Inverse Propensity Scoring & Matching Weights

• Inverse Propensity Scoring is a method that reweights observed samples by the inverse probability of treatment assignment to achieve unbiased estimates of treatment effects.
• Matching weights stabilize IPS by capping extreme weights, thereby reducing estimator variance and improving efficiency in causal inference.
• Augmented matching weight estimators combine outcome regression with weighting to provide double robustness, ensuring consistency even under model misspecification.

Inverse Propensity Scoring (IPS) is a central methodology in causal inference and debiasing from observational or logged data. IPS achieves unbiased estimation of target outcomes by weighting each observed sample by the inverse probability of its observation under known or estimated propensities. This approach forms the backbone of modern techniques in treatment effect estimation, off-policy evaluation, learning-to-rank with implicit feedback, and correction for exposure or position bias. The following sections delineate foundational principles, methodological advances, implementation details, and implications of IPS, with an emphasis on rigorous mathematical connections and empirical comparisons to alternative estimators.

1. Theoretical Foundation of IPS and Matching Weights

At its core, IPS operates by reweighting observed units so that their distribution mimics a target policy or treatment assignment, typically that of a randomized experiment. In traditional causal inference, each subject $i$ receives treatment indicator $Z_i \in \{0,1\}$ according to a propensity score $e_i = P(Z_i = 1|X_i)$. The canonical IPS weights are:

• Treated: $w_i = 1/e_i$
• Control: $w_i = 1/(1-e_i)$

To estimate the average treatment effect (ATE), the standard IPS-form estimator evaluates

$$\hat\Delta_{IPS} = \frac{\sum_i w_i Z_i Y_i}{\sum_i w_i Z_i} - \frac{\sum_i w_i (1-Z_i) Y_i}{\sum_i w_i (1-Z_i)}$$

However, when $e_i$ is close to 0 or 1, these weights become extremely large, leading to instability and inflated estimator variance.

The matching weight estimator modifies the IPS weights to mitigate this instability by introducing a stabilizing numerator: $$W_i = \frac{\min(1-e_i, e_i)}{Z_i e_i + (1-Z_i)(1-e_i)},$$ so that

• Treated: $\eta_1(e_i) = \min(1-e_i, e_i)/e_i$
• Control: $\eta_0(e_i) = \min(1-e_i, e_i)/(1-e_i)$

The matching weight estimator for the treatment effect is thus

$$\hat\Delta_{MW} = \frac{\sum_i W_i Z_i Y_i}{\sum_i W_i Z_i} - \frac{\sum_i W_i (1-Z_i) Y_i}{\sum_i W_i (1-Z_i)}$$

This approach smoothly and optimally trims subjects with extreme propensity scores, creating a "maximal balanced subpopulation" where the propensity score and covariate distributions are identical between weighted treatment groups (Li, 2011).

2. Advantages and Efficiency of Matching Weights over IPS

Matching weights confer several essential practical and theoretical advantages relative to standard IPS:

• Variance Control: Bounded numerators ensure that weights remain finite even when $e_i \to 0$ or $1$, avoiding the domination of the estimator by a few data points with extreme weights.
• Improved Efficiency: The reduction in variance is accompanied by improved estimator efficiency. Matching weights yield efficient estimation even when substantial portions of the sample have extreme $e_i$ values.
• Explicit Target Population: The matching weight approach directly defines the target population with optimal balancing properties, obviating the need to select matching algorithms or caliper sizes.
• Robust Variance Estimation: Variance can be consistently estimated by sandwich estimators: $$\mathrm{var}\left( \hat\Delta_{MW} \right) \approx \frac{E\left[ (\min(1-e_i,e_i)(Y_{1i} - \mu_1))^2/e_i + (\min(1-e_i,e_i)(Y_{0i} - \mu_0))^2/(1-e_i) \right] }{E[\min(1-e_i,e_i)]^2 }$$
• Stability in Extreme Propensity Regimes: Simulation results indicate that, for instances with high proportions of extreme $e_i$, matching weights substantially reduce bias and variance compared with IPS. For example, with severe imbalance (Scenario 3 in Table 1), the matching weight estimator’s variance is approximately 130% of the best estimator, maintaining low bias, while IPS becomes much less stable (Li, 2011).

The matching weights produce optimal covariate balance, as visually confirmed by "mirror histograms," which demonstrate nearly identical propensity score distributions across treatment groups after weighting.

