程序初版完成，目前收敛后结果不太对
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## 实现一下重要抽样
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import numpy as np
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import matplotlib.pyplot as plt
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import scipy as stats
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import seaborn as sns
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from scipy.stats import beta as beta_dist # 导入beta分布用于绘图
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np.random.seed(42)
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# ------------------------------------------------------------------
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# 1. 设置 matplotlib 支持中文显示
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# ------------------------------------------------------------------
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try:
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plt.rcParams['font.sans-serif'] = ['SimHei'] # Windows/Linux
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plt.rcParams['axes.unicode_minus'] = False # 正常显示负号
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except Exception:
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try:
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plt.rcParams['font.sans-serif'] = ['Arial Unicode MS'] # MacOS
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plt.rcParams['axes.unicode_minus'] = False
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except Exception:
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print("未找到中文字体，绘图可能显示异常。请安装'SimHei'或'Arial Unicode MS'字体。")
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def baysian_linear_regression_inportance_sampling():
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# ------------------------------------------------------------------
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# 2. 设定模型参数 (为了模拟您图中的效果)
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# ------------------------------------------------------------------
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n = 20 # X的试验总次数
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# 更改 alpha 和 beta 以匹配图中 ~0.72 的均值
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alpha = 8.0 # Beta分布的先验参数 alpha
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beta = 3.0 # Beta分布的先验参数 beta
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# 理论均值 E[Y] = alpha / (alpha + beta) = 8 / 11 ≈ 0.727
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theoretical_mean = alpha / (alpha + beta)
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# MCMC (Gibbs) 抽样参数
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N_samples = 1000 # 总抽样量 N (同您图中的 N=1000)
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N_burn_in = 200 # 预估的老化期(预热期) N1
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print(f"模型参数: n={n}, alpha={alpha}, beta={beta}")
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print(f"理论均值 E[Y]: {theoretical_mean:.4f}")
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print(f"抽样设置: 总样本 N={N_samples}")
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# ------------------------------------------------------------------
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# 3. 初始化
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# ------------------------------------------------------------------
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# 创建数组来存储所有样本
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samples_X = np.zeros(N_samples, dtype=int)
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samples_Y = np.zeros(N_samples, dtype=float)
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# 设定马尔可夫链的初始状态
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# 故意设置一个远离均值(0.727)的初始值，以观察收敛
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y_t = 0.1
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# ------------------------------------------------------------------
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# 4. 运行 Gibbs 抽样
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# ------------------------------------------------------------------
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print("开始Gibbs抽样...")
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np.random.seed(101) # 使用和您图中一样的随机种子
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for i in range(N_samples):
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x_t = np.random.binomial(n, y_t)
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y_t = np.random.beta(x_t + alpha, n - x_t + beta)
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samples_Y[i] = y_t
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samples_X[i] = x_t
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print("抽样完成。")
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# ------------------------------------------------------------------
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# 5. 定义并计算逐步平均值 (Ergodic Mean)
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# ------------------------------------------------------------------
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def calculate_ergodic_mean(y_samples):
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"""
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使用贝叶斯线性回归方法进行重要抽样
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本例需要估计一个简单的线性回归模型的后验分布
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y = β0 + β1*x + ε, ε ~ N(0, σ²)
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= [1 x] * [β0 β1]' + ε
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计算逐步平均值 (累积平均值)
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y_k_bar = (1/k) * sum(y_i for i=1 to k)
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使用 np.cumsum() 可以高效实现
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"""
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n = len(y_samples)
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# 1. 计算累积和 [y1, y1+y2, y1+y2+y3, ...]
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s = np.cumsum(y_samples)
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# 2. 创建 k 数组 [1, 2, 3, ...]
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k_array = np.arange(1, n + 1)
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# 3. 计算 avg[k] = s[k] / k
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return s / k_array
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# 1. 定义关键数据
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x = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
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y = np.array([2.1, 3.9, 6.2, 8.1, 9.8, 12.3, 13.9, 16.2, 17.8, 20.1])
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n = len(x)
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# 计算所有 Y 样本的逐步平均值 (包括预热期)
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ergodic_mean_Y = calculate_ergodic_mean(samples_Y)
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print(f"数据点数量：{n}")
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print(f"x: {x}")
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print(f"y: {y}")
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# ------------------------------------------------------------------
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# 6. 绘制逐步平均值图 (实现您图片中的效果)
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# ------------------------------------------------------------------
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print("正在绘制逐步平均值图...")
