib.pyplot as plt
import numpy as np
def laplace_function(x, lambda_):
    return (1/(2*lambda_)) * np.e**(-1*(np.abs(x)/lambda_))
x = np.linspace(-5,5,10000)
y1 = [laplace_function(x_,1) for x_ in x]
y2 = [laplace_function(x_,2) for x_ in x]
y3 = [laplace_function(x_,0.5) for x_ in x]

plt.plot(x, y1, color='r', label="lambda:1")
plt.plot(x, y2, color='g', label="lambda:2")
plt.plot(x, y3, color='b', label="lambda:0.5")

plt.title("Laplace distributions")
plt.legend()
plt.show()

$np.random.laplace可以获得拉普拉斯分布的随机值，参数主要如下：$
$loc：就是上面的\mu，控制偏移。$
$scale： 就是上面的\lambda，控制缩放。$
$size： 是产生数据的个数。$
$python生成laplace分布直方图：$

import numpy as np
#print(np.random.laplace(0,1,10)) 生成10个样本
laplace1 = np.random.laplace(0, 1, 10000)
laplace2 = np.random.laplace(0, 2, 10000)

import matplotlib.pyplot as plt
fig, (ax1, ax2) = plt.subplots(1,2, sharex=True, sharey=True)
ax1.hist(laplace1,bins=1000, label="lambda:1")
ax1.legend()

ax2.hist(laplace2, bins=1000, label="lambda:2")
ax2.legend()
plt.show()