Inverse-probability weighting reconstructs a target population in which treatment is independent of measured baseline covariates. Its identification is elementary; its practical quality depends on overlap, propensity estimation, normalization, and the treatment of extreme weights.
Horvitz-Thompson representation
Theorem. Under consistency, conditional exchangeability given \(X\), and positivity, \[ \tau= \mathbb E\left[ \frac{AY}{e(X)}- \frac{(1-A)Y}{1-e(X)} \right]. \]
Proof. Conditional on \(X\), \[ \mathbb E\left(\frac{AY}{e(X)}\middle|X\right) =\mathbb E(Y\mid A=1,X)=m_1(X), \] and similarly the control term has conditional mean \(m_0(X)\). Exchangeability and consistency identify \(\mathbb E\{m_1(X)-m_0(X)\}\) with the ATE.
Definition. The empirical Horvitz-Thompson estimator is \[ \widehat\tau_{\mathrm{HT}} =\frac1n\sum_{i=1}^n \left\{\frac{A_iY_i}{\widehat e(X_i)} -\frac{(1-A_i)Y_i}{1-\widehat e(X_i)}\right\}. \]
Hájek normalization
Definition. The normalized estimator is \[ \widehat\tau_{\mathrm H} = \frac{\sum_iA_iY_i/\widehat e_i}{\sum_iA_i/\widehat e_i} - \frac{\sum_i(1-A_i)Y_i/(1-\widehat e_i)} {\sum_i(1-A_i)/(1-\widehat e_i)}. \]
With known probabilities, Horvitz-Thompson is exactly unbiased while Hájek is a ratio estimator with small bias. Hájek normalization often has lower variance and is invariant to multiplicative weight errors within an arm. The distinction should be stated rather than hidden under the generic label IPW.
The influence function with known propensity
Proposition. If \(e(X)\) is known, the influence function of the IPW functional is \[ \phi_{\mathrm{IPW}}(O)= \frac{AY}{e(X)}-\frac{(1-A)Y}{1-e(X)}-\tau. \] Therefore \[ \sqrt n(\widehat\tau_{\mathrm{HT}}-\tau) \rightsquigarrow N\{0,\mathbb E\phi_{\mathrm{IPW}}^2\} \] when the second moment is finite.
Proof. The estimator is the sample mean of an iid variable whose expectation is \(\tau\). The ordinary central limit theorem applies.
Variance inflation from weak overlap
Proposition. Let \(\sigma_a^2(x)=\operatorname{Var}(Y\mid A=a,X=x)\). Conditional variance of the weighted outcome contrast contains \[ \frac{\sigma_1^2(X)}{e(X)} +\frac{\sigma_0^2(X)}{1-e(X)} +\text{between-covariate terms}. \] Thus variance diverges when outcomes remain noisy where propensity approaches zero or one.
Proof. Conditional on \(X\), only one arm contributes. For the treated term, \[ \mathbb E\left[ \left\{\frac{A(Y-m_1)}{e}\right\}^2\middle|X\right] =\frac{\sigma_1^2(X)}{e(X)}. \] The control calculation gives the second term. Applying total variance adds variation of conditional means.
Stabilized weights
Definition. Stabilized ATE weights are \[ W_{\mathrm{stab}}= \frac{AP(A=1)}{e(X)} +\frac{(1-A)P(A=0)}{1-e(X)}. \] In longitudinal settings, numerator probabilities condition on a restricted treatment history while denominators condition on the full measured history.
Stabilization preserves the target after appropriate normalization and can reduce weight dispersion. It cannot repair genuine nonpositivity.
Weight truncation and its bias
Proposition. If treated weights are clipped at \(c\), the bias in the treated potential-outcome mean is \[ \mathbb E\left[ \{1-c e(X)\}_+m_1(X) \right] \] up to sign and any subsequent normalization convention.
Proof. Conditional on \(X\), \[ \mathbb E[A\min\{1/e(X),c\}Y\mid X] =e(X)\min\{1/e(X),c\}m_1(X) =\min\{1,ce(X)\}m_1(X). \] Subtract \(m_1(X)\) and integrate.
Estimated propensities can improve efficiency
Even in a randomized experiment with known treatment probability, estimating a correctly specified propensity model containing prognostically useful covariates can reduce asymptotic variance by projecting the naive influence function onto the propensity-score tangent space. This is not because the estimated assignment probability is more truthful; it is a calibration effect. In finite samples, misspecification and extreme fitted scores can reverse the gain.
Moment balance
Proposition. If weights solve \[ \sum_i\frac{A_i}{\widehat e_i}g(X_i) = \sum_i\frac{1-A_i}{1-\widehat e_i}g(X_i) \] for a vector of balance functions \(g\), then weighted treated and control empirical means of every linear combination of \(g\) agree.
Proof. Multiply the vector equality by an arbitrary coefficient vector. This motivates covariate-balancing propensity scores and entropy balancing, which target balance equations directly.
Uncertainty with fitted nuisance functions
A standard error that treats \(\widehat e\) as fixed can be conservative or anticonservative depending on the estimator and model. Parametric M-estimation uses the joint sandwich matrix for outcome and propensity equations. Flexible learning is better handled by orthogonal estimators and cross-fitting; plain IPW is not Neyman-orthogonal and generally requires the propensity estimator to converge faster than is available in unrestricted high dimensions.