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# 2. 定义先验分布
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# β0 ~ N(0, 10)
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beta0_prior_mean = 0
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beta0_prior_var = 10
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plt.figure(figsize=(10, 6))
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plt.plot(np.arange(1, N_samples + 1), ergodic_mean_Y, label=r"逐步平均值 $\bar{y}_k$")
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plt.axhline(theoretical_mean, color='red', linestyle='--', label=f"理论均值: {theoretical_mean:.4f}")
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# β1 ~ N(1, 5)
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beta1_prior_mean = 1
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beta1_prior_var = 5
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# 添加一个垂直线来标记我们估计的预热期
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plt.axvline(N_burn_in, color='gray', linestyle=':', label=f"估计的预热期 N1 = {N_burn_in}")
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# σ² ~ Inverse-Gamma(2, 1)
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sigma2_prior_a = 2
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sigma2_prior_b = 1
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# 3. 定义重要抽样分布
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# 使用最小二乘进行估计
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X = np.column_stack([np.ones(n), x])
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beta_estimate = np.linalg.inv(X.T @ X) @ X.T @ y
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y_pred = X @ beta_estimate
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residuals = y - y_pred
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RSS = residuals.T @ residuals
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sigma2_estimate = RSS / (n - 2)
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print(f"最小二乘结果：β0 = {beta_estimate[0]}, β1 = {beta_estimate[1]}, σ² = {sigma2_estimate}")
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print(f"残差平方和 RSS = {RSS}")
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# 4. 定义联合后验分布(未归一化)
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def unnormalized_log_posterior(param):
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"""
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计算后验概率密度的未归一化值
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param: [β0, β1, log_sigma2)]
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"""
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beta0, beta1, log_sigma2 = param
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sigma2 = np.exp(log_sigma2)
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# 计算似然函数值
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y_pred = beta0 + beta1 * x
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residuals = y - y_pred
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log_likelihood = -0.5 * n * np.log(2 * np.pi * sigma2) * -0.5 * np.sum(residuals**2) / sigma2
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# 计算先验概率
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log_prior_beta0 = stats.norm.logpdf(beta0, loc = beta0_prior_mean, scale = np.sqrt(beta0_prior_var))
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log_prior_beta1 = stats.norm.logpdf(beta1, loc = beta1_prior_mean, scale = np.sqrt(beta1_prior_var))
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log_prior_sigma2 = stats.invgamma.logpdf(sigma2, a = sigma2_prior_a, scale = sigma2_prior_b) + log_sigma2
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return log_likelihood + log_prior_beta0 + log_prior_beta1 + log_prior_sigma2
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# 5. 定义重要抽样分布
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proposal_mean = np.array([beta_estimate[0], beta_estimate[1], np.log(sigma2_estimate)])
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proposal_cov = np.diag([1.0, 1.0, 1.0])
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def log_proposal_density(param):
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return stats.multivariable_normal.logpdf(param, mean = proposal_mean, cov = proposal_cov)
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# 6. 进行重要抽样
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num_samples = 10000
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samples = np.zeros((num_samples, 3))
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weights = np.zeros(num_samples)
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log_weights = np.zeros(num_samples)
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for i in range(num_samples):
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samples[i] = np.random.multivariate_normal(proposal_mean, proposal_cov)
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unnormalized_log_posterior = unnormalized_log_posterior(samples[i])
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log_proposal = log_proposal_density(samples[i])
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log_weights[i] = unnormalized_log_posterior - log_proposal
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weights = np.exp(log_weights)
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if __name__ == "__main__":
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baysian_linear_regression_inportance_sampling()
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plt.title("使用逐步平均值图查看预热期")
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plt.xlabel("迭代次数 (k)")
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plt.ylabel(r"逐步平均值 $\bar{y}_k$")
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plt.legend()
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plt.grid(True)
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plt.ylim(0, 1) # Y值在0到1之间
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plt.show()
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