IPW (Inverse Propensity Weighting)

Definition

Estimating treatment effects by using the inverse of the propensity score as weights

$$\hat{\text{ATE}}_{IPW} = \frac{1}{n}\sum_{i=1}^{n} \left[\frac{W_i Y_i}{e(X_i)} - \frac{(1-W_i) Y_i}{1-e(X_i)}\right]$$

Here, $e(X) = P(W=1 \mid X)$ is the Propensity Score.

---

Intuitive Understanding

Why inverse weighting?

“More weight to underrepresented samples”

Situation	Treatment probability	Weight	Meaning
Treatment rare	$e(X) = 0.1$	$1/0.1 = 10$	This person represents 10 individuals
Treatment common	$e(X) = 0.9$	$1/0.9 ≈ 1.1$	Nearly 1:1 representation

Resampling Perspective

IPW is equivalent to:

• “Replicating” each sample according to its weight
• Creating a hypothetical pseudo-RCT (pseudo-randomized experiment)

---

Mathematical Derivation

ATE Identification

Under Strong Ignorability:

$$E[Y(w)] = E\left[\frac{\mathbb{1}(W=w) \cdot Y}{P(W=w \mid X)}\right]$$

Therefore:

$$\text{ATE} = E\left[\frac{WY}{e(X)}\right] - E\left[\frac{(1-W)Y}{1-e(X)}\right]$$

Sample Estimator

$$\hat{\text{ATE}}_{IPW} = \frac{1}{n}\sum_i \left[\frac{W_i Y_i}{e(X_i)} - \frac{(1-W_i) Y_i}{1-e(X_i)}\right]$$

---

Normalized Version

More stable when the PS is estimated:

$$\hat{\text{ATE}}_{IPW}^{norm} = \frac{\sum_i \frac{W_i Y_i}{e(X_i)}}{\sum_i \frac{W_i}{e(X_i)}} - \frac{\sum_i \frac{(1-W_i) Y_i}{1-e(X_i)}}{\sum_i \frac{(1-W_i)}{1-e(X_i)}}$$

Advantages: reduced bias, reduced variance

---

IPW for ATT

$$\hat{\text{ATT}}_{IPW} = \frac{\sum_i W_i Y_i}{\sum_i W_i} - \frac{\sum_i (1-W_i) \frac{e(X_i)}{1-e(X_i)} Y_i}{\sum_i (1-W_i) \frac{e(X_i)}{1-e(X_i)}}$$

See ATT

---

Pros and Cons

Advantages

Advantage	Description
Simple	Intuitive and easy to implement
Nonparametric	No outcome-model assumptions required
Theoretical justification	Guarantees conditional consistency
Flexibility	Applicable to a variety of estimands

Disadvantages

Disadvantage	Description
Dependence on PS estimation	Biased when the PS is misspecified
Sensitivity to extreme PS	Unstable when $e(X) \approx 0$ or $1$
High variance	Especially when overlap is weak
Difficult in high dimensions	PS estimation is difficult

---

Extreme PS Problem

Problem

When $e(X) \to 0$ or $e(X) \to 1$:

• Weights explode: $1/e(X) \to \infty$
• Estimator becomes unstable

Solutions

1. Trimming: remove samples with extreme PS
2. Overlap Weighting: use stable weights
3. Weight clipping: set an upper bound on weights

---

Implementation

Python (EconML)

from econml.dr import LinearDRLearner

# IPW without an outcome model
model = LinearDRLearner(model_propensity=LogisticRegression())
model.fit(Y, T, X)
ate = model.effect(X).mean()

R

library(WeightIt)

# Propensity score weights
weights <- weightit(treat ~ x1 + x2, data = df, method = "ps")

# Weighted outcome regression
lm(y ~ treat, data = df, weights = weights$weights)

---

Related Concepts

• Re-weighting Methods Overview - consolidated overview of reweighting methods
• Propensity Score - the core tool
• Doubly Robust Estimator - IPW + outcome regression
• CBPS - directly optimizing balance
• Trimming - handling extreme PS
• Overlap Weighting - stable weighting

---

Application: Correcting RTB Win Selection Bias

In RTB, training only on won impressions introduces win selection bias. Correct it with IPW:

$$w_i = \frac{1}{p_{\text{win}}(x_i, b_i)}, \quad p_{\text{win}} = P(\text{win} \mid X, \text{bid})$$

The win propensity is estimated via Survival Analysis (Kaplan-Meier) or gradient boosting. Weight stabilization (clipping, normalization) is essential. For details, see Multi-Task Learning (IPW-ESCM²).

---

References

• yaoSurveyCausalInference2021 - Section 3.1.3
• Rosenbaum, P. R., & Rubin, D. B. (1983). The central role of the propensity score
• Horvitz, D. G., & Thompson, D. J. (1952). A generalization of sampling without replacement
• Zhang et al. (2016). Bid-aware Gradient Descent (KDD)