import pandas as pd
# 创建一个扑克数据框示例
df = pd.DataFrame({
    "花色": (["♥"]*13) + (["♠"]*13) + (["♦"]*13) + (["♣"]*13),
    "人头": [1 if i in [10, 11, 12] else 0 for i in range(13)] * 4  # J/Q/K 标记为 1
})

# P(人头)
p_face = df["人头"].mean()
# P(人头 | 红桃)
p_face_given_heart = df[df["花色"] == "♥"]["人头"].mean()
print(f"P(人头) = {p_face:.3f}, P(人头|红桃) = {p_face_given_heart:.3f}")
# 0.231, 0.231 (人头跟花色独立)

# 互斥不独立
P_A = 0.5
P_B = 0.3
P_AB = 0           # 互斥
print(f"独立检查: P(A)*P(B) = {P_A*P_B}, P(A∩B) = {P_AB}  → 不独立!")

# 独立不互斥
P_A = 0.5; P_B = 0.3
P_AB = P_A * P_B    # 独立
print(f"互斥检查: P(A∩B) = {P_AB}  → 不互斥!")

# 例: 工厂 3 条产线, 各产 30%/45%/25%, 不合格率 1%/2%/1.5%
# 求任取一产品是不合格品的概率
P_A = [0.30, 0.45, 0.25]
P_B_given_A = [0.01, 0.02, 0.015]
P_B = sum(p * q for p, q in zip(P_A, P_B_given_A))
print(f"P(不合格) = {P_B:.4f}")  # 0.01575

p_infect = 0.01
p_pos_given_infect = 0.95     # 灵敏度
p_pos_given_healthy = 0.10    # 假阳性率
p_healthy = 1 - p_infect

# 全概率公式
p_pos = p_pos_given_infect * p_infect + p_pos_given_healthy * p_healthy
# 贝叶斯
p_infect_given_pos = (p_pos_given_infect * p_infect) / p_pos

print(f"P(感染 | 阳性) = {p_infect_given_pos:.3f}")  # ≈ 0.088

from sklearn.naive_bayes import MultinomialNB
from sklearn.feature_extraction.text import CountVectorizer
from sklearn.pipeline import make_pipeline

# 训练数据
emails = [
    ("免费 限时 优惠 立即 点击 领取", 1),  # 1 = 垃圾
    ("免费 折扣 立即 抢购", 1),
    ("中奖 通知 现金 领取", 1),
    ("明天下午 3 点 开会", 0),  # 0 = 正常
    ("代码 review 一下", 0),
    ("项目 进度 报告 会议纪要", 0),
]
texts, labels = zip(*emails)

# 训练朴素贝叶斯
model = make_pipeline(CountVectorizer(), MultinomialNB())
model.fit(texts, labels)

# 预测新邮件
new_email = "限时优惠 立即领取 现金"
prob_spam = model.predict_proba([new_email])[0][1]
print(f"新邮件 '{new_email}' 是垃圾邮件的概率: {prob_spam:.3f}")

# 例: 抛 10 次硬币, 8 次正面, MLE 估计正面概率
# L(p) = p^8 × (1-p)^2, 取对数 log L = 8 log p + 2 log(1-p)
# d/dp = 0 → 8/p - 2/(1-p) = 0 → p = 0.8
import numpy as np
from scipy.optimize import minimize_scalar

# MLE: 找让 -log L 最小的 p
def neg_log_likelihood(p):
    return -(8 * np.log(p + 1e-10) + 2 * np.log(1 - p + 1e-10))

result = minimize_scalar(neg_log_likelihood, bounds=(0.01, 0.99), method="bounded")
print(f"MLE 估计 p = {result.x:.3f}")  # 0.800

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

# 场景: 罕见病 (1% 患病率), 检测准确率 95%
# 用贝叶斯更新: 阳性后, 第二次检测, 概率怎么变?

p_disease = 0.01
sensitivity = 0.95
false_positive = 0.10

# 第 1 次检测阳性后
p1 = (sensitivity * p_disease) / (
    sensitivity * p_disease + false_positive * (1 - p_disease)
)
print(f"1 次阳性后: P(患病) = {p1:.3f}")

# 假定条件独立, 2 次都阳性
p2 = (sensitivity**2 * p_disease) / (
    sensitivity**2 * p_disease + false_positive**2 * (1 - p_disease)
)
print(f"2 次阳性后: P(患病) = {p2:.3f}")

# 5 次都阳性
p5 = (sensitivity**5 * p_disease) / (
    sensitivity**5 * p_disease + false_positive**5 * (1 - p_disease)
)
print(f"5 次阳性后: P(患病) = {p5:.4f}")

1 次阳性后: P(患病) = 0.088
2 次阳性后: P(患病) = 0.473
5 次阳性后: P(患病) = 0.9875