Shoki Okubo
arXiv 10 Sep 2026 · Statistics — Methodology
arXiv:2609.11222 · PDF · Extracted main text
Graphical causal inference supplies a complete theory of efficient covariate adjustment for the average treatment effect: one adjustment set, computable from the graph, is optimal under every compatible distribution. We show that this is a property of the average treatment effect's inverse-prevalence weights, not of causal estimands in general. For the average treatment effect on the treated we index the efficiency bound by the adjustment set and derive exact identities for its change under treatment-side and outcome-side extensions of a valid set. Covariates that predict only the treated-arm outcome are exactly efficiency-neutral, and covariates that predict the control-arm outcome can strictly increase the bound when the propensity is below one half -- a reversal of the supplementation lemma whose source is an arithmetic-geometric-mean inequality that holds for the average treatment effect and fails for the treated-population estimand. A construction with two faithful distributions on one graph proves that no graphical optimality criterion exists for the treated-population estimand; under no effect modification the ATE-optimal set is nonetheless optimal among the graphically valid sets, with an exact expression for its advantage. The results extend to weighted average treatment effects with propensity-dependent weights, yielding symmetric thresholds for overlap weights, an estimand-drift phenomenon under instrument adjustment, and a characterization of constant weights as the only smooth positive weights for which outcome-side supplementation never increases the bound. Simulations and the LaLonde data provide illustrations.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Rotnitzky, Andrea, and Ezequiel Smucler (2020) Efficient Adjustment Sets for Population Average Causal Treatment Effect Estimation in Graphical Models | 1.000 | 12 | 6 | 100% |
| 2 | Hahn, Jinyong (2004) Functional Restriction and Efficiency in Causal Inference | 1.000 | 5 | 3 | 100% |
| 3 | Wang, Yiming, Yi Liu, and Shu Yang (2025) Rate Doubly Robust Estimation for Weighted Average Treatment Effects | 1.000 | 5 | 3 | 100% |
| 4 | de Luna, Xavier, Ingeborg Waernbaum, and Thomas S. Richardson (2011) Covariate Selection for the Nonparametric Estimation of an Average Treatment Effect | 0.928 | 4 | 3 | 100% |
| 5 | White, Halbert, and Xun Lu (2011) Causal Diagrams for Treatment Effect Estimation with Application to Efficient Covariate Selection | 0.928 | 4 | 3 | 100% |
| 6 | Henckel, Leonard, Emilija Perković, and Marloes H. Maathuis (2022) Graphical Criteria for Efficient Total Effect Estimation via Adjustment in Causal Linear Models | 0.843 | 3 | 3 | 100% |
| 7 | Kitagawa, Toru, and Chris Muris (2016) Model Averaging in Semiparametric Estimation of Treatment Effects | 0.843 | 3 | 3 | 100% |
| 8 | Dehejia, Rajeev H., and Sadek Wahba (1999) Causal Effects in Nonexperimental Studies: Reevaluating the Evaluation of Training Programs | 0.811 | 5 | 2 | 80% |
| 9 | Li, Fan, Kari Lock Morgan, and Alan M. Zaslavsky (2018) Balancing Covariates via Propensity Score Weighting | 0.811 | 4 | 2 | 100% |
| 10 | Dehejia, Rajeev H., and Sadek Wahba (2002) Propensity Score-Matching Methods for Nonexperimental Causal Studies | 0.644 | 3 | 2 | 67% |
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