 Inverse probability weighting

Inverse probability weighting weights observations by reciprocals of the probabilities that generated their observed assignment or selection. For binary treatment with propensity score $e(X)$, the average-treatment-effect observation weight isIt uses the probability of the treatment actually received. Under conditional exchangeability, consistency in causal inference and positivity, the identification identity isTo prove it, condition on $X$: the assignment probability cancels its denominator, leaving $\mathbb{E}(Y\mid A=a,X)$. Average over $X$ and apply the causal identification assumptions. The same cancellation with a function $g(X)$ instead of $Y$ proves population balance of the measured covariates under correctly specified assignment probabilities.

If a stratum has treatment probability $0.2$, each treated observation there receives weight five: it represents units with the same measured baseline characteristics who usually receive the other treatment. Very small assignment probabilities give large weights and unstable estimates. Estimated assignment models, trimming and normalization can change finite-sample behavior and, for trimming, the target population. Overlap weights deliberately define a different target rather than estimating the same full-population contrast by another name.