# Variable elimination, graph reduction and efficient g-formula

arXiv: https://arxiv.org/abs/2202.11994 · HTML: https://mlsanity.com/paper/2202.11994/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Feb 2022.

## Built on by (most cited first, 4 of 4)

- [SNAP: Sequential Non-Ancestor Pruning for Targeted Causal Effect Estimation With an Unknown Graph](https://mlsanity.com/paper/2502.07857/) (2025)
- [Linear-Time Primitives for Algorithm Development in Graphical Causal Inference](https://mlsanity.com/paper/2506.15758/) (2025)
- [Integrating Background Knowledge for Scalable Causal Discovery](https://mlsanity.com/paper/2607.10456/) (2026)
- [Verifying formulas for interventional distributions](https://mlsanity.com/paper/2607.13883/) (2026)

## Abstract (arXiv)

We study efficient estimation of an interventional mean associated with a point exposure treatment under a causal graphical model represented by a directed acyclic graph without hidden variables. Under such a model, it may happen that a subset of the variables are uninformative in that failure to measure them neither precludes identification of the interventional mean nor changes the semiparametric variance bound for regular estimators of it. We develop a set of graphical criteria that are sound and complete for eliminating all the uninformative variables so that the cost of measuring them can be saved without sacrificing estimation efficiency, which could be useful when designing a planned observational or randomized study. Further, we construct a reduced directed acyclic graph on the set of informative variables only. We show that the interventional mean is identified from the marginal law by the g-formula (Robins, 1986) associated with the reduced graph, and the semiparametric variance bounds for estimating the interventional mean under the original and the reduced graphical model agree. This g-formula is an irreducible, efficient identifying formula in the sense that the nonparametric estimator of the formula, under regularity conditions, is asymptotically efficient under the original causal graphical model, and no formula with such property exists that only depends on a strict subset of the variables.