Inverse probability weighting

Within a baseline stratum, the expectation of the treated outcome indicator is the propensity times the observed treated conditional mean. Dividing by that same propensity cancels the selection probability. Averaging over the baseline distribution returns the standardized intervention-one mean under consistency and conditional exchangeability. The control identity uses one minus the propensity. Subtracting gives ATE. The formula requires the relevant denominators to be positive wherever the target has support and enough integrability for the expectations to exist.

This is a statement about the observed probability law and its relation to potential outcomes under assumptions. It does not mean that a large weight creates information about a never-observed treatment alternative. At a structural zero score, the treatment numerator is always zero and division cannot recover the missing mean. With a very small score, a rare treated observation can dominate an estimate. The algebra correctly describes the target while leaving a challenging variance and finite-sample problem.

A Horvitz–Thompson mean divides the weighted outcome total by the original sample size. In a fixed finite population with known randomized assignment probabilities, the corresponding inverse-probability estimator is exactly design-unbiased for the finite intervention mean. That claim concerns random assignment and known probabilities. Substituting estimated observational propensities generally does not preserve exact finite-sample unbiasedness, even if the model is correctly specified. Consistency and uncertainty then depend on the fitting method and regularity assumptions.

A Hájek estimator instead divides each weighted outcome total by its arm’s observed weight total. It often has useful finite-sample stability and keeps a binary-outcome arm mean within the observed outcome range, but it is a ratio estimator and is generally not exactly unbiased. Its ATE subtracts two separately normalized arm means. The two methods can agree asymptotically under appropriate conditions while differing substantially in a small sample. Naming the normalization makes an estimate reproducible and prevents a probability identity from being mistaken for every possible weighting implementation.

For an ATE pseudo-population, treated units receive reciprocal propensity weights and controls receive reciprocal complement weights. For an ATT comparison, treated units retain unit weight while controls are tilted by the propensity odds, followed by the appropriate normalization. Applying one formula and labeling the result as the other target is incorrect when effects vary. Stabilizing weights by arm constants leaves separately normalized means unchanged, but does not magically restore support or remove confounding.

A common descriptive diagnostic is the squared sum of weights divided by the sum of squared weights, often called a weight effective sample size. It measures concentration of weights under a specific arm and normalization; it is not a universal replacement for the actual sampling variance or dependence structure. Truncating extreme weights may improve precision at the cost of bias for the original target. Report the chosen rule, overlap, and target interpretation rather than presenting a narrow standard error as proof that the weighted intervention contrast is identified.