import numpy as np
from scipy.optimize import minimize
# 定义负对数似然函数
def negative_log_likelihood(params, data):
mu, sigma2 = params
n = len(data)
term1 = n / 2 * np.log(2 * np.pi * sigma2)
term2 = 1 / (2 * sigma2) * np.sum((data - mu) ** 2)
return term1 + term2
# 生成一些正态分布的数据
np.random.seed(0)
data = np.random.normal(0, 1, 100)
# 初始参数估计
initial_params = [np.mean(data), np.var(data)] # 注意：这里使用方差作为sigma2的初始估计
# 极大似然估计（通过最小化负对数似然函数）
result = minimize(negative_log_likelihood, initial_params, args=(data,))
# 输出结果
mu_est, sigma2_est = result.x
sigma_est = np.sqrt(sigma2_est)
print(f"Estimated mean: {mu_est}")
print(f"Estimated variance: {sigma2_est}")
print(f"Estimated standard deviation: {sigma_est}")
