Inverse Probability Weighting
Inverse probability weighting creates a pseudo-population in which observed treatment assignment is independent of measured covariates, provided the propensity model and identification assumptions are adequate.
Horvitz-Thompson ATE estimator
With estimated propensity scores $\hat e(X_i)$, the inverse-probability-weighted estimator is
\[\boxed{ \hat\tau_{IPW} = \frac{1}{N}\sum_{i=1}^N \left[ \frac{W_iY_i^{obs}}{\hat e(X_i)} - \frac{(1-W_i)Y_i^{obs}}{1-\hat e(X_i)} \right] }.\]Treated units with low treatment probability receive large treated weights; control units with high treatment probability receive large control weights. Each observed unit represents similar units whose opposite treatment outcome is missing.
Weight definitions
For the ATE, the raw weights are
\[\omega_i= \frac{W_i}{\hat e(X_i)}+ \frac{1-W_i}{1-\hat e(X_i)}.\]Stabilized weights replace the numerators by the marginal treatment probabilities, reducing variability while preserving the target under the corresponding normalization.
Normalized form
A Hájek-style estimator normalizes weights within each arm:
\[\hat\tau_H= \frac{\sum_i W_iY_i^{obs}/\hat e_i}{\sum_i W_i/\hat e_i} - \frac{\sum_i(1-W_i)Y_i^{obs}/(1-\hat e_i)} {\sum_i(1-W_i)/(1-\hat e_i)}.\]It is not algebraically identical to the Horvitz-Thompson form in finite samples, although both target the ATE under suitable conditions.
Assumptions and diagnostics
IPW requires consistency, Unconfoundedness and Overlap, and a sufficiently accurate propensity-score model. After weighting:
- inspect weighted covariate balance;
- inspect maximum and tail weights;
- report effective sample size;
- investigate sensitivity to trimming or alternative specifications;
- use uncertainty estimates that account for estimated weights.
Near-zero or near-one propensity scores produce extreme weights. Trimming changes the target population, so the resulting estimand should be described explicitly.