# Complete Causal Identification from Ancestral Graphs under Selection Bias

arXiv: https://arxiv.org/abs/2603.26301 · HTML: https://mlsanity.com/paper/2603.26301/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 91 references that are in the map.
- First posted on arXiv 27 Mar 2026.

## Builds on (most cited first, 7 of 7)

- [When does the ID algorithm fail?](https://mlsanity.com/paper/2307.03750/) (2023)
- [Towards Complete Causal Explanation with Expert Knowledge](https://mlsanity.com/paper/2407.07338/) (2024)
- [Transitional Conditional Independence](https://mlsanity.com/paper/2104.11547/) (2021)
- [Foundations of Structural Causal Models with Latent Selection](https://mlsanity.com/paper/2401.06925/) (2024)
- [Causal Effect Identification in a Sub-Population with Latent Variables](https://mlsanity.com/paper/2405.14547/) (2024)
- [Effect-Level Validation for Causal Discovery](https://mlsanity.com/paper/2602.08340/) (2026)
- [Notes on Forr'e's Notion of Conditional Independence and Causal Calculus for Continuous Variables](https://mlsanity.com/paper/2603.24333/) (2026)

## Built on by (most cited first, 1 of 1)

- [Verifying formulas for interventional distributions](https://mlsanity.com/paper/2607.13883/) (2026)

## Abstract (arXiv)

Many causal discovery algorithms, including the celebrated FCI algorithm, output a Partial Ancestral Graph (PAG). PAGs serve as an abstract graphical representation of the underlying causal structure, modeled by directed acyclic graphs with latent and selection variables. This paper develops a characterization of the set of extended-type conditional independence relations that are invariant across all causal models represented by a PAG. This theory allows us to formulate a general measure-theoretic version of Pearl's causal calculus and a sound and complete identification algorithm for PAGs under selection bias. Our results also apply when PAGs are learned by certain algorithms that integrate observational data with experimental data and incorporate background knowledge.