Overlap weight ($w_1=1-e(X),w_0=e(X)$)

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.