3. Double Robustness: Augmented Matching Weight Estimator

The augmentation of the matching weight estimator with outcome regression models yields a "double robust" estimator: $$\begin{aligned} \hat\Delta_{MW,DR} &= \frac{ \sum_i W_i \left\{ m_1(X_i, \alpha_1) - m_0(X_i, \alpha_0) \right\} }{ \sum_i W_i }\ & \qquad + \frac{ \sum_i W_i Z_i \left\{ Y_i - m_1(X_i, \alpha_1) \right\} }{ \sum_i W_i Z_i } - \frac{ \sum_i W_i (1-Z_i) \left\{ Y_i - m_0(X_i, \alpha_0) \right\} }{ \sum_i W_i (1-Z_i) } \end{aligned}$$ where $m_1$, $m_0$ are regression models for the outcome under treatment and control.

This estimator is consistent if either the propensity score model or the outcome model is correctly specified. Proposition 3 (Li, 2011) formally establishes that at least one correct model suffices for consistency:

• If the propensity score model or both outcome models are correct, then $\hat\Delta_{MW,DR}$ consistently estimates the target parameter $\Delta_0$.

This double robustness property reduces the risk of bias from model misspecification, providing two opportunities for valid inference. Furthermore, Proposition 4 demonstrates that among regular asymptotically linear estimators utilizing matching weights, the augmented estimator achieves minimal asymptotic variance, making it the most efficient member of this class.

4. Empirical Comparison: Numerical Studies

The empirical results in (Li, 2011) systematically compare IPS, matching weights (MW), augmented matching weights, and alternative approaches:

Scenario	IPS Bias	MW Bias	IPS Var	MW Var	MW MSE	Comments
1 (well-behaved)	~0.1%	~0.1%	similar	low	low	IPS and MW similar under strong overlap
2 (moderate imbal.)	higher	very low	higher	lower	lower	MW outperforms IPS on variance and MSE
3 (severe imbal.)	high	very low	much higher	reasonable	lower	MW remains stable; IPS variance inflates rapidly

• Augmented matching weights demonstrate superior type I error control and power, especially with smaller sample sizes (n = 200), relative to the double robust IPS estimator.
• Mirror histogram analysis confirms optimal balance of covariate distributions post-matching weight application, contrasting with IPS where extreme weights may distort the empirical distribution.

5. Practical Considerations and Implementation Guidance

• Estimator Formulae: Practitioners should deploy MW or MW-DR estimators using the explicit weighting formula: $$W_i = \frac{ \min(1 - e_i, e_i) }{ Z_i e_i + (1 - Z_i)(1 - e_i) }$$ with $e_i$ typically estimated from a logistic regression or nonparametric model.
• Variance Estimation: Prefer robust (sandwich) variance estimators, as their simple closed forms follow from the MW estimator's boundedness.
• Data Regimes: MW estimators offer particular advantages in datasets with poor propensity overlap or where overlap cannot be assured across all covariate strata.
• Augmentation and Double Robustness: The MW-DR estimator is recommended whenever plausible outcome models can be constructed, due to its double robustness and minimum variance among regular estimators.
• Target Population: Matching weights define an estimand for the "maximal balanced subpopulation"—the subset in which inference is most robust due to indistinguishable treated and control covariate distributions post-weighting.

In summary, modifying IPS via matching weights stabilizes finite-sample estimators, controls variance, and enhances inference robustness, particularly when extreme propensity scores are present. Augmented matching weight estimators further improve efficiency and deliver double robustness.

6. Connections to Alternative Weighting Schemes and Future Directions

The matching weight approach is part of a larger class of propensity score weighting methods aimed at optimizing bias-variance tradeoffs and efficiency:

• Overlap Weights (OW): $h(X) = e(X)[1-e(X)]$—emphasizes individuals with propensity near 0.5, analogous in stabilizing effect to matching weights (Zhou et al., 2020).
• Entropy Weights (EW): Derived from minimizing (cross-)entropy criteria, these similarly downweight tail units.
• Trimming Approaches: Discarding units with extreme propensities, at the cost of altering the target estimand and sacrificing information. Numerical and simulation evidence across studies demonstrates that matching weights consistently outperform IPS and even OW and EW under severe overlap violations or model misspecification.

A future research direction is the integration of matching weights with high-dimensional and machine-learning-based propensity score estimation, as well as further exploration of model selection and augmented methods that preserve double robustness under flexible nonparametric modeling.

---

The detailed mathematical framework and empirical comparisons provided in (Li, 2011) rigorously establish matching weights and their augmented, double robust counterpart as principled, efficient, and practical alternatives to classical IPS weighting in observational causal inference. Matching weights yield robust inference even under limited overlap and bolster both efficiency and stability—central concerns in applying propensity score methods to complex or high-dimensional datasets.