Inverse probability weighting

Inverse probability weighting reweights each observed outcome by the reciprocal probability of receiving its observed treatment, creating a weighted population in which treatment is balanced across measured covariates.

The propensity score is the conditional probability $\pi (X)=\mathbb{P}(A=1\mid X)$ of receiving a binary treatment given observed covariates. Under conditional exchangeability, conditioning on the propensity score balances the covariate distribution between treatment groups.

For binary treatment, outcomes $Y_i$, and estimated propensity score $\hat{e}(X_i)$, the inverse-probability-weighted estimator isIt is consistent under conditional exchangeability, consistency of potential outcomes, positivity in causal inference, and a consistent propensity-score estimator.

For binary treatment with propensity score $\pi (X)$, the overlap weight $\pi (X)\{1-\pi (X)\}$ emphasizes covariate strata in which both treatments occur with appreciable probability.