# Causal Calculus in the Presence of Cycles, Latent Confounders and Selection Bias

arXiv: https://arxiv.org/abs/1901.00433 · HTML: https://mlsanity.com/paper/1901.00433/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jan 2019.

## Built on by (most cited first, 17 of 17)

- [Causal models in string diagrams](https://mlsanity.com/paper/2304.07638/) (2023)
- [Dynamic Structural Causal Models](https://mlsanity.com/paper/2406.01161/) (2024)
- [Detecting and Identifying Selection Structure in Sequential Data](https://mlsanity.com/paper/2407.00529/) (2024)
- [Identifying Macro Conditional Independencies and Macro Total Effects in Summary Causal Graphs with Latent Confounding](https://mlsanity.com/paper/2407.07934/) (2024)
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- [Identifying Macro Causal Effects in a C-DMG over ADMGs](https://mlsanity.com/paper/2504.01551/) (2025)
- [Identifying Macro Causal Effects in C-DMGs over DMGs](https://mlsanity.com/paper/2506.19650/) (2025)
- [Causal reasoning in difference graphs](https://mlsanity.com/paper/2411.01292/) (2024)
- [Notes on Forr'e's Notion of Conditional Independence and Causal Calculus for Continuous Variables](https://mlsanity.com/paper/2603.24333/) (2026)
- [Towards a Transportable Causal Network Model Based on Observational Healthcare Data](https://mlsanity.com/paper/2311.08427/) (2023)
- [Regression-Based Estimation of Causal Effects in the Presence of Selection Bias and Confounding](https://mlsanity.com/paper/2503.20546/) (2025)
- [Causal identification with Y0](https://mlsanity.com/paper/2508.03167/) (2025)
- [Beyond the Black Box: Identifiable Interpretation and Control in Generative Models via Causal Minimality](https://mlsanity.com/paper/2512.10720/) (2025)
- [Data-Driven Covariate Selection for Nonparametric and Cycle-Agnostic Causal Effect Estimation](https://mlsanity.com/paper/2605.06385/) (2026)
- [Characterizing and Identifying Separable Graphical Models](https://mlsanity.com/paper/2607.01057/) (2026)
- [Equilibrium Causal Games: Separation, Identification, and the Identifiability of Cyclic Latent States](https://mlsanity.com/paper/2607.19531/) (2026)
- [Beyond Directed Acyclic Graphs: Causal Zeros and Causal Differential Equations](https://mlsanity.com/paper/2607.22910/) (2026)

## Abstract (arXiv)

We prove the main rules of causal calculus (also called do-calculus) for i/o structural causal models (ioSCMs), a generalization of a recently proposed general class of non-/linear structural causal models that allow for cycles, latent confounders and arbitrary probability distributions. We also generalize adjustment criteria and formulas from the acyclic setting to the general one (i.e. ioSCMs). Such criteria then allow to estimate (conditional) causal effects from observational data that was (partially) gathered under selection bias and cycles. This generalizes the backdoor criterion, the selection-backdoor criterion and extensions of these to arbitrary ioSCMs. Together, our results thus enable causal reasoning in the presence of cycles, latent confounders and selection bias. Finally, we extend the ID algorithm for the identification of causal effects to ioSCMs.