01 - 概率论基础¶

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

# 设置随机种子，保证可重复
np.random.seed(42)

# ========== 1. 一维高斯分布 ==========
print("=" * 50)
print("一维高斯分布")
print("=" * 50)

# 定义参数
mu = 0      # 均值
sigma = 1   # 标准差

# 采样
samples = np.random.normal(mu, sigma, 10000)

# 绘制直方图和理论曲线
plt.figure(figsize=(12, 4))

plt.subplot(1, 3, 1)
plt.hist(samples, bins=50, density=True, alpha=0.7, label='Samples')

# 理论曲线
x = np.linspace(-4, 4, 100)
y = stats.norm.pdf(x, mu, sigma)
plt.plot(x, y, 'r-', linewidth=2, label='Theory')
plt.title(f'Gaussian: μ={mu}, σ={sigma}')
plt.xlabel('x')
plt.ylabel('Probability Density')
plt.legend()
plt.grid(True, alpha=0.3)

# ========== 2. 不同参数的比较 ==========
plt.subplot(1, 3, 2)
params = [(0, 0.5), (0, 1), (0, 2), (2, 1)]
colors = ['blue', 'red', 'green', 'orange']

for (mu, sigma), color in zip(params, colors):  # zip按位置配对
    y = stats.norm.pdf(x, mu, sigma)
    plt.plot(x, y, color=color, linewidth=2,
             label=f'μ={mu}, σ={sigma}')

plt.title('Different Parameters')
plt.xlabel('x')
plt.ylabel('Probability Density')
plt.legend()
plt.grid(True, alpha=0.3)

# ========== 3. 二维高斯分布 ==========
plt.subplot(1, 3, 3)

# 生成二维高斯样本
mean = [0, 0]
cov = [[1, 0.5], [0.5, 1]]  # 协方差矩阵
samples_2d = np.random.multivariate_normal(mean, cov, 1000)

plt.scatter(samples_2d[:, 0], samples_2d[:, 1], alpha=0.5, s=10)
plt.title('2D Gaussian Distribution')
plt.xlabel('x1')
plt.ylabel('x2')
plt.axis('equal')
plt.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('gaussian_distribution.png', dpi=150)
plt.show()

# 打印统计信息
print(f"样本均值: {np.mean(samples):.4f} (理论: {mu})")
print(f"样本标准差: {np.std(samples):.4f} (理论: {sigma})")

# 在扩散模型中，加噪过程就是不断添加高斯噪声
x_0 = np.random.normal(0, 1, 1000)  # 原始数据
epsilon = np.random.normal(0, 1, 1000)  # 噪声

# x_t = sqrt(1-beta) * x_{t-1} + sqrt(beta) * epsilon
beta = 0.1
x_t = np.sqrt(1-beta) * x_0 + np.sqrt(beta) * epsilon

print(f"x_0: mean={np.mean(x_0):.4f}, std={np.std(x_0):.4f}")
print(f"x_t: mean={np.mean(x_t):.4f}, std={np.std(x_t):.4f}")
# x_t 仍然是高斯分布！

# 用简单的例子理解条件概率

# 假设有两个盒子
# 盒子A：3红球，2蓝球
# 盒子B：1红球，4蓝球

# 随机选择一个盒子，然后抽一个球
P_A = 0.5  # 选择盒子A的概率
P_B = 0.5  # 选择盒子B的概率

# 条件概率：P(红|A) = 3/5, P(蓝|A) = 2/5
P_red_given_A = 3/5
P_blue_given_A = 2/5

# 条件概率：P(红|B) = 1/5, P(蓝|B) = 4/5
P_red_given_B = 1/5
P_blue_given_B = 4/5

# 联合概率：P(红, A) = P(红|A) * P(A)
P_red_and_A = P_red_given_A * P_A
P_red_and_B = P_red_given_B * P_B

# 边缘概率：P(红) = P(红, A) + P(红, B)
P_red = P_red_and_A + P_red_and_B

print(f"P(红) = {P_red:.4f}")

# 贝叶斯定理：P(A|红) = P(红|A) * P(A) / P(红)
P_A_given_red = P_red_given_A * P_A / P_red
print(f"P(A|红) = {P_A_given_red:.4f}")

# 验证：P(A|红) + P(B|红) = 1
P_B_given_red = P_red_given_B * P_B / P_red
print(f"P(B|红) = {P_B_given_red:.4f}")
print(f"Sum = {P_A_given_red + P_B_given_red:.4f}")

import numpy as np
import matplotlib.pyplot as plt

# ========== KL散度计算 ==========
def kl_divergence(p, q):
    """
    计算KL散度 D_KL(P||Q)
    p, q: 概率分布（已归一化）
    """
    # 避免log(0)，添加小量
    epsilon = 1e-10
    p = np.clip(p, epsilon, 1)
    q = np.clip(q, epsilon, 1)

    return np.sum(p * np.log(p / q))

# 创建两个高斯分布
x = np.linspace(-5, 5, 1000)

# 分布P：标准正态
mu_p, sigma_p = 0, 1
p = stats.norm.pdf(x, mu_p, sigma_p)
p = p / np.sum(p)  # 归一化

# 分布Q：不同的均值
mu_q, sigma_q = 1, 1
q = stats.norm.pdf(x, mu_q, sigma_q)
q = q / np.sum(q)

# 计算KL散度
kl_pq = kl_divergence(p, q)
kl_qp = kl_divergence(q, p)

print(f"D_KL(P||Q) = {kl_pq:.6f}")
print(f"D_KL(Q||P) = {kl_qp:.6f}")
print(f"不对称！D_KL(P||Q) != D_KL(Q||P)")

