= Propensity score
{title2=$\pi(X)$}
{wiki}
The <propensity score> for binary <treatment> $A$ and pretreatment <covariates> $X$ is $e(X)=P(A=1\mid X)$. It is a balancing score:
$$
A\mathrel\perp X\mid e(X).
$$
This distributional fact does not require <conditional exchangeability> of the <potential outcomes>. It follows from the definition of the true assignment <probability>.<The central role of the propensity score in observational studies for causal effects by Paul R. Rosenbaum and Donald B. Rubin>[Section 2: the propensity score as a balancing score]{ref}
For a bounded function $g$, condition successively on $X$ and on $e=e(X)$:
$$
\mathbb E[A g(X)\mid e]
=\mathbb E[e(X)g(X)\mid e]
=e\,\mathbb E[g(X)\mid e].
$$
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 $(Y(0),Y(1))\mathrel\perp A\mid X$, 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.
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