Propensity score ($\pi (X)$)

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.