# 可视化
plt.figure(figsize=(12, 4))

plt.subplot(1, 2, 1)
plt.plot(x, p, 'b-', linewidth=2, label=f'P: μ={mu_p}, σ={sigma_p}')
plt.plot(x, q, 'r-', linewidth=2, label=f'Q: μ={mu_q}, σ={sigma_q}')
plt.fill_between(x, p, alpha=0.3)
plt.fill_between(x, q, alpha=0.3)
plt.title(f'KL(P||Q) = {kl_pq:.4f}')
plt.xlabel('x')
plt.ylabel('Probability')
plt.legend()
plt.grid(True, alpha=0.3)

# 比较不同差异程度的KL散度
plt.subplot(1, 2, 2)
mu_q_values = np.linspace(-3, 3, 50)
kl_values = []

for mu_q in mu_q_values:
    q = stats.norm.pdf(x, mu_q, sigma_q)
    q = q / np.sum(q)
    kl = kl_divergence(p, q)
    kl_values.append(kl)

plt.plot(mu_q_values, kl_values, 'g-', linewidth=2)
plt.axvline(x=0, color='r', linestyle='--', alpha=0.5, label='P=Q')
plt.xlabel('μ_Q (Q的均值)')
plt.ylabel('KL Divergence')
plt.title('KL散度随分布差异的变化')
plt.legend()
plt.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('kl_divergence.png', dpi=150)
plt.show()

print("\n观察：当两个分布越接近时，KL散度越小")
print("当P=Q时，KL散度为0")

def kl_gaussian(mu1, sigma1, mu2, sigma2):
    """
    计算两个高斯分布之间的KL散度 D_KL(N1||N2)
    """
    return (np.log(sigma2 / sigma1) +
            (sigma1**2 + (mu1 - mu2)**2) / (2 * sigma2**2) - 0.5)

# 验证
mu1, sigma1 = 0, 1
mu2, sigma2 = 1, 2

kl_closed = kl_gaussian(mu1, sigma1, mu2, sigma2)
print(f"闭合形式计算: D_KL = {kl_closed:.6f}")

# 数值计算验证
x = np.linspace(-10, 10, 10000)
p = stats.norm.pdf(x, mu1, sigma1)
q = stats.norm.pdf(x, mu2, sigma2)
p = p / np.sum(p)
q = q / np.sum(q)
kl_numerical = kl_divergence(p, q)
print(f"数值计算: D_KL = {kl_numerical:.6f}")

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(42)

# 目标分布参数
mu = 2.0
sigma = 1.5

# 方法1：直接从目标分布采样
samples_direct = np.random.normal(mu, sigma, 10000)

# 方法2：重参数化技巧
epsilon = np.random.normal(0, 1, 10000)
samples_reparam = mu + sigma * epsilon

# 比较两种方法
plt.figure(figsize=(12, 4))

plt.subplot(1, 2, 1)
plt.hist(samples_direct, bins=50, density=True, alpha=0.5,
         label='Direct Sampling', color='blue')
plt.hist(samples_reparam, bins=50, density=True, alpha=0.5,
         label='Reparameterization', color='red')
plt.title('Sampling Comparison')
plt.xlabel('x')
plt.ylabel('Density')
plt.legend()
plt.grid(True, alpha=0.3)

# 统计比较
plt.subplot(1, 2, 2)
stats_data = {
    'Direct': [np.mean(samples_direct), np.std(samples_direct)],
    'Reparam': [np.mean(samples_reparam), np.std(samples_reparam)],
    'Theory': [mu, sigma]
}

x_pos = np.arange(2)
width = 0.25

for i, (method, values) in enumerate(stats_data.items()):  # enumerate同时获取索引和元素
    plt.bar(x_pos + i*width, values, width, label=method)

plt.xlabel('Statistic')
plt.ylabel('Value')
plt.title('Mean and Std Comparison')
plt.xticks(x_pos + width, ['Mean', 'Std'])
plt.legend()
plt.grid(True, alpha=0.3, axis='y')

plt.tight_layout()
plt.savefig('reparameterization.png', dpi=150)
plt.show()

print("统计比较:")
print(f"直接采样: mean={np.mean(samples_direct):.4f}, std={np.std(samples_direct):.4f}")
print(f"重参数化: mean={np.mean(samples_reparam):.4f}, std={np.std(samples_reparam):.4f}")
print(f"理论值: mean={mu:.4f}, std={sigma:.4f}")

# 扩散模型中的前向过程
# x_t = sqrt(alpha_t) * x_0 + sqrt(1 - alpha_t) * epsilon

def q_sample(x_0, t, alphas):
    """
    从 q(x_t | x_0) 采样
    使用重参数化技巧

    参数:
        x_0: 原始数据
        t: 时间步
        alphas: 累积alpha值
    """
    # 计算均值和方差
    alpha_t = alphas[t]
    mean = np.sqrt(alpha_t) * x_0
    std = np.sqrt(1 - alpha_t)

    # 重参数化采样
    epsilon = np.random.normal(0, 1, x_0.shape)
    x_t = mean + std * epsilon

    return x_t, epsilon

# 示例
x_0 = np.array([1.0, 2.0, 3.0])  # 原始数据
alphas = np.linspace(0.99, 0.01, 1000)  # 随时间递减的alpha
alphas_cumprod = np.cumprod(alphas)

# 在不同时间步采样
for t in [0, 250, 500, 750, 999]:
    x_t, eps = q_sample(x_0, t, alphas_cumprod)
    noise_level = np.sqrt(1 - alphas_cumprod[t])
    print(f"t={t:4d}: x_t={x_t}, noise_level={noise_level:.4f}")