所属模块：
M1 · 数学与统计基础 (Mathematics & Statistics Fundamentals)| 专题分类：因果推断 (Causal Inference)| 难度等级：Medium
一、核心一句话结论 (One-Sentence Summary)
PSM 用倾向得分匹配相似样本；IPW 用 1/倾向得分重加权，构造伪随机化总体。
The propensity score $e(X) = P(T=1mid X)$ condenses high-dimensional confounders into a scalar; PSM matches treated and control units with identical scores, while IPW weights units by $1/e(X)$ to create a synthetic pseudo-randomized population.
二、核心考点要义 (Key Insights)
- 📌 倾向得分 e(X)=P(T=1|X)
- 📌 重叠性假设：0<e(X)<1
- 📌 IPW 对倾向得分模型误设敏感
English Insights:
– Propensity Score: By Rosenbaum & Rubin (1983), conditioning on scalar $e(X)$ achieves conditional independence: $(Y(0), Y(1)) perp T mid e(X)$.
– Propensity Score Matching (PSM): Pairs each treated unit with one or more nearest-neighbor control units having similar $e(X)$.
– Inverse Probability Weighting (IPW): Weights treated units by $frac{1}{e(X)}$ and control units by $frac{1}{1 – e(X)}$; Horvitz-Thompson estimator.
– Positivity Assumption: $0 < e(X) < 1$ for all $X$; every unit must have non-zero probability of receiving either treatment.
三、核心数学原理与机理推导 (Mathematical Principles & Derivation)
$$mathrm{IPW}: hattau=frac1nsumBig[frac{T_iY_i}{e(X_i)}-frac{(1-T_i)Y_i}{1-e(X_i)}Big]$$
倾向得分 e(X)=P(T=1|X) 是’在给定协变量下接受处理的概率’，它把高维协变量压缩为一维——这是 Rosenbaum & Rubin (1983) 的核心定理：若控制 X 后处理可忽略，则控制 e(X) 同样可忽略。PSM 通过匹配倾向得分相近的处理/对照个体来构造可比组，直观但丢弃未匹配样本且匹配质量难控。IPW 的思路完全不同：给每个观测赋予权重 1/e(X)（处理组）或 1/(1−e(X))（对照组），使得加权后的样本在协变量分布上模拟随机化——直觉是’倾向得分低却接受了处理的人代表了很多没被处理的人，故应放大其权重’。IPW 估计量可写为加权均值差，且有两组正确性条件：倾向得分模型正确，或结果模型正确（后者对应 DR）。
📖 查看英文严格数学推导 (English Mathematical Derivation)
Proof of balancing score property: $P(T=1 mid X, e(X)) = P(T=1mid X) = e(X)$. Thus, $T$ and $X$ are conditionally independent given $e(X)$. For IPW Horvitz-Thompson estimator: $hat{mu}_1 = Eleft[frac{T Y}{e(X)}right] = E_Xleft[Eleft[frac{T Y(1)}{e(X)} mid Xright]right] = E_Xleft[frac{E[Tmid X] E[Y(1)mid X]}{e(X)}right] = E_Xleft[frac{e(X) E[Y(1)mid X]}{e(X)}right] = E_X[E[Y(1)mid X]] = E[Y(1)]$. Similarly, $Eleft[frac{(1-T)Y}{1-e(X)}right] = E[Y(0)]$. The IPW ATE estimator is $hat{tau}_{text{IPW}} = frac{1}{N}sum_{i=1}^N left( frac{T_i Y_i}{e(X_i)} – frac{(1-T_i)Y_i}{1-e(X_i)} right)$.
四、工业级落地权衡与工程考量 (Industrial Trade-offs)
实践要点：① 假设与诊断——需满足条件可忽略性（无未观测混淆，不可检验）、重叠性/正性（0<e(X)<1，即每个个体都有机会接受两种处理）、SUTVA。诊断手段：检查倾向得分的重叠区域（画两组的 e(X) 分布）、用标准化偏差（standardized bias）检查加权/匹配后的协变量平衡、检查有效样本量（ESS）。② IPW 的方差问题——权重可能极大（当 e(X)→0 或 1 时），导致方差爆炸；缓解手段是稳定化权重（乘以边际处理概率，SNIPS）、权重截断（trimming，如限制到 99 分位）、或用 overlap weights（一种自带降方差性质且不需截断的加权方案）。③ PSM 的常见误用——只匹配倾向得分而不检查协变量平衡、用匹配后样本直接做 t 检验（未考虑匹配的不确定性）、丢弃大量样本导致外部效度下降（估的是 ATT 而非 ATE）。④ PSM vs IPW 选择——IPW 用全部样本、理论基础更清晰（可推导渐近方差）、更易扩展（DR、TMLE），现代实践中更受推荐。
⚙️ 查看英文落地权衡分析 (English Systems & Trade-offs)
Comparison: (1) PSM drawbacks: Discards unmatched control units (reducing effective sample size), suffers from caliper tuning sensitivity, and cannot easily handle continuous treatments. (2) IPW drawbacks: Extreme propensity scores ($e(X) 0.99$) blow up weights, causing massive estimator variance. In practice, scientists use Doubly Robust Estimators (AIPW), which combine propensity scores with outcome regression models; if EITHER model is correctly specified, the estimator remains consistent.
五、常见面试避坑陷阱 (Common Pitfalls & Traps)
- ⚠️ 只匹配倾向得分但不检查协变量平衡
- ⚠️ 忽视重叠性假设（e(X) 接近 0 或 1 时权重爆炸）
English Pitfalls:
– Violating positivity (common support): Trying to match treated units in regions of feature space where zero control units ever exist.
– Unnormalized IPW weights creating massive variance spikes when propensity scores approach zero or one (mitigated by weight trimming / clipping).
六、高频深度面试追问与预测 (Follow-Up Questions)
- IPW 方差很大怎么办？（稳定化权重/trimming）
- How does the Augmented Inverse Probability Weighting (AIPW) estimator achieve double robustness?
- 双重稳健（DR）为什么更稳？
- Why is covariate balance checking (standardized mean difference < 0.1) mandatory after propensity score matching?
七、知识图谱对齐 (Knowledge Graph Anchor)
- 🔗 关联底层卡片：
因果推断框架：潜在结果模型、倾向评分匹配与双重差分(Causal Inference: Potential Outcomes, PSM & DiD) - 🗺️ 知识图谱模块：
数据科学与因果实验导图
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