The propensity score compresses a potentially high-dimensional treatment mechanism into a scalar balancing score. Its role is identification and design; it is not itself a causal effect model.
Balancing scores
Definition. A function \(b(X)\) is a balancing score if \[ A\perp\!\!\!\perp X\mid b(X). \] The propensity score is \[ e(X)=P(A=1\mid X). \]
Theorem. The propensity score is a balancing score: \[ A\perp\!\!\!\perp X\mid e(X). \]
Proof. For any \(x\) with \(e(x)=t\), \[ P(A=1\mid X=x,e(X)=t)=e(x)=t. \] On the other hand, iterated expectation gives \(P(A=1\mid e(X)=t)=\mathbb E\{e(X)\mid e(X)=t\}=t\). Thus the conditional treatment law does not depend on \(X\) once \(e(X)\) is fixed.
Coarsest balancing score
Proposition. If \(b(X)\) is any balancing score, then \(e(X)\) is measurable with respect to \(b(X)\): \[ e(X)=P(A=1\mid b(X)). \]
Proof. Balancing gives \(P(A=1\mid X,b(X))=P(A=1\mid b(X))\). Because \(b(X)\) is a function of \(X\), the left side is \(P(A=1\mid X)=e(X)\).
Exchangeability given the score
Theorem. If \(\{Y(0),Y(1)\}\perp\!\!\!\perp A\mid X\), then \[ \{Y(0),Y(1)\}\perp\!\!\!\perp A\mid e(X). \]
Proof. This is the balancing-score theorem. Integrate the conditional potential-outcome law over \(X\mid A,e(X)\). Exchangeability removes \(A\) from the potential-outcome factor, and balancing removes it from the distribution of \(X\).
Propensity-score weighting creates a pseudo-population
Proposition. Define the ATE weight \[ W=\frac{A}{e(X)}+\frac{1-A}{1-e(X)}. \] Under the weighted law, \[ P_W(X\in B\mid A=1)=P_W(X\in B\mid A=0)=P_X(B). \]
Proof. For treatment, \[ \mathbb E\left[\frac{A}{e(X)}\mathbf1(X\in B)\right] =P_X(B), \qquad \mathbb E\left[\frac{A}{e(X)}\right]=1. \] Normalize to obtain the treated weighted distribution. The control calculation is identical.
Targets encoded by different weights
Definition. For a nonnegative target function \(h(X)\), balancing weights take \[ W_h(A,X)=h(X) \left\{\frac{A}{e(X)}+ \frac{1-A}{1-e(X)}\right\}. \] They target the covariate density proportional to \(h(x)p_X(x)\).
Examples. \(h(x)=1\) targets the ATE population. \(h(x)=e(x)\) targets the treated population. \(h(x)=e(x)\{1-e(x)\}\) gives overlap weights: treated units receive \(1-e(X)\), controls receive \(e(X)\), and the target emphasizes equipoise.
Estimation of the score
Logistic regression estimates \(e(x)\) by modeling treatment assignment. Flexible machine learning can reduce model misspecification, but predictive accuracy alone is not the target. A propensity model should support covariate balance, overlap, and stable weights in the population relevant to the estimand.
Proposition. If a fitted score \(\widehat e(X)\) equals the true propensity score, then for every integrable function \(g(X)\), \[ \mathbb E\left[\frac{A}{\widehat e(X)}g(X)\right] = \mathbb Eg(X) = \mathbb E\left[\frac{1-A}{1-\widehat e(X)}g(X)\right]. \]
Proof. Condition on \(X\) and use \(\mathbb E(A\mid X)=e(X)=\widehat e(X)\).
Empirical imbalance after weighting estimates violations of these moment equations. Balance on measured functions is necessary for a fitted weighting design to mimic its target, but it cannot reveal unmeasured confounding.
Extreme weights and effective sample size
Definition. For nonnegative weights \(w_i\), the Kish effective sample size is \[ n_{\mathrm{eff}}= \frac{(\sum_iw_i)^2}{\sum_iw_i^2}. \]
Proposition. \(1\le n_{\mathrm{eff}}\le n\), with equality \(n_{\mathrm{eff}}=n\) exactly when all nonzero weights are equal.
Proof. Cauchy-Schwarz gives \((\sum_iw_i)^2\le n\sum_iw_i^2\). The lower bound follows from \((\sum_iw_i)^2\ge\sum_iw_i^2\). Equality conditions are standard.
Trimming changes the estimand
Restricting analysis to \(\{\alpha\le\widehat e(X)\le1-\alpha\}\) can stabilize estimation, but it targets an overlap subpopulation rather than the original ATE population. This is scientifically legitimate when stated explicitly. Silently trimming and retaining the label “ATE” confuses design with estimand.
Propensity score versus prognostic score
Definition. A prognostic score \(q(X)\) satisfies \(Y(0)\perp\!\!\!\perp X\mid q(X)\); for example, \(q(X)=\mathbb E\{Y(0)\mid X\}\) under suitable conditions. The propensity score balances treatment assignment; a prognostic score balances outcome risk. Combining both can improve precision and robustness, but they solve different reductions.