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demo.py

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+## 实现一下重要抽样
+import numpy as np
+import matplotlib.pyplot as plt
+import scipy as stats
+import seaborn as sns
+
+np.random.seed(42)
+
+def baysian_linear_regression_inportance_sampling():
+    """
+    使用贝叶斯线性回归方法进行重要抽样
+
+    本例需要估计一个简单的线性回归模型的后验分布
+
+    y = β0 + β1*x + ε,  ε ~ N(0, σ²)
+      = [1 x] * [β0 β1]' + ε
+    """
+
+    # 1. 定义关键数据 
+    x = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
+    y = np.array([2.1, 3.9, 6.2, 8.1, 9.8, 12.3, 13.9, 16.2, 17.8, 20.1])
+    n = len(x)
+
+    print(f"数据点数量：{n}")
+    print(f"x: {x}")
+    print(f"y: {y}")
+
+    # 2. 定义先验分布
+    # β0 ~ N(0, 10)
+    beta0_prior_mean = 0
+    beta0_prior_var = 10
+
+    # β1 ~ N(1, 5)
+    beta1_prior_mean = 1
+    beta1_prior_var = 5
+
+    # σ² ~ Inverse-Gamma(2, 1)
+    sigma2_prior_a = 2
+    sigma2_prior_b = 1
+
+    # 3. 定义重要抽样分布
+    # 使用最小二乘进行估计
+    X = np.column_stack([np.ones(n), x])
+    beta_estimate = np.linalg.inv(X.T @ X) @ X.T @ y
+    y_pred = X @ beta_estimate
+    residuals = y - y_pred
+    RSS = residuals.T @ residuals
+    sigma2_estimate = RSS / (n - 2)
+
+    print(f"最小二乘结果：β0 = {beta_estimate[0]}, β1 = {beta_estimate[1]}, σ² = {sigma2_estimate}")
+    print(f"残差平方和 RSS = {RSS}")
+
+    
+
+    # 4. 定义联合后验分布(未归一化)
+    def unnormalized_log_posterior(param):
+        """
+        计算后验概率密度的未归一化值
+
+        param: [β0, β1, log_sigma2)]
+        """
+
+        beta0, beta1, log_sigma2 = param
+
+        sigma2 = np.exp(log_sigma2)
+
+        # 计算似然函数值
+        y_pred = beta0 + beta1 * x
+        residuals = y - y_pred
+        log_likelihood = -0.5 * n * np.log(2 * np.pi * sigma2) * -0.5 * np.sum(residuals**2) / sigma2
+
+        # 计算先验概率
+        log_prior_beta0 = stats.norm.logpdf(beta0, loc = beta0_prior_mean, scale = np.sqrt(beta0_prior_var))
+        log_prior_beta1 = stats.norm.logpdf(beta1, loc = beta1_prior_mean, scale = np.sqrt(beta1_prior_var))
+        log_prior_sigma2 = stats.invgamma.logpdf(sigma2, a = sigma2_prior_a, scale = sigma2_prior_b) + log_sigma2
+
+        return log_likelihood + log_prior_beta0 + log_prior_beta1 + log_prior_sigma2
+    
+    # 5. 定义重要抽样分布
+    proposal_mean = np.array([beta_estimate[0], beta_estimate[1], np.log(sigma2_estimate)])
+    proposal_cov = np.diag([1.0, 1.0, 1.0])
+
+    def log_proposal_density(param):
+        return stats.multivariable_normal.logpdf(param, mean = proposal_mean, cov = proposal_cov)
+    
+    # 6. 进行重要抽样
+    num_samples = 10000
+    samples = np.zeros((num_samples, 3))
+    weights = np.zeros(num_samples)
+    log_weights = np.zeros(num_samples)
+
+    for i in range(num_samples):
+        samples[i] = np.random.multivariate_normal(proposal_mean, proposal_cov)
+        unnormalized_log_posterior = unnormalized_log_posterior(samples[i])
+        log_proposal = log_proposal_density(samples[i])
+        log_weights[i] = unnormalized_log_posterior - log_proposal
+ 
+    weights = np.exp(log_weights)
+
+
+
+if __name__ == "__main__":
+    baysian_linear_regression_inportance_sampling()
+
+