# Necessary and sufficient graphical conditions for optimal adjustment sets in causal graphical models with hidden variables

arXiv: https://arxiv.org/abs/2102.10324 · HTML: https://mlsanity.com/paper/2102.10324/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 26 references that are in the map.
- First posted on arXiv 20 Feb 2021.

## Builds on (most cited first, 4 of 4)

- [Separators and Adjustment Sets in Causal Graphs: Complete Criteria and an Algorithmic Framework](https://mlsanity.com/paper/1803.00116/) (2018)
- [Efficient adjustment sets in causal graphical models with hidden variables](https://mlsanity.com/paper/2004.10521/) (2020)
- [Markovian acyclic directed mixed graphs for discrete data](https://mlsanity.com/paper/1301.6624/) (2013)
- [Efficient adjustment sets for population average treatment effect estimation in non-parametric causal graphical models](https://mlsanity.com/paper/1912.00306/) (2019)

## Built on by (most cited first, 16 of 16)

- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [CUTS: Neural Causal Discovery from Irregular Time-Series Data](https://mlsanity.com/paper/2302.07458/) (2023)
- [Data-Driven Causal Effect Estimation Based on Graphical Causal Modelling: A Survey](https://mlsanity.com/paper/2208.09590/) (2022)
- [Identifiability of total effects from abstractions of time series causal graphs](https://mlsanity.com/paper/2310.14691/) (2023)
- [Identifying Drift, Diffusion, and Causal Structure from Temporal Snapshots](https://mlsanity.com/paper/2410.22729/) (2024)
- [Local Discovery by Partitioning: Polynomial-Time Causal Discovery Around Exposure-Outcome Pairs](https://mlsanity.com/paper/2310.17816/) (2023)
- [Separation-based distance measures for causal graphs](https://mlsanity.com/paper/2402.04952/) (2024)
- [Local Causal Discovery for Structural Evidence of Direct Discrimination](https://mlsanity.com/paper/2405.14848/) (2024)
- [An Overview of Causal Inference using Kernel Embeddings](https://mlsanity.com/paper/2410.22754/) (2024)
- [Structural Causal Bottleneck Models](https://mlsanity.com/paper/2603.08682/) (2026)
- [Local Learning for Covariate Selection in Nonparametric Causal Effect Estimation with Latent Variables](https://mlsanity.com/paper/2411.16315/) (2024)
- [Unveiling the Structure of Do-Calculus Reasoning via Derivation Graphs](https://mlsanity.com/paper/2606.03719/) (2026)
- [A note on efficient minimum cost adjustment sets in causal graphical models](https://mlsanity.com/paper/2201.02037/) (2022)
- [Projecting infinite time series graphs to finite marginal graphs using number theory](https://mlsanity.com/paper/2310.05526/) (2023)
- [Spatio-Temporal Graphical Counterfactuals: An Overview](https://mlsanity.com/paper/2407.01875/) (2024)
- [On Efficient Adjustment for Micro Causal Effects in Summary Causal Graphs](https://mlsanity.com/paper/2512.18315/) (2025)

## Abstract (arXiv)

The problem of selecting optimal backdoor adjustment sets to estimate causal effects in graphical models with hidden and conditioned variables is addressed. Previous work has defined optimality as achieving the smallest asymptotic estimation variance and derived an optimal set for the case without hidden variables. For the case with hidden variables there can be settings where no optimal set exists and currently only a sufficient graphical optimality criterion of limited applicability has been derived. In the present work optimality is characterized as maximizing a certain adjustment information which allows to derive a necessary and sufficient graphical criterion for the existence of an optimal adjustment set and a definition and algorithm to construct it. Further, the optimal set is valid if and only if a valid adjustment set exists and has higher (or equal) adjustment information than the Adjust-set proposed in Perkovi'c et al. [Journal of Machine Learning Research, 18: 1--62, 2018] for any graph. The results translate to minimal asymptotic estimation variance for a class of estimators whose asymptotic variance follows a certain information-theoretic relation. Numerical experiments indicate that the asymptotic results also hold for relatively small sample sizes and that the optimal adjustment set or minimized variants thereof often yield better variance also beyond that estimator class. Surprisingly, among the randomly created setups more than 90% fulfill the optimality conditions indicating that also in many real-world scenarios graphical optimality may hold. Code is available as part of the python package urlhttps://github.com/jakobrunge/tigramite.