贝叶斯方法与数字普惠金融的概率决策

import numpy as np
import matplotlib.pyplot as plt

y, n = 9, 60                 # 已观察到：60 笔中有 9 笔逾期
p = np.linspace(0.001, 0.45, 600)

# 对数形式更稳定；减去最大值后，曲线最高点仍为 1
log_likelihood = y * np.log(p) + (n - y) * np.log(1 - p)
relative_likelihood = np.exp(log_likelihood - log_likelihood.max())
p_mle = y / n

print(f"极大似然估计 p_MLE = {p_mle:.2%}")
plt.plot(p, relative_likelihood, color="#1f6b65", linewidth=2.5)
plt.axvline(p_mle, color="#b65a2b", linestyle="--")
plt.xlabel("候选真实逾期率 p")
plt.ylabel("相对似然")
plt.title("60 笔中观察到 9 笔逾期时的似然函数")
plt.show()

from pathlib import Path
import pandas as pd

DATA_PATH = Path("data/credit_nb_teaching.csv")
df = pd.read_csv(DATA_PATH)

features = [
    "AGE", "INCOME", "EXP_YRS", "SAVINGS", "DEBT_TOTAL",
    "CC_UTIL", "LATE_30", "LATE_60", "NUM_CC", "ACCT_AGE",
]
y = df["high_risk"]

print("数据维度：", df.shape)
print("高风险样本比例：", f"{y.mean():.2%}")
print(df[["applicant_id", *features[:4], "high_risk"]].head())

import matplotlib.pyplot as plt

counts = y.value_counts().sort_index()
labels = ["低风险（0）", "高风险（1）"]
plt.bar(labels, counts.values, color=["#6f9e8d", "#b65a2b"])
plt.ylabel("样本数")
plt.title("教学数据中的类别不平衡")
for i, value in enumerate(counts.values):
    plt.text(i, value, f"{value:,}（{value / len(y):.2%}）", ha="center", va="bottom")
plt.show()

from sklearn.model_selection import train_test_split
from sklearn.naive_bayes import GaussianNB
from sklearn.metrics import accuracy_score, recall_score, roc_auc_score

X_train, X_test, y_train, y_test = train_test_split(
    df[features], y, test_size=0.30, random_state=42, stratify=y
)
model = GaussianNB(var_smoothing=1e-9)
model.fit(X_train, y_train)

pred_default = model.predict(X_test)
prob_high_risk = model.predict_proba(X_test)[:, 1]

print("训练集 / 测试集：", X_train.shape[0], "/", X_test.shape[0])
print("类别先验：", model.class_prior_.round(4))
print("AUC：", round(roc_auc_score(y_test, prob_high_risk), 4))
print("默认阈值下的 accuracy：", round(accuracy_score(y_test, pred_default), 4))
print("默认阈值下的 recall：", round(recall_score(y_test, pred_default, zero_division=0), 4))

from sklearn.metrics import roc_curve
import matplotlib.pyplot as plt

fpr, tpr, _ = roc_curve(y_test, prob_high_risk)
fig, axes = plt.subplots(1, 2, figsize=(12, 4))
axes[0].plot(fpr, tpr, color="#1f6b65", label="GaussianNB")
axes[0].plot([0, 1], [0, 1], "--", color="#9aa1a8")
axes[0].set(xlabel="假阳性率", ylabel="真阳性率", title="ROC 曲线")
axes[0].legend()
axes[1].hist(prob_high_risk[y_test == 0], bins=30, alpha=.7, label="低风险")
axes[1].hist(prob_high_risk[y_test == 1], bins=20, alpha=.8, label="高风险")
axes[1].set(xlabel="预测的高风险后验概率", ylabel="样本数", title="后验概率分布")
axes[1].legend()
plt.tight_layout()
plt.show()

new_application = pd.DataFrame([{
    "AGE": 32, "INCOME": 75000, "EXP_YRS": 6, "SAVINGS": 120000,
    "DEBT_TOTAL": 28000, "CC_UTIL": 0.28, "LATE_30": 0,
    "LATE_60": 0, "NUM_CC": 2, "ACCT_AGE": 54,
}])
posterior = model.predict_proba(new_application)[0]
print(f"低风险后验概率：{posterior[0]:.2%}")
print(f"高风险后验概率：{posterior[1]:.2%}")
print("默认标签：", "高风险" if model.predict(new_application)[0] == 1 else "低风险")
print("10% 复核规则：", "进入人工复核" if posterior[1] >= 0.10 else "常规处理")

import matplotlib.pyplot as plt

labels = ["低风险后验", "高风险后验"]
colors = ["#6f9e8d", "#b65a2b"]
plt.bar(labels, posterior, color=colors)
plt.axhline(0.10, color="#8b4f24", linestyle="--", label="10% 复核阈值")
plt.ylim(0, 1)
plt.ylabel("后验概率")
plt.title("单个申请人的模型后验输出")
plt.legend()
plt.show()

from sklearn.metrics import confusion_matrix, precision_score, recall_score

review_threshold = 0.10
review_flag = (prob_high_risk >= review_threshold).astype(int)
tn, fp, fn, tp = confusion_matrix(y_test, review_flag).ravel()

print("进入人工复核：", int(review_flag.sum()), "人")
print("precision：", round(precision_score(y_test, review_flag, zero_division=0), 4))
print("recall：", round(recall_score(y_test, review_flag, zero_division=0), 4))
print("混淆矩阵 [[TN, FP], [FN, TP]]：")
print([[int(tn), int(fp)], [int(fn), int(tp)]])

thresholds = [0.05, 0.10, 0.15, 0.20, 0.30]
review_counts = [(prob_high_risk >= t).sum() for t in thresholds]
recalls = [recall_score(y_test, prob_high_risk >= t) for t in thresholds]

fig, ax1 = plt.subplots(figsize=(8, 4.5))
ax1.plot(thresholds, review_counts, marker="o", color="#1f6b65")
ax1.set(xlabel="人工复核阈值", ylabel="进入复核的人数")
ax2 = ax1.twinx()
ax2.plot(thresholds, recalls, marker="s", color="#b65a2b")
ax2.set_ylabel("高风险召回率")
plt.title("阈值改变时的筛查规模与召回率")
plt.show()