PSM Matching & Doubly Robust Estimator

Propensity Score Matching (PSM), Inverse Probability Weighting (IPW) & Doubly Robust Estimators (AIPW, Robins et al.) eliminate confounding and selection bias from high-dimensional observational data; Core theory: the Propensity Score is the conditional treatment assignment probability $e(X) = P(T=1 | X)$ (modeled via Logistic Regression or GBDT); Rosenbaum-Rubin Theorem: under unconfoundedness, conditioning on the 1D scalar propensity score $e(X)$ is mathematically sufficient to balance all multi-dimensional covariates $X$; Matching Algorithms include Nearest Neighbor, Caliper thresholding, and Common Support trimming; The Doubly Robust (AIPW) estimator unifies the propensity score weighting model with an outcome regression model: if either the propensity model OR the outcome model is correctly specified, the causal estimate remains asymptotically unbiased and consistent.

Measuring non-randomized feature adoption (e.g. true causal lift of VIP subscription purchasing) and retention campaign attribution.

Active users naturally retain higher (selection bias), inflating perceived feature value; PSM/AIPW constructs synthetic twin control users to estimate genuine causal increments.

Prove the Rosenbaum-Rubin dimension reduction theorem establishing conditional independence given 1D propensity score $e(X)$?

Derive Inverse Probability Weighting (IPW) estimator and explain extreme variance explosions when propensity scores approach 0 or 1?

Derive the AIPW estimator formula and prove why misspecification in either the propensity score or outcome model preserves unbiasedness?

How to audit propensity score overlap histograms and enforce common support trimming to prevent dangerous extrapolation?

Why is Standardized Mean Difference (SMD $< 0.1$) preferred over $p$-values to evaluate post-matching covariate balance?