# Foundations of Structural Causal Models with Latent Selection

arXiv: https://arxiv.org/abs/2401.06925 · HTML: https://mlsanity.com/paper/2401.06925/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jan 2024.

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

- [Detecting and Identifying Selection Structure in Sequential Data](https://mlsanity.com/paper/2407.00529/) (2024)
- [Identifying Selections for Unsupervised Subtask Discovery](https://mlsanity.com/paper/2410.21616/) (2024)
- [Complete Causal Identification from Ancestral Graphs under Selection Bias](https://mlsanity.com/paper/2603.26301/) (2026)
- [Revisiting the Berkeley Admissions data: Statistical Tests for Causal Hypotheses](https://mlsanity.com/paper/2502.10161/) (2025)

## Abstract (arXiv)

Three distinct phenomena complicate statistical causal analysis: latent common causes, causal cycles, and latent selection. Foundational works on Structural Causal Models (SCMs), e.g., Bongers et al. (2021, Ann. Stat., 49(5): 2885-2915), treat cycles and latent variables, while an analogous account of latent selection is missing. The goal of this article is to develop a theoretical foundation for modeling latent selection with SCMs. To achieve that, we introduce a conditioning operation for SCMs: it maps an SCM with explicit selection mechanisms to one without them while preserving the causal semantics of the selected subpopulation. Graphically, in Directed Mixed Graphs we extend bidirected edge--beyond latent common cause--to also encode latent selection. We prove that the conditioning operation preserves simplicity, acyclicity, and linearity of SCMs, and interacts well with marginalization, conditioning, and interventions. These properties make those three operations valuable tools for causal modeling, reasoning, and learning after abstracting away latent details (latent common causes and selection). Examples show how this abstraction streamlines analysis and clarifies when standard tools (e.g., adjustment, causal calculus, instrumental variables) remain valid under selection bias. We hope that these results deepen the SCM-based understanding of selection bias and become part of the standard causal modeling toolbox to build more reliable causal analysis.