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.

The propensity score for binary treatment $A$ and pretreatment covariates $X$ is $e(X)=P(A=1\mid X)$. It is a balancing score:This distributional fact does not require conditional exchangeability of the potential outcomes. It follows from the definition of the true assignment probability.^[ref]

For a bounded function $g$, condition successively on $X$ and on $e=e(X)$:Also $P(A=1\mid e)=e$, so division by $e$ gives the same conditional covariate distribution among treated units as in the full score stratum; the analogous argument uses $1-e$ for controls. Comparison of both arms requires $0<e<1$.

Identifying a causal effect is an additional step. If $(Y(0),Y(1))\perp A\mid X$, then the balancing property lets the scalar propensity score replace the full measured $X$ in that exchangeability statement. Without this causal assumption, balancing measured covariates leaves possible unmeasured confounding. An estimated score also need not deliver exact finite-sample covariate balance; its estimation and the subsequent matching in causal inference or weighting procedure must be checked.

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 and propensity score $e(X)=P(A=1\mid X)$, the observed-unit overlap weights areEach unit receives the probability of assignment to the opposite group. These weights are bounded between zero and one.^[ref]

The common target-population tilting function is instead $h(X)=e(X)\{1-e(X)\}$. If the original covariate density is $f(X)$, then the unnormalized weighted densities in the two arms are equal:Thus $h$ describes the target population, while $w_1,w_0$ are the arm-specific observation weights. Multiplying every observed unit by $h(X)$ would not give this balance identity.

Under conditional exchangeability, consistency in causal inference and nonzero total overlap, the weighted contrast identifies the overlap-weighted average treatment effectUnits with deterministic treatment assignment have $h=0$ and contribute nothing to this target. The estimand therefore generally differs from the full-population average treatment effect.